Sine pattern geometry, formation and recovery

Relative patterns and origins, composition, equilibrium and symmetry, prepared formation, recovery/capture, controlled slow references and budget exclusions.

Part of Relational dynamics contract index. Section links remain stable; hypotheses and model changes remain local to each result.

Resolution budgets without an observed response

assess_sine_aperture_budget(*, probe_duration, form_radius, phase_radius, readout_error_bound, averaged_reading_halfwidth_bound, clock_rate_derivative_bound) in relational_sine_aperture_budget.py returns SineApertureBudget. All six primitive arguments are mandatory. The resolution theorem retains the complete source and law of the aperture inverse. It fixes the original angle rectangle, cumulative phase inputs (1/4,3/4), gain prior [1,2], global clock-rate prior [1/2,2] and four normalized boxcars. These are model premises, not quantities estimated by the report.

Require 0<H=probe_duration<=1/4 and nonnegative finite source radii X=form_radius, Y=phase_radius, per-average sensor error delta=readout_error_bound, average-band halfwidth bound t=averaged_reading_halfwidth_bound and clock derivative bound Lambda=clock_rate_derivative_bound. The positive C1 rate obeys its global prior and abs(rho')<=Lambda throughout [0,2*H]; gain and offset are held. Source norms use the existing degree metric, phases are radians, H uses observation time, delta,t use recorded units and Lambda uses structural time per observation time squared. Shared exact/represented admission preserves rational inputs and rejects Booleans, nonfinite values and nonzero materialization loss before arithmetic.

The input t bounds each supplied recorded-average enclosure's complete halfwidth. It can contain propagated source uncertainty and numerical enclosure error; it is not automatically a pure solver-error estimate. The four additive sensor errors retain their separate bound delta. The report exposes their separate virtual radii, clock-transfer and reconstruction radii, as well as the combined error allowance. It consumes no observed readings, forward trajectory, inverse result or cached verdict, and does not search for a favorable horizon.

sufficient_bound_eligible requires strict source-chart, transformed-radius and curvature-quotient guards. The corresponding width _upper_bound fields are None if those guards fail; diagnostic candidates and ineligibility_reasons remain available. Eligibility supplies conditional ideal exact-arithmetic widths for data from an admitted state. It does not prove compatible observations, joint realizability or availability of a later outward numerical inverse.

sufficient_resolution_certified additionally requires the actual initial long-arc mean width below 1/1024, effective-gain width below 1/2048, sensor-gain width below 1/16 and first-window mean-rate width below 1/64. status is certified_sufficient_budget or not_certified; unmet_resolution_targets distinguishes failed targets from failed guards. A failed sufficient certificate is not an information impossibility.

The separate noise_overlap_witness_admitted flag compares sensor error alone with noise_overlap_sensor_error_threshold. Its complete-history witness uses the same ideal source and held offset with (G,rho)=(3/2,1) and (1,3/2). At or above the sufficient overlap threshold, some common four-average record can arise from either history using bounded additive errors. Uniformly valid gain and mean-rate marginals for that record must cover their separation 1/2. This is an existential bounded-error obstruction, not a claim about every record or stochastic sensor behavior. Zero source residuals belong to every nonnegative supplied source ball, so this flag can hold even when the sufficient whole-ball chart guard fails. Numerical halfwidth t never substitutes for physical sensor error in this witness.

Export uses tnfr.sine-aperture-budget.v1 with shared SDK projection and atomic writing. The workflow checks a declared budget without evaluating a response. Neither result establishes achievable laboratory uncertainty, a physical unit conversion or independent selection of the complete nodal law.

Validated boxcar observations under an explicit affine clock

bound_sine_aperture_readout(*, initial_form_bounds, initial_phase_bounds, phase_increments, probe_duration, initial_clock_rate, clock_slope, order) in relational_sine_aperture_readout.py returns SineApertureReadout. All seven primitive arguments are mandatory. The producer proof retains the fixed two-port support and complete sine law of the full-state readout, with an explicit observation clock and passive integral. It does not consume an inverse, target root, cached source verdict, sensor gain, offset or noise realization.

initial_form_bounds and initial_phase_bounds each supply eighteen primitive endpoint pairs in the existing node order. Form is signed and phase entries are continuous lifts in radians. The complete smooth law admits general source boxes; producing their averages does not establish the separately restricted inference source family or an acute sector. Shared exact/represented scalar and interval admission precedes arithmetic. Malformed pairs, Booleans, nonfinite values and materialization loss reject.

phase_increments=(a1,a2) gives cumulative amplitudes with 0<=a1<=a2<=1. The first phase-only jump occurs at zero; the second, of size a2-a1, occurs at H=probe_duration>0. Zero jump size is passive. The known affine clock is rho(s)=r0+k*s, with r0=initial_clock_rate and k=clock_slope. Require r0>0, r0+2*k*H>0 and max(r0,r0+2*k*H)*H<=1/2. There is no separate upper bound on H. order is an ordinary integer from 1 through 16. Rate is in structural time per observation time; slope is in structural time per observation time squared.

The shared source-box Picard/Taylor kernel advances four fixed steps over [0,H/3], [H/3,2H/3], [2H/3,H] and [H,2H]. Its 38 coordinates are all eighteen forms, all eighteen phases, the clock rate and a cumulative passive integral z. Both nodal rows multiply by rho; the additional rows are rho'=k and z'=q.T*x, with q=e4-e5. The integral is in observation time, so its row has no extra clock-rate factor. It starts at zero once. All 38 coordinates and their uncertainty carry to the next step; only the declared phase entries jump. Neither nodal state nor instrument state is reset.

Each average is step.increment[37] divided by its exact aperture width. The increment excludes the common initial coordinate symbolically; it is not obtained by subtracting two unrelated integral endpoint boxes. The step retains the full initial box, strict Picard tube, Taylor coefficients, remainder and endpoint. The constant domain margin records the globally smooth extended field, not positive rates throughout its interval tube. clock_positivity_certified instead follows from the exact affine endpoints; a broad tube may include zero without negating that primitive clock premise.

steps and averaged_readout_bounds retain the successfully completed prefix. window_initial_boxes retains every attempted window source, including the first failed attempt when present. completed_window_count, completed_observed_time and completed_endpoint_box state how far that certificate extends. The complete 38-coordinate final_state_bounds is available only after all four windows pass. A failed fixed step returns unavailable with its failed_window_index, failed_tube and reasons. It supplies no uncertified average or retry with a different budget. A completed prefix is not a complete four-reading observation. clock_rate_bounds and cumulative_structural_exposures retain the exact affine-clock evidence.

These are true model-average bounds before any sensor transformation. Their widths contain both the supplied source uncertainty and numerical enclosure error; a width is not automatically a solver-error estimate or an additive sensor-error allowance. Applying held gain/offset and additive errors requires a separately declared observation record. The producer does not itself supply physical acquisition, calibration, identity retention or a reserved evaluation. Export uses tnfr.sine-aperture-readout.v1 with shared SDK projection and atomic writing. The workflow illustrates an independent stationary control.

The retained finite-aperture evaluation keeps its separately frozen preparation, sensor and first outcome. Its read-only audit reconstructs consumed arithmetic without invoking this producer or the inverse.

Necessary inference from four finite-aperture averages

infer_sine_geometry_gain_clock_aperture(*, bulk_angle_bounds, receiver_short_angle_bounds, form_radius, phase_radius, phase_increments, probe_duration, averaged_reading_bounds, readout_error_bound, readout_gain_bounds, clock_rate_bounds, clock_rate_derivative_bound) in relational_sine_aperture_inference.py returns SineApertureInference. All eleven primitive arguments are mandatory. The aperture theorem retains the clock-drift source, event and clock domains, including H=probe_duration>0, rho_plus*H<=1/2, the complete original source uncertainty and one held gain and offset on [0,2*H].

averaged_reading_bounds contains four ordered finite endpoint pairs. They enclose recorded scalar averages over [0,H/3], [H/3,2H/3], [2H/3,H] and [H,2H]. Each sensor kernel is nonnegative and constant on its declared interval, with integral one in observation time. Thus the first three averages use factor 3/H and the last uses 1/H. The fixed kernel, normalization and event alignment are independent premises, not quantities fitted or calibrated by this API. No configurable aperture, unknown timing shift or kernel perturbation is admitted.

Each average has one additive error bounded by readout_error_bound in recorded units. This is not an error per integration sample. The four interval widths retain their separate numerical/observation provenance. The offset remains common and normalization preserves it. The phase events remain at 0,H; form is continuous through them, while both full state rows continue evolving. The sensor is passive and uses no unspecified prehistory or post-horizon state. Shared exact/represented scalar and interval admission rejects malformed pairs, Booleans, nonfinite values and nonzero materialization loss.

The adapter first bounds the averaged clock discrepancy against the constant first-window mean reference. The first three reference averages determine a unique quadratic moment reconstruction at virtual times 0,H/2,H. The last virtual reading is twice the fourth average minus the reconstructed H value. The first-window cubic remainder and the separate second-window interpolation remainder respect the event at H; a smooth Taylor remainder is not continued across that jump. The derivative bounds are in structural time on each continuous window; the positive C1 observation clock is not assumed to have a third derivative.

reconstruction_matrix, aperture_windows, virtual_observation_times, averaged_reading_midpoints and averaged_reading_radii retain the exact observation map. projected_numerical_radii, projected_sensor_error_radii, projected_clock_discrepancy_candidates and reconstruction_error_candidates keep the four error sources separate. The original-error coefficients of the curvature and embedded inverse are exposed as original_curvature_average_coefficients and embedded_inverse_average_coefficients. They retain the common reading history before the conservative rectangular projection; that projection does not assert independent errors or joint realizability of its corners.

point_reference_reading_bounds encloses auxiliary point constraints for one fresh reference_envelope from the curvature owner. Its sensor error is exactly zero because the four original sensor errors are already included in those bands. These virtual points are not newly measured samples, an evaluated constant-clock trajectory or a reset of the source. Do not add the original sensor allowance to the child a second time.

Check source_admitted, clock_drift_transfer_certified, aperture_reconstruction_certified, finite_curvature_bound_certified and inverse_enclosure_available before consuming bounds. Form, readout speed, second/third derivatives, clock discrepancy and reconstruction have explicit candidate fields; certified _upper_bound or _upper_bounds fields require the original source premise. nominal_bulk_angle_outer_bounds, actual_long_arc_mean_outer_bounds, effective_gain_outer_bounds, readout_gain_outer_bounds and first_window_mean_clock_rate_outer_bounds are necessary marginals of the unchanged original source and J=G*tau(H)/H. They do not recover an instantaneous clock profile or certify a compatible parameter tuple.

Malformed inputs raise. bounded_candidate means necessary projections are available; incompatible is a strict necessary exclusion; unavailable retains failed source, rank or arithmetic evidence without proving physical nonidentifiability. Zero drift removes only the clock discrepancy, not the finite-aperture reconstruction error. Equal sample/event clock exposures alone do not imply equal interval averages. This detached calculation evaluates no trajectory or reserved response. Export uses tnfr.sine-aperture-inference.v1 with the shared SDK projection and atomic writer. See the workflow for input admission without a response campaign.

Mean clock rate with bounded drift and sampled exposure

infer_sine_geometry_gain_clock_drift(*, bulk_angle_bounds, receiver_short_angle_bounds, form_radius, phase_radius, phase_increments, probe_duration, recorded_reading_bounds, readout_error_bound, readout_gain_bounds, clock_rate_bounds, clock_rate_derivative_bound) in relational_sine_clock_drift_inference.py returns SineClockDriftInference. All eleven primitive arguments are mandatory. The clock-drift theorem retains the four-reading source, input and observation domains, including H=probe_duration>0, rho_plus*H<=1/2, all original coordinate uncertainty and the four ordered reading pairs at 0,H/2,H,2*H.

The clock premise is now a positive C1 function rho(s) on [0,2*H], with global pointwise bounds clock_rate_bounds=(rho_minus,rho_plus) and abs(rho'(s))<=L=clock_rate_derivative_bound. The derivative bound is a finite nonnegative exact/represented scalar; shared admission rejects Booleans, nonfinite values and nonzero materialization loss. Its units are structural time divided by observation time squared. Both continuous state rows multiply by the same rho(s). Gain and offset remain held, events stay at 0,H, and the intermediate observation is passive. No capacity, support, input or structural-law coefficient is changed.

The target is the first-window mean rate rho_bar=tau(H)/H, where tau(s)=integral_0^s rho(v) dv. The effective gain is J=G*rho_bar. The API consumes no clock profile and does not verify a supplied function or empirical drift bound. Equal exposures on [0,H/2], [H/2,H] and [H,2*H] give identical full sampled states under this autonomous law and fixed events. Even perfect observations cannot recover an arbitrary instantaneous profile from those samples.

The finite transfer uses g=1/3069, the centered-form bound Q0=X+14*g*rho_plus*H, and readout-speed bound U=2*Q0+7*g. The four structural exposure error bounds relative to the constant-rate reference are (0,min(L*H**2/8,(rho_plus-rho_minus)*H/4),0, min(L*H**2,(rho_plus-rho_minus)*H)). Multiplying by G_plus*U bounds the associated recorded-reading differences. Widen only those primitive reading pairs, leaving the original sensor-error bound unchanged, then construct one fresh reference_envelope through the curvature owner. The source and the first-window endpoint match exactly; no state is reset.

exposure_discrepancy_bounds retains the four clock-exposure allowances; recorded_discrepancy_candidates and comparison_reading_bounds retain the exact transfer calculation. whole_window_form_norm_candidate and structural_readout_speed_candidate are algebraic bounds. Their certified _upper_bound counterparts and recorded_discrepancy_upper_bounds require original source admission; otherwise they are None. Check clock_drift_transfer_certified separately from the inherited curvature and inverse availability flags.

These transferred bands concern auxiliary reference observations. They are not new measurements, additional independent sensor errors or an observed constant-clock trajectory. The reference keeps all four reading associations and the inherited finite-difference error map; interval projection may conservatively lose further correlations. L=0 or a singleton global rate prior gives zero transfer and reduces exactly to the held-clock calculation.

first_window_mean_clock_rate_outer_bounds, effective_gain_outer_bounds, readout_gain_outer_bounds, nominal_bulk_angle_outer_bounds and actual_long_arc_mean_outer_bounds are necessary marginals. The actual angle remains the original-source long-arc mean with the original phase-radius allowance. There is no instantaneous-rate output, profile recovery or assertion that every retained parameter tuple is jointly realizable.

Malformed inputs raise. The inherited status distinguishes an available bounded_candidate, a strict necessary exclusion incompatible and unsupported source, rank or arithmetic evidence unavailable. Inspect the parent availability flags and retained reference scope before using its bounds; an unavailable result is not a clock-instability theorem. This detached inverse evaluates no trajectory or reserved response. Export uses tnfr.sine-clock-drift-inference.v1 and the shared SDK projection and atomic writer. The workflow illustrates input admission without a new campaign.

Finite-curvature constraints from four associated readings

infer_sine_geometry_gain_clock_curvature(*, bulk_angle_bounds, receiver_short_angle_bounds, form_radius, phase_radius, phase_increments, probe_duration, recorded_reading_bounds, readout_error_bound, readout_gain_bounds, clock_rate_bounds) in relational_sine_curvature_inference.py returns SineCurvatureInference. All ten primitive arguments are mandatory. The finite-curvature owner retains the clock-inference model and domains, but requires four ordered reading pairs at observation times 0,H/2,H,2*H, where H=probe_duration. Admit every pair before selecting any subset, including the additional half-window reading. Invalid, nonfinite, Boolean, reversed or unordered values cannot be hidden in a reading omitted from the older three-reading calculation.

The two phase events remain at 0,H; the extra reading installs no event and resets no coordinate. One source, positive constant clock conversion, held sensor gain and offset govern the full trajectory. All source, capacity, law, angle and unit premises remain unchanged, including H>0 and rho_plus*H<=1/2. Original source acuteness remains required; whole-window acuteness is separate optional evidence in the clock report.

The report constructs a fresh clock_envelope from reading columns (0,2,3). It computes exact reading midpoints and radii before the curvature difference C=c_H-2*c_half+c_0. The shared four-reading error map retains curvature_reading_coefficients=(1,-2,1,0) and the embedded inverse coefficients. Its curvature observation radius is the sum of the corresponding absolute coefficients times each numerical radius plus the per-reading sensor-error allowance. Curvature is not an independently measured channel; separate interval projections may conservatively lose cross-constraint correlations. No error is divided by an unknown clock.

sensor_corrected_curvature_bounds retains the recorded finite difference with its reading uncertainty. Full-state bounds on initial curvature and the third derivative then account for original form/phase residuals and continuous motion during the first window. The midpoint difference is not an exact derivative. Inspect initial_curvature_error_candidate, third_derivative_bound_candidate and normalized_curvature_error_candidate as algebraic diagnostics; normalized_curvature_error_upper_bound requires the finite-curvature certificate. The complete continuous sine law supplies these bounds, not a fitted curvature or a reset to the nominal source.

Positive effective-gain and curvature-coefficient bounds admit the necessary clock_rate_constraint_bounds. One projection intersects that constraint with the coarse clock bounds and projects the same J=G*rho relation back to sensor gain. nominal_bulk_angle_outer_bounds, actual_long_arc_mean_outer_bounds, effective_gain_outer_bounds, readout_gain_outer_bounds and clock_rate_outer_bounds remain necessary marginals. The actual angle concerns the original source and retains the original phase-radius expansion. There is no iteration, joint realization claim or promise that every available interval becomes narrower.

base_inverse_enclosure_available, finite_curvature_bound_certified, curvature_coefficient_positive and curvature_refinement_available separate the coarse inverse from the new bound and its divisions. Unsupported source, rank or arithmetic admission yields unavailable, retaining the coarse child while refined outer fields remain unavailable. A strictly empty necessary intersection yields incompatible; bounded_candidate exposes the admitted outer marginals. Inspect the status and reason fields rather than treating a diagnostic candidate as certified evidence.

The theorem's local information comparison fixes receiver geometry and zero source residuals with noiseless readings. It does not establish global identification for the full uncertain family, or remove the exact clock/common-law-rate equivalence. This detached calculation runs no trajectory, calibration or reserved assessment. Export uses tnfr.sine-curvature-inference.v1, shared SDK projection and atomic writing. See the workflow.

Geometry and gain with a bounded observation-clock conversion

infer_sine_geometry_gain_clock(*, bulk_angle_bounds, receiver_short_angle_bounds, form_radius, phase_radius, phase_increments, probe_duration, recorded_reading_bounds, readout_error_bound, readout_gain_bounds, clock_rate_bounds) in relational_sine_clock_inference.py returns SineClockInference. All ten arguments are mandatory primitives. The clock-inference owner extends the two-input source and observation contract while keeping the same complete law, fixed gamma, support, capacities, source-angle meanings and three-reading error association.

Here probe_duration=H uses observation time s, with one held positive conversion tau=rho*s. clock_rate_bounds=(rho_minus,rho_plus) is an independently supplied positive ordered interval. Require H>0 and rho_plus*H<=1/2; both observed-time evolution rows multiply by rho. The events occur at s=0,H, with jumps a1*q and (a2-a1)*q, and readings at 0,H,2*H. One clock rate, gain and offset apply throughout the uninterrupted trajectory. No state-dependent clock, clock drift, different second-window rate or unknown law coefficient is admitted.

Every scalar and pair retains shared exact/represented-real admission. Reject Boolean or nonfinite values, lost nonzero represented values, unordered or malformed pairs, nonpositive rates/gains, and unsupported durations. The original source guard and all other source, event and readout domains are inherited. Angles remain radians; rho converts observation-time units to structural-time units and does not calibrate either against laboratory time.

The adapter reuses the joint inverse at h_star=rho_plus*H and auxiliary gain L=G*rho/rho_plus, whose prior is [G_minus*rho_minus/rho_plus,G_plus]. Monotonicity of the full-flow remainder divided by clock rate makes this an outer constraint for every admitted clock. The nested constraint_envelope is an auxiliary necessary inequality: its gain fields refer to L, and its upper duration is not the actual unknown trajectory duration. It provides no source reset or known-clock trajectory certificate. Recorded readings, their numerical widths and sensor-error allowances remain in recorded units; none is divided by an unknown clock rate.

The parent exposes the original nominal and actual-angle bounds, effective_gain_outer_bounds for J=G*rho, readout_gain_outer_bounds for G, and clock_rate_outer_bounds for rho. The latter two project the same product relation onto their original priors. Their Cartesian product need not be jointly compatible or realizable, and the separate marginals do not imply point identification. Exact products and positive quotients precede outward interval materialization.

structural_probe_duration_bounds, auxiliary_gain_prior_bounds and effective_gain_prior_bounds retain the conversion. The diagnostic finite_remainder_over_clock_candidate bounds the response remainder divided by rho; its _upper_bound counterpart is available only when finite_response_certified passes. Source, finite-response, rank and optional whole-window acute flags keep their separate meanings. bounded_candidate supplies necessary outer marginals, incompatible excludes the joint premises, and unavailable records a failed source, rank or arithmetic admission. Inspect the corresponding reason fields.

The exact clock/common-rate equivalence and the gain/clock curvature counterexample belong to the theorem. An extra multiplier of every law row is not an input to this fixed-law API. Equal leading G*rho does not assert equality of full responses, and an ideal-source derivative control is not uniform recovery from noisy finite observations. No trajectory, calibration or reserved assessment runs. Export uses tnfr.sine-clock-inference.v1, shared SDK projection and atomic writing. See the workflow.

Joint geometry and held-gain constraints from successive inputs

infer_sine_two_pulse_geometry_gain(*, bulk_angle_bounds, receiver_short_angle_bounds, form_radius, phase_radius, phase_increments, probe_duration, recorded_reading_bounds, readout_error_bound, readout_gain_bounds) in relational_sine_two_pulse_inference.py returns SineTwoPulseInference. Every keyword is required. The complete source family, two-port support, held capacities, sine law and fast clock are those of the single-response inverse, without a critical-state or acquisition premise. The two-pulse theorem owns the retained-history bound and the joint inverse construction.

phase_increments=(a1,a2) gives cumulative nominal phase amplitudes: the actual jumps are a1*q at zero and (a2-a1)*q at h, with q=e_4-e_5 and h=probe_duration. Neither jump changes form. The second window begins with the complete first endpoint; no coordinate, common mean or residual is reset. Admit 0<a1<=a2<=1 and 0<h<=1/2. The three ordered primitive recorded_reading_bounds pairs refer to times 0,h,2*h. The phase-only events leave the corresponding instantaneous form readings unchanged. A single positive gain in readout_gain_bounds and one held arbitrary offset apply to all three readings. Each individual reading has additive error bounded by the nonnegative readout_error_bound. Input interval widths and sensor-error allowances are distinct.

The nominal priors lie in b=[11/8,3/2], c=[2/3,1] radians. Radii are nonnegative full degree-norm source bounds. Every scalar and endpoint uses shared exact/represented-real admission before outward arithmetic. Reject Boolean or nonfinite physical scalars, lost nonzero represented values, unordered containers, malformed or reversed pairs, nonpositive gain, out-of-domain priors, and invalid event/time declarations. There is no caller-supplied cached report or verdict.

The report rebuilds source geometry and its acute guard. Its full-window bounds cover mean-removed form and phase displacement from the original nominal lifts after removing the held common phase. They include both inputs and the intermediate state. The finite remainder includes continuous form relaxation and phase motion under the complete law. Whole-window acuteness is separate optional evidence; the inverse concerns the original pre-probe angle. An event schedule alone does not establish maintenance or post-probe recovery.

Exact primitive reading midpoints are differenced before interval coefficient arithmetic, so a large held offset cancels without subtracting rounded floating readings. Halfwidths and sensor errors propagate through the same three-reading incidence map. In particular, the middle reading's error radius is multiplied by abs(w1-w2) in each inverse row, retaining its single error realization. Coordinate projections can still discard correlations between the two inferred coordinates; the output is a necessary outer constraint.

The leading matrix acts on (G*cos(b), G*sin(b)). Its determinant uses the factored sine of the amplitude separation. Rank admission checks both positive response scales and the positive sine factor separately. The inverse cancels redundant scale factors and divides successively, avoiding needless rounding of small products. rank_certified may therefore be true when the diagnostic determinant_bounds has a zero lower endpoint. Equal amplitudes or unresolved positivity of an individual factor yield unavailable, not a conclusion that the full nonlinear response has no information. Failed source or numerical chart admission also yields unavailable. Strictly empty necessary intersections yield incompatible; touching boundaries are retained. Nonempty nominal angle and gain projections yield bounded_candidate, without asserting that their Cartesian product, or any complete state, realizes the readings. The actual initial long-arc mean expands the nominal angle interval by phase_radius/8 on either side, without clipping to the nominal prior.

The output fields are nominal_bulk_angle_outer_bounds, actual_long_arc_mean_outer_bounds and readout_gain_outer_bounds. raw_transformed_bounds precedes prior intersection; transformed_coordinate_bounds retains the resulting coordinate box. Separate transformed_observation_error_radii and transformed_flow_error_radii expose the two error budgets. Inspect unavailable_reasons and incompatibility_reasons with the status.

The report installs no events and integrates no trajectory. A broad positive gain prior remains a premise; model-based gain inference is distinct from independent sensor calibration. Law, clock, input and observation assumptions are supplied. Export uses tnfr.sine-two-pulse-inference.v1, shared exact SDK projection and atomic writing. See the workflow. The separate reserved software result retains its complete source-to-reading evidence; this generic inverse API does not itself establish that preparation or generate those observations.

Independent complete-flow readout from a primitive source box

bound_sine_two_port_readout(*, initial_form_bounds, initial_phase_bounds, phase_increment, probe_duration, order) in relational_sine_two_port_readout.py returns SineTwoPortReadout. All five keyword inputs are required. Each state channel supplies exactly eighteen ordered pairs of finite real endpoints, in node order 0..17. Forms remain signed; phases are continuous radian lifts. Shared exact/represented-real admission precedes outward interval arithmetic. Boolean, nonfinite, nonzero-underflow, malformed and reversed primitive bounds reject; existing interval objects do not bypass that boundary. The amplitude lies in [0,1] and fast duration in (0,1]; order is an ordinary non-Boolean integer from one through sixteen.

The support is the two unit C9 rings with contacts (0,9) and (1,10). The held capacities are all one. The complete smooth sine law has beta=1, e=1023/1024, w=1/1024 and gamma=1/(1023*pi). Both shared rate rows are transformed to tau=e*t. The supplied event adds phase_increment*(e_4-e_5) to phase and leaves every form unchanged. Zero amplitude means no event: every source coordinate remains unchanged before the same complete flow starts. It does not freeze either rate row. All thirty-six original state coordinates, including their common means, then evolve for probe_duration. A primitive source box is an admitted conditional preparation, not evidence of its acquisition or physical origin. For successive segments, re-admit all thirty-six previous endpoint bounds as primitive source pairs. A zero-amplitude segment continues that full state without a reset or an additional phase jump.

The producer uses validated_box_taylor_step, with one declared step and order. Strict Picard inclusion has its fixed sixteen-iteration budget. Every lower Taylor coefficient is evaluated on the full initial box, and the order-plus-one remainder on the admitted whole-time tube. The source-box Taylor argument distinguishes these two sets. No center substitution, comparison-radius propagation, adaptive split, changed precision or retry is performed. The direct-box kernel's sixty-four-coordinate work cap is separate from the unchanged twenty-four-coordinate comparison-flow cap.

The smooth-domain callback applies globally to this complete sine field. Its positive domain bound does not certify an acute chart, winding identity, maintenance or native Arg-flow continuation. A failed tube or Taylor calculation returns status unavailable, failed_tube and unavailable_reasons; step, true_increment_bounds and endpoint_readout_bounds are then None. Status admitted, also exposed by .admitted, requires the complete requested step. Invalid declarations raise before flow evaluation.

baseline_readout_bounds retains the pre-event local form difference x_4-x_5. The admitted step retains its initial_box, tube, series, local_remainder_bounds, increment, endpoint, positive Picard margin, domain bounds, elapsed time and order. true_increment_bounds projects step.increment with q=e_4-e_5. That increment removes the common order-zero coordinate symbolically; subtracting independently enclosed endpoint and initial readings would discard this correlation. endpoint_readout_bounds remains a separate raw endpoint observation. Only coordinate endpoints may be intersected with the Picard tube; that intersection does not shrink the direct increment.

No inverse error formula, equilibrium root, acquired-source report or cached forward verdict is consumed. Gain, offset and sensor-error realizations are absent from this API. A separate observation model must apply them to the retained raw readings and correlated increment before any inverse request. Export uses tnfr.sine-two-port-readout.v1, shared exact SDK projection and atomic writing. See the workflow and the separately frozen reserved protocol.

Conditional geometry inference from one calibrated local increment

infer_sine_two_port_geometry in relational_sine_two_port_inference.py returns SineTwoPortInference. The theorem owner defines a complete affine family on the same eighteen-node two-port graph: donor bulk angle b, donor short angle 4*pi-8*b, receiver short angle c, receiver bulk angle (2*pi-c)/8, and opposed contact gaps. The family does not impose criticality. Its continuous unknown geometry is therefore not the unique equilibrium already selected by the compatibility theorem. Supplied initial form and phase norm errors retain every node.

All keyword inputs are mandatory: bulk_angle_bounds, receiver_short_angle_bounds, form_radius, phase_radius, phase_increment, probe_duration, recorded_increment_bounds, readout_error_bound, readout_gain_bounds and refinements. Bounds contain two primitive real scalars; scalar admission uses the shared exact/represented-real boundary before interval arithmetic. Booleans, nonfinite values, malformed or reversed bounds and unsupported domains reject. The prior lies within 11/8 <= b <= 3/2 and 2/3 <= c <= 1 in radians; the retained source errors must preserve the strict acute pre-probe chart. This is a declared preparation domain, not a fitted range. Radii and readout errors are nonnegative; amplitude and duration lie in (0,1], gain endpoints are positive, and refinements is an ordinary integer from one through sixty-four. Strict source acuity is a sufficient admission test; failure leaves the report unavailable rather than asserting that every source state is invalid. The complete law fixes gamma=1/(1023*pi), held unit capacities and fast clock tau=(1023/1024)*t. Input amplitude is in radians; elapsed time is in that fast clock. These declarations do not establish laboratory calibration or infer the law from the supplied response.

The event is theta_plus=theta_minus+a*(e_4-e_5), with form unchanged. The observation uses two scalar readings of the same local form difference, with one constant positive gain selected from readout_gain_bounds. The bound readout_error_bound applies to each scalar reading in recorded units. The inverse retains both errors and the gain interval, then the complete-flow remainder. A constant shared observation offset cancels; an unbounded offset drift does not. The input amplitude, elapsed clock, law and gain calibration must be admitted independently of the response.

The new remainder includes the nonstationary nominal port currents. Their first three readout moments vanish by graph locality, yielding a fourth-order time bound for their background contribution. Form relaxation, phase-source errors, finite pulse transport and nonlinear phase motion are also retained. The old equilibrium-dipole remainder cannot be reused without this addition.

The inverse returns a necessary outer enclosure on the declared monotone branch. A retained candidate does not prove that any complete state jointly realizes the observation; a wide enclosure is not a constructed ambiguity. An excluded band contradicts the joint family, law, calibration and error premises, without selecting which premise failed. The nominal family coordinate and the actual pre-probe mean donor long-arc angle differ under residual phase uncertainty; the latter requires the proved additional phase_radius/8 allowance. The API reads no equilibrium root, frozen evidence file or incoming cached verdict. The report installs no event and executes no trajectory. Acquisition, post-event maintenance, full-state reconstruction and physical identification are outside this inverse contract.

Status bounded_candidate returns nominal_bulk_angle_outer_bounds and actual_long_arc_mean_outer_bounds; incompatible certifies strict exclusion under the combined premises. Status unavailable retains unavailable_reasons when source acuity or positive interval division is unresolved, including an exact positive scale too small for the interval grid. finite_remainder_upper_bound requires source admission. whole_window_acute_certified is optional evidence, not an inverse gate. boundary_refinement_counts and refinement_precision_limited retain numerical work and unresolved signs; unresolved comparisons cannot exclude candidates. The direct schema is tnfr.sine-two-port-inference.v1, with shared exact SDK projection and atomic export.

See the usage guide and the single execution queue for the distinction between this conditional theorem and the separately retained inference assessment. That first three-case evaluation passes its forty-three frozen conditions using full-law response enclosures and independently supplied software calibration references. Its public-packet boundary and posterior truth checks establish only the declared computational result. They do not admit an arbitrary source or replace an independent physical measurement bridge.

Interior phase dipole after a finite complete-law warmup

assess_sine_two_port_dipole(*, warmup_duration, form_radius, phase_radius, phase_increment, probe_duration, readout_error_bound, contrast_threshold, work_allowance) in relational_sine_two_port_dipole.py returns SineTwoPortDipole. All eight scalars are mandatory. Shared admission preserves exact rationals and otherwise applies finite represented-real conversion; Boolean, nonfinite and nonzero-underflow inputs reject. Warmup duration, both radii, readout error, contrast threshold and work allowance are nonnegative. Phase increment and probe duration lie in (0,1]. Durations use the fast structural clock tau=e*t, phase uses radians and form retains its declared structural unit. The public decay budget requires floor(kappa*warmup_duration)<=4096, with exact kappa=1/2233865700000; unsupported domains reject before target evaluation.

The source premise is two already forward-trapped full sine-law families inside their admitted relative radius 1/12, with coarse form and phase norm caps 1/12. Those caps do not admit every point of an independent product box. The joined state uses its full degree metric and one conserved form/phase mean. The unjoined state removes each C9's own means, then uses the combined eighteen-node degree norm. Independent per-ring caps cannot replace that combined norm. Both actual state channels, all source errors and every consumed complete-law row remain present.

form_radius and phase_radius are desired bounds after the declared warmup, not a new initial preparation. The shared modified-energy kernels apply through the degree-metric isometry proved in the dipole owner. Each returned squared norm must lie strictly below its requested squared radius. Exact dyadic decay bounds avoid losing small residuals at a fixed interval square-root floor. This quantitative continuation neither establishes earlier acquisition nor resets the reached state to its target. No incoming report or cached acquisition flag supplies the source premise.

The supplied event is theta_plus=theta_minus+a*(e_4-e_5), with form unchanged, on each actual support. Nodes four and five have degree two in both models; the event preserves every relevant weighted phase mean, and all form means are unchanged. The scalar observation is q.T*(x_after-x_before), with q=e_4-e_5. It uses the identical coefficient vector in both supports, with degree-metric dual norm one. readout_error_bound applies to one scalar q.T*x reading, not to each node; the recorded joined-minus-unjoined increment includes all four independently bounded reading errors.

Fresh compatibility admission uses the fixed thirty-two outer and sixty-four inner root refinements. The joined bulk-angle interval and the unjoined twist determine the correlated ideal contrast. Retain that shared coefficient before adding the two complete-flow remainder bounds and four readout errors. Equal local degrees and adjacency do not imply equal finite-time propagators: the remainder includes each full support, pre-probe form and phase residuals, form relaxation and nonlinear phase motion. An initial derivative alone is insufficient.

The separate phase-blind alternative declares both rows, x_tau=-A*x, theta_tau=gamma*A*x, on each actual support with the same capacities, clocks and original source family. Its form is unaffected by the phase intervention, but its recorded background is not silently replaced by zero. A separate heat warmup must place its residual below form_radius; the recorded contrast is then bounded above by 4*probe_duration*form_radius+4*readout_error_bound. This comparator has its own complete law. Sine-law acquisition and recovery certificates are not transferred to it.

The frozen protocol fixes warmup_duration=285934809600000, form_radius=phase_radius=2^-40, phase_increment=2^-12, probe_duration=2^-10, readout_error_bound=2^-50, contrast_threshold=2^-38 and work_allowance=2^-21. The warmup starts at the unchanged capture handoff sigma=1025, so its end is at tau=1025*1023**2*pi**2+warmup_duration. The actual readout adds probe_duration in that same fast clock. The original thirty-six source errors remain 1/65536. This is a new continuation from that preparation; the earlier uniform form pulse is not inserted into its history. The retained capture audit and the unjoined original-family trapping argument supply separate source obligations. No laboratory clock or measurement calibration follows from these values.

target_admitted, warmup_certified, geometry_certified, response_certified, heat_warmup_certified, heat_control_excluded, work_certified, identity_certified and recovery_certified expose distinct obligations. Status certified_dipole requires the complete chain; otherwise it is unavailable. recorded_contrast_bounds includes full-law and readout errors; response_margin and phase_blind_exclusion_margin must be strictly positive. Work upper bounds may equal the supplied allowance, while post-event radius and storage margins must be strictly positive for both sine models. Signed work intervals and positive-work flags are separate from passivity or autonomous event selection.

Delivered response, work and identity bounds require certified warmup. Candidate fields remain distinguishable from those conditional bounds; unavailable delivered fields are None. model_order fixes joined then unjoined in per-model fields. The direct schema is tnfr.sine-two-port-dipole.v1, with shared SDK exact projection and atomic export. The assessment installs no event and runs no trajectory or earlier capture producer. Failed sufficient margins do not demonstrate a failed physical response.

The execution plan owns the gate's status. The retained first assessment passes all thirteen fixed conditions, including both original-source handoffs, finite warmup, response, the heat comparison and post-event identities. Its protocol, proof and producing source were preserved before evaluation; no source, threshold or horizon was changed afterward. The earlier form-probe export failure and recovery remain separate historical evidence. The certified contrast concerns local geometric response under the declared preparation, supports and complete laws. It excludes the specified phase-blind comparator; it does not select a unique constitutive law or establish a topology-independent intrinsic or physical property. See the inspection guide.

Supplied donor pulse, finite transmission and identity recovery

assess_sine_two_port_probe(*, form_radius, phase_radius, pulse_amplitude, probe_duration, readout_error_bound, contrast_threshold, work_allowance) in relational_sine_two_port_probe.py returns SineTwoPortProbe. All seven primitives are mandatory. Shared scalar admission preserves exact rationals and otherwise applies the finite represented-real contract; Boolean, nonfinite and nonzero-underflow inputs reject. The form and phase radii, readout error, contrast threshold and work allowance are nonnegative. Pulse amplitude is strictly positive and 0<probe_duration<=1 in the fast structural clock tau=e*t. The phase radius uses radians; form, readout and work retain their model units.

The endpoint API is conditional: form_radius=X bounds the full degree-metric norm of centered form, and phase_radius=Y bounds the phase distance from the implicit joint equilibrium on the same mean leaf. It accepts neither a saved capture report nor a source-acquisition flag. Supplying these numbers does not establish that an arbitrary trajectory reached the endpoint ball. The probe theorem and the separately matched capture handoff give the declared original preparation its finite source-to-probe chain. The target geometry is freshly admitted with the fixed compatibility proof.

The actual intervention adds one uniform form increment to all nine donor nodes, leaving every phase and support edge unchanged. It is a supplied hybrid jump, with a separate storage-work allowance. The same complete unforced normalized-sine law then resumes on the two-port support, with held capacities, e=1023/1024, w=1/1024, gamma=1/(1023*pi) and sigma=gamma**2*tau. The jump changes the global degree-weighted form mean by half the pulse amplitude and preserves the phase mean. Recovery therefore concerns the unchanged joint shape on the new form-mean leaf.

The observation is the receiver's degree-weighted mean form increment between the pre-pulse state and the fixed elapsed endpoint. The matched unjoined control consists of two separate C9 rings evolving independently from the same original preparation under the same coefficient and clock rules, followed by the same donor pulse. Its receiver mean is exactly conserved. The same observation rule uses receiver degree mass 20 in the joined model and 18 in the unjoined model; its coefficient vectors are consequently different. No contact is removed from an acquired state, and no mean, residual or hidden coordinate is reset. Four scalar readings form the contrast: before and after in each model. Their independently bounded errors contribute at most four times readout_error_bound; no error cancellation is assumed.

The reserved protocol uses the actual capture endpoint at sigma=1025, equivalently tau=1025*1023**2*pi**2, with endpoint allowances X=1/8192, Y=1/1024. It supplies pulse 1/2048, elapsed fast time 1/4, per-readout error 1/67108864, strict recorded contrast above 1/262144 and work allowance 1/2000000. The preceding full family retains all thirty-six independent original errors 1/65536; the endpoint allowances do not redefine that preparation. The disconnected control uses the same original elapsed structural time. No laboratory clock or sensor calibration is supplied by these numerical choices.

target_admitted, response_certified, work_certified, joined_identity_certified and recovery_certified expose the separate proof obligations. Joint status certified_probe requires every requested obligation; otherwise it is unavailable. Failed sufficient margins do not demonstrate trajectory failure. recorded_contrast_bounds includes all four readout errors; response_margin must be strictly positive. joined_work_bounds retains both actual event-work endpoints, while work_allowance_margin may equal zero. Post-event radius and capture storage margins must both be strictly positive. positive_supplied_work_certified is a separate statement about the lower work bound, not the passivity of the event. The whole-window form and phase candidates become certified bounds only after target admission; unavailable conditional fields remain None. The direct report schema is tnfr.sine-two-port-probe.v1, with shared SDK exact projection and atomic export. Projection alone neither authenticates source acquisition nor installs the declared intervention on a live network.

The original frozen assessment attempt failed during control-report export after executing the primary assessment; it saved no complete response. The retained result comes from a separately archived deterministic export recovery and passes all ten fixed conditions. It preserves the original failure and scientific inputs; it is not a successful first attempt. The correction changes only projection of the auxiliary control mapping and does not change the scientific runtime. The execution plan owns its status. The source-evidence audit must re-admit primitive source, law, geometry and retained numerical evidence and rebuild consumed handoff bounds. It must state which obligations rely on the retained execution: matching hashes and a consistent chain are not independent numerical revalidation. The earlier producer and artifacts remain unchanged.

The shared research owner sine_two_port_handoff.py provides audit_sine_two_port_capture_record(record, *, protocol) for the fixed capture-v1 mapping and audit_sine_two_port_capture_handoff(evidence_directory) for its retained artifact directory. Incomplete, malformed or mismatched records reject. The mapping audit reconstructs the source preparation and support, checks the admitted root-bracket signs, rebuilds the implicit target, checks the complete reference metric chain and whole-tube margins, and recomputes the analytic original-form/phase handoff. Cached verdicts and derived report bounds are ignored. The directory wrapper additionally checks exact artifact sizes and hashes, archived inventory, protocol bytes, source revision and runtime-overlay associations. Mapping admission alone checks none of those file associations.

Both return SineTwoPortHandoffAudit. Its direct to_dict() schema is tnfr.sine-two-port-handoff-audit.v1; this research audit is not registered as a generic SDK relational report. numerical_execution_replayed and provenance_authenticated remain false. The compact retained reference omits full Taylor coefficients, so the archived validated execution remains an explicit conditional premise for Picard inclusion, local remainders and computed endpoint centers. Root-sign and tube checks do not silently replace that premise with a fresh integration. The rebuilt endpoint radii must fit the fixed probe allowances 1/8192 and 1/1024; no previous capture producer or target root search is run by these readers.

This gate tests transmission on supplied support together with post-pulse identity maintenance. A pure heat law also transmits the ideal donor pulse, so the response is not a unique signature of winding, acquired deformation or sine coupling. Autonomous support selection, passive attachment, organization-specific response and physical identification remain separate. The inspection guide keeps the conditional API separate from the original-family source handoff.

Validated reference transit and complete-state capture

assess_sine_two_port_capture(*, form_error_radius, phase_error_radius, reference_duration, time_step, order, max_steps) in relational_sine_two_port_capture.py returns SineTwoPortCapture. All six primitives are required; no earlier capture flag or endpoint report substitutes for them. Shared admission preserves exact rationals and otherwise applies represented-real conversion; Boolean, nonfinite, negative-radius and nonzero-underflow inputs reject. The radii use the signed form coordinate and phase radians. reference_duration and time_step use the slow clock. The public reference duration is positive and at most 1024, and 0<time_step<=1/4; Taylor order is an ordinary non-Boolean integer in [1,16], and the step budget an ordinary non-Boolean integer in [1,4096]. The ceiling of duration divided by step must fit that budget; the final step is truncated to the exact requested endpoint when necessary. Unsupported domains raise, while failed proof margins or Taylor inclusion return unavailable evidence without an adaptive retry. The public handoff time is reference_duration+1; the reserved protocol below selects one fixed set of these primitives.

The capture owner defines a fixed-budget test for the same prepared family as the finite transit: undeformed midpoint-aligned C9 classes (2,1), nominal zero form and independent nodewise form and phase errors 1/65536. The eighteen-node support, complete positive-loss law, capacities, actual degrees and structural clocks are unchanged. All thirty-six source errors and their actual conserved means remain present. The joint equilibrium supplies a capture target, not a replacement source.

Only the nominal gradient reference is folded to eight coordinates. The folding retains an exact reconstruction and the inherited full-network metric; its symmetry does not restrict actual preparation errors. The shared retained-metric Picard/Taylor kernel must admit every whole-time tube and its acute metric-growth premise. A stored endpoint without the complete validated prefix cannot certify the reference's passage.

The declared reference horizon is slow time 1024, with fixed step 1/4, Taylor order eight and at most 4,096 steps. The target enclosure is freshly rebuilt with thirty-two outer and sixty-four inner refinements. The strict whole-reference acute-margin requirement is 1/2048 radians, and the endpoint distance allowance in the full degree metric is 1/2048 radians. These are distinct from the original full-source uncertainty and from the complete-flow comparison error.

The reference is then continued analytically for one more slow unit, while the actual full flow continues without reset. At slow time 1025, the handoff needs both its refined original-form bound and its phase bound, with complete excess storage strictly below 1/648000 and every local chart premise proved. The invariance principle gives subsequent convergence only if that entire chain passes. A small reference phase distance does not establish actual capture or remove the retained form coordinate.

The execution plan owns the gate's status. Before the first assessment, preserve its protocol, proof and complete producing source; keep the frozen horizon, precision, step, order, root and preparation budgets unchanged afterward. An incomplete reference, unresolved sign or failed sufficient handoff remains unavailable evidence, not a trajectory failure or permission to retune the protocol. Earlier compatibility, obstruction and finite-transit results retain their original scope and evidence.

The retained first assessment passed all ten frozen conditions at the declared inputs, including every reference step and the complete-state capture handoff. Its certified later convergence concerns the entire original uncertainty family on each member's conserved-mean leaf, not only the numerical reference.

preparation_admitted, target_admitted, reference_validated, reference_endpoint_certified, full_comparison_certified and capture_certified expose the separate stages. The joint status is certified_capture only when the complete chain passes; otherwise it is unavailable. A validated shorter prefix remains distinguishable from the requested complete reference. validated_reference_duration, failed_tube and unavailable_reasons retain that stopping evidence.

Each compact reference_steps entry retains its center and metric radius at both endpoints, the whole tube, Picard inclusion margin, local metric error and all twenty edge-domain margins. Those domain margins are the excess over 1/2048; reference_minimum_acute_margin instead reports the absolute acute margin in radians. The metric radius is carried forward, not replaced by a coordinate-box radius. These records project the shared step certificate; they do not independently authenticate its execution.

reference_target_distance_upper_bound includes the retained numerical radius and fresh target-enclosure uncertainty. When the complete comparison passes, endpoint_phase_distance_upper_bound, endpoint_relative_form_norm_upper_bound and endpoint_excess_storage_upper_bound describe the actual handoff at full_slow_horizon=reference_duration+1. The phase comparison uses each member's conserved phase mean and the form norm is for its centered P_M*x. capture_storage_margin subtracts that excess bound from 1/648000; strict positivity and the local phase chart are required. Candidate bounds are not certified errors, and unavailable conditional fields remain None.

The direct schema is tnfr.sine-two-port-capture.v1, with the shared SDK exact projection and atomic export. A current-source assessment and an archived first response have distinct provenance; projection alone does not authenticate either.

This assessment installs no contact event, forcing or state reset. A successful capture certificate would concern the supplied source and fixed support; it would not derive contact occurrence, passive attachment work, laboratory time or physical binding. The inspection guide keeps prospective setup and the subsequent retained response separate.

Finite directional deformation under the complete two-port law

assess_sine_two_port_transit(*, form_error_radius, phase_error_radius) in relational_sine_two_port_transit.py returns SineTwoPortTransit. Both required radii are finite nonnegative reals. Shared admission preserves exact rationals and applies represented-real conversion to other admitted scalars, rejecting Boolean, nonfinite or negative values and nonzero values lost during conversion. The form radius uses the model's signed form units; the phase radius is in radians, not turns.

The fixed support has two increasing-index C9 rings, classes (2,1) and unit contacts D0--R0, D1--R1. Both rows use actual joined degrees, held unit capacities, beta=1, e=1023/1024, w=1/1024 and tau=e*t. The declared slow clock is sigma=gamma**2*tau, with gamma=1/(1023*pi). The complete window is 0<=sigma<=1/4; no larger horizon is certified. Its endpoint is original structural time t=261888*pi**2, without a laboratory-unit interpretation.

Nominal form is zero. The nominal phase lifts are theta_Dj=(j-1/2)*4*pi/9 and theta_Rj=(j-1/2)*2*pi/9, giving undeformed internal twists and opposite contact gaps +pi/9, -pi/9. Each of the eighteen form and phase coordinates may vary independently within its radius, with no imposed reflection or zero-sum constraints. Each member retains its actual conserved means. This is a conditional theorem about that complete family, not admission of an observed array or a target chosen from the implicit equilibrium. The midpoint alignment is a supplied source premise; it does not reorient an earlier experiment after its response. Subtract the single full degree-weighted mean from the displayed nominal lifts to obtain the report's nominal phase gauge. This common shift leaves every edge gap unchanged and does not recenter the components separately.

The proof uses a gradient reference, the full storage loss and a mixed-coordinate comparison whose error grows with the square root of the slow horizon. It includes the initial form, fast transient and phase uncertainty, then reconstructs both full rows. The reference's common phase origin matches each member's conserved mean; the report bounds those source means without requiring them to be zero. The storage floor uses the isolated ring minima, not a solved joint equilibrium. A first-exit argument admits the reference, mixed and actual phases in one acute chart throughout the window. Neither the reference nor the detached assessor installs a flow, contact event, forcing or reset. In particular, the nominal full phase velocity is initially zero because the nominal form is zero; the nonzero initial reference velocity cannot replace it. The comparison retains this fast transient.

initial_excess_storage_upper_bound is the sufficient bound 10/81 + 40*phase_error_radius + 40*form_error_radius**2; admission requires this sufficient upper bound to be strictly below 1/8, with equality unavailable. energy_budget_margin and energy_budget_admitted retain that premise separately. joint_error_candidate, scaled_form_norm_candidate, phase_error_candidate and acute_margin_candidate are intermediate proof bounds, not certified trajectory observations. In particular, a positive acute candidate alone does not bypass the complete bootstrap.

Only successful complete admission makes error_certified and whole_window_acute_certified true and supplies phase_error_upper_bound and relative_form_norm_upper_bound. The phase and mixed-coordinate errors use the nominal reference shifted to each member's conserved phase mean. The form norm bounds the centered form P_M*x; initial_form_mean_bounds separately retains its common mode. The reference's common phase shift cancels from every gap change. The preserved integer periods on that window are (2,1,0). short_arc_change_lower_bound then bounds both the donor's decrease and the receiver's increase, each relative to that member's own initial short gap. direction_margin subtracts the fixed 1/32 radian threshold; strict positivity is required for direction_certified.

The joint status is certified_directional_transit when the direction passes, transit_bound_only when the full comparison is admitted without the required direction, and unavailable when the comparison premises fail. Conditional output bounds are None when the comparison is not certified. Unavailable or failed sufficient margins do not demonstrate a reverse motion or a failed trajectory. The direct schema is tnfr.sine-two-port-transit.v1, with shared SDK exact projection and atomic export.

This finite directional result does not establish basin entry, final convergence or all-future acute retention. The earlier scalar-storage obstruction remains valid, as do the frozen compatibility reports. The new theorem executes no numerical trajectory or frozen producer. Contact work, support birth and physical identification remain separate questions. The usage example assesses the full declared uncertainty family.

A boundary witness excludes direct scalar-storage handoff

assess_sine_two_port_handoff_obstruction(*, phase_error_radius) in relational_sine_two_port_compatibility.py returns SineTwoPortHandoffObstruction. The required radius is a finite nonnegative real in radians. Exact rationals are preserved without binary64 materialization; other admitted real scalars use the shared represented-real contract, rejecting nonfinite conversion or loss of a nonzero value. Boolean, nonfinite and negative values reject. It bounds each node's continuous phase-lift error around an undeformed isolated uniform twist. It is not a turn radius, a target-equilibrium tolerance or a flow duration.

The fixed support and complete positive-loss law are those of the two-port compatibility assessment, with ordered classes (2,1) and sector (2,1,0). This source class allows every relative component origin and arbitrary finite signed form coordinates. Their full form storage is nonnegative. No symmetry constraint, independent recentering after contact, original formation certificate or particular trajectory is an input to this assessment. The report is conditional evidence for that family, not admission of an observed live state.

The proof constructs a realizable state on the acute-sector boundary whose storage is below every source's total storage whenever its strict gap passes. The explicit path to this witness is a geometric proof device, not a solution of the evolution rows. Its full edge reconstruction and integer periods preserve the correlations required by the sector. It follows that no valid lower bound on every boundary face can put these sources below the boundary, as required by the target-free capture theorem. For an initially nonacute source that theorem also lacks its initial-domain premise. Neither exclusion implies dynamical escape or failed capture. The displayed zero-form, zero-mean witness represents any source conserved mean leaf after a uniform form shift and common phase shift, both of which preserve its storage and edge geometry. These proof representatives are not state resets.

ideal_storage_gap_lower_bound=17/13824 is the conservative rational consequence of the proved ideal gap 17*pi/41472 and pi>3. phase_storage_lipschitz=40 retains all twenty edges, each with phase-gap error at most twice the nodal radius. Thus phase_storage_error_allowance is forty times the radius and storage_gap_lower_bound is the ideal lower bound minus that allowance. No omitted form coordinate can reduce the source's lower storage bound. The strict conservative threshold is max_certifying_phase_error_radius=17/552960.

handoff_obstruction_certified is true and status is certified_handoff_obstruction only when storage_gap_lower_bound>0. Otherwise the report is unavailable with strict_storage_gap_not_certified; equality at the conservative threshold is unavailable rather than an obstruction verdict. This does not erase the stronger analytic inequality or establish capture for larger errors. The report rebuilds its fixed support, degrees and boundary witness directly; it neither consumes a cached equilibrium verdict nor reruns the nested root enclosure or any frozen producer.

The direct schema is tnfr.sine-two-port-handoff-obstruction.v1; shared SDK projection and atomic export preserve exact rational evidence. This exact method obstruction is separate from the retained static compatibility experiment. Actual source-to-basin evolution, attachment work and autonomous contact occurrence remain separate obligations. The usage example assesses a declared radius without performing a simulation.

Joint equilibrium through two distinct component ports

assess_sine_two_port_compatibility(*, classes, outer_refinements, inner_refinements) in relational_sine_two_port_compatibility.py returns SineTwoPortCompatibility. All three primitives are required. classes supplies exactly two ordinary non-Boolean integers in {1,2}, in donor/receiver order; all four ordered pairs are admitted. Both refinement budgets are ordinary non-Boolean integers in [1,64]. The budgets control finite enclosure work, not physical evolution time or a new law parameter.

The support is fixed: donor nodes 0..8, receiver nodes 9..17, both increasing-index C9 cycles, and unit contacts 0--9 and 1--10. Both rows use the actual joined degrees. The bridge_* fields name these contact edges; neither is a graph cut edge on the joined support. The complete positive-loss sine law retains e=1023/1024, w=1/1024, beta=nu_i=1 and tau=e*t. The phase geometry has the supplied ring periods and zero period on the oriented interface cycle D0,R0,R1,D1,D0. Uniform form is fixed at zero; a single global degree-weighted phase gauge fixes its displayed lifts. Independently recentered components or a supplied favorable relative origin do not replace the joint balance equations.

The theory owner derives the enclosure, reusing the existing circulation/sector uniqueness and full-law local recovery arguments. Root existence and strict acute admission must be proved; a small residual at a represented midpoint is not an exact equilibrium. Unresolved strict signs yield an unavailable numerical certificate, without promoting that midpoint or selecting a new budget.

canonical_root_turn_bracket always describes the (2,1) construction; the reversed class order uses exact component exchange. Matched classes use analytic uniform twists and have no root brackets. The short, bulk, bridge and nodal turn enclosures are correlated images of that construction. nodal_turn_affine_coefficients and edge_turn_affine_coefficients retain rows (constant, donor-short coefficient, receiver-short coefficient), with edge_integer_offsets preserving the closing-edge branches. Their shared variables are constrained by the root equations. Independent corners of these intervals are not a family of equilibria or an admitted preparation.

The exact sine-symbol factorization reconstructs every nodal balance. balance_equation_bounds and nodal_current_residual_bounds separately check numerical consistency; full_nodal_residuals_consistent does not assert that a midpoint is critical. phase_hessian_gap_lower_bound combines strict edge cosines with an independently checked full-support Laplacian bound. implicit_equilibrium_certified, acute_geometry_certified and local_attraction_certified retain their separate meanings; status is certified_compatible only when the complete numerical admission succeeds, otherwise unavailable with explicit reasons. An unresolved root leaves target numerical fields None; a later consistency or margin failure can retain diagnostic enclosures. Neither failure refutes the analytic existence theorem.

uniform_pair_compatible and uniform_pair_excluded concern the undeformed isolated twists. uniform_pair_nodal_current_bounds displays the reference with its first contact aligned. Exclusion for every relative rotation uses the theory's two simultaneous contact-balance conditions, not that one displayed residual alone.

This detached assessor admits a joint equilibrium and local attraction on its conserved-mean leaf. The original isolated class-formation certificates are not inputs or a proof that their contact trajectories reach this basin. Neither the bridge-tree decomposition nor the central-port reduced model applies unchanged to this two-port support. Balanced stationary sine currents can be nonzero while every nodal rate vanishes; they do not establish ongoing nodal motion, dissipated power or a physical current.

The direct schema is tnfr.sine-two-port-compatibility.v1; shared SDK projection and atomic export preserve exact enclosures and availability. No trajectory, support event, work payment, autonomous preparation or physical binding is supplied by this assessment. The usage guide reads the saved primary and matched-control certificates.

Ordered heat comparison for uniform form tracking

assess_sine_port_form_tracking in relational_sine_port_form_tracking.py returns SinePortFormTracking. It requires the same fourteen keyword primitives and applies every domain, source, support, gap and work admission of assess_sine_port_relaxation. It accepts no prior report. baseline_certificate is freshly rebuilt from those primitives, including its original phase/form envelopes, separate verdicts and limitations. This is a new comparison method for the unchanged full and reduced trajectories. The proof and protocol own the ordered convolution and finite-prefix gain argument.

The fixed reference phase Hessian is a proof device: the surrogate's nonlinear bridge remains unchanged. bridge_hessian_variation_upper_bound and even_bridge_forcing_upper_bound retain its time-dependent deviation from that reference. Odd port values vanish, so this additional bridge term belongs only to the even comparison. Both odd_heat_bounds and even_heat_bounds expose the admitted spectral bounds, exact dyadic band count, heat-integral and filter gains, initial-state contributions, forcing contribution and loop margin. Each loop margin must be strictly positive. The argument retains operator order, initial phase-to-form exchange and bounded time-varying forcing; it assumes neither commuting operators nor a derivative of that forcing.

all_time_form_error_upper_bound combines the two centered parity estimates and restores the original form_mean_error_floor once. all_time_phase_error_upper_bound, its margin and its verdict retain the fresh baseline values unchanged. Both *_mean_error_floor fields remain upper uncertainty budgets for possible constant offsets, not measured lower errors. The same origin-span phase and form allowances apply, with strictly positive outward lower margins required for resolution. No allowance is enlarged.

Missing baseline envelopes or a failed new loop margin leave the new form bound unavailable and add heat_form_envelopes_not_certified to unavailable_reasons. The phase evidence can still remain available. all_time_envelopes_certified requires the new comparison prerequisites; resolution_limitations separately records either failed channel test. With every theorem and supplied-work prerequisite admitted, status is full, phase_only, form_only or envelopes_only; otherwise it is unavailable. Mathematical channel flags do not override a failed work policy. joint_resolution_certified is the conjunction of the two channel flags.

The direct schema is tnfr.sine-port-form-tracking.v1. Shared SDK projection and atomic export retain exact bounds, optional fields and the fresh nested baseline. The earlier frozen phase_only assessment and its false joint stopping criterion remain unchanged. A separately passing new method is not a correction to that historical verdict, a measured trajectory error or physical validation. The usage guide reads both certificates without invoking their assessors.

Uniform full and reduced tracking through recovery

assess_sine_port_relaxation(*, classes, contacts, phase_origins, formation_time, relaxation_duration, form_error_bound, phase_error_bound, endpoint_radius, radius, decay_power, work_allowance, normalized_gap_lower_bound, phase_resolution_fraction, form_resolution_fraction) in relational_sine_port_relaxation.py returns SinePortRelaxation. All fourteen primitives are required. This detached assessment compares the unchanged degree-aware surrogate with the actual fine trajectories for all uninterrupted postcontact times. It neither accepts a prior report nor extends a finite-window budget by changing its horizon. The proof and frozen protocol own the invariant charts, parity comparison and channel resolution rule.

Support, classes, origins, original source errors and formation/relaxation budgets retain the composition admission, including its sixteen-component implementation cap and held unit law. There is no contact-duration argument. The proposed normalized gap satisfies 0<normalized_gap_lower_bound<=7/30; the reader checks an exact rational positive-semidefinite inequality in the actual degree-weighted even quotient, rather than trusting the supplied number or an unnormalized graph gap. Failure of that check yields unavailable premises, not a substitute gap. Each resolution fraction lies in (0,1]. All scalar admission precedes arithmetic, retaining exact rationals and rejecting Boolean/nonfinite values.

unprobed_handoff and joined_bounds are freshly rebuilt from the original source families. Both full and nominal surrogate trajectories must remain in their admitted acute charts. The fixed curvature refinement and normalized gap feed independent even/odd comparison bounds. Initial errors, their exchange between the mixed comparison coordinates, and odd modes generated by the full interior sine law all remain present; target convergence alone does not establish the comparison.

all_time_phase_error_upper_bound and all_time_form_error_upper_bound bound every fine coordinate against the lifted surrogate, with the declared twists and component origins restored. phase_mean_error_floor and form_mean_error_floor are upper budgets for possible constant global weighted-mean offsets. Despite their field names, they are not measured positive errors or lower bounds: an actual member can have zero offset. These budgets remain in the final upper bounds; separately frozen component means are not assumed. all_time_envelopes_certified identifies valid uniform comparison bounds. Unmet source, joined-identity, gap or comparison prerequisites remain explicit in unavailable_reasons; unavailable bound fields are None.

With origin_span=max(phase_origins)-min(phase_origins), the phase allowance is phase_resolution_fraction*origin_span, and the form allowance is gamma*form_resolution_fraction*origin_span, where gamma=1/(1023*pi). The form allowance retains its interval enclosure. Separate phase_resolution_certified and form_resolution_certified flags require strictly positive outward lower margins; joint_resolution_certified requires both. A valid uniform envelope can fail either resolution policy. resolution_limitations reports these failed sufficient comparisons separately from missing theorem prerequisites. Zero origin span is valid but supplies no positive resolution allowance. These are mathematical coordinate scales, not sensor precision or a guarantee of resolving the earlier donor contrast.

With all source, theorem and supplied-work prerequisites admitted, status is full, phase_only, form_only or envelopes_only, according to the two independent channel tests. Otherwise it is unavailable. Mathematical envelope/channel flags can remain available when only the declared work allowance fails, but they do not override that joint admission failure. An envelopes_only result retains proved error bounds without resolving either requested channel; a single-channel status must retain its qualifier.

The direct schema is tnfr.sine-port-relaxation.v1; shared SDK projection and atomic export retain exact bounds, optional fields and each channel's verdict. Failure of a sufficient resolution bound is not an observed trajectory error or a proof that no sharper bound exists. No solver, new support selection, physical identification or practical-time claim is supplied. The usage guide shows how to inspect the retained bounds without rerunning their assessment.

Validated full-state observations of four probe histories

bound_sine_class_four_history_readout returns SineClassFourHistoryReadout on the fixed three-C9 support and complete normalized-sine law. Its admission proof uses the shared source-box Picard/Taylor kernel with all 54 coordinates. It consumes nine mandatory keyword primitives:

Primitive	Admission
initial_form_bounds, initial_phase_bounds	Exactly 27 ordered primitive real endpoint pairs each; phases are continuous lifts
first_probe_amplitude, second_probe_amplitude	Finite signed form jumps at the selected first/second nodes
delay, total_duration	Structural times with 0<=delay<=total_duration<=2
time_step	Strictly positive, at most 2
order	Ordinary integer 1 through 16
max_steps	Ordinary integer 1 through 4096, counting all unique kernel attempts, including a failed attempt

Optional first_probe_node=4, second_probe_node=4 and readout_node=22 select the two event coordinates and the observed form coordinate. Each must be an ordinary integer in 0 through 26, admitted before field construction. The defaults preserve the original same-donor protocol. Selecting other nodes does not transfer its target-specific work, identity or response theorems.

Shared interval and exact-or-represented-real admission rejects Boolean, nonfinite, reversed or unrepresentable inputs before flow construction. No class label, sensor error, prediction, inverse or source-report verdict is accepted. Class is encoded in the primitive phases. The one input box may cover a correlated acquired family; acquisition, original zero sums, identity and work remain external premises. The proof's symbolic source recipe preserves exact target enclosures and covers all actual family members without asserting that every independent box corner is acquired.

source_box is enclosed once in form-then-phase node order. The six segments are prefix_unprobed, prefix_first, neither, first_only, second_only, both. history_segment_indices retains the four ancestry pairs (0,2), (1,3), (0,4), (1,5). Every event changes only its selected form coordinate; all other forms, phases and hidden coordinates carry. Every suffix starts from its parent's complete endpoint. A zero-length segment applies its event without a flow step. The fixed step size is clipped at declared event/final boundaries; there is no adaptive retry.

Each successful step retains the complete initial box, strict Picard tube, Taylor coefficients, remainder and full endpoint. The smooth-law domain guard does not certify an acute chart, winding identity or event work. planned_unique_step_count, attempted_step_count and completed_step_count distinguish requested work, attempts and validated steps. A global first failure stops subsequent attempts and events. Segment statuses distinguish admitted, unavailable, budget_exhausted and not_attempted; the report preserves completed prefixes, failed source/time/tube, reasons and unattempted ancestry. Failure before an event leaves its preevent/initial boxes unavailable rather than applying that event silently.

Global status="admitted" requires all four suffixes to reach the common final time; otherwise it is unavailable. Completed individual receiver readings remain in completed_history_readout_bounds. The complete endpoint_readout_bounds, raw_endpoint_mixed_bounds, suffix_receiver_increment_bounds and mixed_readout_bounds are all unavailable until every history completes. A partial endpoint is not the requested final endpoint.

The primary mixed_readout_bounds combines suffix observed-form increments with coefficients (1,-1,-1,1). Each suffix sums step.increment[readout_node]; shared prefix values cancel exactly. If the second jump acts at the observed node, its equal contributions to second_only and both also cancel. raw_endpoint_mixed_bounds instead combines the four endpoint bands and remains separate evidence. Both are safe outer bounds; remaining cross-history or step correlations can be relaxed, and arbitrary marginal choices need not be jointly realizable.

Interval widths include supplied-state uncertainty, parameter rounding, truncation and accumulated enclosure widening. They are neither sensor noise nor automatically pure numerical error. No response-sign, model discrimination, formation or physical-readiness verdict is inferred. The schema is tnfr.sine-class-four-history-readout.v1; shared SDK projection and atomic export retain the full evidence and availability. New reports retain all three selectors. Legacy donor_node and receiver_node alias the first event and observed node respectively; receiver-named interval fields use the selected observer. Reconstruction requires explicit fields to agree with admitted selectors and aliases. Older reports missing all three selectors are accepted only under legacy defaults; partial, invalid or conflicting metadata rejects reconstruction. The complete interval/jet field uses _sine_flow.py and snapshots held law/degree/capacity factors without changing nodal arithmetic order. Changed factors require a new evaluator. The shared Taylor solver uses reconstruct_box_taylor_arithmetic to check exact interval shapes, source containment in its tube and retained Horner/remainder/endpoint arithmetic. That reusable operation does not regenerate derivatives or prove Picard inclusion; a detached reader still owns its law, source, event and stored-certificate premises. The distinct-neighbor protocol uses this owner with selectors (4,22,13) and its own common-source transfer. The guide uses an unrelated stationary fixture, not the reserved nonlinear design. The separately frozen evaluation protocol associates the canonical source cover and its acquisition premises with fixed inputs, observations and numerical policy. Its freeze receipt is not a response or attempt record. Forward intervals remain independent of the analytic prediction; source association, finite completion, open-band consistency, containment and strict zero/four-error/eight-error margins are separate checks. No reserved flow is part of freeze inspection. The retained first evaluation adds a separate attempt and byte-preserving response archive; it does not change this producer's domains or turn its source cover into an acquisition certificate.

Finite nonlinear separation with the complete event history

bound_sine_class_nonlinear_protocol returns SineClassNonlinearProtocol from the same eleven mandatory class/probe/source/work primitives. Its horizon domain is 0<=delay<=total_duration<=2; the older superposition API retains its 1/4 ceiling. Signed amplitudes, ordinary integer class admission, radius and budget domains are otherwise unchanged. Shared exact-or-represented-real admission occurs before model arithmetic. The new proof replaces the short-time comparison with heat contraction; it does not extend an invalid 1/(1-3*T) bound.

Each of the four histories begins at the same complete actual source, with componentwise form/phase Euclidean error bounds and separate zero sums as conditional premises. All nodal and hidden coordinates carry through the delayed event. The calculator consumes no measured readings, source report, fitted coefficients or cached source verdict. It certifies no formation, physical precision or autonomous preparation.

heat_polynomial_order=32 is fixed. heat_cubic_channel_coefficients contains exact rational integrals for the outer-cycle cosine, mediator-cycle cosine and contact channel. The method preserves exact history cancellation before adjoining the ideal trigonometric factors. heat_uniform_tail_upper_bound and heat_coefficient_truncation_error_upper_bound bound the truncated heat polynomials. This is an analytic coefficient calculation, not an integration of the complete nonlinear trajectories.

gamma_fourth_scaled_heat_bounds encloses the resulting finite cubic contribution. per_history_ideal_remainder_upper_bounds, ideal_mixed_remainder_upper_bound and source_mixed_error_upper_bound retain higher nonlinear orders and all arbitrary original residuals. true_mixed_bounds and recorded_mixed_bounds are authoritative exact rational endpoint pairs. Their *_interval fields are outward dyadic representations, not substitute inputs for exact sign decisions. A finite response orientation need not match the earlier formal joint-short-time coefficient.

predicted_orientation records the sign of the enclosed heat contribution when available. oriented_true_mixed_lower_bound then subtracts the full nonlinear and source errors in that orientation; only a strictly positive bound certifies the true sign. recorded_sign_margin subtracts four scalar errors. disjoint_record_margin subtracts eight: the tangent comparator also permits four reading errors. four_record_sets_disjoint requires this latter margin to be strictly positive. strict_disjoint_noise_ceiling exposes a sufficient per-reading error ceiling; it is not an optimal information limit or instrument calibration. If the heat contribution has unresolved sign, the orientation is zero and its oriented bound and margins are unavailable. If its sign is available but the complete error bound is too large, the margins remain nonpositive and no true sign is certified. The noise ceiling is unavailable unless the complete true-sign bound is strictly positive.

Status precedence is record_sets_disjoint, recorded_sign_certified, true_sign_certified, exact_mixed_zero, then bounds_only. A failed sufficient separation bound proves neither linearity nor record overlap. Exact null inputs or an observation at the delayed event are distinct from an unresolved enclosure. The existing work and strict identity guards remain separate from these observation statuses. Each history's second work uses its actual preevent envelope and its mean shifts are retained. tangent_endpoint_discrepancy_upper_bound is unavailable in these histories: this report bounds the mixed response, not individual nonlinear/tangent endpoint distances.

The schema is tnfr.sine-class-nonlinear-protocol.v1; shared SDK projection and atomic export preserve normalized primitives, exact bounds and scope. The guide evaluates the declared response-free design. Earlier frozen outcomes and the short-window record-overlap theorem remain unchanged.

Two complete source readouts and independent source transfer

bound_sine_class_comparison_readout returns SineClassComparisonReadout. Its nine mandatory inputs retain the single-source names: each of initial_form_bounds and initial_phase_bounds now contains two ordered 27-node covers. The other inputs are the shared two amplitudes, delay, duration, time step, Taylor order and total max_steps. Single-source scalar domains remain unchanged; the global ordinary-integer attempt cap is 1..8192, with at most 4096 attempts per child.

Both sources are admitted before the first execution. Children run in source order, receiving the remaining global budget up to their per-child limit. The first incomplete child stops all later execution. Budget exhaustion before the next child applies no event to that source. Reports retain both child slots, attempted/completed counts, failed-source association and unattempted source indices. The planned count is distinct from an attempt.

Each child observation is reconstructed from its admitted complete law, source, branch ancestry and retained Taylor arithmetic through the shared evidence reader. Stored derivatives and strict Picard evidence retain their execution premise; arithmetic reconstruction does not regenerate or authenticate them. Rebuilt suffix increments provide contrast_readout_bounds as source zero minus source one. The separate raw_endpoint_contrast_bounds retains the wider endpoint construction. Both remain unavailable unless all four histories in both children complete. Shared SDK projection uses tnfr.sine-class-comparison-readout.v1.

The forward API accepts no class, acquisition, prediction or sensor verdict. For the fixed comparison protocol, its covers enclose mathematical nominal references. The separate assess_reference_contrast uses exact primitive bounds to extend a reconstructed reference contrast to the original acquired families by 8*epsilon/(1-2*gamma_upper*T). Its caller must establish the matching law, reference and actual-source distance. The extension applies once, without adding a cubic approximation error or identifying every outer-cover corner as an acquired state.

That helper reports both widths and tests the transported width against the declared allowance. Reference and actual open predictions are separate consistency checks; neither is intersected with a forward band. Missing observations retain unavailable decisions. Status precedence is unavailable, consistency_conflict, resolution_not_certified, discrimination_not_certified, then discrimination_certified. Individual sign/separation flags remain visible even if a width policy fails. The selected step/order policy promises neither completion nor resolution.

Conditional feasibility over a common amplitude interval

bound_sine_class_amplitude_feasibility returns SineClassAmplitudeFeasibility from four mandatory scalar inputs: amplitude_scale_lower, amplitude_scale_upper, base_cubic_lower and base_cubic_upper. Admit finite represented reals before arithmetic; require an ordered strictly positive scale interval and an ordered base coefficient interval. Booleans and nonfinite values reject admission.

The coefficient interval is an explicitly supplied mathematical premise for the complete central cubic coefficient at the original fixed design. The calculator does not authenticate it, consume a saved report verdict, or establish its model association. The proof and retained-evidence audit provide that association for the documented example. A consumer reusing other reports must re-admit their primitive model and coefficient evidence and reconstruct every consumed derived enclosure.

Both donor impulses are scaled from 1/2000; the two source families, support, complete law, clock, delay 1, duration 2, source radius 1e-32, per-reading error 1e-30, identity radius 1/12 and original work ceilings are fixed. Common-scale homogeneity multiplies the entire base cubic interval, including its time error, by the cube of the scale. The higher-amplitude error uses the upper scale and the source allowance remains unchanged. The shared eight-reading decision distinguishes true sign, recorded sign and separation from a separately noisy null requiring sixteen errors. A failed sufficient margin establishes no impossibility.

The strict complex-amplitude guard applies throughout the scale interval. Outside its domain, full-response bounds remain unavailable; a negative remainder denominator cannot produce a certificate. Work and identity guards are independent of observation resolution.

The new delayed-work estimate bounds the carried donor Laplacian through the reversible heat spectrum and a full-law source defect. It retains the exact work q*(L*x_pre)[4] + 3*q*q/2. It credits no continuous loss as an event reserve and leaves earlier conservative ledger defaults intact. upper_scale_history_bounds contains the four histories at the upper scale; monotonicity makes their work, storage and radius ceilings uniform. Their pointwise mean shifts are not uniform means: separate first_form_mean_bounds, final_form_mean_bounds and phase_mean_bounds cover the full scale interval and original source uncertainty.

Shared SDK projection uses tnfr.sine-class-amplitude-feasibility.v1. This detached arithmetic report generates no coefficient, nonlinear trajectory or acquired source. It is not a frozen evaluation protocol. The guide demonstrates the conditional interval calculation.

Spatial receiver observation with sixteen scalar reading errors

bound_sine_class_spatial_observation returns SineClassSpatialObservation from the same ten mandatory primitives as the cubic comparison. It keeps the complete law, source families, support, clock, events and horizon domain. The observation is fixed: x[23]-x[21], followed by the four-history mixed difference in each class and class one minus class two. Two independently bounded nodal readings per history give reading_count=16; this is not a directly calibrated difference sensor.

The proof uses nominal reflection parity and receiver locality. It retains the full quadratic hierarchy and tangent phase feedback. The heat-only quadratic term is class blind; neither a lower amplitude degree nor a formal short-time term certifies a finite response.

class_parameters and class_segments reuse the shared fixed order-64, dyadic128 coefficient owner, retaining all coordinates and event ancestry. complete_quadratic_contrast_bounds projects the second amplitude level, subtracts classes and then applies the shared gamma**3 enclosure. The scaled_class_quadratic_bounds are the separate class coefficients. Finite time tails are included in these intervals.

Each per_history_higher_amplitude_contrast_remainder_upper_bounds entry already compares both nominal classes. Their sum supplies the higher_amplitude_contrast_error_upper_bound, with no extra factor two. The separate source_contrast_error_upper_bound is 16*endpoint_radius/(1-2*gamma_upper*total_duration) and admits arbitrary actual-source residuals. The complex-amplitude guard and exact event-pairing zeros retain the cubic owner's availability boundaries.

decision uses sixteen errors for the recorded sign and thirty-two to exclude a separately noisy zero-contrast alternative by this statistic. Strict separation, true sign and a compatible scalar cancellation witness remain different statements; a failed sufficient inequality does not prove record overlap. Work and identity guards remain independent. Model-specific SDK projection uses tnfr.sine-class-spatial-observation.v1. Compact retained assessments reference the complete coefficient evidence explicitly; they are not interchangeable with a full report or checkpoint.

Complete cubic-amplitude class response and recording limits

bound_sine_class_cubic_response returns SineClassCubicResponse from the same ten mandatory primitives as the organization contrast. Shared admission retains signed amplitudes, 0<=delay<=total_duration<=2 and 0<radius<=1/12. The class pair, complete law, support, clock and observation remain fixed. The proof retains the full first, second and third amplitude equations; its complex parameter is an analytic device for a remainder bound, not a complex physical EPI.

The coefficient uses fixed time order 64 and shared dyadic128 arithmetic. class_parameters and class_segments retain all 54 coordinates at each of three amplitude levels, oriented-edge coefficients, normalized time series, initial norms, majorant tails and carried endpoints. Each class has one first-probe prefix and three suffixes. The delayed event changes only the first level's donor form; higher levels and every phase coordinate carry unchanged. This triangular polynomial calculation neither changes the general integrator's limits nor evaluates a full nonlinear response.

scaled_class_cubic_bounds includes the factor gamma**4 converting the conditioned third-level coordinates back to form. Their difference is enclosed by complete_cubic_contrast_bounds. Time truncation is included in those coefficient bounds. The separate higher_amplitude_contrast_error_upper_bound uses the nominal central parity and the Cauchy estimate with fixed g=1/3000; the original-coordinate source_contrast_error_upper_bound retains both actual source families. The shared tighter gamma enclosure is sufficient for the latter.

cauchy_admitted requires strict bootstrap and amplitude-radius margins. If they fail, the full-response decision remains unavailable with explicit reasons. Exact zero-input or final-event pairing identities remain available separately, without a coefficient evaluation. Missing remainders are None, not zero. Acquisition and full-family source association remain external matched premises.

decision uses the shared exact class-contrast rules. A true sign, a recorded sign above eight reading errors, and exclusion of a separately noisy zero-contrast alternative above sixteen errors are distinct. decision.scalar_cancellation certifies that every allowed true contrast admits some allowed error vector making the mixed recorded contrast zero; it asserts neither one error vector for all states nor overlap of raw reading vectors. This flag can coexist with true_sign_certified status. Work and identity guards remain independent. Shared SDK projection under tnfr.sine-class-cubic-response.v1 preserves the retained coefficient evidence, exact decision bounds and unavailable fields.

Only immutable class coefficients enter the bounded cache; primitive admission precedes it and source, observation and work decisions rebuild on every call. The usage example shows the separate true-sign and scalar-cancellation decisions.

Nonlinear contrast between acquired mediator organizations

bound_sine_class_nonlinear_organization returns SineClassNonlinearOrganization from the ten mandatory primitives of the finite nonlinear protocol excluding mediator_class. The comparison is fixed: classes (1,1,1) minus (1,2,1) on the same complete law, support, clock and two-event design. Shared scalar admission retains 0<=delay<=total_duration<=2 and 0<radius<=1/12. Each class has its own full source; exact source pairing applies only within its four histories. Acquisition, component norms and separate zero sums remain conditional premises, and original preparation costs need not be equal.

The proof cancels the outer-cycle and bridge heat channels before enclosing the mediator coefficient and cosine difference. The shared order-32 heat tail also bounds the restricted mediator channel with actual full-graph degrees. gamma_fourth_scaled_heat_contrast_bounds is this analytic contribution, not the complete nonlinear contrast. ideal_contrast_remainder_upper_bound retains two full mixed remainders; source_contrast_error_upper_bound retains independent errors from both classes. Their sum widens the heat contribution into true_contrast_bounds. recorded_contrast_bounds adds eight scalar reading-error allowances. Exact endpoint pairs determine decisions; interval fields are outward dyadic projections.

predicted_orientation refers to the heat contribution. A strictly positive oriented_true_contrast_lower_bound is additionally needed for a true-sign certificate. recorded_sign_margin subtracts 8*readout_error_bound. The separate zero_contrast_comparator_separation_margin subtracts sixteen errors to exclude an alternative with zero true contrast and its own eight readings. This null condition does not supply that alternative's complete law. Status precedence is zero_contrast_record_sets_disjoint, recorded_sign_certified, true_sign_certified, exact_contrast_zero, then bounds_only. Failure certifies neither overlap nor class equivalence. The sufficient noise ceiling is unavailable without a true-sign certificate.

uniform_history_bounds applies the shared event ledger separately to each class, without equating their states or actual work. Observation status, work policies and strict identity guards are independent. The fixed design in the guide retains bounds_only even with source and reading errors removed: its independent nonlinear remainder dominates the isolated mediator signal. This limits that estimate, not the complete-law response. The calculator consumes no saved report or response and executes no formation or nonlinear trajectory. Shared SDK projection preserves exact bounds and scope under tnfr.sine-class-nonlinear-organization.v1.

Two form probes and a four-history superposition test

bound_sine_class_superposition returns SineClassSuperposition for the fixed three-C9 complete law. The proof owns the symmetry, nonlinear onset, finite error and observation-overlap claims. This calculator evaluates bounds; it generates no trajectory and admits no formation source.

All eleven keyword primitives are mandatory:

Primitive	Admission
mediator_class	Ordinary integer 1 or 2; donor and receiver remain class 1
first_probe_amplitude, second_probe_amplitude	Finite signed form jumps at the donor central node
delay, total_duration	Structural times with 0<=delay<=total_duration<=1/4
endpoint_radius, readout_error_bound	Nonnegative conditional source and per-reading error bounds
radius	Strictly positive identity radius, at most 1/12
contact_work_allowance, first_probe_work_allowance, second_probe_work_allowance	Separate nonnegative work ceilings

Shared exact-or-represented-real admission preserves exact rationals and rejects Boolean, nonfinite or underflowing represented scalars before model construction. Zero amplitudes, endpoint radius, noise or duration are admitted. The source premise is one complete reached state whose componentwise form and target-phase Euclidean errors are at most endpoint_radius, with separate unweighted zero sums in each component and channel. It is not reconstructed from a supplied report or verdict.

The four histories are ordered neither, first_only, second_only, both. They share that initial full state, mediator class, support, capacity and clock. The delayed-only history follows its unprobed flow; the both-probe history retains its first-probe state. The common final readout is receiver central form, with coefficients (1,-1,-1,1). Four scalar errors contribute 4*readout_error_bound. Neither a shifted single-input solution nor a reset at the delayed event is admitted.

formal_mixed_coefficient_rational_factor and its sign retain the exact amplitude/time polynomial separately from the outward gamma**4 factor. It is cubic in amplitudes and fourth order in a joint rescaling of delay and final time. Its sign is not a finite endpoint lower bound. oriented_instantaneous_curvature_lower_bound concerns the right second derivative just after the second event, with orientation given by the second amplitude's sign. It is None when the sufficient gap bound does not resolve a sign, and zero for a proved exact-null case. The associated certificate neither measures that derivative nor infers the final mixed readout's sign.

true_mixed_upper_bound bounds the absolute mixed statistic, including arbitrary original residuals; recorded_mixed_bounds also includes all four reading errors. Nominal odd symmetry does not impose symmetry on actual sources. scalar_cancellation_admitted means the true-statistic bound is at most 4*readout_error_bound: each true value can be hidden by some admissible error vector. It does not assert common individual records.

maximum_tangent_endpoint_discrepancy_upper_bound compares each full nonlinear endpoint with the tangent endpoint initialized at the same actual full state. If it is at most the per-reading error, four_record_overlap_admitted establishes a common vector of all four final readings, using admissible nonlinear-model errors and zero tangent errors. This is existential endpoint compatibility, not equality of the continuous histories or indistinguishability of every possible record. The statuses four_record_overlap, scalar_cancellation_only and bounds_only retain those separate conclusions. A failed sufficient overlap bound does not establish nonlinear discrimination.

Each history retains its actual pre-second-event state envelope, both work intervals, radius/storage margins and mean shifts. The second work uses that history's carried form, not another single-probe certificate. Work allowances admit equality; identity needs strict entry-radius and storage margins after each event. all_work_within_allowances and all_identities_certified are separate conditional guards. They do not control the observation status: finite observation bounds remain valid when a policy or sufficient identity guard fails. No formation, physical precision or autonomous event selection follows from any flag.

The direct schema is tnfr.sine-class-superposition.v1; shared SDK projection and atomic export preserve the exact primitives and report scope. The guide shows a response-free analytic budget, leaving the frozen single-probe class-response assessment unchanged.

Effective memory of the acquired mediator

relational_sine_class_memory.py provides a structural reader and a separate conditional finite-error calculator. The proof owns the exact causal identities, matched-source nonlinear error and scoped instantaneous-comparator obstructions. Neither reader runs a trajectory, admits a formation source or mutates a graph.

derive_sine_class_mediated_memory() takes no inputs and returns SineClassMediatedMemory for the fixed three-C9 comparison. It retains all 36 donor/receiver coordinates as visible and all 18 mediator coordinates as hidden, with forms before real target-phase deviations. Actual central degrees remain (3,4,3) and the clock is tau=e*t. The full hidden dimension is not asserted minimal. No phase rescaling is used.

normalized_laplacian_vv, _vh, _hv and _hh are exact rational spatial blocks. Separate phase-interior/contact blocks and named gamma and class cosine factors define the ideal full tangent generator. The shared rational coordinate-memory owner partitions only the spatial matrix; its result is not a rational replacement for the transcendental full law. The report keeps outward scalar coefficient enclosures separately. kernel_at_zero_spatial_matrix and kernel_at_zero_scalar_bounds assemble the common two-channel zero-lag kernel; the corresponding kernel_derivative_difference_* fields retain its class-dependent first derivative. No exponential or fitted memory timescale is evaluated.

The effective tangent law retains both B*exp(D_k*t)*h0 and its convolution kernel. Exact nonlinear elimination also retains the visible residual and the propagated hidden residual; dropping them requires the finite-error bound. The supplied class does not determine h0. Its cancellation in a paired tangent probe response requires the same full initial state in the probe and baseline trajectories.

hidden_midpoint_map and quasistatic_* fields describe declared stationary comparators. They start from the same visible projection but generally replace the actual hidden initialization. Visible mobility is inherited; no degree renormalization follows from eliminating a node. The midpoint section is not generally invariant, and its reconstructed total charge, storage and identity are not certified by the original full-law result. It is neither an installed effective law nor an admitted fast limit.

bound_sine_class_mediated_memory(*, probe_amplitude, contact_duration, endpoint_radius, readout_error_bound) returns SineClassMediatedMemoryBound. All four primitives are mandatory. Shared exact-or-represented-real admission preserves exact rationals and rejects Boolean, nonfinite or underflowing represented inputs. All budgets are nonnegative and 0<=contact_duration<=1/4; zero endpoint radius is admitted. The conditional source premise is that each reached component's form and target-phase Euclidean errors are at most endpoint_radius. It is not verified from a source, report, observation or saved verdict by this API.

probe_tangent_error_upper_bound and baseline_tangent_error_upper_bound compare each nonlinear trajectory with its tangent trajectory started at the same full state. paired_class_reduction_error_upper_bound is twice their sum. recorded_class_contrast_bounds adds the correlated class-tail bound and four scalar readout errors. No extra ideal-initialization error is inserted after the proved paired cancellation. whole_window_kernel_norm_upper_bound and hidden_initial_source_norm_upper_bound are norm ceilings, not evaluated kernel histories, memory-decay statements or reconstructed hidden states.

The tangent stationary comparator has recorded null bounds ±4*delta. Its nonlinear counterpart also retains the independently uncertain visible initial states, giving ±(4*epsilon/(1-3*h)+4*delta). quasistatic_exclusion_margin_bounds uses this latter, stronger obligation. status="certified_conditional_contrast" requires a strictly positive recorded contrast and exclusion of that declared comparator; otherwise status="bounds_only" retains all finite bounds. These are conditional flags, not acquisition, work or identity verdicts. A failed sufficient margin does not establish response equality or exclude another error bound.

The direct schemas are tnfr.sine-class-mediated-memory.v1 and tnfr.sine-class-mediated-memory-bound.v1. Shared SDK projection and atomic export retain exact values and scope. Applying the bound to an acquired family requires independently matching the existing formation/contact/probe obligations. The guide shows structural inspection without replaying the frozen assessment.

Acquired mediator class and a transmitted form response

assess_sine_class_mediation returns SineClassMediation from twelve mandatory keyword primitives: formation_time, relaxation_duration, contact_duration, probe_amplitude, form_error_bound, phase_error_bound, endpoint_radius, radius, decay_power, readout_error_bound, contact_work_allowance and probe_work_allowance. The theorem owns the complete-law response, work and identity proof; this detached assessment installs neither a support event nor a probe in a live graph.

All scalar arguments use shared exact-or-represented-real admission before arithmetic; exact rationals retain their value. Booleans, nonfinite values and nonzero values lost during represented-real conversion reject. Times, amplitude, errors and allowances are nonnegative, endpoint_radius>0, 0<radius<=1/12, and contact_duration<=1/4. decay_power is an ordinary integer in [0,4096]; the shared formation/dwell exponential caps apply. Zero amplitude or duration admits a calculation without a positive-response certificate. Invalid domains reject; a failed sufficient certificate reports unavailable and its unmet obligations.

The fixed comparison has three C9 components, with classes (1,1,1) and (1,2,1), aligned origins, unit capacities and central contacts on a path. It retains the actual central degrees (3,4,3), all 54 form/phase coordinates, tau=e*t, e=1023/1024 and gamma=1/(1023*pi). Within each class, the same reached source is continued with and without the donor central form jump. The receiver central form difference is R_k; the reported contrast is R_1-R_2. Cross-class source residuals need not agree. Formation sources and their different preparation costs are retained, so this is not an equal-cost comparison or an ideal-target reset.

source_handoff rebuilds both class formations and unprobed handoffs from original primitive errors and budgets. No previous report is an input. leading_contrast_bounds, linear_tail_upper_bound and nonlinear_contrast_error_upper_bound concern an ideal reference used in the proof. actual_contrast_bounds adds the four full-history preparation errors, and recorded_contrast_bounds adds four scalar readout errors. The response begins to depend on the mediator class at derivative order three; common earlier terms do not supply a class distinction. Without the fresh source handoff, actual response, work, means and post-probe family bounds are None, even though ideal coefficients remain available.

joined_bounds admits the unprobed family using full sine storage. post_probe_radius_squared_upper_bound, post_probe_excess_storage_upper_bound and their strict margins separately admit post-probe identity and recovery. The degree-weighted form mean shifts by 3*probe_amplitude/58; recovery retains the resulting mean leaf. contact_work_bounds and probe_work_bounds refer to distinct supplied events. Their allowance comparisons are non-strict and use exact rational upper-bound arithmetic; a zero allowance margin may pass equality. Neither earlier loss nor positive response substitutes for work admission.

The complete phase-blind comparison x'=-KLx; theta'=gamma*KLx applies only after the same source handoff and contact. Its paired form response is class independent. phase_blind_recorded_contrast_bounds retains a conservative source/readout allowance; phase_blind_alternative_excluded requires a strictly positive separation margin from that null bound. This does not certify the alternative's formation or identity, or exclude other constitutive laws.

status="certified_class_mediation" requires fresh source handoff, strictly positive recorded contrast, phase-blind separation, both baseline and probe identity, and both work allowances. A frozen protocol may demand a stronger positive threshold than the reusable API. The direct schema is tnfr.sine-class-mediation.v1; shared SDK projection and atomic export retain exact and unavailable fields. This is an inherited collective interaction property conditional on the supplied law, preparation and interface. No autonomous event selection or physical constituent identity follows. The guide distinguishes assessment from reading retained evidence. The first frozen assessment applies its separately declared stronger threshold; its report does not broaden these reusable domains or establish physical measurement precision.

Degree-aware assembly of reduced component ports

relational_sine_port_composition.py extends the single-contact description to finite simple undirected unit central-contact networks. The theorem and protocol own its exact surrogate balance, full-space defect and actual-family identity proof. These detached readers install no graph, event, forcing or solver.

evaluate_sine_port_composition(*, classes, contacts, phase_origins, forms, phase_deviations) returns SinePortCompositionState. Admit one through sixteen components, with ordinary non-Boolean integer classes in {1,2}. The cap is an implementation budget, not a theorem about possible component counts. Contacts contain two distinct ordinary component indices; reversal is canonicalized, but duplicate, self, weighted or out-of-range contacts reject. All sequences admit single-pass iterables with bounded length. Each form and phase-deviation row has exactly five signed real entries per component, in component-major layer order (4), (3,5), (2,6), (1,7), (0,8).

There is one finite real phase_origins[i] per component. The reconstructed phase is Theta_i + phase_origins[i]*1 + T*y_i, where Theta_i[j]=2*pi*classes[i]*(j-4)/9. Deviations exclude both the reference twist and origin and are not wrapped. Edge gaps derive from these component origins and port deviations; independently prescribed edge offsets are not admitted. Origins define fixed coordinates, not separately conserved component means after contact. Physical scalars use shared exact/represented-real admission before arithmetic; Boolean and nonfinite values reject.

The geometry retains actual central degree 2+contact_degree and layer masses (2+contact_degree,4,4,4,4). Both rows use this mobility in tau=e*t, with e=1023/1024. Internal phase currents use each class tangent; all bridge currents retain the exact nonlinear sine. The report exposes matrices, rates, internal/bridge storage, component charges and their exchange. storage_rate=-dissipation and the two zero network_*_charge_rate values are algebraic identities. storage_rate_from_rows_bounds separately encloses the derivative evaluated from interval rows. Port supply uses internal gradient-conjugate efforts; bare port form/phase values are not declared physical power. Disconnected networks and an isolated component admit rows.

assess_sine_port_composition(*, classes, contacts, phase_origins, formation_time, relaxation_duration, contact_duration, form_error_bound, phase_error_bound, endpoint_radius, radius, decay_power, work_allowance, approximation_allowance) returns SinePortComposition. All thirteen primitives are required; no report is an input. Times and budgets are nonnegative, endpoint_radius>0, 0<radius<=1/12, and contact_duration<=1/4. decay_power is an ordinary integer in [0,4096]. The shared formation and relaxation exponential work caps remain applicable. Zero duration and zero allowances are valid inputs, without a promised certificate. Phase origins remain arbitrary admitted signed reals.

unprobed_handoff freshly rebuilds the two-class source library, retaining the original independent component errors, their separate zero-sum channel constraints and source-storage ceilings. Both class handoffs are required even when the selected network uses only one class. The nominal surrogate starts at zero form/deviation; its comparison with actual formation images includes their uncertainty and does not reset those images. ideal_surrogate_discrepancy_upper_bound includes nonlinearly generated odd modes. preparation_error_upper_bound separately includes all initial even and odd errors. Their sum bounds every fine form and lifted phase coordinate throughout the whole contact window; approximation_certified requires the outward approximation_margin_bounds.lo>0. This absolute mathematical error allowance is neither a sensor budget nor a class-response contrast.

Only connected networks with at least two components have the supplied global identity theorem. It recomputes the full graph gap, actual sine storage, quotient radius and reweighted conserved means. identity_certified requires strict outward radius and storage margins and supplies subsequent uninterrupted recovery; surrogate storage is not its proof. Contact work is separate from continuous dissipation: work_within_allowance uses the exact non-strict upper-bound comparison, so a rounded zero work margin does not by itself reject an admitted equality. Without both source handoffs, actual approximation, work, means and initial joined-family bounds remain None. A disconnected assessment can still expose these observations after handoff, but its global identity flag is false and its gap/barrier are None; its total weighted means are not claimed to become one common recovered state.

status="certified_sine_port_composition" requires fresh formation and handoff, connected multi-component identity, work admission and strict whole-window accuracy. Otherwise unavailable_reasons names insufficient obligations; it does not prove the reduction impossible. Direct schemas are tnfr.sine-port-composition-state.v1 and tnfr.sine-port-composition.v1; shared SDK projection and atomic export retain exact and unavailable fields. The usage guide distinguishes instantaneous assembly from inspecting an actual-family certificate. No exact nonlinear quotient, runtime speedup, autonomous support selection, laboratory clock or physical identification follows.

Reduced component ports and controlled receiver transfer

relational_sine_reduced_class_ports.py separates evaluation of a twenty-coordinate surrogate from its conditional comparison with the original nonlinear C9 families. Both public readers admit original primitives and rebuild coefficients; neither consumes a prior report, installs an event, propagates a trajectory or changes the complete fine law. The protocol and proof own the projection, port convention and approximation bounds.

evaluate_sine_reduced_class_ports(*, donor_class, receiver_class, forms, phase_deviations, phase_origin_difference) returns SineReducedClassPortState. Classes are ordinary non-Boolean integers in {1, 2}. Each ordered state row contains exactly ten signed real values: five donor orbits followed by five receiver orbits. Single-pass iterables are consumed once with bounded length. The orbit order is (4), (3,5), (2,6), (1,7), (0,8); multiplicities are (1,2,2,2,2), and live node degrees are (3,2,2,2,2) in each component. Phase coordinates are real deviations from the selected winding target, not substituted absolute circular phases. Writing T for the layer lift, Theta_j[i] = 2*pi*j*(i-4)/9, and splitting the phase row into y_D, y_R, the reconstructed phases are theta_D = Theta_k + T*y_D and theta_R = Theta_l + phi*1 + T*y_R, where phi=phase_origin_difference. The receiver deviations therefore exclude the declared common origin; adding phi to them again would count it twice. At the ports the consumed difference is exactly phi + y_R[0] - y_D[0]. This instantaneous reader admits any finite signed phase-origin difference; it has no preparation or horizon certificate. All scalar coordinates use shared exact/represented-real admission before interval construction, retaining exact fractions and rejecting Boolean or nonfinite physical values.

The report exposes exact rational normalized matrices, class-cosine bounds, the actual bridge phase difference and sine enclosure, and both reduced rate rows in tau=e*t, with e=1023/1024. Internal phase exchange is the class's linear tangent; the bridge keeps its exact nonlinear sine current and the degree-three port mobility. These are ten coordinates per component instead of eighteen. The original nonlinear interior can generate discarded odd modes, so this is not an exact nonlinear invariant quotient. Arbitrary admitted surrogate coordinates alone certify no corresponding fine trajectory.

assess_sine_reduced_class_ports(*, formation_time, relaxation_duration, phase_origin_difference, contact_duration, form_error_bound, phase_error_bound, endpoint_radius, readout_error_bound, radius, work_allowance, decay_power, error_fraction) returns SineReducedClassPorts. All twelve arguments are required. This assessment fixes receiver winding two and compares donor windings one and two, using the same class coefficients as the instantaneous reader. It retains the actual contact scalar domains, formation/relaxation work caps and original source families, and requires 0 < error_fraction < 1. In particular, radius is in (0,1/12], endpoint radius is positive, phase origin is in [0,1], contact duration is in [0,1/4], and decay_power is an ordinary integer from zero through 4096. Zero phase origin or duration is valid but cannot certify positive response.

unprobed_handoff freshly rebuilds both formation families and their exact tiny squared-norm recovery bounds. Their original even and odd preparation errors and separate zero-sum constraints remain present. Receiver two retains its original class-two source cost under the common per-component ceiling; this is not an equal-cost experiment. Once both handoffs pass, joined_contact_bounds recomputes full-law identity, weighted means and supplied bridge work for the actual eighteen-node families. Surrogate stability does not replace these obligations. No target reset or prior phase probe is used.

receiver_derivative_bounds_by_donor encloses orders one through four derived from the reduced matrices, including the changed receiver cosine. The correlated donor contrast is evaluated through those matrices, rather than importing the earlier contact's coefficient or verdict. ideal_leading_contrast_bounds encloses the exact fourth-order expression; reduced_ideal_contrast_bounds also includes its finite reduced-flow errors. The report keeps the semigroup tail, reduced nonlinear remainder, pairwise surrogate/full discrepancy, actual preparation and readout errors separate. total_error_upper_bound adds all five once. It bounds error about the exact leading expression enclosed by interval arithmetic, not about an arbitrary chosen midpoint without its additional rounding error.

recorded_contrast_bounds encloses actual donor-two minus donor-one records at receiver node 13. response_certified requires its lower endpoint to be strictly positive. Only then are error_ratio_upper_bound and error_fraction_margin_bounds available: the denominator is the final outward recorded lower endpoint, and approximation_certified requires the outward margin's lower endpoint to be strictly positive. Thus the denominator is a certified full-law separation, not the formal leading amplitude. Independent preparation and readout errors also remain in the disconnected control band. Without both actual source handoffs, actual response, joined bounds, total error, ratio and fraction margin remain None; ideal surrogate quantities retain their explicitly hypothetical scope.

status="certified_reduced_class_ports" requires fresh formation, both handoffs, actual joined identity, admitted work, positive recorded response and the strict fractional-error bound. Otherwise unavailable_reasons identifies insufficient obligations; failed bounds prove neither an exact closure obstruction nor instability. The generic response rule is positivity; the frozen research protocol additionally requires its larger contrast threshold. Direct schemas are tnfr.sine-reduced-class-port-state.v1 and tnfr.sine-reduced-class-ports.v1; shared SDK projection and atomic export retain exact and unavailable fields. This is a conditional finite-horizon reduction, not a measured speedup, autonomous interaction selector, laboratory realization or physical identification. The usage guide separates instantaneous row evaluation from inspecting the saved certificate.

Receiver discrimination after contact of actually formed classes

assess_sine_formed_class_contact(*, formation_time, relaxation_duration, phase_origin_difference, contact_duration, form_error_bound, phase_error_bound, endpoint_radius, readout_error_bound, radius, work_allowance, decay_power) in relational_sine_formed_class_contact.py returns SineFormedClassContact. Its proof and protocol use the original unprobed C9 source families. The receiver has winding one; donor cases are winding one and two. The same receiver common phase origin relative to the donor is supplied from initial preparation, not imposed as a state rotation at contact. Original form/phase errors keep their separate exact zero sums within each component; uncertain port values are not forced to match between the two experiments.

All eleven primitives are required. Scalar inputs use shared original exact/represented-real admission before arithmetic. They are nonnegative, with 0 < radius <= 1/12, positive endpoint_radius, phase origin at most one, and contact_duration <= 1/4. decay_power is an ordinary non-Boolean integer from zero through 4096. The formation assessment retains its own time cap. The additional slow-rate budget requires lyapunov_decay_rate*relaxation_duration <= 4096, whenever that rate exists. Boolean/nonfinite scalars, unsupported domains and arithmetic bounds reject.

The reader freshly rebuilds SineFormedClassPair through the shared actual-family handoff and report-free Lyapunov kernels. It consumes no phase-probe result. After the admitted formation checkpoint, the actual family relaxes without interventions for the supplied duration. If the slow decay exponent is at least decay_power, the exact bound 2**(-decay_power) applies. The returned norm squares remain rational even below the interval grid; no rounded square root or vanishing materialized value replaces them. Both form and target-phase norms of both classes must fit endpoint_radius before contact bounds become available. Failed formation or an insufficient relaxation budget leaves those actual-state bounds unavailable.

At contact_time=formation_time+relaxation_duration, one supplied unit edge joins donor node 4 to receiver node 13. No coordinate is reset. The changed degrees, memberwise weighted means, full-support radius/storage barrier and event work are recomputed for eighteen nodes. identity_certified covers both whole joined families and implies future winding retention and convergence modulo common origins under the unchanged joined law. work_within_allowance compares the contact work bound to an independently supplied allowance. Continuous loss does not supply that allowance or select the event.

The readout is actual receiver form at node 13 after contact_duration. recorded_contrast_bounds encloses donor-two minus donor-one readings with independent observation errors. The ideal fourth-order term, correlated semigroup tail, nonlinear remainder and actual preparation error remain separate. Ideal targets are comparison states, never substituted for reached states. The disconnected control has donor-independent receiver dynamics; equality of connected initial currents alone decides no later contrast.

response_certified requires strictly positive recorded contrast. Joint status="certified_formed_class_contact" also requires the fresh formation, both handoffs, full joined identity and supplied work allowance. Otherwise status="unavailable" retains named reasons; insufficient bounds do not prove instability or response equality. The frozen protocol has its own larger threshold. The direct schema is tnfr.sine-formed-class-contact.v1; shared SDK projection and atomic export retain exact quantities and unavailable fields. This analytic path does not enlarge the generic forecast's coordinate cap or claim a practical preparation time, sensor accuracy or physical identity.

Reusing formed classes in a supplied contact experiment

The contact-admission proof concerns the complete normalized-sine law and actual unprobed nominal C9 preparations. With aligned origins and central ports, their symmetry makes the joined receiver evolution independent of the donor class. Since the original uncertain families contain those nominal members, a strictly positive whole-family receiver contrast is excluded for that protocol. This is an exact obstruction, not an unavailable numerical bound. General perturbed members need not have the same symmetry or zero contact currents.

An explicit common relative phase offset is a different preparation. With nonzero sine, its instantaneous port current is nonzero and identical for both nominal classes; this alone supplies neither a finite class discriminator nor identity retention. Contact work and occurrence remain supplied premises. The separate contact assessment supplies a finite comparison and actual-family handoff under its additional declared premises; it executes no live support event. The execution plan owns task status and the freeze boundary.

Existing consumers have distinct reuse boundaries:

Consumer	What can be reused and what must be established
Formed-class pair	The actual image of the original sixteen-dimensional relative source families, including their two exact zero-sum residual constraints. Target geometries and nominal states cannot replace those reached families; any endpoint enclosure needs a proof from the original source and must retain every correlation its consumer uses.
Prepared composition	Primitive reassessment, synchronous contact, declared relative origins, full-support storage and event accounting. Its inputs are SinePreparedEntry reports from SineRelativePattern, not SineFormedClassPair. The separate contact assessment derives its unprobed handoff from original primitives and retains the source correlations; it does not cast a pair report into a prepared-entry report.
Probe response and repeated maintenance	Their analytic methods retain value, but their coefficients, contrast, return neighborhoods and dwell concern an isolated C9 with supplied phase jumps. They do not certify a joined eighteen-node system.
Validated sine forecast	The current layout has 2*n+1 coordinates and a shared limit of 24. Two C9 components require 37 and are outside this implementation domain. The contact assessment uses a full-support analytic bound; the generic cap remains unchanged. Increasing it or discarding coordinates alone supplies no certificate.

For the joined graph, recompute endpoint degrees (two to three), mobility, memberwise weighted means, internal and bridge storage, and every consumed capture margin. Retain all fine coordinates, both original winding identities and the declared structural clock. A detached composition assessment installs no edge, and a passing storage test alone certifies no transmitted response.

Quantitative return under repeated supplied probes

assess_sine_formed_class_maintenance(*, formation_time, probe_time, probe_duration, phase_increment, form_error_bound, phase_error_bound, readout_error_bound, radius, common_dwell) in relational_sine_formed_class_maintenance.py returns SineFormedClassMaintenance. It freshly rebuilds the formed-class response assessment from its eight primitive inputs; it accepts no incoming report, cached verdict or state reset. The mathematical owner owns the fixed hybrid experiment and uniform return proof.

All nine arguments are required and use shared original exact/represented-real admission. The first eight retain the response reader's domain. The additional common_dwell must be strictly greater than probe_duration, so every readout precedes the next jump. When the exponential return bound is evaluated, its work cap is lyapunov_decay_rate*common_dwell <= 4096; the initial fast semigroup rate does not impose this cap. The clock remains tau=e*t, with original_common_dwell=common_dwell/e. Structural times have no supplied laboratory-unit interpretation.

The freshly rebuilt reference_certificate defines each pre-probe set K_k by Euclidean form and target-phase norm radii, on the same zero-mean leaf. The actual first pre-probe families belong to these sets. Each supplied jump has the original phase increment and preserves form and both means. The complete unforced law then evolves for common_dwell. A strict nonlinear Lyapunov estimate bounds every member of J(K_k), not sampled trajectories. The returned form and phase-error norms must both be strictly less than half their respective pre-probe radii. This establishes an invariant return neighborhood, not an exact periodic orbit or a unique driven attractor.

The response and signed event-work intervals in reference_certificate then apply to every cycle. The class contrast retains its common heat-response factor. Readout errors are bounded separately at every reading; they are observations, not state feedback or accumulating state disturbances. Original preparation uncertainty and intervening nonlinear state evolution remain in the uniform set inclusion. Additional perturbations, event-amplitude errors, timing errors and adaptive probes are outside this contract.

For N events, cumulative work is enclosed by N times the per-event work interval for each class, without assuming independent events. Continuous storage loss is accounted for separately. A positive uniform work lower bound therefore requires unbounded cumulative supplied work as the event count grows. The initial source-storage ceiling is not a reservoir for this later work. Finite-time recovery means return to an admitted pre-probe neighborhood; asymptotic target convergence applies when interventions stop.

Missing positive common acute geometry or failed post-jump trapping leaves the return bounds unavailable (None), rather than supplying a decay estimate. return_certified_by_class requires strict positive outward form and phase half-radius margins. Joint status="certified_repeated_probe_maintenance" also requires the fresh formation/response/recovery prerequisite; otherwise status="unavailable" retains named reasons. The nested generic response criterion is strict positive contrast; the frozen experiment separately requires its larger stated threshold. An insufficient bound does not prove instability or impossible maintenance.

The direct schema is tnfr.sine-formed-class-maintenance.v1. Shared SDK projection and atomic export preserve the nested prerequisite, exact budgets, return bounds and availability. No cycle-count simulation, minimum-dwell optimization, practical-speed guarantee, finite-total-work maintenance or autonomous probe-selection claim follows.

A common supplied probe of two actually formed classes

assess_sine_formed_class_response(*, formation_time, probe_time, probe_duration, phase_increment, form_error_bound, phase_error_bound, readout_error_bound, radius) in relational_sine_formed_classes.py returns SineFormedClassResponse. It rebuilds the pair-formation prerequisite from primitive inputs, then bounds the continuation of those same complete source families. It accepts no incoming formation report, cached verdict or ideal-state reset. The mathematical owner declares the full probe protocol and its proof obligations.

The law, C9 support, capacities, rational source preparations and separate zero-sum initial error constraints remain those of the pair certificate. All times use tau=e*t. At probe_time, a supplied simultaneous phase jump adds phase_increment*(e_0-(1/9)*1) to the lifted phases and leaves form unchanged. It preserves the conserved means. The same unforced law resumes immediately, with no later inverse jump. The readout is actual signed form at node zero after probe_duration, with the supplied additive scalar error. This intervention is neither an autonomous event-selection law nor execution of a native semantic operator.

All eight arguments are required and use shared exact/represented-real admission. Times, phase increment and all error widths are nonnegative; radius is positive. Require formation_time <= probe_time, probe_time/5 <= 4096 and 2*probe_duration <= 4096. Booleans and nonfinite physical scalars reject; exact fractions are retained. Zero increment or duration is admitted and does not by itself certify a response. Unsupported arithmetic domains reject; insufficient outward margins give unavailable evidence, not a proof of response equality or failed recovery.

formation_certificate is freshly rebuilt. The warmup_* bounds retain original source errors and nonlinear history at the actual probe time. warmup_scaled_form_remainder_upper_bounds and warmup_phase_error_upper_bounds use the weighted norm M=2*I; the phase bound measures distance from the nominal proxy v=gamma*x0, not the target. warmup_form_norm_upper_bounds is the Euclidean form norm, while warmup_target_phase_radius_upper_bounds is the Euclidean phase distance from the target, including the proxy-to-target distance. ideal_readout_bounds_by_class is an analytic comparison, whose error is bounded in response_error_upper_bounds. recorded_readout_bounds_by_class includes that error and readout uncertainty. The class-two-minus-class-one recorded_contrast_bounds retains one common heat-response factor; subtracting the two marginal readout intervals generally loses this correlation and gives a weaker enclosure.

recovery_certified_by_class requires strict acute-chart, post-jump radius and post-jump excess-storage margins for each actual family. These conditions invoke the existing positive-loss recovery theorem under the unchanged law. probe_work_bounds_by_class separately encloses the signed phase-potential jump about its nominal target value. The probe is not assumed passive: its work is distinct from continuous storage loss. No uncertainty in the supplied event amplitude, clock or timing is included in the initial-state widths.

response_certified requires recorded_contrast_bounds.lo > 0. Joint status="certified_formed_class_response" additionally requires the fresh pair certificate and both recovery flags. Otherwise status="unavailable" retains named unavailable_reasons. The frozen research protocol has its own larger response threshold; that threshold is not the generic API's status rule. The direct schema is tnfr.sine-formed-class-response.v1; shared SDK projection and atomic export retain primitive inputs, separate margins and availability. This is a conditional mathematical probe of supplied preparations and law, not a physical measurement bridge or a proof of autonomous preparation.

Two formed attracting classes under one complete law

assess_sine_formed_class_pair(*, scaled_time, form_error_bound, phase_error_bound, radius) in relational_sine_formed_classes.py returns SineFormedClassPair. This detached analytic assessment concerns the two source families in the mathematical owner. It accepts primitive budgets, not a captured graph, an observed endpoint or a cached formation verdict. It evaluates no probe or receiver response.

The fixed support is a simple nine-node cycle, with unit held capacities and beta, effective coefficients e=1023/1024, w=1/1024, and structural clock tau=e*t. For target winding k equal to 1 or 2, nominal initial phases are zero and forms are k*m*(j-4), where m=(2046/9)*(355/113)**2 is exact rational preparation data. The target phases are mathematical 2*pi*k*(j-4)/9; the rational source declaration does not replace pi in that target or in the complete law.

Every initial form and phase lift admits an error bounded by the respective primitive width, subject to separate exact zero sums in the two channels. Errors may otherwise vary independently between classes. Thus every actual source has the same zero conserved form and phase means. Positive widths give a full sixteen-dimensional relative neighborhood while all eighteen fine coordinates remain represented. This is not an independent Cartesian box in the ambient coordinates, and it admits no additional unknown origin. preparation_dimension is 0, 8 or 16 according to the number of positive error widths; full_relative_preparation requires both to be positive. Nominal source storage is 36*k**2*m**2, so the two costs differ; the common declared storage ceiling 10**9 is an upper budget, not an equal-energy constraint.

All four arguments use shared original exact/represented-real admission. Time and error widths must be nonnegative and radius must be positive. Boolean and nonfinite physical scalars reject; exact fractions are retained. The analytic horizon also requires scaled_time/5 <= 4096, the shared exponential work limit. An unresolved sufficient inequality produces unavailable evidence, not a proof that formation fails.

Capture concerns the actual full-law image of the original source set. The positive-loss recovery theorem then retains the attained sector and gives convergence to its attracting geometry, with no reset or change of law. The two targets must be inequivalent under the declared support, common-origin and sign/reflection symmetries. Opposite winding alone would not satisfy this obligation. Their distinct attracting geometries do not by themselves certify distinguishable responses to a common physical or mathematical probe.

Class-indexed evidence follows class_order=("winding_one","winding_two"). formation_certified_by_class combines initial zero winding, the acute target chart and strict endpoint-radius and excess-storage margins. source_budget_certified_by_class is separate: each whole source set must fit the common storage budget. Joint status="certified_two_formed_classes" requires both formation flags, both source-budget flags and symmetry_inequivalent. Otherwise status="unavailable" retains named unavailable_reasons. Zero-containing outward margins do not certify.

Export retains exact primitive data, separate class evidence and availability through the shared report projection and atomic writer; the direct schema is tnfr.sine-formed-class-pair.v1. Serialization neither authenticates an actual preparation nor selects the supplied support, source organization or positive-loss law. The guide shows the primitive call. Any subsequent interaction experiment needs its own declared probe, relaxation, work, readout and recovery obligations.

Prepared acquisition and actual receiver form

assess_sine_formation_response(*, scaled_time, form_error_bound, phase_error_bound, readout_error_bound, radius) in relational_sine_formation_response.py assesses the two exact integer preparations and positive-loss doubled-C5 law declared by the mathematical owner. It is a detached analytic calculation. It accepts no observed graph, cached endpoint, unknown form origin, fitted coefficient or event schedule.

All five inputs use shared exact/represented-real admission before arithmetic. Time and the three error radii must be nonnegative; the geometric radius must be positive. Booleans, nonfinite values and nonzero values lost during scalar materialization reject. The shared exponential budget requires (2/3)*scaled_time <= 4096. An admitted input whose sufficient inequalities fail produces unavailable evidence, not a claim that its trajectory fails.

The complete law has fixed effective e=1023/1024, w=1/1024, unit held capacity and beta. Its clock is tau=e*t, distinct from the conservative pair certificates. Nominal forms have pair means 4039*(a-2) and an antisymmetric (+96,-96) increment at pair 0 or pair 3; nominal phases are zero. Every fine form and phase has its own independent initial error. The source declaration bounds common origins too. Shared preparation machinery rebuilds the consumed weighted norms and coefficients from these primitive rows after one fresh domain admission for both sources. This avoids constructing temporary observation reports or evaluating discarded rates. Each call still admits every public budget, and the absolute receiver retains its form-mean uncertainty. Report-consuming preparation readers separately retain their original source re-admission boundary; the fixed-source path does not trust an external report's derived fields.

The actual endpoint Y=(x_2+x_3)/2 is enclosed using a mean-free Poisson predictor plus a full-law Duhamel remainder. This keeps the exponentially decaying initial state, nonlinear phase history, initial coordinate errors, conserved form mean and independent additive readout error. The Poisson expression is an approximation with an explicit remainder; it is not an installed reduced law or a substitution of instantaneous phase current for the total form response. Exact matrix identities validate the fixed support's inverse on its mean-free subspace.

Formation admission separately requires initially zero winding, endpoint distance strictly inside the declared radius, an acute target neighborhood and full endpoint storage strictly below its forward first-exit barrier. These bounds concern the actual reached family, with no reset. Positive loss then retains its winding-one geometric identity for all subsequent time. The nominal initial storage agrees between branches; uncertain members and their endpoint storage need not agree. Geometric retention does not promise an indefinitely nonzero receiver contrast or internally active pairs.

initial_zero_winding_certified, acute_radius_certified, endpoint_radius_certified_by_preparation and storage_barrier_certified_by_preparation retain those separate obligations; formation_certified_by_preparation combines them for each source. recorded_readout_bounds encloses both absolute records and recorded_difference_bounds their A-minus-B contrast. status="certified_formation_response" requires both formation admissions and a strictly positive lower contrast bound. Otherwise it is "unavailable" with named reasons, even if one component succeeds. None of these fields authenticates an external source or observed record.

The direct SineFormationResponse export and generic SDK envelope retain primitive budgets, separate margins and availability, through the shared exact projection and atomic JSON writer. Its direct schema is tnfr.sine-formation-response.v1. The frozen research protocol has the additional fixed response threshold stated by its owner. A mathematical error budget is not a calibrated apparatus tolerance; the report neither selects the preparation or loss mechanism nor identifies a physical object.

Relative sine patterns and moving references

bound_relational_sine_pattern(graph, *, reference_node, reference_model, form_error_bounds, phase_error_bounds) in relational_sine_pattern.py admits a complete supplied network under the same separate smooth sine law. Shared sine staging validates signed represented form, phase, exact held nonnegative capacities, simple connected unit support and absent Gamma. It does not execute the native Arg law.

The two error inputs are required ordered sequences of nonnegative radii, one per captured node. Exact rationals are retained; Booleans and nonfinite values reject. The observation premise is actual_i = nominal_i + common_offset + residual_i at one actual time, with each residual bounded by its supplied radius. Phase centers and errors must use supplied consistent real lifts. No phase unwrapping, sensor calibration, missing-node inference or clock authentication is performed. Common offsets can be arbitrarily unknown, but cannot be assumed common across asynchronous samples without another observation law.

All nodes, including intermediaries, remain. Relative state boxes enclose x_i-x_ref and theta_i-theta_ref; the reference rows are identically zero. For every other row the radius is the sum of its own radius and the reference's. Direct edge bounds use only their two endpoint radii. The shared sine rate/work kernel supplies full rates, reference rates, relative rates, storage and loss. Relative rates subtract the reference rate; this is not an intervention that freezes the reference node.

The returned SineRelativePattern retains both uncertainty descriptions. Its static edge, rate and storage bounds apply to the original residual class. The Cartesian relative box forgets the correlation from the shared reference error and can contain additional states. The tighter static bounds must not be used to certify every point of that larger box. The mathematical owner defines pattern identity, winding limits and a sufficient analytic recovery criterion. This report does not itself certify a recovery basin.

pattern.forecast(observation_time=..., end_time=..., time_step=..., order=6) uses chained-report admission and reconstructs the initial relative box from nominal coordinates, the reference node and original residual radii. The returned pattern retains those rebuilt boxes. The method selects an initial zero-reference representative and passes the full outer box to bound_sine_flow. Every node then evolves, including the reference; only the returned endpoint is projected. The solver's final held-capacity coordinate is a storage layout and does not designate a physically hidden node in this adapter. The same size, time and numerical-budget restrictions apply. Its 2*n+1<=24 layout limits this forecast adapter to eleven nodes; static reports are not restricted to the solver's state dimension.

SineRelativeForecast retains the source, entire full_forecast, relative endpoint intervals and reference displacement from its initial representative. Its status forwards the solver's status. Endpoint bounds refer to full_forecast.validated_end_time, even if the requested horizon is unavailable. Initial absolute offsets remain unknown. The method neither fits an observation history nor proves that relaxed sample bounds came from a common trajectory.

Both reports use shared exact projection and label admission through to_dict(), with schemas tnfr.relational-sine-relative-pattern.v1 and tnfr.relational-sine-relative-forecast.v1, suitable for the SDK atomic JSON writer. The existing hidden inverse consumes absolute nodal rates; feeding it moving-reference differences is unsupported and can produce an incorrect hidden form even with exact, full-rank observations.

Joining relative sine patterns with declared origins

assess_sine_pattern_composition(left, right, *, bridge, observation_time, edge_turn_offsets, form_origin_difference=None, phase_origin_difference=None, work_allowance=None) in relational_sine_composition.py accepts two SineRelativePattern sources. The equivalent left.compose_with(right, ...) delegates to that owner. This is a detached preparation and capture assessment; it does not add a live edge or choose an event time.

Both sources retain disjoint ordered node labels, complete connected simple unit supports, positive held capacities and identical admitted effective loss/exchange/storage coefficients under the regular sine reference model. Positive form loss is required. Shared primitive admission runs before any derived field is consumed. Cached source storage, rates and verdicts are rebuilt. The joint support retains the existing geometry budget of 32 nodes and 50 edges. bridge is one ordered pair, left node then right node.

The two origin differences are exact or represented finite real scalars, defined as right common additive origin minus left common additive origin in the source's nominal + common + residual representation. They are not measured reference-node gaps. The phase difference belongs to the declared consistent real lifts. Interval origin inputs are unsupported here; no midpoint or zero replaces an uncertain or absent origin. Original per-node residual radii remain in the whole joined family. Its single absolute common origin remains unknown.

observation_time declares the nonnegative synchronous structural time of both sources and the hypothetical event; the reports do not authenticate that declaration. Model equality does not prove clock calibration or sample synchrony. Boolean, nonfinite or malformed authoritative scalars reject. Missing origin differences return an explicitly unavailable preparation, with no joined state or capture certificate; unrelated malformed declarations still reject instead of being hidden by that unavailability.

When both differences are supplied, the shared exact-row pattern builder shifts right nominal coordinates and retains every original residual radius. It recomputes joined degrees, mobility, rates, storage and weights without a graph roundtrip that would round admitted rationals. The existing certify_sine_sector_capture checks the actual joined family and all full support faces in the independently supplied integer edge sector. Internal certificates and the bridge cost alone cannot replace this full-state check.

SinePatternComposition.status describes preparation availability; the separate capture.admitted describes sufficient maintenance after the hypothetical event. A failed sufficient bound is not a destruction verdict. Representative weighted-mean bounds are relative to the left source's unknown common origin; the capture's absolute weighted means remain None. Support reweighting is distinct from continuous weighted-form transport.

The report separates internal storage, the bridge storage jump and complete joined storage. Optional nonnegative work_allowance is a declared event budget: within_allowance, exceeds_allowance and unresolved distinguish its interval comparison, while not_supplied leaves it unevaluated. Neither capture nor continuous loss supplies an allowance or an occurrence law. The mathematical owner owns the exact family map, event-weight identity and missing-frame obstruction.

The owner and generic SDK relational_report_to_dict use shared exact JSON projection and node-label admission. Export preserves independent availability, source declarations and nested capture evidence; it does not authenticate them.

Composing analytically acquired patterns

assess_sine_prepared_composition(left, right, *, bridge, left_initial_time, right_initial_time, edge_turn_offsets, form_origin_difference=None, phase_origin_difference=None, work_allowance=None) in the composition owner accepts two SinePreparedEntry reports whose original sources are SineRelativePattern, including zero-radius families. The equivalent left.compose_with(right, ...) delegates to this owner. Standalone absolute comparison sources are outside this handoff contract; no implicit conversion changes their frame interpretation.

The source primitives, scaled durations and original integer sector declarations are re-admitted. Both entries are then rebuilt by their owner; cached endpoint intervals, horizons, means, status and storage are not evidence. The sources need disjoint labels, positive held capacities and identical admitted effective coefficients. The shared joint 32-node/50-edge budget is checked before recomputing the analytic bounds.

Each finite nonnegative initial time is explicit. The endpoint time is initial_time + scaled_time / effective_epi_weight; the two exact results must agree. No caller-supplied observation-time override or silent waiting is accepted. Different durations are allowed only when the declared starting times give the same endpoint. The report records these preparation declarations, not authenticated timestamps or measurements.

Exact relative common origins keep the meaning of the static composition contract. Original residual mean intervals and memberwise conserved means propagate those origins to the analytic endpoints. Tighter internal edge bounds are retained from each rebuilt entry. Bridge bounds use the centered port enclosures and the conserved mean intervals. The prefix uses isolated degrees; post-event dynamics uses joined degrees. The post-event representative mean uses conserved component inventories plus the two port reweightings.

SinePreparedCompositionSource records the actual analytic endpoint provenance. It is not a synthetic observation or an input accepted by the generic relative-state reader. The shared sector core receives the corresponding whole-state enclosure, preserving its internal correlations. Replacing it by independent endpoint-box corners would require recomputing the bounds for that larger set.

SinePreparedComposition.status describes frame availability. Missing origins leave capture and joined edge/storage bounds unavailable. The separate acquisition_and_capture_certified property requires both rebuilt entries to certify acquisition and the new joint capture to be admitted. Joint capture alone does not establish earlier formation. budget_status independently compares the bridge-work enclosure with an optional supplied allowance. An unavailable sufficient bound does not assert destruction or impossibility.

Direct and generic SDK export retain exact fractions, independent availability and nested source/capture evidence, with shared label admission. Neither export nor rebuilding authenticates the preparation. No live edge, reset, solver trajectory or occurrence law is installed. The mathematical owner provides the containment, clock and charge identities and the frozen control.

Whole-set sine cycle recovery

certify_sine_cycle_recovery(source, *, cycle, winding, radius, phase_turns=None) in relational_sine_recovery.py accepts a SineRelativePattern or SineRelativeForecast. Both expose the same operation through certify_cycle_recovery(...). It observes the supplied state set without running a solver or changing a graph.

Cycle recovery, general-target recovery, conservative identity and sector capture share chained-report admission. Specialized cycle certificates retain their larger support domain; general-target and sector consumers keep their declared geometry budgets.

The ordered cycle must cover every node and every edge of the complete simple unit support, with at least three nodes. Extra chords, attached environmental nodes and a selected cycle inside a larger graph reject. The exact target has phase turns winding*j/n in this cycle order and uniform form. Integer winding must satisfy 4*abs(winding)<n; consensus is zero. This analytically specifies a strictly acute critical phase, rather than recognizing an equilibrium from a small floating residual.

radius is a finite strictly positive proof-domain parameter in the declared form/phase coordinates. It is neither a physical threshold nor a tolerance for replacing equality. Optional phase_turns are integers in captured node order, added to the observed phase lifts. Their default is zero. Booleans, noninteger turns and malformed ordered inputs reject. Lifts are not guessed from the observations to obtain a successful result.

The certificate uses the whole-set proof: pairwise differences bound the centered form/phase-deviation norm; exact criticality cancels the linear phase-storage term before its quadratic remainder is enclosed. The cycle spectral gap, radius cosine margin and storage barrier use shared rational pi, square-root and cosine bounds. The declared phase target and its storage retain mathematical pi.

The two input scopes remain distinct:

• An original pattern uses its nominal differences and the two endpoint residual radii. Common origins cancel before interval evaluation.
• A forecast uses differences of the full endpoint coordinates at validated_end_time. It consumes neither the source's tighter static bounds nor the twice-projected relative-coordinate box.

Positive dissipation and strictly positive held capacities are separate theorem premises. exact_held_capacity retains exact scalar constraints where present, including positive values smaller than the interval grid; capacity_bounds also records outward arithmetic enclosures. A forecast's augmented final capacity has no such exact constraint and its entire interval must have a positive lower endpoint. Admission supplies no uniform decay rate as capacities approach zero.

SineCycleRecovery retains the source, exact target turns, ordered support, input uncertainty scope, norm/storage bounds and both strict margins. hypothesis_failures identifies missing capacity/dissipation premises; unresolved_conditions identifies unproved geometric or numerical bounds. Only an empty pair gives status="admitted". Otherwise status is "unavailable", which is not an instability verdict or proof that the state cannot recover. Invalid target, support and input contracts raise.

For original observations, observation_time is unavailable (None), not invented as zero. For forecasts it is the actual validated endpoint time. input_forecast_admitted and input_forecast_requested_end_time retain the original solver result. A partial endpoint can pass the recovery theorem without changing that solver's unavailable requested-horizon verdict.

Admission guarantees recovery to the declared twist modulo common origins for every state in the stated set, under the supplied unforced sine law, fixed support and held positive capacities. It does not establish formation from a different winding, survive arbitrary future inputs or support events, select the law, or identify matter. to_dict() uses shared exact projection and label admission with schema tnfr.relational-sine-cycle-recovery.v1, suitable for the SDK atomic JSON writer. A projected public dataclass does not authenticate its source or proof history.

Whole-set sine recovery with a retained environment

certify_sine_pattern_recovery(source, *, target_phase_turns, radius, phase_turns=None) extends the same recovery owner to the full connected simple unit support. SineRelativePattern and SineRelativeForecast expose it through certify_pattern_recovery(...). The cycle method above retains its more specific target and spectral formula.

target_phase_turns supplies one exact integer or Fraction per node in captured order, in units of mathematical 2*pi. Booleans and floating-point target turns reject. The target has uniform form, with a free common form and phase origin. No target is fitted to the observed response.

The owner reuses exact phase-cycle reconstruction on every supplied edge. Principal target edge turns must be strictly acute. Opposite signed sine currents must cancel symbolically at every node. This is a sufficient exact criticality check, not a complete decision procedure for trigonometric identities. A target failing that check rejects; a near-zero floating residual or an interval containing zero cannot certify its equilibrium. The geometry owner's 32-node/50-edge evaluation budget remains explicit. It is a software budget, not a restriction derived for the underlying law.

The spectral gap is bounded on the full exact combinatorial Laplacian by the existing rational quotient owner. Common-origin removal, source uncertainty, capacity admission, pairwise centered norm, quadratic storage remainder and strict barrier checks share the cycle certificate's implementation. Target edge angles may differ. Adding an environmental edge changes criticality and the spectral problem; two isolated cycle certificates cannot be composed.

The complete-network proof admits two C5 patterns and one intermediary with nonzero observation errors. A positive-capacity intermediary transmits a donor perturbation while the whole network remains in a common recovery domain. A zero-capacity intermediary is a separate no-influence control and does not pass the positive-capacity recovery theorem.

The returned SinePatternRecovery retains exact target and geometric evidence, full-support spectral provenance and the source/time distinctions specified above. Its schema is tnfr.relational-sine-pattern-recovery.v1. An unavailable radius or storage inequality is not an instability verdict. No trajectory, node deletion, support event or native-law dispatch occurs. In particular, static admission of eleven nodes does not extend the existing validated forecast solver's coordinate budget or certify an unevaluated forecast horizon. Public dataclass export does not authenticate evidence.

Sine formation eligibility and timed exclusion

assess_sine_mediated_formation(*, model, amplitude, capacity_contrast, exclusion_time=None, profile="localized", donor_epi=None) in relational_sine_formation.py assesses declared eleven-node preparations, without executing a trajectory. Its donor is an exact winding-one C5, its receiver has uniform phase and form, and the intermediary has form amplitude and phase zero. The default localized profile sets every other form to zero. The balanced profile instead sets every donor form to 2*amplitude, retaining receiver form zero. Both profiles cost amplitude**2 in form storage; balanced initial loss is one quarter of localized loss. This restricted comparison is not an optimum over arbitrary preparations. profile="explicit" instead requires exactly five ordered signed donor_epi coordinates for donor nodes 0..4, with amplitude still supplying intermediary form. Receiver forms remain zero. The complete edge differences determine storage; it need not equal amplitude**2. A donor vector with either named profile is rejected rather than silently overriding that profile. Capacities are one except receiver nodes 6 and 9, which have capacities 1+capacity_contrast and 1-capacity_contrast. Both contact edges and every state coordinate remain supplied premises.

An explicit regular reference model with positive dissipation is required. Amplitude, donor coordinates and contrast use shared represented-real admission; abs(capacity_contrast)<1 preserves positive capacities. Optional exclusion_time must be strictly positive and denotes a proof horizon, in the declared sine-model clock. It is not a new dynamics parameter, detection threshold, numerical time step or laboratory duration. Invalid domains raise. Phases are retained as exact rational turns; a materialized radian approximation is not substituted for the preparation.

The proof distinguishes three obligations:

• Exact symmetry and equilibrium controls: zero contrast preserves receiver reflection; an identically zero full form gradient gives equilibrium. Zero intermediary form alone does not establish equilibrium for an explicit donor. silent_donor_subspace identifies intermediary form and donor port zero with D1=-D4,D2=-D3: donor sign/reflection symmetry holds the receiver fixed even with nonzero capacity contrast. Its exclusion reason remains separate from equilibrium. A nonzero reflection-odd initial acceleration is proved from an exact coefficient, without a numerical cutoff; its absence does not exclude a later causal response.
• Storage eligibility: entering the joint acute target requires strictly more storage than its entry-face barrier, which is higher than the final target's minimum storage. The donor need not retain its winding earlier in the path for this argument to apply.
• Timed entry exclusion: either global phase-speed bounds or the necessary phase-action time exclude entry through a supplied horizon. A lower bound on accumulated full-network dissipation must also exhaust the entry budget. Initial loss alone is not a finite-time proof.

phase_action_bound retains the actual maximum of nu_i/d_i+nu_j/d_j over receiver edges, the allowable-loss interval F-beta*B5, and a rational necessary_entry_time_lower_bound. When that allowance is resolved strictly positive, the latter bounds e*beta**2*pi**4/(w**2*k_max*(F-beta*B5)) from below using the shared outward constants. A nonpositive allowance is already a storage obstruction; an unresolved sign does not permit division. Both leave this finite time bound unavailable with explicit reasons. entry_through_horizon_excluded requires the supplied horizon to be strictly below the certified lower bound. The bound is conditional on target entry, not a predicted event or a global speed estimate after exhausting the budget.

Nodal affine bounds use each profile's actual initial form gradients and clip at zero before integrating. early_loss_lower_bound retains the intermediary-only bound for compatibility; full_early_loss_lower_bound sums disjoint nodal contributions. The balanced intermediary contribution is zero, although the full loss is positive for nonzero amplitude. The directional proof additionally retains exact weighted-Laplacian moments of the initial gradient. directional_loss_bound exposes those moments, its polynomial norm bound and explicit availability. A missing horizon, zero gradient, excessive hyperbolic argument or nonpositive polynomial endpoint makes that sufficient certificate unavailable. Its squared bound is never used outside the admitted positive window. The certificate retains the short-window cosh<=11/10 majorant and admits wider windows through cosh<=1/(1-v/2) for squared argument bound v<2. The reported cosh_upper_bound and cosh_bound_method retain which proof was consumed; these are bounds, not a replacement evolution law. combined_early_loss_lower_bound takes the maximum of the nodal and directional bounds, never their sum: both bound the same total dissipation.

The separate maintained_target_obstruction applies the global auxiliary-function theorem. It requires the exact effective ratio 0<w/e<3/2, storage scale one and positive capacity contrast 1/2 on the existing eleven-node preparation. Its form_storage_coefficient=4/5 and mobility spectrum enclosure [1/44,3] belong to the proof, not the evolution law. The mixed coefficient is w/(2*pi*e), which recovers 1/(2*pi) at equal weights. It retains initial/target functional bounds and the correlated gap V5-(4/5)*F, rather than subtracting independent enclosures of the shared V5 term.

exchange_to_loss_ratio uses exact rationals of the model's stored normalized coefficients; sufficient_ratio_upper_bound records 3/2. These metadata remain available when the ratio fails admission, without fabricated functional bounds. The open interval is a conservative sufficient proof domain, not a physical transition threshold or a classification of every ratio. At or above its endpoint this certificate is unavailable, not evidence of formation or instability. The proof identifier is eleven_node_sine_ratio_mixed_lyapunov. The ratio fields append optional defaults so older positional construction remains valid; a missing field does not establish the extended proof's premises.

For the analytical clock change tau=k*t, k>0, dividing both complete evolution rows by k preserves the ratio, mixed functional and target verdict; necessary entry times multiply by k. This is not an extra clock setting in the report. RelationalExchangeModel normalizes supplied channel weights: ideal common rescaling of those raw weights is removed by normalization, not converted into a time change. Rounded inputs can alter the represented ratio. Admission always follows the represented effective pair, including near the ratio boundary; raw decimal labels are not exact normalized coefficients.

A positive gap sets maintained_target_excluded and adds the distinct maintained_target_lyapunov_obstruction reason to the outer report. This excludes convergence to the maintained target and entry into a valid basin that guarantees it. It does not set early_loss_exclusion_certified or exclude every transient visit to the joint acute region. The nested status is excluded, not_excluded, unresolved or unavailable; the latter retains law/capacity mismatch reasons and no invented functional bounds. Neither an insufficient gap nor an unavailable certificate proves formation. No artificial F<=4 admission cutoff is used: the actual source gap decides the certificate, while the theorem proves it positive for the whole class with F<=4 under the stated premises.

Both new nested observations have default None for older or manually constructed reports; the assessment always supplies them. None means absent evidence. The outer tnfr.relational-sine-mediated-formation.v1 schema, existing constructor positions and shared exact JSON projection are retained. No frozen response archive is regenerated by these readers.

The assessment separates excluded, passes_necessary_conditions and unresolved_bound. Passing these checks does not establish formation, capture or existence of a successful trajectory. An absent or insufficient timed bound cannot certify a future crossing. The target is its geometric orbit modulo common origins; it does not fix an incompatible final mean. The held-capacity weights degree/capacity define conserved form and continuously lifted phase sums, retained separately from circular means. The report retains the complete preparation, law coefficients, exact and outward bounded quantities, proof time and reasons. Shared JSON projection supports SDK export without authenticating provenance.

With the plan's default coefficients, analytic proofs exclude both profiles throughout abs(amplitude)<=2, abs(capacity_contrast)<1. At amplitude=2, capacity_contrast=1/2, computational exclusion passes with proof horizon 1/5 for localized and 3/5 for balanced. These are restrictions on the supplied preparation families, not a general impossibility of pattern formation. The theory's constant-donor corollary also covers arbitrary constant donor form D and intermediary form H at ((D-H)**2+H**2)/2<=4 under those coefficients. Explicit donor vectors extend preparation admission without changing the target, phase state, support or receiver initialization. The internal-form analysis derives the full six-coordinate quadratic forms and causal response maps. Its nonuniform example has donor forms (10,13,14,14,13)/3, intermediary 4/3 and contrast 1/2. It passes initial and short-window necessary checks, but the wider certificate excludes its joint entry by 6/5. Independently, the global functional excludes maintained-target convergence for every six-coordinate preparation with F<=4 throughout the sufficient ratio domain, at unit storage scale and positive half contrast. The earlier entry exclusions retain their own scopes; transient acute-region visits are not generally excluded.

Source geometry and the necessary nonlinear receiver correction

SineMediatedFormation.receiver_excitation() returns SineReceiverExcitation from the existing formation owner. Under chained-report admission, it rebuilds the preparation before decomposing full form into donor-reflection-even and odd parts. Both eleven-coordinate vectors are retained; their storage and dissipative norms use the actual support, degrees and capacities. source_coordinates has order (H, D0-H, s1, s2, a1, a2), as defined in the source theorem. The decomposition remains available for valid preparations outside the more restricted tangent theorem.

For effective half weights, unit storage scale and capacity contrast +1/2, the report bounds the receiver's tangent quadratic phase cost for every future time by 3*even_form_storage/4. The inequality is strict for positive even storage; zero even storage gives exactly zero tangent receiver output. This is the full damped tangent about the prepared phase equilibrium, with its noncommuting matrices retained. It is not an upper bound on the nonlinear receiver potential or a forecast of a particular time.

If the actual receiver reaches potential 7/2, its phase-edge deviation from the matching tangent must have norm at least sqrt(7)-sqrt(3*even_form_storage/2). The interval field encloses this analytic threshold, not an observed correction. A strictly positive outward lower endpoint gives nonlinear_correction_status="positive_bound"; an unresolved interval or nonpositive analytic margin leaves the positive bound unavailable. Unsupported laws give status="unavailable" with reasons and None tangent quantities. Corrupted consumed source data reject rather than certify.

For the same law and total initial form storage F<=4, the separate full_phase_budget_status="available" certifies that the actual full-network phase potential stays strictly below actual_full_phase_storage_upper_bound=139/20 for all future time. Any receiver barrier crossing must follow a donor potential-barrier crossing, and requires a donor potential decrease greater than V5-69/20. The report exposes the conditional order flag and a fresh outward lower bound on that decrease. This decrease is not identified receiver work or a reusable energy reserve. Above the total form budget the status is outside_original_form_budget; outside the admitted law it is unavailable_law. In both cases the nonlinear fields remain None, independently of any available tangent evidence.

The odd component is silent in the tangent and when prepared alone under the nonlinear law. It cannot be removed from nonlinear mixtures: equal even coordinates, energy and quadratic spectral moments do not fix receiver work. No value in this report establishes passage or installs an excitation law. to_dict() uses the shared exact SDK projection with schema tnfr.relational-sine-receiver-excitation.v1.

Nonlinear receiver localization from regional storage

SineMediatedFormation.receiver_localization() returns SineReceiverLocalization. It reuses source admission and the original half-weight sine law, unit storage scale and positive-half capacity contrast. The auxiliary function U=F+V_D+7*V_R/5+7*V_bridges/6-q.T*K*S/3 is globally decreasing along that complete nonlinear law. Its coefficients are proof weights, not configured evolution parameters. The regional proof uses exact positive-definite matrices and full-state asymptotic convergence.

For total initial form storage F<=(5-sqrt(5))/2, every nonflat receiver equilibrium is excluded and relative receiver phase converges to consensus. The report exposes receiver_nonflat_equilibria_excluded and relative_receiver_consensus_certified. It does not select the donor's final state, prohibit transient winding or barrier passage, or supply a recovery deadline. This is an endpoint result; receiver_excitation() owns the distinct tangent and whole-network phase bounds.

status is certified, not_certified or unavailable. Admission uses the exact conjunction 5-2*F>=0 and (5-2*F)**2-5>=0; a positive squared margin alone cannot admit the wrong algebraic branch. Fresh shared cycle constants produce critical_form_storage_bounds and target_minus_initial_margin_bounds. Those intervals display the threshold and comparison, even when too narrow a distinction is unresolved numerically; they do not control the exact decision. Unsupported laws leave the functional coefficients and endpoint bounds None, with explicit reasons. Invalid consumed source data reject. Above the sufficient threshold there is no formation verdict.

to_dict() uses shared exact projection with schema tnfr.relational-sine-receiver-localization.v1. receiver_transfer() reuses one validated preparation, retains this report in its optional trailing receiver_localization field and adds weighted_receiver_localization as an endpoint-exclusion reason when certified. Necessary passage/work fields retain their conditional meaning; legacy reports leave the new field None.

Receiver identity transfer with donor unwinding

SineMediatedFormation.receiver_transfer() assesses a different endpoint: the donor becomes flat, the receiver retains winding +1, both bridge phase gaps vanish and form becomes uniform. It retains the original formation report as its source and does not reinterpret its coexistence verdict. Under chained-report admission, static symmetry flags, conserved means, mobility and phase-storage constants are reconstructed from the admitted preparation and shared mathematical owners. Cached equilibrium, silence or barrier fields cannot supply an exclusion. The transfer theorem fixes effective e=w=1/2, beta=1, capacity contrast +1/2 and the same eleven-node preparation. Other coefficients or contrasts leave this transfer certificate unavailable; no alternative law is installed. SineReceiverTransferAdmission.status separates excluded, passes_necessary_conditions, unresolved_bound and unavailable. The latter two distinguish unresolved numerical budget signs from failed theorem premises. Its export schema is tnfr.relational-sine-receiver-transfer.v1.

The exact target_phase_turns and target_geometry reuse shared phase reconstruction. The target is an acute full-support critical point. Its local basin can be checked with certify_sine_pattern_recovery on a supplied full-state observation or validated forecast. Target criticality alone does not put the original preparation, or its later evolution, in that basin. The compatible uniform form equals the original conserved weighted form mean. The reported common phase offset selects a representative with no additional integer turns; actual lifted history can require a weighted integer correction. It is not a prediction of the final absolute phase.

At this target, storage and the mixed functional both equal V5. Their source-minus-target margins are respectively F and (4/5)*F: the shared phase term cancels exactly. These are endpoint budget margins, not positive certificates of transfer. Zero form and the proved silent donor subspace remain distinct exclusions.

First entry into the joint acute donor-zero/receiver-one region requires storage at least B5. The remaining loss allowance is therefore A=F+V5-B5, not the coexistence allowance F-B5. Both rings must cross an antipodal edge before that entry. The donor's minimum angular excursion is 3*pi/5, the receiver's is pi; their disjoint nodal loss sums give the combined action cost 29/25 at the admitted capacities. For resolved A>0, the necessary entry time satisfies T>58*pi**4/(25*A) in the original structural clock. The report provides a conservative rational lower bound. Nonpositive or unresolved allowance cannot be divided by, and no entry time is predicted. The supplied source horizon is used only for a strict necessary-time comparison.

Any reused full-flow early-loss bound is compared with this transfer's own allowance; the coexistence deficit and its exclusion flag are not copied. Its horizon must be finite and strictly positive, and its retained loss bound finite and nonnegative. This optional bound remains conditional evidence: the caller must retain its justification for the same preparation, law and clock through that horizon. The reader does not regenerate or authenticate the timed-loss certificate. An early-loss exclusion needs both the new entry-time condition and a strictly positive recomputed deficit. Passing static conditions leaves accessibility unresolved. Export retains the source, target, proof scope and unavailable evidence through the shared exact JSON projection.

The optional receiver-passage fields retain a separate potential barrier: receiver_first_barrier_phase_storage=7/2. This is the minimax cost between the flat and twisted receiver wells, not the B5 acute face or an antipodal edge test. If transfer occurs, a finite first crossing must occur and the signed accumulated incoming port work there must strictly exceed necessary_receiver_running_work_strict_lower_bound=7/2. receiver_barrier_time_loss_product_lower_bound is a conservative rational lower bound for 14*pi**2/3: crossing at time tau requires tau*D_receiver(tau) at least this value. It uses full nodal gradients and original incidence degrees in the receiver's share of continuous loss. Neither field evaluates an integral or predicts a crossing time.

For initial form storage F<=4, donor_barrier_first_required=True means that any such receiver crossing requires an earlier donor potential barrier crossing. It does not assert an actual crossing, a winding change at that instant or a donor endpoint. The distinct simultaneous_barrier_passage_excluded flag excludes both ring potentials being at least 7/2 together. Both flags reuse the full nonlinear phase budget from receiver_excitation(), strengthening the energy-only sufficient conditions F<=7/2 and F<=7-V5. Admission uses exact source storage, not a rounded display interval. A false flag leaves the condition unresolved; it does not predict simultaneous passage or reverse the order. These fields are None outside this reader's existing theorem domain or in legacy records. They never become general transfer-exclusion reasons. The port-work and order proof has broader explicitly stated premises; this adapter retains its own scope.

Frozen receiver barrier exclusion

relational_receiver_barrier.py prepares and assesses one full eleven-node H=2 source under the smooth sine law. prepare_receiver_barrier() declares its exact phase turns, held capacities, original clock, T=32, step 1/8 and Taylor order 16. evaluate_receiver_barrier() delegates once to bound_sine_flow; it supplies no independent solver or altered pressure. All 23 augmented coordinates are retained. The held last capacity is an uncertainty coordinate, not another evolving physical variable.

assess_receiver_barrier_forecast() checks the frozen declaration and complete step coverage. Its conditional theorem requires receiver potential strictly below 7/2 in every whole-time tube and full twelve-edge endpoint storage strictly below 7/2. Together, these exclude the receiver potential barrier for all future time. Safe endpoints alone, receiver-only endpoint storage or an unfinished horizon cannot pass. An upper bound failing either comparison is unresolved, not a proved passage. No all-time winding constancy or selected donor endpoint follows.

The producer freezes a protocol and source archive before evaluation, refuses overwrites, and retains evaluation errors and unsuccessful responses. Post-evaluation source, runtime and archive checks guard consistency. Public forecast records and archive hashes do not independently authenticate execution or chronology. This is a source-specific research certificate, not a native runtime event or new law.

Donor-well recovery from the full prepared form state

SineMediatedFormation.donor_well_retention() returns the separate SineDonorWellRetention, reusing the original exact donor-twist/flat-receiver preparation and the auxiliary-well theorem. It validates the consumed preparation, support, phase and capacity data and rederives form storage rather than trusting an altered report's budget or verdict. It does not recompute a trajectory or a timed loss producer.

The law domain is the existing beta=1, delta=1/2, e>0, 0<w/e<3/2 domain of the global auxiliary function. On that domain, the exact critical level 7/2 separates the donor's sublevel component from the other relative equilibria. The source belongs to that component, or immediately enters it at the boundary, whenever F<=F_c=(25*sqrt(5)-55)/16. The complete-law convergence theorem then selects donor winding +1 and receiver winding 0, modulo common phase origin. This is a terminal identity certificate; it does not assert an unchanged winding or acute chart at every intermediate time.

Admission uses the exact rational sign of exact_retention_polynomial_margin=3125-(16*F+55)**2, with nonnegative F. The reported threshold and escape_margin_bounds are outward intervals for display; the latter is critical-minus-source, 7/2-W(0). Interval overlap or a rounded threshold cannot override the polynomial decision.

Within the law domain, status="certified" sets relative_donor_pattern_convergence_certified and receiver_only_targets_excluded. A larger form budget gives status="not_certified" with both flags false; it proves neither escape nor receiver formation. Unsupported law premises give status="unavailable" and explicit reasons. There is no research ceiling F<=4 in the reader's admission; the actual sufficient condition is checked against the source.

receiver_transfer() reuses this certificate in its optional donor_well_retention field and adds a distinct exclusion reason when certified. Its older acute-face allowance F+V5-B5, phase-action time and early-loss evidence retain their own definitions: the minimax level is not substituted into a bound derived for another passage. The original coexistence report is unchanged. The new export schema is tnfr.relational-sine-donor-well-retention.v1, using shared exact JSON projection and the SDK atomic writer. A public report remains distinct from authenticated provenance or a checkpoint.

Dissipative capture above the initial donor-well barrier

SineMediatedFormation.donor_dissipative_capture() returns SineDonorDissipativeCapture. It reuses the source validation of the donor-well reader and recomputes the full initial gradient q=Lx and N=q.T*K*q, with K=diag(nu_i/d_i). Stored gradient or loss verdicts do not replace that calculation. Its proof requires the original effective half weights, beta=1 and delta=1/2.

This is an analytic finite-window bound, not a propagated trajectory. At the fixed structural horizon T=1/4, the proof gives

W(T) <= V5 + A,             A = (4/5)*F - N/25
V(theta0 + s*(theta(T)-theta0)) <= V5 + B,  B = 121*N/38400,  0<=s<=1

The sufficient admission is A < (5*sqrt(5)-11)/4, evaluated exactly through endpoint_exact_polynomial_margin=125-(11+4*A)**2 > 0. On this admitted support, N<=6*F proves A>=14*F/25>=0; it also makes the phase-path inequality automatic whenever the endpoint condition passes. The retained phase-path margin explains how the endpoint is connected to the donor's component; it is not an independent adjustable gate. Outward display intervals do not decide either exact algebraic comparison.

donor_component_entry_certified means the actual state at this horizon belongs to the donor component below W=7/2. relative_donor_pattern_convergence_certified and receiver_only_targets_excluded then express its asymptotic consequences. The certificate supplies no point-valued future state, convergence deadline, claim of earlier escape, all-time winding history or physical identification. Its fixed horizon and conservative coefficients are proof choices, not new terms in the law or empirical constants. Strict admission includes an open neighborhood of any certified preparation within the six-form source slice.

An admitted source outside the sufficient bound is not_certified; it is not predicted to transfer. Unsupported law premises are unavailable, and invalid consumed source data reject. receiver_transfer() appends this report in its optional donor_dissipative_capture field and uses a distinct exclusion reason when certified, preserving the earlier static and action-time contracts. The export schema is tnfr.relational-sine-donor-dissipative-capture.v1, using shared exact projection and the SDK atomic writer.

Complete sine equilibria on the two-C5/intermediary support

classify_c5_sine_critical_set accepts exact support geometry and two ordered, labeled C5 cycles. The graph must consist of those disjoint rings and one remaining intermediary joined to one node of each ring, with no additional edge. It retains the supplied node order and ring orientations; only a common phase origin is removed.

The completeness proof gives 30 exact configurations per labeled ring and two choices on each bridge, or 3,600 full relative geometries. The catalog is factorized: it stores local options and reconstructs an individual requested combination. Nonacute cycle branches and antipodal bridges remain present. Its rational turns denote exact multiples of mathematical 2*pi, not rounded radian measurements. cycle_choices follows the two declared cycles; bridge_turns follows the catalog's bridge_edge_indices in full geometry order. The reconstructed nodal representative fixes the first stored node's phase to zero.

The separate CircularPhaseState and reconstruct_circular_phase_state share exact cycle closure and nodal reconstruction with the existing geometry owner. They do not weaken PhaseCycleState or the strict acute admission of reconstruct_phase_cycle_state and recovery readers. An antipodal branch requires its declared turn representative; no branch-independent winding is inferred from it. Geometric criticality alone is not a stability certificate.

SineExchangeComparison.asymptotic_equilibria(cycles=...) delegates to assess_sine_asymptotic_equilibria. It retains the captured complete sine source, its model, support, state and held capacities. With positive loss, exchange, storage scale and every capacity positive, the theorem certifies convergence to one catalog member and uniform form fixed by the conserved weighted form mean. It also proves finite limits for continuous phase lifts, while their terminal integer turns and the selected relative geometry remain unknown. No initial form budget or acute-phase restriction is needed.

Zero loss or a frozen node leaves this stronger theorem unavailable; it does not prove nonconvergence. Invalid source or support rejects. This detached reader executes no trajectory, assigns no convergence rate or deadline and installs no new runtime law. In particular, one exact sine equilibrium has a zero native neighbor resultant, so this catalog cannot establish native Arg-law continuation. Source provenance and unavailability remain explicit in the shared exact JSON projection; an exported report is not authenticated proof history or a checkpoint. The report schema is tnfr.relational-sine-asymptotic-equilibria.v1.

Exact geometric inertia and full-law stability

The general compose_bridge_tree_hessian_inertia(component_inertias, bridges) owner returns BridgeTreeHessianInertia. Each component triple is an independently justified relative (positive, negative, zero) Hessian inertia. Each bridge triple contains two component indices and sign +1 or -1. Ordered collections, nonnegative non-Boolean integer counts, valid distinct indices, unique pairs and a connected tree are required. An intercomponent cycle is rejected. Singleton components use (0,0,0); additional relative zero modes remain in the output rather than becoming an attraction verdict.

The shared algebra retains bridge order/orientation, total_nodes, relative inertia and one removed common-phase direction. Its export schema is tnfr.bridge-tree-hessian-inertia.v1. These are conditional geometric results: the reader does not infer component inertias or bridge signs from a live state, authenticate their premises, classify a dynamical law or execute an attachment. A one-node report is valid algebra, but is outside the nonisolated row-normalized dynamical theorem. The composition proof states the full-law consequences and the additional equilibrium premises.

C5SineCriticalSet.phase_hessian_inertia(cycle_choices=..., bridge_turns=...) returns C5PhaseHessianInertia for one revalidated exact catalog member. It derives both component inertias and actual bridge signs before calling that shared composition owner; its existing report and schema remain unchanged. relative_inertia records (positive, negative, zero) dimensions of the cosine-weighted phase Hessian after removing one common phase origin. common_phase_nullity=1 retains that removed direction separately. The report includes the exact reconstructed state, ring negative-edge counts and bridge signs. It uses constrained cycle algebra, not a floating eigenvalue threshold. Its schema is tnfr.c5-phase-hessian-inertia.v1.

phase_hessian_index_counts[m] counts catalog members with m negative relative directions. This compact histogram comes from the local branch counts and two independent bridge signs. No full Cartesian list is stored. The stability proof shows that every relative Hessian is nonsingular; a global neutral origin must not be reported as an unresolved relative degeneracy.

SineAsymptoticEquilibria.classify_equilibrium(cycle_choices=..., bridge_turns=...) returns a separate SineEquilibriumStability. It revalidates the captured source through the shared asymptotic admission and checks the supplied catalog against that source before applying the full reciprocal-law theorem. Altered public flags cannot override the actual loss or capacity premises.

For positive loss, exchange, storage scale and held capacities, a negative Hessian index m gives relative_unstable_modes=m, relative_stable_modes=20-m and relative_center_modes=0 on the two conserved-mean tangent spaces. The full 22-dimensional tangent retains two neutral common-origin directions. These counts follow from the complete form-phase pencil; they do not assume commuting form and phase Hessians. For m=0, status is locally_exponentially_attracting and local_exponential_attraction_certified is true. Otherwise status is nonlinearly_unstable and nonlinear_instability_certified is true. Missing law hypotheses give unavailable, false certification flags and None mode counts, while exact geometry remains available.

There are nine locally attracting relative geometries under these premises: each ring independently has winding -1, 0 or +1, with both bridge gaps zero. Strictly positive coefficient or capacity changes preserve the local classification, but may change rates, responses and basins. Classification of a named equilibrium does not assert that the captured source is near it, that it will reach it, or that its basin has a supplied numerical radius. The export schema is tnfr.relational-sine-equilibrium-stability.v1 and uses the same exact projection and source-label controls as the parent report.

Joint acute cycle-period obstruction

assess_acute_cycle_periods(geometry, cycle_periods=..., cycle_combination=...) in phase_cycle_geometry.py returns AcuteCyclePeriodAssessment. The reader rederives and validates the supplied PhaseCycleGeometry; its existing support and evaluation budgets remain unchanged. Ordered periods must be non-Boolean integers in the stored fundamental-cycle order. The ordered combination must have that same length and be nonzero. Rational coefficients remain exact; other supported real coefficients follow shared represented-real admission. A tree has no nonzero cycle combination and cannot supply this witness.

The reader combines the actual signed edge chains before taking their absolute sum. combined_period is the same linear combination of supplied periods; strict_period_bound is one quarter of the combined chain's L1 norm in turns. strict_bound_margin subtracts the absolute combined period from that bound. A nonpositive margin gives status="obstructed" and obstruction_certified=True: no strictly acute circular phase configuration has those periods. Equality is excluded because the phase domain is open.

A positive margin gives status="necessary_bound_passed" and no obstruction certificate. It does not establish phase feasibility, sine balance, equilibrium or recovery. The function tests one supplied witness; it neither enumerates all separating directions nor solves a nonlinear equilibrium. Periods and chains retain their specified cycle basis. No live graph or state is changed. The export schema is tnfr.acute-cycle-period-assessment.v1, with shared exact projection and node-label validation. The joint-compatibility theorem owns the full criterion, complete-law transfer and counterexamples.

Target-free acute-sector capture

certify_sine_sector_capture(source, edge_turn_offsets=...) in the shared relational_sine_recovery.py returns SineSectorCapture. It accepts a captured SineExchangeComparison, SineRelativePattern or SineRelativeForecast. Pattern and forecast reports also expose .certify_sector_capture(...). No equilibrium target, local target radius or reference phase profile is supplied.

Offsets are exact non-Boolean integers in the canonical PhaseCycleGeometry edge order. Each declares delta_e=theta_head-theta_tail+2*pi*offset_e; the resulting periods are the fundamental-cycle sums of those integers. Independent edge offsets can express nonzero winding. They are not the node-wise phase_turns argument of the target-centered recovery readers. Support retains the shared 32-node/50-edge evaluation budget, all nodes and all unit edges. Supplied graph, coefficients, capacity and sector remain premises rather than autonomous selections.

The reader uses chained-report admission and recomputes form and phase storage from source coordinates and uncertainty. It requires positive held capacities and positive loss, exchange and storage scale. A zero loss or frozen node cannot receive the convergence theorem.

The complete source set must be strictly acute in its declared cell. boundary_face_lower_bounds covers every canonical edge with signs -1,+1, including possibly empty faces. Convex supporting planes, shared exact linear algebra and outward rational trigonometry provide each bound. The minimum is boundary_phase_storage_lower_bound; multiplication by the storage scale gives boundary_storage_lower_bound. The strict energy_margin subtracts storage_upper_bound from this barrier. These are sufficient bounds, not claims that a numerical optimizer found the exact boundary minimum.

With all premises and both strict comparisons satisfied, status="admitted" certifies existence of one interior equilibrium phase geometry, uniqueness in the sector, forward sector retention and convergence of every represented initial state under the complete sine law. Its coordinates and a convergence time are not produced. Otherwise status="unavailable" retains separate hypothesis_failures and unresolved_conditions; no instability or absence of equilibrium follows. The theorem owns the compactness, invariance and full nodal dissipation proof.

Comparison sources retain exact captured degree/capacity-weighted form and declared lifted-phase means. Relative observations leave absolute origins unobserved. Forecasts follow the full-endpoint and held-capacity requirements of shared admission. The schema is tnfr.relational-sine-sector-capture.v1 and uses shared exact JSON projection and label checks. No graph, event or trajectory is executed. Capture within this supplied sector does not certify formation of the sector from another preparation or admission of a preceding support/reset event.

Analytic prepared entry into a captured sector

certify_sine_prepared_entry(source, scaled_time=..., edge_turn_offsets=...) in relational_sine_entry.py returns SinePreparedEntry. An exact captured SineExchangeComparison retains its identical-real-phase admission contract. A SineRelativePattern instead supplies its original per-node form and phase residual radii, consistent real lifts and arbitrary common origins; its .certify_prepared_entry(...) adapter uses this same owner. Both paths require strictly positive exact held capacities and positive loss. The complete normalized sine law, all support and all coefficients remain unchanged through transit and capture. Unsupported source domains raise. The supplied nonnegative scaled_time=tau corresponds to horizon=tau/e in the declared original clock. The shared support budget is 32 nodes/50 edges; the exact exponential work budget is lambda*tau<=4096. The reader never adjusts the supplied horizon or law.

The transit theorem uses the exact degree/capacity-weighted mean, reversible diffusion gap and a global bound on the nonlinear sine current. It retains both complete evolution rows through nonacute and antipodal passage. Weighted centering contracts the initial form residual norm. Initial phase and form errors remain in the phase endpoint; phase_radius_upper_bound describes only its dynamic correction. Cached source storage, projected relative rectangles and nominal means cannot replace the original residual set. Initial storage is recomputed with both form and phase uncertainty.

centered_endpoint_form_bounds and centered_endpoint_phase_bounds enclose every node after subtracting that member's own conserved weighted means. Tighter edge intervals retain the same correlations. For an exact source, endpoint_form_bounds and endpoint_phase_bounds also enclose absolute coordinates. For a relative pattern they remain None, as do the absolute weighted means, even at zero residual error. Nominal mean fields describe only the supplied centers. The family is a union of distinct conserved mean leaves; arbitrary corners of the outer node rectangle need not satisfy the correlated edge constraints. No forecast or observation is fabricated.

The existing sector-capture interval kernel assesses that enclosed endpoint at the actual horizon, using the same complete law. Its source field retains the initial source; uncertainty_scope explicitly identifies the analytic endpoint. input_forecast_admitted and input_forecast_requested_end_time remain unavailable. initial_zero_winding_certified requires every original raw phase-edge interval to be strictly acute, with no automatic unwrapping. Their differences then telescope, proving zero initial periods. If this sufficient initial comparison fails, initial_cycle_periods is None and acquisition is unavailable. Admission additionally requires endpoint capture and a nonzero final period vector. An early, loose or wrong-sector comparison remains unavailable, not proof of failed formation. No exact crossing time or acuteness during transit is claimed.

initial_form_storage is exact only without form uncertainty; otherwise it is None. The additive initial_form_storage_bounds, initial_phase_storage_bounds and initial_storage_bounds retain the full preparation budget. A structured form preparation can be very costly, and the small-ratio existence family has diverging initial storage. This is conversion of supplied form information into maintained phase winding, not autonomous selection of a preparation, support birth or physical identity. The exact export schema is tnfr.relational-sine-prepared-entry.v1 and uses the shared JSON projection and label admission.

Controlled slow-phase comparison

bound_sine_slow_phase(source, slow_time=...) in relational_sine_reduction.py returns SineSlowPhaseBound; SineRelativePattern.bound_slow_phase(...) delegates to the same owner. Sources are exact SineExchangeComparison preparations or original SineRelativePattern residual families, with all primitive data revalidated by the shared preparation owner. Forecast reports are not accepted. The fixed connected simple unit support, positive exact held capacities, positive loss, exchange and storage scale belong to the complete normalized sine law. Initial phases may be nonuniform and nonacute on their supplied real lifts. The identical-phase restriction of exact prepared entry does not apply to this comparison.

The caller supplies a finite nonnegative slow_time=sigma through shared exact/represented-real admission; Booleans and nonfinite inputs reject. With alpha=w/(beta*pi*e), eta=beta*alpha^2 and tau=e*t, the comparison instant satisfies sigma=eta*tau. The report retains the supplied exact slow_time and outward intervals scaled_time_bounds=sigma*beta*e^2*pi^2/w^2 and horizon_bounds=sigma*beta*e*pi^2/w^2. These generally irrational clock conversions enclose one declared instant; they are not independently uncertain times or midpoint substitutes. No horizon is selected or shortened.

The comparison theorem, Section 29 uses K=diag(nu_i/d_i), M=K^-1, A=KL and z=alpha*P_M*x, where P_M subtracts the weighted mean. Its reference obeys dpsi/dsigma=K*S(psi) with psi(0)=theta(0)+z(0). Initial form information therefore remains in the reference preparation. Write D(tau)=exp(-A*tau). The reported bounds distinguish these weighted M-norm comparisons at the declared instant:

Report field	Quantity bounded
composite_phase_error_upper_bound	theta(tau)+D(tau)*z(0)-psi(sigma)
phase_error_upper_bound	theta(tau)-psi(sigma), including the fast initial transient
joint_error_upper_bound	theta(tau)+z(tau)-psi(sigma)
scaled_form_remainder_upper_bound	z(tau)-D(tau)*z(0)
form_remainder_upper_bound	P_M*x(t)-D(tau)*P_M*x(0) in original form units

uniform_composite_phase_error_upper_bound covers the corrected comparison over the entire interval from zero through the supplied slow horizon. The uncorrected phase has initial discrepancy -z(0); uniform small error from time zero does not follow for a fixed nonzero scaled preparation. scaled_form_norm_upper_bound and form_norm_upper_bound also retain their homogeneous transients. The scalar exponential_decay_bounds controls the matrix semigroup's norm; it is not a componentwise replacement for D(tau)*z(0). No matrix exponential or reference trajectory is evaluated.

phase_edge_error_upper_bounds follow the canonical edge order and bound differences between actual and reference edge gaps. The separate phase_potential_error_upper_bound concerns V(theta)=sum_edges(1-cos(theta_j-theta_i)), without a storage-scale factor. initial_form_norm_upper_bound and initial_storage_bounds preserve the original preparation cost. Small feedback strength with fixed scaled form does not imply a small original form or storage budget.

Every member of a relative residual family is compared with its own reference initialized from that member's actual phase and form. The common error bound does not compare the entire family to one nominal reference trajectory. reference_initial_phase_bounds, weighted_form_mean and weighted_phase_mean are None for relative sources, including zero-error sources with unobserved origins. centered_reference_initial_phase_bounds retains correlated outer bounds after subtracting each member's own weighted phase mean. reference_scope records this distinction. Cached storage and projected relative rectangles do not replace the original residual set.

Shared geometry admission retains the 32-node/50-edge budget. The global Lipschitz bound is ell=2*max(nu_i); the rational growth enclosure requires ell*slow_time<=4096. Unsupported domains and larger growth exponents raise. Decay uses the shared exact exponential owner through exponent 4096 and a monotone tail enclosure beyond it, recorded by decay_tail_enclosure_used. This preserves the requested comparison time. The tail enclosure and outward 128-bit dyadic rounding can impose a numerical floor on very small errors; they do not weaken the analytic theorem or certify a rounded value as zero.

The report supplies error bounds without an accuracy threshold or an admitted verdict. A large bound remains a bound. It supplies no solver execution, endpoint capture, basin match, infinite-time limit exchange or event selection. Those claims require their own full-state evidence. Its schema is tnfr.relational-sine-slow-phase.v1. Direct to_dict() export and relational_report_to_dict() delegation retain the same exact report body and node-label admission, with their respective schema envelopes. The latter also supports SinePreparedEntry and SineSlowCapture; neither route authenticates source provenance.

Full-state capture from a controlled phase reference

certify_sine_slow_capture(source, slow_time=..., target_phase_turns=...) in relational_sine_reduction.py returns SineSlowCapture. SineRelativePattern.certify_slow_capture(...) delegates to this owner. An existing SineSlowPhaseBound also exposes .certify_capture(target_phase_turns=...), which recomputes preparation and comparison bounds from its source and slow_time under chained-report admission. The same exact/relative source, positive complete-law coefficients, held capacities, finite slow clock and numerical budgets apply as in the preceding comparison contract.

target_phase_turns supplies one exact integer or Fraction per captured node, in source node order. Floats, Booleans, incomplete vectors and unsupported targets reject. The shared target owner reconstructs the circular geometry, requires strictly acute principal edge turns and proves exact sine criticality by its symbolic odd-cancellation test. An unresolved symbolic identity is an unsupported target, not a proved absence of equilibrium. target_geometry, target_phase_turns and target_edge_turns retain this evidence; the edge fields use the canonical geometry order. No numerical equilibrium solve or target search is performed.

The handoff theorem, Section 30 first establishes an independent enclosure of the reference phase flow. target_centered_phase_bounds subtracts the target's degree/capacity-weighted mean before enclosing mathematical pi. Each actual source member compares with the stationary target shifted to that member's conserved phase mean. nominal_initial_reference_distance_bounds and initial_reference_uncertainty_upper_bound retain the initial mismatch and the original form/phase residuals. Their combined bound propagates with the global Lipschitz growth factor into reference_distance_upper_bound. This bounds each member's own reference; it does not replace the family by one nominal flow or assign the target to an actual state.

actual_phase_distance_upper_bound adds the complete law's uncorrected phase error, including its fast initial transient. The remaining original form norm is retained from slow_phase.form_norm_upper_bound. These two norm bounds supply correlated outer edge intervals for the actual endpoint. The target's supplied real lifts determine integer edge offsets exactly; the reader does not unwrap the source to reduce a residual. Arbitrary common origins cancel, while unknown absolute means of relative sources remain unavailable.

For ell=2*max(nu_i), phase distance bound r and original form norm bound X, the report also proves correlated_form_storage_upper_bound=ell*X^2/2 and correlated_phase_storage_upper_bound=V(target)_upper+ell*r^2/2. The phase expression uses criticality to cancel the linear term and bounds the unscaled potential V; the complete storage includes its factor beta. The shared capture core intersects these producer-proved upper bounds with its edgewise storage intervals, preserving their lower endpoints. These private intersections are not public caller-supplied certificate inputs. The enclosed endpoint retains both edge boxes and the proved correlations; arbitrary corners of the larger Cartesian edge box need not satisfy its storage upper bounds.

The nested capture requires every enclosed endpoint phase-edge interval to be strictly acute and total form-plus-phase storage to lie strictly below all certified sector boundary faces. SineSlowCapture.admitted and .status delegate to this full-state result. Good phase geometry with excessive form storage, or an unresolved acute margin, remains unavailable. Such a failed sufficient comparison does not prove instability or impossibility. Passing certifies the actual source family's subsequent retention and convergence in that sector under the unchanged complete law.

source remains the original preparation. The analytic endpoint time is enclosed by slow_phase.horizon_bounds; nested capture.observation_time and forecast metadata remain None. No rounded timestamp, observed endpoint, supplied response or solver trajectory is inserted. Initial winding is not assumed or certified, and a zero target period is allowed: this capture verdict alone is not an acquisition or winding-creation certificate. The target remains a proof reference, without a state assignment, changed law or event. Exact export uses tnfr.relational-sine-slow-capture.v1 and the shared SDK projection and label admission, retaining the nested preparation, comparison, target and capture provenance.

Consensus for a phase-flat preparation budget

certify_sine_budget_consensus(geometry, reference_model=..., capacity=..., form_storage_budget=...) in relational_sine_budget.py returns SineBudgetConsensus. Its family is a SinePhaseFlatBudgetFamily: every signed form vector with x.T*L*x/2<=B, initially identical real phase lifts, and arbitrary independent common form and phase origins. No particular EPI vector, phase record, reference response or representative state is consumed. Independent nonzero phase errors are outside this family.

The supplied PhaseCycleGeometry is rebuilt from its primitive support; forged derived fields reject. The connected simple support declares unit sine coupling and retains the 32-node/50-edge budget. capacity contains one strictly positive exact/represented scalar per geometry node, held throughout the complete unforced sine law. The regular reference model supplies positive loss, exchange and storage scale. form_storage_budget=B must be finite and nonnegative. Shared scalar admission rejects Booleans and nonfinite values. This reader has no horizon argument or trajectory solver.

The all-time theorem uses the shared weighted quotient gap lambda, K=diag(nu_i/d_i), M=K^-1 and alpha=w/(beta*pi*e), with eta=beta*alpha^2. It bounds the initial auxiliary function W=(||P_M*theta+z||_M^2+||z||_M^2)/2, z=alpha*P_M*x, by Wbar=2*eta*B/(beta*lambda). This function is distinct from the model's original form-plus-phase storage. For B>0, the sufficient premises are eta<=1 and, on every edge, 2*sqrt(K_i+K_j)*sqrt(Wbar)<pi. feedback_margin and branch_margins retain their outward interval checks. The feedback premise is a sufficient proof condition, not an instability threshold when it fails.

Passing proves consensus_certified and zero_winding_certified for every family member and all nonnegative structural times (tau=e*t). Each member converges to its own conserved weighted form mean and common phase lift. No convergence rate or common absolute origin is supplied. The natural invariant strip is |theta_j-theta_i|<pi, so consensus and zero winding do not require every edge to remain acute. acute_margins and the stronger acute_trapping_certified flag separately test the pi/2 condition. If B=0, connectedness forces uniform form as well as flat phase: the whole family is exactly stationary for every admitted positive coefficient ratio, independently of the eta<=1 premise.

initial_form_norm_upper_bound and initial_lyapunov_upper_bound retain preparation bounds. candidate_phase_edge_upper_bounds describes the proposed invariant sublevel and is not a trajectory bound unless the certificate passes. Then lyapunov_upper_bound, phase_norm_upper_bound, scaled_form_norm_upper_bound, form_norm_upper_bound and phase_edge_upper_bounds supply uniform all-time bounds on the centered coordinates and canonical edge gaps. When sufficient inequalities remain unresolved, status="unavailable", unresolved states why, and those certified-bound fields are None. Initial and candidate fields remain visible; their availability does not certify motion, formation or instability. Fixed dyadic rounding can leave a mathematically valid strict condition unresolved.

When the feedback premise is certified, nonzero_winding_budget_lower_bound gives a conservative necessary original form cost for nonzero winding acquisition from exactly flat phases. With r=w/e and kappa=max_edges(K_i+K_j), it encloses from below beta^2*lambda*pi^4/(8*kappa*r^2). This inverse-square requirement is neither sufficient for formation nor a sharp threshold. The field is None when its feedback premise is unavailable. The result covers the declared fixed-budget family; it does not extend to arbitrary initial phase records or select a preparation, constitutive law or support.

The exact schema is tnfr.relational-sine-budget-consensus.v1. to_dict() and the generic SDK exporter reuse the shared rational projection and label admission. Generic export wraps the same report payload in the standard tnfr.relational-report.v1 envelope; SDK file export uses the shared atomic writer. They retain the explicit family and theorem scope without authenticating a physical preparation. See the C5 usage example.

Full-state symmetry and an oriented cycle

assess_sine_cycle_symmetry(source, permutation_indices=..., cycle=...) in relational_sine_symmetry.py returns SineCycleSymmetryAssessment for one supplied node permutation under the complete unforced sine law. source.assess_cycle_symmetry(...) delegates to the same owner. The source must be an exact SineExchangeComparison and uses chained-report admission. Nonnegative form loss and capacities are allowed; exchange and storage scale must be positive. No inverse capacity, spectral gap or phase-geometry solver budget is needed.

permutation_indices[i] gives the destination index of source node i. The ordered sequence must contain each index exactly once, with nonboolean integer entries. cycle gives at least three distinct source labels in orientation order, without repeating its first vertex at the end. Every successive edge, including the closing edge, must belong to the support. The cycle may cover only part of a larger graph, but the permutation must preserve the complete support and all capacities. A symmetry of the cycle alone does not establish equivariance of the full law.

support_automorphism, capacity_preserved, form_preserved and phase_lift_preserved separately record the support and exact full-state checks, including x[p(i)]=x[i] and theta[p(i)]=theta[i]. cycle_orientation_reversed compares the mapped signed cycle chain with the negative original chain. Phase equalities concern the captured real lifts, without a tolerance or an inferred nodewise wrapping. The proof uses full-field equivariance and uniqueness to preserve a fixed initial state: trajectory_symmetry_certified does not itself require cycle reversal. Reversal additionally certifies zero_winding_when_nonantipodal: the cycle's principal winding is zero at every time when none of its edges is antipodal. nonzero_winding_limit_excluded also rules out a circular phase limit with nonzero winding and no antipodal cycle edges. In particular, the source cannot acquire a strict acute nonzero winding on that cycle.

The full obstruction gives status="certified"; otherwise status="unavailable" and unresolved identifies each failed premise. Malformed source, permutation or cycle inputs raise. Exact form, phase-lift and capacity residuals remain visible. initial_form_storage is recomputed from primitive edge differences, independently of cached source storage.

This conditional exclusion does not certify avoidance of antipodal edges, assign a principal period on that boundary, identify a limiting equilibrium or prove consensus. A failed symmetry check does not prove acquisition. The reader supplies no trajectory, horizon or event and does not enumerate an automorphism group. Common form and phase shifts preserve the tested equalities. A permutation-invariant uncertainty box need not have invariant individual members, so relative patterns and forecasts are unsupported. Combined transformations such as a permutation followed by global sign reversal are also outside this pure-permutation test.

to_dict() uses schema tnfr.relational-sine-cycle-symmetry.v1 with the shared SDK rational projection and label admission. Generic SDK export wraps the same report payload in tnfr.relational-report.v1; file export uses the shared atomic writer. These exports retain the revalidated source, supplied permutation and oriented cycle without authenticating a physical preparation. The equal-budget example compares this obstruction with the separately certified acquisition source.

Conservative cycle identity and recurrent families

assess_sine_cycle_identity(source, cycle=..., winding=..., radius=..., excess_ceiling=..., form_mean_bounds=(lower, upper), phase_turns=None) in the shared recovery/geometry owner assesses the complete conservative sine law on an acute cycle. source is a SineRelativePattern or SineRelativeForecast. The same source admission, exact target, centered norm, Taylor remainder and coercive barrier serve the existing recovery certificate. Positive-loss recovery still requires e>0; this new assessment requires exactly e=0 and positive held capacities. No model coefficient is changed to make either theorem apply.

The target has integer winding k with 4*abs(k)<n. The supplied radius must certify abs(2*pi*k/n)+sqrt(2)*radius<pi/2. With the shared lower coercivity bound kappa, the family is

Z² < radius²,
E - E_target < excess_ceiling < kappa*radius²,
lower < sum_i(d_i/nu_i)*x_i / sum_i(d_i/nu_i) < upper.

excess_ceiling is strictly positive and the mean interval has positive width. These are caller-declared analysis restrictions, not new parameters in the evolution law. Shared scalar admission rejects Booleans, nonfinite values and malformed bounds. The family is open, invariant in both time directions and has finite positive ambient volume; its closure is compact. Every member retains its acute cycle geometry and winding. Almost every member is nonstationary and recurrent in full form/circular-phase state.

SineCycleIdentityAssessment keeps distinct conclusions:

• family_admitted and family_almost_everywhere_recurrence_certified concern the invariant family, not the supplied observation's individual future.
• source_set_trapping_certified concerns every state consistent with the source's relative uncertainty and the declared held law. It requires the strict radius and barrier tests, independently of a smaller family ceiling.
• relative_source_family_membership is certified_inside only when the source bounds also satisfy the family ceiling. Otherwise it is unresolved: failure of a sufficient upper bound is not an outside-state proof.
• Absolute family membership remains unavailable because these relative sources discard an arbitrary common form origin. The supplied mean slab restricts a family; it does not measure that missing origin. Recurrence of a chosen state remains unavailable_for_chosen_state.

Forecast reports retain their full endpoint box, actual validated time and original admission status. When a held capacity is interval-valued, the theorem applies separately to each admitted fixed positive realization, using that realization's weighted mean. It does not define a flow of uncertain or changing capacities. A source set may be trapped even when a requested family ceiling fails admission; an admitted family may coexist with unresolved source trapping. Their margins and unresolved conditions are kept separate.

The proof uses conserved excess storage and the shared finite-volume recurrence argument. It supplies no return time, single frequency, attraction, pattern formation, coarse-scale closure or physical identification. A finite grid, singleton or fixed-energy surface does not inherit an ambient almost-everywhere result automatically.

to_dict() uses tnfr.relational-sine-cycle-identity.v1; the generic SDK export delegates to the same exact projection and label admission. This detached assessment evolves no graph and installs no new runtime law.