Retained sine pair state, identification and pulses

Sufficient unordered/mixed pair state and cancellation inversion, capacities and support symmetry, pairing observations and windows, emission/transfer and doubled-C5 pulse response.

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

Retained interaction state of a moving pair

The theory contract combines existing owners on the fixed conservative unit doubled C5:

• comparison.regional_transfer(region=pair) retains per-edge currents and the weighted cut sum. Each pair has weighted form 8*X; divide a selected neighboring block's four-edge sum by eight to obtain its contribution to mean form. Opposite nonzero block contributions can cancel in the net sum.
• assess_sine_replica_scale retains the realized internal R,U,Q, means, environment and fine source, and derives the exact full-law pushforward within its nonantipodal chart. This supplies sufficient state under its symmetry premises; the instantaneous current alone does not.
• assess_sine_mixed_pair_state retains ordered constituent information where support breaks an independent swap symmetry. Its actual attachment-sensitive field cannot be replaced by complete-replica formulas.

At zero pair resultant, keep the full comparison and regular cut/kinematics readers, or use the global pair coordinates under their fixed support and unit-law premises. The midpoint scale reader still rejects its chart boundary. Node-relative resultant_kinematics is not a pair midpoint or normalized pair phasor; its rates and the regional ledger retain their own normalizations. None of these observations drops constituents, advances a graph or supplies a boundary law.

A fully retained joint network is closed under its supplied law. A pattern considered alone still consumes its surroundings; a held environment is a new premise, and hidden-state elimination must retain initialization and causal memory. Prepared phase-offset families and tangent memory kernels do not provide an all-state replacement. The existing SDK delegates/exporters remain the implementation boundary; no duplicate interface report is needed. The guide connects the current and retained-state workflows.

Global unordered pair state, including phase cancellation

The pair owner exposes SineGlobalPairState through two exact evaluators:

• derive_sine_global_pair_state(forms, phasors) consumes ten signed forms and ten Cartesian unit phasors in fine-node order 0,...,9.
• evaluate_sine_global_pair_state(*, form_means, resultants, phase_products, internal_form_squared, form_phase_moments) consumes five entries in each primitive vector, in base-cycle order 0,...,4.

The pair-state, cancellation and finite receiver APIs are defined in tnfr.physics.relational_sine_pair. Their established imports from tnfr.physics.relational_sine_scale remain aliases to the same objects; signatures, report fields and JSON schemas are unchanged.

The support is fixed: consecutive pairs (0,1),...,(8,9), all four unit cross edges between adjacent pairs around C5, and no within-pair edges. The complete normalized-sine law has e=0, w=beta=nu=1, held capacities, original structural time t, and no forcing or events. These evaluators neither capture an arbitrary graph nor select another model.

For unit phasors z_plus,z_minus, set u=(x_plus-x_minus)/2. The five coordinates are X=mean(x), Z=mean(z), P=z_plus*z_minus, U=u**2 and W=u*(z_plus-z_minus)/2. Complex coordinates use (real, imaginary) pairs, not Python complex values or complex EPI. Shared exact/represented-real admission normalizes scalars to fractions and rejects Boolean, nonfinite, unordered or incorrectly sized inputs. Unit norms and realizability are exact tests; represented decimal phasors are not silently normalized.

The invariant evaluator requires abs(P)**2=1, P*conj(Z)=Z, abs(Z)**2<=1, U>=0, and W**2=U*(Z**2-P), and checks the implied abs(W)**2=U*(1-abs(Z)**2). These conditions admit a fine representative up to whole-member exchange, including at zero resultant or zero variance. Rational invariant coordinates may require irrational fine representatives. The global-state proof owns reconstruction, closure and continuation; arbitrary independent boxes or unconstrained Euler updates are not admitted by this API.

Every *_rate_pi_numerators coordinate field contains pi times its original-clock derivative. directed_block_edges orders (receiver, source); block_current_pi_numerators contains pi times each contribution to receiver mean-form rate, while block_current_rate_pi_squared_numerators contains pi squared times its derivative. Weighted cut contributions are eight times the mean-form contributions. form_storage, phase_storage, storage and the three *_storage_rate_pi_numerator fields retain independently computed component work and its total. pair_strata distinguishes split_phase, coincident_phase and antipodal_phase.

No cached report rate, storage or verdict is an evaluator input. Reevaluate the five primitive vectors to obtain a fresh report; export alone does not authenticate edited dataclasses. The direct schema is tnfr.sine-global-pair-state.v1, also supported by generic SDK projection and atomic export. This is a global symmetry representation and instantaneous field evaluation, not a new integrator, native execution mode, grouping rule, free parameter or physical identification. Existing midpoint adapters keep their chart restrictions. See the standalone example and full-node controls.

Recover internal information from an exactly cancelled pair

observe_sine_pair_cancellation(*, form_means, resultants, pair_index, resultant_first_tau_derivative, resultant_second_tau_derivative) in the pair owner returns SinePairCancellationObservation. It retains the fixed support and complete unit-coefficient conservative law of the global pair state above. Its clock is explicitly tau=t/pi; input derivatives are pi*d_t Z and pi**2*d_t**2 Z, respectively, not both multiplied by the same factor.

Supply all five signed form means and Cartesian resultants in base-cycle order, an integer index 0..4, and the two exact Cartesian derivatives of that selected resultant. Shared exact/represented-real admission rejects Boolean, nonfinite, unordered and incorrectly sized inputs. Every resultant must lie in the unit disk, and the selected resultant must be exactly zero. The function derives h and B from the supplied means and adjacent resultants using the same support owner as the forward evaluator. It accepts no hidden phase products, internal variances or cached verdicts as evidence.

The report always reconstructs form_phase_moment=-i*Z' and internal_form_squared=abs(Z')**2. Its phase_product is:

• Z'**2/abs(Z')**2 when Z'!=0, with reconstruction order 1;
• (2*Z''-B)/conj(B) when Z'=0,B!=0, with order 2;
• None when Z'=B=Z''=0, with order None and unavailable_reason="higher_order_or_persistent_environment_evidence_required".

Recovered products must have exact unit norm and satisfy Z''=2*i*h*Z'+(B+P*conj(B))/2. Contradictory evidence, including nonzero Z'' with Z'=B=0, raises ValueError. No tolerance, derivative fitting, phase repair or guessed hidden neighboring state is supplied. The report retains admitted input evidence, form_contrast=h, neighbor_resultant=B and phase_product_reconstruction_order alongside the reconstructed state.

This is exact local two-derivative compatibility and conditional state recovery, not authentication of a time series or independent validation of the supplied law. A finite zero record does not certify permanent silence; the proof separates delayed revelation from genuine environmental indistinguishability. No noisy-data admission, trajectory integration, graph mutation or physical identification follows. Direct schema tnfr.sine-pair-cancellation-observation.v1 and generic SDK export preserve exact fractions and the unavailable None value. See the example and fine-row tests.

Finite directed exchange near an invisible pair family

assess_sine_pair_finite_exchange(*, phase_rotation, horizon_tau) in the pair owner returns SinePairFiniteExchange. It evaluates the analytic finite-exchange theorem on the fixed conservative unit doubled C5. The law, held capacities and support are those of the global pair state above; there are no inputs or events.

Supply an exact Cartesian unit phasor phase_rotation=(a,b) and a strictly positive finite real horizon_tau. Shared admission retains represented-real values as fractions and rejects Boolean scalars, malformed/unordered phasors, nonfinite values and a norm different from exactly one. No phase normalization or horizon adjustment is performed. The clock is tau=t/pi; the original-clock endpoint is T=pi*horizon_tau.

All ten initial forms are zero. In orientation_order, selected pair 0 has phasors (1,-1) or (i,-i). Both members of each surrounding pair share its phasor. The comparison uses surrounding pairs (q,1,-1,-1), whereas its symmetry control uses (q,q,-q,-q), with q=a+i*b. These are separately prepared autonomous networks. The report retains their full initial phasors and fresh SineGlobalPairState assessments, including component storage and internal rates. The control is an exact stationary family; its selected-edge integrated currents and their difference are zero for every admitted horizon.

directed_block_edge=(0,1) uses (receiver, source) order. The observable is the difference, real-antipodal minus imaginary-antipodal, of integral_0^T I_(1->0)(t) dt, where I_(1->0)=Im(conj(Z_0)*Z_1)/(2*pi). It is a contribution to mean form; the corresponding weighted-cut integral is eight times larger. The report does not substitute the net change in pair 0, which also consumes the other neighbor's current, or interpret form exchange as energy transfer.

The report retains the exact rational integrated_current_difference_leading_term and integrated_current_difference_remainder_upper_bound. Its integrated_current_difference_bounds encloses their difference and sum using the shared outward rational interval method. The bounds hold for the evolving complete nonlinear law, including environmental backreaction. Strictly negative/positive reported intervals yield certified_negative or certified_positive; otherwise status="unavailable" retains the unresolved interval and its reason. A zero leading coefficient or an interval containing zero does not certify equality of the two responses. The evaluator neither searches for a shorter horizon nor retries with a different rotation.

This is an analytic finite-time certificate, not an executed trajectory, an inverse observation, a new propagator or a noisy-data inference. The direct schema tnfr.sine-pair-finite-exchange.v1 and generic SDK projection retain exact inputs, source states, bounds and unavailable values; export does not authenticate an edited report. See the usage example and independent fine-law controls.

One receiver endpoint with preparation and readout uncertainty

assess_sine_pair_receiver_readout(*, phase_rotation, horizon_tau, form_error_bound, phase_error_bound, readout_error_bound) in the pair owner returns SinePairReceiverReadout. It reuses the finite-exchange preparation family above, with the same complete conservative sine law and all ten fine nodes. Its new observation is the absolute mean form of pair 1, fine nodes (2,3), at T=pi*horizon_tau. Both incoming connections and the complete environment evolve. No phase derivative, hidden-state reading, baseline sample or selected-edge integral is an observation input.

The phase rotation is an exactly unit Cartesian phasor. The horizon must satisfy 0<horizon_tau<1/2. This upper restriction belongs to the rational uncertainty majorant, not a physical lifetime or loss of existence of the fine flow. All three error bounds are required finite nonnegative scalars; zero is admitted, Boolean/nonfinite/negative values reject. Shared exact and represented-real admission occurs before arithmetic, with no normalization, float materialization of exact fractions, missing-budget default or adaptive horizon selection.

form_error_bound bounds each fine node's initial deviation from its zero nominal form. phase_error_bound bounds each initial circular phase distance from its nominal phase, in radians. The budgets cover the hidden pair, receiver and environment independently. The proof chooses initial lifts within that circular bound and follows the actual periodic field; no arbitrary unwrapping is inferred from data. The form budget also bounds the common form origin. An additional unbounded offset is outside the contract. Support, capacities, law coefficients and clock are held exact. The final scalar mean-form reading has additive error at most readout_error_bound.

For each orientation, nominal_readout_centers combines the common linear term and its own cubic coefficient. nominal_readout_bounds includes the complete-law remainder. The two nonnegative amplification coefficients propagate separate form/phase preparation errors into propagated_preparation_error_upper_bound; adding final reading error gives readout_uncertainty_radius. expanded_readout_bounds encloses the complete recorded reading in orientation_order=("real_antipodal","imaginary_antipodal"). These are outward intervals; all primitive budgets, coefficients, nominal phasors and freshly derived full pair states are retained as exact fractions.

readout_gap_lower_bound is the larger of the two ordered differences of one interval's lower endpoint and the other's upper endpoint. Strictly positive gap yields status="certified_disjoint". Overlap or a shared endpoint yields status="unavailable", retaining the intervals and reason receiver_readout_intervals_overlap_or_touch. An unresolved sufficient bound does not prove that the actual responses coincide. Membership in an enclosing interval alone does not prove existence of a compatible complete trajectory, identify an arbitrary hidden state or authenticate an observed sample.

The separate symmetry controls have exact nominal receiver center zero. Their symmetry_control_readout_bounds retain preparation and readout errors: perturbed controls need not remain stationary or give an exact zero reading. Every admitted trajectory conserves its own complete storage; perturbed preparations need not share the nominal storage value or exact antipodality.

This is a forward analytic discrimination certificate for two declared preparation families. It contains no measured sample, fitted pressure or executed numerical trajectory. Direct schema tnfr.sine-pair-receiver-readout.v1, generic SDK projection and the shared atomic writer retain its uncertainty and unavailable evidence. Export does not revalidate an edited report. The theorem, usage and independent controls own the proof, execution example and tests.

Joint law and orientation ambiguity of the receiver endpoint

assess_sine_pair_receiver_confounding(*, phase_rotation, horizon_tau, epsilon_upper) in the pair owner returns SinePairReceiverConfounding. It retains the doubled-C5 nominal preparations and the sole observation Y=(x_2+x_3)/2 at t=pi*horizon_tau. Orientation A now evolves under the supplied family j_epsilon(delta)=sin(delta)+epsilon*sin(delta)^3, with constant 0<=epsilon<=epsilon_upper; orientation B evolves under epsilon=0. Both complete models retain the same phase row, support, held unit capacities, zero loss and exact structural clock. Each conserves its own declared storage. This is a comparison of possible law/state pairs, not an executor or a fitted correction to the observed pressure.

The exactly unit Cartesian phase_rotation uses shared scalar admission. horizon_tau and epsilon_upper must be finite and strictly positive; Boolean, nonfinite and nonpositive inputs reject. Exact fractions remain exact. There is no normalization, numerical trajectory, root search, adaptive coefficient range or fitted observation. The finite Taylor bound has no horizon_tau<1/2 restriction: that restriction belongs to the separate preparation-error majorant of the preceding report.

The report reconstructs the nominal primitive preparations and evaluates Y_A^epsilon-Y_B^0 at the two declared coefficient endpoints. Its intervals include the separate nonlinear remainders from both complete laws and round outward. coefficient_endpoints=(0,epsilon_upper) orders the evidence in endpoint_difference_bounds; endpoint_difference_signs uses 1, -1 or 0 for positive, negative or unresolved enclosures. Strictly opposite endpoint signs yield status="certified_collision_exists": continuous dependence on epsilon then proves that at least one coefficient strictly between zero and the upper endpoint gives exactly equal actual receiver readings. The certificate does not supply a root value, uniqueness or an observed collision. An interval containing zero, a shared endpoint or unresolved signs yield "unavailable"; these outcomes neither prove nor exclude a collision. collision_parameter_open_interval is the exact pair (0,epsilon_upper) only when certified, and otherwise None; both endpoints are excluded.

The preparations and readings here are exact. Their equal-output witness is also admissible when a containing preparation/readout error family permits zero error. That fact does not transfer the earlier fixed-law separation certificate to an unknown law. The stationary cancellation controls and the consensus linearization remain shared by this whole constitutive family; neither resolves the ambiguity. Different laws may have different storage values even for the same preparation, and no conservation of the reference sine storage is imposed on the cubic law.

The cubic alternative consumes phase information beyond the first circular moment. An independently justified first-moment sufficiency premise excludes it. The collision does not contradict that conditional sine-selection theorem or disprove the sine law; this one endpoint and the shared controls cannot independently establish the missing selection premise.

Direct schema tnfr.sine-pair-receiver-confounding.v1 and the generic SDK projection retain the inputs, bounds and unavailable evidence. Export does not authenticate an edited report. The proof, usage and independent controls own the result and its checks. No physical constitutive range, measurement bridge, formation claim or universal law selection is established.

Two receiver readings with one constant coefficient

assess_sine_pair_receiver_two_time(*, phase_rotation, horizons_tau, epsilon_upper, form_error_bound, phase_error_bound, readout_error_bound) in the pair owner returns SinePairReceiverTwoTime. It compares the preceding A-epsilon and B-zero families at two times using the same receiver Y=(x_2+x_3)/2. Each branch has one initial state throughout; A has one constant coefficient in the closed interval [0,epsilon_upper]. The full doubled-C5 support, unit capacities, phase row, own-law storage, exact clock and zero inputs, loss and events retain the preceding contract's meaning.

The exact unit Cartesian rotation q=a+ib must satisfy a>1/2 and b>0. The ordered pair horizons_tau must satisfy 0<h0<h1; epsilon_upper is positive and all three error bounds are nonnegative. Shared scalar admission rejects Boolean and nonfinite inputs before arithmetic and preserves exact fractions. The sufficient preparation bound requires 4*(1+3*epsilon_upper)*h1**2<1; violation rejects the domain, without asserting a physical instability. Every initial fine-node form and phase may vary independently within its budget, including the absolute form origin and environment. Phase errors mean initial continuous-lift offsets in radians; no phase-mean reconstruction is performed. Each scalar reading of each branch has the declared readout error. No independence of errors over time is assumed.

The report rebuilds the nominal primitive preparations, cubic contrast and full-law Taylor remainder. A coupled form/phase comparison propagates each branch's initial uncertainty. Convex coefficient chords bound these errors and yield affine inequalities N-A*epsilon <= recorded_A-recorded_B <= U-B*epsilon at each time. Strictly positive slopes give the necessary coefficient interval [N/A,U/B]. necessary_coefficient_bounds retains both outward dyadic intervals in time order, without clipping to the separately reported coefficient domain. Matching at both times requires membership in both intervals with the same coefficient. Interval membership alone proves neither a realizable collision nor an estimate of the coefficient.

Strictly disjoint outward intervals give status="certified_disjoint". coefficient_gap_lower_bound is the maximum of the two directed endpoint gaps; a positive value certifies separation. Overlap or touching yields status="unavailable" and unavailable_reason="necessary_coefficient_intervals_overlap_or_touch". Nonpositive affine slopes yield "coefficient_interval_slope_not_positive", with coefficient bounds and gap unavailable (None). These sufficient bounds need not resolve every distinguishable family.

On certification, joint_readout_weights gives exact signed weights in time order, with absolute values summing to one. They eliminate the common coefficient between the two affine bounds. The outward joint_readout_gap_lower_bound bounds their weighted A-minus-B record difference in form units, including all declared errors. An extremely small positive exact readout margin may round down to zero; coefficient-interval separation still determines status. Weights and margin are None when the coefficient test is unavailable. No response, fitted parameter, numerical trajectory or time-varying coefficient is consumed.

Direct schema tnfr.sine-pair-receiver-two-time.v1 and generic SDK projection retain primitives, coefficient endpoints, error bounds, affine evidence and unavailable fields. Export does not authenticate a report. The frozen protocol and proof, usage and independent controls own this conditional separation. It resolves the named two-family ambiguity; general state recovery, physical law selection and sensor/clock calibration remain outside its scope.

Two receiver readings with an unknown coefficient in either hypothesis

assess_sine_pair_receiver_two_law(*, phase_rotation, horizons_tau, epsilon_upper, form_error_bound, phase_error_bound, readout_error_bound) in the pair owner returns SinePairReceiverTwoLaw. Both nominal orientations now evolve under the complete cubic-sine family. A and B may use different coefficients, each in [0,epsilon_upper]; each branch retains one coefficient and one initial state throughout both readings. The complete support, phase row, capacities, clock and error semantics are those of the preceding two-time contract. Neither branch imposes the reference sine law or its storage on the other. phase_exchange_beta=1 records the held coefficient of the phase row; coefficient_endpoints bounds the unknown cubic-current coefficient of either hypothesis.

Primitive admission retains the exact unit Cartesian rotation a+ib with a>1/2,b>0, two strictly increasing positive horizons, a positive coefficient upper bound, and three nonnegative error bounds. Boolean and nonfinite inputs reject before arithmetic; fractions remain exact. The sufficient preparation domain is 4*(1+3*epsilon_upper)*h1**2<1. Initial phase errors are bounded lift offsets in radians. All ten form and phase coordinates, including the environment and absolute form origin, may be perturbed. Readout errors may be correlated across times and hypotheses.

The fixed readout_weights=(-h1,h0)/(h0+h1) have absolute values summing to one. Applied to the two receiver values, they cancel each nominal law's linear Taylor term separately. cubic_time_factor=h0*h1*(h1-h0) multiplies the remaining cubic coefficient. nominal_cubic_coefficients stores each orientation's quadratic polynomial in its own law coefficient, in ascending power order. nominal_cubic_coefficient_ranges gives its exact extrema on the declared interval; positivity of the polynomial coefficients makes these the endpoint values in the admitted rotation domain.

nominal_initial_rate_bounds bounds the maximum fine-node form rate at initialization for each orientation over its whole coefficient family. The full-law continuation bound then supplies form_growth_bounds_by_time, form_rate_bounds_by_time and remainder_upper_bounds_by_time, with time outermost and orientation innermost. These retain the evolving environment; they do not hold the initial pressure fixed. The cubic remainder applies to nominal zero-form trajectories. Separate preparation_error_bounds_by_time propagate every admitted nonzero initial error under its own law.

joint_error_radii combines the complete-law remainder, preparation error and scalar readout error using the absolute weights, separately for A and B. joint_difference_bounds is an outward dyadic enclosure of every admitted weighted record difference Z_A-Z_B over the entire two-coefficient square. A strictly positive lower endpoint gives status="certified_disjoint". Otherwise the report returns status="unavailable" with unavailable_reason="joint_readout_difference_not_strictly_positive". A zero endpoint cannot certify; an interval containing zero does not prove a realizable collision. This is a sufficient test of positive separation, not a complete classifier of all admitted inputs.

The report rebuilds all consumed quantities from primitives; it consumes neither a measured record nor another report's verdict. The weights do not cancel an arbitrary unknown constant form origin, which remains bounded by the preparation budget. Coefficients varying between readings, unknown clocks and continuous model defects require separate bounds. Certification distinguishes these orientation families without identifying either coefficient or selecting a unique physical law.

Direct schema tnfr.sine-pair-receiver-two-law.v1 and the generic SDK retain exact rational evidence and unavailable verdicts. Export is not authentication. The frozen protocol and proof, usage and independent controls own the result. Earlier single-reading and fixed-reference certificates keep their original inputs, remainder bounds and verdicts.

Continuous defects in both complete evolution rows

assess_sine_pair_receiver_defect in the pair owner takes the same primitive arguments as assess_sine_pair_receiver_two_law plus mandatory form_rate_defect_bound and phase_rate_defect_bound. It returns SinePairReceiverDefect. The added primitives are finite nonnegative bounds on the absolute additive residual in every fine-node form or phase row, respectively, almost everywhere over the entire observation window. Their units are form per unit tau and radians per unit tau. Boolean, nonfinite, non-real and negative inputs reject through shared original-scalar admission; exact fractions remain exact. The reference rotation, horizon and preparation admission domains remain unchanged.

Each hypothesis retains one initial state, one constant coefficient in the declared family and one residual history across both readings. Histories may differ between hypotheses and may be arbitrarily correlated across time and nodes. The theorem concerns absolutely continuous paths satisfying the full rows almost everywhere. It needs no derivatives of the residuals and installs no runtime input, pressure law, event or controller.

reference_certificate is a fresh two-law report rebuilt internally from the supplied primitives; this assessor accepts no report argument. Its conservative-model statements apply to the reference comparison. Residuals in the expanded family generally change total form and own-law storage; the proof owns their explicit balances. Perturbed controls need not be stationary.

For L=1+3*epsilon_upper, the separate response factors are h/(1-4*L*h**2) and L*h**2/(1-4*L*h**2). form_defect_response_factors_by_time and phase_defect_response_factors_by_time retain these in reference time order. defect_error_bounds_by_time combines them with their respective residual budgets. The result is additional form uncertainty, separate from the reference preparation, Taylor remainder and readout budgets.

joint_defect_radius is the absolute-weighted defect error for each branch; joint_error_radii adds it to the two reference radii in orientation order. joint_difference_bounds combines those expanded errors with the exact reference cubic ranges before outward materialization. It does not reuse a cached verdict or inflate already-rounded endpoints. Zero residual budgets reproduce the reference interval and verdict exactly.

Status is "certified_disjoint" only if the expanded interval's lower endpoint is strictly positive. Otherwise it is "unavailable", with unavailable_reason="joint_readout_difference_not_strictly_positive". Touching zero or losing a positive margin to outward rounding cannot certify. Unavailable evidence does not prove a realizable collision. Arbitrary time-dependent residuals are not canceled by the fixed nominal time weights.

Direct schema tnfr.sine-pair-receiver-defect.v1 and generic SDK projection retain exact inputs, nested reference evidence and the new verdict. Export does not authenticate a source or verify that an actual residual history satisfies its declared budget. Physical use needs independent whole-window defect bounds and a justified state, sensor and clock mapping. The frozen protocol, usage and independent controls retain this conditional scope.

Internal allocation, finite response and persistent geometric identity

assess_sine_pair_persistent_response(delta_bounds=..., horizon_tau=..., form_error_bound=..., phase_error_bound=..., readout_error_bound=..., radius=...) in the pair owner assesses the declared comparison. It is a detached analytic family certificate, not a captured graph, a numerical trajectory or an executor. relational_sine_scale retains a compatibility re-export of the same implementation.

The fixed model is the ten-node, degree-four, complete doubled C5 with unit held capacities, unit exchange/storage scales, zero form loss and clock tau=t/pi. Mathematical target phases are a*2*pi/5. In preparation order phase_split then form_split, pair 0 has either phase offsets (+delta,-delta) or form offsets (+u,-u), with u^2=2*cos(2*pi/5)*(1-cos(delta)). All other nominal coordinates retain the target. Both nominal centers have the same collective means and storage for the same delta. The independently perturbed full-state boxes need not be energy matched. Neither symbolic pi nor the square root is replaced by a captured binary64 source coordinate.

Shared represented-real admission precedes arithmetic. delta_bounds contains exactly two scalars satisfying 0<lower<=upper<=1 in radians. The horizon is strictly positive and satisfies 2*horizon_tau**2<1, the domain of this rational comparison bound. Radius is strictly positive; all three error budgets are nonnegative. Boolean and nonfinite physical values reject. These restrictions describe this assessor, not a stability limit of the law. Positive form and phase widths give full-dimensional open box interiors; zero widths remain valid lower-dimensional conditional preparations.

The scalar reading is the absolute mean form of nodes 2 and 3 at the declared horizon. Analytic remainder bounds include the evolving nominal environment. The nominal synchronized-pair reduction is used only to derive those bounds; the preparation-error comparison and storage barrier retain all twenty fine coordinates. Arbitrarily correlated initial errors and independently bounded additive readout errors are covered.

SinePairPersistentResponse retains nominal_readout_bounds, recorded_readout_bounds and recorded_difference_bounds separately from initial_radius_margin_bounds and storage_barrier_margin_bounds. persistence_certified_by_preparation describes the two geometric admissions; response_separation_certified describes only the recorded contrast. full_dimensional_preparation is true exactly when both initial widths are positive. The symbolic target, support, complete law and clock remain explicit. status="certified_persistent_response" requires both preparations to pass strict initial-radius and conserved-storage barrier tests inside the acute target chart, together with a strictly positive outward A-minus-B recorded contrast. Failure of a sufficient inequality yields status="unavailable" with its reason; it establishes neither a realizable response collision nor loss of identity. A response certificate alone cannot substitute for the geometric admission. The frozen protocol's stronger numerical stopping margin is checked against the resulting bound, rather than hidden in this API status.

Persistence concerns the specified winding/acute geometry for all time under the unchanged unforced law. It does not promise five internally active pairs, formation, a selected microscopic law, physical energy units or a particle identity. No result from the preceding defect or cubic-law certificates is transferred to this conservative family. The usage and independent controls retain these boundaries. SDK export preserves exact premises and availability; the direct schema is tnfr.sine-pair-persistent-response.v1. Serialization does not authenticate an actual preparation or observation.

Inherited sine dynamics with both internal nodes retained

assess_sine_replica_scale(graph, reference_model=..., pairs=..., phase_turns=None) in relational_sine_scale.py is a detached observer of the complete conservative sine law. It consumes an ordered partition into two-node pairs. Every fine node must occur exactly once; there are no edges inside a pair, and each adjacent pair of blocks must retain all four unit edges. The resulting base graph is connected. Capacities must be held, strictly positive and equal within each pair; different pairs may have different capacities. The model explicitly has zero EPI loss weight and the existing absent-input/event premises.

The caller supplies consistent phase lifts, optionally corrected by integer phase_turns in captured fine-node order. Each pair must admit a certified half-difference of magnitude less than pi/2. This selects its nonantipodal circular midpoint; no phase unwrapping is guessed. Common turn offsets cancel before mathematical-pi enclosure. A failed chart test does not mean that the underlying smooth sine field is undefined, nor is initial chart admission a future trapping certificate.

At an antipodal pair the global phasor is zero and its angle is unavailable, while the full sine law remains regular. Its magnitude can have a cusp through the zero, so extending signed cos(delta) does not extend this nonnegative midpoint-magnitude contract. Use the complete fine comparison and regional current observation there; do not relax chart admission.

For each pair the report retains mean form/phase and their half-differences, along with the complete captured fine comparison. These four coordinates retain both original nodes. The internal phase half-difference delta_i gives a resultant amplitude R_i=cos(delta_i). The inherited phase-driven mean form response contains R_i*R_j, while the mean phase row keeps the original base Laplacian. The amplitude's derivative additionally consumes internal form contrast. These are calculated effects of the retained state, not tunable coupling strengths or controller scores.

The report compares averaged full fine rates, their factorized mean/internal identities and the naive base-law rates. It also retains the complete storage decomposition. Computed residual intervals check the implementation; the proof owner establishes the all-state identities. The internal phase storage correction need not be positive outside an acute coarse chart.

Exact synchronization requires zero internal form and phase differences. That supplied invariant submanifold inherits the original nodal rows, capacity and clock, with fine storage four times coarse storage. It is not attracting synchronization or autonomous formation. A zero instantaneous coarse-rate defect does not prove this membership or general closure: internal form can subsequently create phase dispersion. Nearby equal-mean preparations can already have different mean rates.

SineReplicaScaleAssessment separates synchronized_submanifold_invariant from source_in_synchronized_submanifold and same_law_reduced_flow_certified_for_source. The latter certifies the supplied source's inherited reduced flow only on exact synchrony. The all_state_coarse_closure_obstructed flag concerns a description using only block means across the full admitted family, not the impossibility of every specially prepared coarse trajectory or an augmented collective state. resultant_magnitude_bounds, resultant_magnitude_rate_bounds and phase_current_factors retain the derived internal effect separately from coarse_form_defect and coarse_phase_defect. Intervals containing zero are not automatically exact equalities.

The same report also carries an exact collective description of unordered pairs. Alongside mean form X and midpoint phase Theta, it retains R=cos(delta), U=u^2 and Q=u*sin(delta). Here Q is a form/phase correlation, not nodal pressure. Their domains are 0<R<=1, U>=0 and Q^2=U*(1-R^2). These invariants identify the complete pair up to exchanging its two constituents. They remove a redundant label, not continuous internal degrees of freedom. The unordered-state proof derives their closed evolution and covers the synchronized singular point.

internal_form_squared stores exact U; form_phase_correlation_bounds encloses Q; the existing resultant field encloses R. Their direct rates are pushforwards of the full fine-node rates. The closed_*_rates independently evaluate the inherited invariant equations, using the derived internal_restoring_coefficients and internal_phase_coefficients. The three rate-closure residuals and internal_constraint_residual retain computed interval evidence. internal_constraint_rate_residual differentiates the constraint with those closed rates; it is not overwritten with zero. unordered_storage_bounds and unordered_storage_rate_bounds compute full fine storage and its derivative from the invariant description, with separate comparison residuals against the original fine calculation.

The inherited Poisson identity provides a further independent comparison within this report. poisson_coordinate_order declares (X,Theta,R,U,Q); poisson_coefficient_pairs declares the four independent bracket entries (X,Theta), (R,U), (R,Q), (U,Q). Each pair's poisson_coefficients encloses those entries; the opposite entries are their negatives and all others are zero. The coefficient uses the base degree: kappa=b*nu/(4*d). Capacities and support are held as required above.

unordered_storage_gradients and poisson_generated_rates retain the full storage gradient and its bracket-generated five rates. poisson_rate_residuals compare these rates with the direct pushforward of the captured fine-node field. poisson_casimir_residuals evaluate the tensor on the gradient of Q^2-U*(1-R^2); they retain computed interval bounds, not forced zeros. These fields expose an inherited identity, not an added evolution law, physical energy, a trajectory or a new arbitrary-state admission API.

The separate resultant-derivative inverse is an exact observation theorem with known law, capacities, clock and neighboring collective state. This forward assessment does not implement an inverse solver for noisy samples. Independent derivative intervals or a positive resultant alone do not establish a jointly realizable observation.

unordered_pair_state_closure_certified and unordered_pair_state_identifies_swap_orbits apply to realized states in the declared pair chart. They coexist with the means-only obstruction. The assessment consumes a captured fine state; it does not admit arbitrary independent interval tuples as realizable invariant data or authenticate a manually constructed report. At R=1,U>0,Q=0, the correlation rate is nonzero: equal phases alone do not justify discarding internal form. The polynomial rows divide by neither U nor 1-R^2, and remain defined at the synchronized point. At R=0 the chosen midpoint chart ends; a smooth fine field does not make this particular collective coordinate chart globally valid.

Generic SDK export delegates exact projection and label admission to the report owner; to_dict() uses tnfr.relational-sine-replica-scale.v1. No node is removed, no edge weight or runtime law is replaced, and no trajectory is executed. A collective description does not establish that the supplied pair is an emergent NFR, a universal fractal hierarchy or a physically identified constituent.

Collective identity with active internal pairs

assess_sine_replica_persistence(graph, reference_model=..., pairs=..., radius=..., excess_ceiling=..., form_mean_bounds=..., winding=1, phase_turns=None) reuses the single captured replica comparison and the shared full-support sine geometry. The five ordered pairs must describe a complete doubled C5, with common strictly positive held capacity, exactly zero form loss, and the existing absent-input/event premises. winding accepts exactly the nonboolean integer 1 or -1. Both members of pair j have exact target turns winding*j/5; the captured phases remain their actual represented values, with explicit optional lift turns in node order.

The caller declares a positive radius, positive excess-storage ceiling and finite open form-mean interval of positive width. These choose an analysis family; they do not modify the law. The joint theorem requires a strict acute radius and an excess ceiling below the full-state coercive barrier. The engine reuses the general exact rational Laplacian gap lower bound; it need not match the sharper symbolic doubled-C5 gap. Failure of a sufficient numerical bound is explicit abstention, not instability.

SineReplicaPersistenceAssessment keeps the following decisions separate:

• family_admitted certifies the open invariant family with all five synchronized pair tips removed. It has finite positive volume in the fine form/phase state; family_almost_everywhere_recurrence_certified applies to that measure, not an arbitrary preparation surface.
• source_set_trapping_certified checks the captured whole-state norm and excess against the acute barrier. It does not require membership in the smaller declared excess/mean family.
• source_family_membership additionally checks the captured absolute mean, strict family ceiling and every pair's nontip status. It is certified_inside, outside for a proved mean/tip exclusion, or unresolved when upper bounds are insufficient. An unresolved upper bound is not a proved dynamical exclusion.
• pair_source_status distinguishes exact_nontip from synchronized_tip. pair_all_time_activity_certified and pair_all_time_circulation_certified require source trapping and that pair's exact nontip status. source_joint_persistence_certified requires trapping and all five nontip pairs. A synchronized pair can coexist with protected geometry and activity in the other pairs.
• individual_recurrence_status remains unavailable_for_chosen_state. Family membership supplies neither a selected-state recurrence certificate nor a deadline for a full-state return.

The internal angle belongs to (delta,u/sqrt(beta)), not to primitive nodal phase or a new clock. For strict acute geometry, the report retains internal_phase_radius_bounds, normalized_angular_speed_lower_bound, internal_angular_speed_bounds and internal_full_turn_time_bounds. The numerical lower factor uses c_r*cos(D), with D=radius/sqrt(2), which is safely below the theorem's c_r*sinc(D). These are positive-turn traversal bounds for admitted active sources, not equal periods, phase locking, amplitude floors or bounds on full-state recurrence. An unresolved positive numerical speed bound for an extremely small coefficient does not refute the strict mathematical rotation theorem. Uncertified geometry withholds the associated numeric bounds.

The report preserves exact mean, divergence, target provenance, computed geometry/storage bounds and explicit unresolved conditions. Its to_dict() uses tnfr.relational-sine-replica-persistence.v1; the generic SDK exporter delegates to that owner. No simulation, native runtime dispatch, graph write, event, attracting waveform or autonomous grouping is introduced. The two-sided invariant family cannot certify capture from outside itself.

Unequal capacities and sufficient collective state

assess_sine_replica_capacity(graph, reference_model=..., pairs=..., phase_turns=None, radius=None, excess_ceiling=None, form_mean_bounds=None, winding=1) observes the same conservative sine law with independently held, strictly positive fine capacities. It captures once and reuses the paired coordinate and support kernels. Complete bipartite replica support is required over a connected simple unit base graph, with no within-pair edges. Pair lifts must admit |delta|<pi/2; the reader does not infer a branch. No graph state, capacity, event or runtime law is changed.

The derivation retains pair mean capacity v, half-difference eta, and the original (X,Theta,u,delta) coordinates. ordered_direct_rates are pushed forward from the actual fine field; ordered_generated_rates use the derived rows. Their differences appear in ordered_rate_residuals. The inherited storage gradients and two Poisson coefficients expose the same-law cross terms, with separate ordered_poisson_rate_residuals.

The unordered coordinates additionally retain P=eta*u, T=eta*sin(delta) and E=eta**2, where E here is squared capacity contrast, not total storage. capacity_aware_coordinate_order identifies (X,Theta,R,U,Q,P,T,E). Independent fine-rate pushforwards, generated rows and residuals have that order. capacity_aware_constraint_names identifies the six reported Gram constraint residuals. These observations come from one realizable fine state; containing zero in separate interval residuals does not admit arbitrary invariant tuples. Forward formulas require no division by eta and recover the symmetric case exactly. Swapping whole members, including capacity, is a relabeling symmetry; swapping states while leaving unequal capacities fixed generally changes the rates.

normalized_form_mean_weights and weighted_form_mean use fine degree-over-capacity weights. Their balance residual is separate from arithmetic_mean_rate_bounds, which need not contain zero. All identities refer to captured represented values: shared staging may round two distinct raw rational capacities to the same float. A lost raw contrast is not evidence about an asymmetric captured system. Existing equal-capacity scale and persistence readers keep their exact captured-equality requirements.

Supply all three family arguments together or leave all absent. Without them, family is None. With them, five ordered pairs must form doubled C5 and winding must be the nonboolean integer 1 or -1. The nested SineReplicaCapacityFamily reuses the full-state geometry, exact target turns and rational spectral lower bound of the persistence owner. Its finite-volume family includes synchronized tips and uses the conserved weighted mean slab. family_form_coordinate_bounds safely expands that slab by sqrt(2)*radius.

• family.family_admitted and family.family_almost_everywhere_recurrence_certified concern an open invariant fine-state family at the supplied fixed capacities.
• family.source_set_trapping_certified concerns the captured whole-state norm and excess barrier, independently of the smaller declared mean/ceiling family. family.source_family_membership additionally checks those strict bounds, retaining outside or unresolved when appropriate.
• family.individual_recurrence_status stays unavailable_for_chosen_state. No selected-state return or return deadline is inferred.

all_time_internal_activity_status and angular_circulation_status stay not_certified_by_this_reader, even when geometric trapping succeeds. Exact unequal-capacity nontip stalls and tip crossings prevent transferring the earlier all-state claim; they do not establish permanent cessation or loss of collective identity. The report's to_dict() schema is tnfr.relational-sine-replica-capacity.v1, supported by the generic SDK exporter.

Phase-only partners and independent collective-law admission

observe_phase_pairs(nodes=..., phases=...) in the shared scale owner takes only ordered labels and their circular phase representatives. Form, capacity, edges, a model and a proposed partition are not inputs. It compares squared chords 2*(1-cos(theta_j-theta_i)) using the shared certified cosine bounds. This is strictly monotone in circular distance on [0,pi], so it needs no inverse angle, phase unwrapping or fitted proximity threshold. Rational inputs are retained exactly; other real inputs use shared represented-real admission. The 64-node limit is an explicit quadratic-work budget, not a physical scale.

PhasePairObservation retains the admitted nodes and phases, pairwise distance_indices and squared_chord_bounds, and each node's nearest_partner_indices, nearest_separation_margins and node_statuses. A partner is certified only when its distance upper bound is strictly below every competitor's lower bound. The complete candidate_pairs is available only if every choice is unique and mutual. Otherwise status is unavailable, candidate_pairs is None, and reasons preserve the abstention. An overlap of enclosures does not prove an exact tie. Equal absolute represented gaps can separately establish an exact nearest tie when its group is strictly closer than every outside competitor. Node order determines serialization, never tie-breaking or greedy rematching.

The separation theorem proves that the protected doubled-C5 tube identifies its constituent pairs this way, including unequal positive held capacities. Common circular shifts, phase reversal and relabeling preserve the mathematical partition. A rounded common shift that discards differences before capture is a changed state; numerically unresolved comparisons still abstain. An instantaneous match on a generic state has no future guarantee.

assess_sine_state_pairing(graph, reference_model=..., pair_order=None, phase_turns=None, radius=None, excess_ceiling=None, form_mean_bounds=None, winding=1) captures the sine comparison once, projects its phases into the same observer, then independently applies the existing paired-support and capacity contracts to the observed partition. It does not recapture the graph. SineStatePairingAssessment retains comparison, observation, the declared pair_order and its source, supplied phase_turns, optional capacity assessment and explicit capacity_admission_status and reasons.

When a partition is observed, optional pair_order may only reorder or reverse members of those already observed pairs. If observation abstains, a supplied order remains an unverified declaration and capacity admission is not attempted. Without it, capture order supplies serialization for the instantaneous collective rows. All three family bounds must be supplied together, and target-dependent family assessment additionally requires explicit pair_order. Neither cycle orientation nor a favorable target fit is selected automatically. Lifts are forwarded to the existing chart admission: a valid circular match across a raw branch boundary can require declared turns before its collective coordinates are admitted.

Known support/law/chart refusal leaves the observed partition intact, with capacity=None and an explicit rejection. Invalid caller declarations remain errors; numerical or programming failures are not converted into scientific counterexamples. In particular, a phase-only permutation on fixed support can yield strict mutual partners that contain live internal edges and fail the replica contract. The reader never replaces them with graph twin classes.

all_time_pairing_persistence_certified requires successful capacity assessment and its captured-source trapping certificate in the protected doubled-C5 tube. It concerns the persistence of the identified partition, independently of membership in the narrower declared mean/excess family. It gives no formation, internal-circulation, selected-state recurrence or physical-identity certificate. The standalone schemas are tnfr.phase-pairs.v1 and tnfr.relational-sine-state-pairing.v1; the generic SDK exporter delegates to their owners while retaining these distinct verdicts.

Local onset and loss of observed pairing

assess_sine_pairing_transition(graph, reference_model=...) uses one shared sine comparison capture and its unchanged phase-only observer. It differentiates the observation under the complete normalized-sine law, without introducing a pairing force or changing the native runtime. Fixed unit support, held capacity, the comparison's coefficients and structural clock remain declared premises.

SineExchangeComparison.phase_rate_numerators() supplies the exact rational row N_i=w*nu_i*(L*x)_i/(beta*d_i), so theta'_i=N_i/pi. The same owner serves resultant kinematics without computing unused trigonometric drift bounds when only the phase row is needed. For the observer's squared chords, D'_ij=2*sin(theta_j-theta_i)*(N_j-N_i)/pi. SinePairingTransitionAssessment retains comparison, observation, these phase_rate_numerators, and squared_chord_rate_bounds in the observation's distance_indices order. These are ideal law rates, not measured secants.

The shared nearest-group admission returns exact_nearest_groups and outside_group_distance_margins. A singleton needs strict separation from every competitor. An exact tie needs equal absolute rational phase gaps and strict separation from outsiders; overlapping distance enclosures alone remain unresolved. Within a tie of absolute gap d, derivatives share the factor 2*sin(d)/pi; the signed numerator differences are exact rational coefficients. This preserves their dependence instead of subtracting separate rounded derivative bounds. A zero or unresolved factor, or tied first-order coefficients, gives no split certificate.

The backward and forward sides each retain:

• time_direction (-1 or 1), nearest_partner_indices and node_statuses;
• tied_rate_factor_bounds, tied_rate_coefficient_margins and tied_derivative_margin_bounds, with unavailable quantities explicit;
• candidate_pairs, status and reasons: certified_matching requires all choices to be strict and mutual; certified_nonmutual requires all choices to be determined and at least one to be nonmutual; otherwise the result is unavailable;
• separate support_admission_status and reasons, checked on the captured fixed graph only when a complete partition is established. Admission uses the existing paired-support and positive-capacity contract; it supplies no future phase chart, collective closure or protected-family membership.

local_interval_existence_certified certifies the reported nearest choices on some sufficiently small one-sided interval by smoothness and strict transversality. certified_time_horizon stays None: no numerical endpoint, uniform finite-precision resolution or lifetime follows. A finite-precision observer may abstain arbitrarily close to a tie even when the mathematical choice is strict. Complete grouping can be established or excluded locally while its separate support admission fails.

The proof and frozen direction control use the same phases with opposite internal forms. The theorem also gives qualitative robustness on time windows away from the crossing, but the report does not supply a numerical preparation radius or force perturbed switches to occur simultaneously. No graph edge, controller, trajectory, asymptotic capture or physical identification is created. The schema is tnfr.relational-sine-pairing-transition.v1, supported by the generic SDK exporter.

Pairing response under a declared alternative mobility

The constitutive-scope proof separates a local grouping mechanism from a uniquely selected microscopic law. On its fixed doubled-C5 preparation, both initially tied preference margins have positive derivatives for every finite strictly positive, phase-only reciprocal mobility with a locally well-defined flow. Reversing the initial form reverses these derivatives. This supplies a local interval for each admitted law; it does not transfer the sine law's numerical window.

SineExchangeComparison.with_current_squared_mobility(epsilon=...) evaluates the existing counterfamily on its unchanged source, without another capture: f_i=1+epsilon*(S_i/d_i)**2, m_i=f_i/(pi*d_i), x_dot_i=w*nu_i*m_i*S_i, theta_dot_i=(w/beta)*nu_i*m_i*q_i. Here q=L*x and S_i is the captured neighbor sine sum. Epsilon must be a finite nonnegative admitted real scalar; form loss must be explicitly zero. The graph, signed state, held nonnegative capacity and clock retain the comparison's admission. No native law is installed or dispatched. The constructor and relative_balance() share fresh reconstruction from admitted primitive state and the declared mobility coefficient. They do not consume cached gradients, resultants or rate rows as independent evidence.

SineMobilityComparison retains its original comparison, exact epsilon, mobility_factors, mobility_corrections, changed inverse_phase_metric, phase source, pressure, both full rate rows and computed storage-work residuals. Its law is current_squared_reciprocal_mobility; at epsilon zero it recovers the reference field. The actual source storage stays in the nested comparison. The shared kernel applies the factor to both exchange rows, with no derivative reconstructed from an observed response. Work cancellation follows from their matched reciprocal structure; interval residuals alone do not prove it. State-dependent skew exchange does not by itself preserve a Poisson bracket, invariant volume, recurrence or the prepared sine pulse.

assess_sine_pairing_mobility(graph, reference_model=..., epsilon=..., margin_triples=...) evaluates two caller-declared squared-chord margins. Each triple is (observer, alternative, preferred), defining M=D_observer,alternative-D_observer,preferred with D_ij=2-2*cos(theta_j-theta_i). It retains the source and alternative full fields, the chosen margins and their initial rates. No partner, epsilon or observation is optimized after evaluating the response.

SinePairingMobilityAssessment contains mobility, the independent phase observation, margin_triples and their captured margin_indices. initial_exact_ties distinguishes algebraic equal distances from overlapping initial_margin_bounds. sine_margin_rate_bounds, mobility_margin_rate_bounds and mobility_margin_correction_bounds retain the two source rates, changed rates and their separately calculated corrections. The sine_ and mobility_ contrast fields retain numerator, denominator, normalized bounds and status. Status is available or denominator_not_separated_from_zero; unavailable normalized bounds are None. source_margin_rate_difference_exact_zero records the exact shared-sine coefficient cancellation. certified_time_horizon remains None.

The dimensionless contrast (M1_dot-M2_dot)/(M1_dot+M2_dot) uses those declared rate rows. A denominator not separated from zero leaves it unavailable. Common positive clock scaling cancels, and common phase-origin motion does not change chord derivatives. A separated contrast can therefore distinguish phase-space directions without fitting a clock. Exact zero requires algebraic cancellation, not an interval that merely contains zero.

The fixed comparison uses triples ((1,2,0), (2,1,3)), the original twenty-edge source, and epsilon zero versus one. The baseline contrast is exactly zero; the alternative is strictly between 1/500 and 1/400. Both local preference directions agree. Form reversal reverses the raw rates and retains this ratio. This is a mathematical discriminator under declared laws, not a physical measurement or a unique-law selection. Neither a complete future matching, protected family nor numerical time horizon is inferred by this reader. The schemas are tnfr.relational-sine-mobility-comparison.v1 and tnfr.relational-sine-pairing-mobility.v1, supported by the generic SDK exporter.

Relative geometry and balance under alternative mobility

The relative-geometry theorem extends the existing acute storage barrier to the current-squared reciprocal law. The full field is invariant under common form and phase translations. Removing these two origins gives a sufficient autonomous relative state; it does not remove internal coordinates, freeze a moving reference or change the declared clock. Conservation of storage protects this relative geometry. It does not by itself establish an invariant measure or recurrence.

SineMobilityComparison.relative_balance(reference_node=...) consumes its existing capture and requires strictly positive held capacities. The node must belong to that capture. SineMobilityRelativeBalance retains mobility, the reference label/index, the nonreference relative_indices, and exact relative_form and relative_phase differences. Phase coordinates are circular coordinates supplied as raw real representatives, not a globally single-valued chart. The corresponding rate intervals subtract the actual moving reference's rates. No graph is read again.

normalized_mean_weights are normalized d_i/nu_i weights. The report keeps the weighted form mean, the local lifted phase mean, their rate bounds both from full-node sums and after exact baseline cancellation, and the difference between those evaluations. The lifted phase mean is not a scalar on the phase torus. node_divergence_bounds sum to divergence_bounds; the relative divergence is identical because the discarded origins do not enter the field. The exact formulas and their scope belong to the linked proof. At epsilon zero, constant_mobility_mean_and_volume_identities_certified is true. A zero rate or divergence at a single positive-epsilon source proves no identity.

assess_sine_mobility_geometry(graph, reference_model=..., pairs=..., radius=..., excess_ceiling=..., epsilon=..., winding=1, phase_turns=None) uses one capture, the existing doubled-C5 support admission and the shared exact target/storage barrier. It requires common positive held capacity, zero form loss, positive phase exchange and storage scale, and winding -1 or 1. Radius, ceiling, epsilon and phase lifts retain shared real/integer admission; a positive ceiling is required. Unsupported support or capacity raises. Insufficient strict interval margins produce explicit unresolved conditions.

SineMobilityGeometryAssessment keeps its balance and thus both full rate rows and the original source. The first member of the first pair selects the relative reference. phase_turns independently supplies the barrier chart; no absolute form-mean bounds are consumed. Its target, spectral bound, norm, cancelled excess storage and barrier margins retain the original meanings.

• family_admitted checks the geometric barrier and strict family ceiling. The bounded relative open family excludes synchronized pair tips. Their complement is invariant by twin symmetry and uniqueness.
• source_set_trapping_certified checks the source's strict norm and barrier margins independently of that smaller family's ceiling or tip exclusion.
• source_relative_family_membership is certified_inside, outside for an exact pair tip, or unresolved; pair_source_status and reasons remain available. Family admission and represented-source membership are separate.
• relative_coordinate_deviation_bound encloses the uniform sqrt(2)*radius bound for anchored form differences and lifted phase deviations from target along the trapped flow. It is not a bound on absolute means or raw phases.
• At epsilon zero, an admitted family has relative_family_recurrence_status="certified_almost_everywhere_in_relative_family". At positive epsilon it is unavailable_invariant_measure_unproved. An unadmitted family instead reports unavailable_family_not_admitted.

For positive epsilon, the old mean and Euclidean-volume identities fail in general, including arbitrarily close to the same exact target. To recover recurrence for almost every relative preparation in the volume sense, one must supply a finite invariant measure equivalent to relative volume, such as a positive integrable invariant density. A singular equilibrium measure does not discharge that obligation. Nonzero divergence does not rule out such a density or recurrence. No return claim for the selected state or discarded common origins, internal circulation, pulse period, attraction or native runtime dispatch follows from this report.

The schemas are tnfr.relational-sine-mobility-relative-balance.v1 and tnfr.relational-sine-mobility-geometry.v1; both use the generic SDK exporter. Projection validates nested source, pair and reference labels but does not authenticate a manually constructed public dataclass.

Whole-window pairing under a full initial uncertainty box

assess_sine_pairing_window(graph, reference_model=..., form_error_bounds=..., phase_error_bounds=..., window_start=..., window_end=...) retains one comparison capture. Both error arguments are mandatory vectors of nonnegative radii in captured node order. They describe independent errors in every initial form and real phase representative at time zero. The captured center, fixed unit support, held nonnegative capacities and structural clock are retained; the reader requires exactly zero form loss and no input or event. Time admission requires 0<=window_start<window_end. Boolean, nonfinite and malformed inputs are rejected through shared admission. The 64-node ceiling is a work budget. The exact captured centers and rational radii define the admitted box. Displayed interval endpoints may be rounded outward; their enclosing intervals do not authorize additional initial states beyond the declared exact radii.

Let a=w/pi, b=w/(beta*pi) and let avg_i average the actual neighbors. The complete conservative law gives global bounds M_i=a*nu_i on abs(x'_i) and B_i=b*nu_i*(M_i+avg_i(M)) on abs(theta''_i). Initial form errors give a phase-rate error bound V_i=b*nu_i*(r_x_i+avg_i(r_x)). With the exact captured numerator N_i, every admitted solution satisfies abs(theta_i(t)-theta_i(0)_center-N_i*t/pi)<=r_theta_i+V_i*t+B_i*t^2/2. The implementation uses rational upper bounds based on the shared lower bound for mathematical pi. This controls the nonlinear feedback for the entire window, without a Taylor-step solver or reduced hidden state.

SinePairingWindowAssessment preserves:

• The comparison, input error vectors, initial_form_bounds, initial_phase_bounds and exact window endpoints.
• phase_rate_numerators, form_speed_upper_bounds, initial_phase_rate_error_bounds, phase_acceleration_upper_bounds and phase_remainder_upper_bounds; the last quantity uses the window end and bounds deviation from each affine center throughout the window.
• phase_window_bounds and pairwise phase_gap_window_bounds in distance_indices order. Gaps use the exact numerator difference directly, preserving common-rate cancellation rather than subtracting separate phase windows.
• squared_chord_window_bounds and chord_bound_methods. Absolute gap ranges proved within [0,pi] use monotone cosine endpoint bounds; otherwise the shared interval cosine supplies a conservative bound. No phase lift is inferred or imposed to improve a verdict.
• nearest_partner_indices, nearest_separation_margins, candidate_pairs, status, reasons and whole_window_nearest_relation_certified.

Each nearest choice requires its chord upper bound to lie strictly below every competitor's lower bound. All nodes must have certified choices before the report states certified_matching or certified_nonmutual; the former also requires mutuality. These claims hold for every source in the full box and every time in the closed window. Wide or overlapping bounds yield unavailable, not a refuted physical prediction. No exact synchronization of initially equal-phase pairs is assumed under independent errors.

A complete matching receives the shared, independent support_admission_status and reasons. This checks the existing fixed graph and positive-capacity paired-support contract; it supplies no future pair chart, arbitrary collective closure or permanent trapping. The initial box is a declared analysis premise, not an authenticated measurement uncertainty. The result does not guarantee that a separately rounded sensor resolves the mathematical margins.

The frozen whole-box proof uses errors 2^-20 in all twenty initial coordinates and the window [1/128,1/64], with opposite nearest maps for the two form-reversed centers. The earlier local-transition reader retains its absent numerical horizon; this certificate has separately admitted finite-time evidence. No graph is mutated, no integration trajectory is fabricated, and no support birth or unbounded lifetime is claimed. The schema is tnfr.relational-sine-pairing-window.v1, supported by the generic SDK exporter.

Compare phase grouping with joint form-phase grouping

SinePairingWindowAssessment.joint_observation() returns a SineJointPairingProjection from the same captured comparison, preparation errors and time window. It does not reread a graph, change the law or replace the phase-only verdict. This general projection is distinct from the special synchronized K2,2 reference certificate below.

The projection re-admits primitive source state, preparation radii and window, then rebuilds all consumed phase/chord bounds, speed bounds and distance indices through the same owner as initial window construction. Altered cached rates, outcomes or verdicts cannot replace those premises. An equivalent original report association can be retained only after the calculation uses normalized state and rebuilt evidence; this is not provenance authentication.

initial_observation applies the shared joint observer to the exact captured center. The initial full error box receives separate joint-distance bounds and strict nearest-partner admission; the center's matching alone does not certify that box. On the whole future window, the form tube follows |x_i(t)-x_i(0,center)| <= form_error_i + form_speed_upper_i*window_end. The projection adds its squared pair differences to beta times the existing sharp phase-chord bounds, preserving the original nonlinear phase remainder and every competing partner. Zero-crossing form intervals must be squared with a zero lower bound. Joint distances use their known nonnegative domain.

The source joint-distance rates include both terms 2*dx*dx_dot + 2*beta*sin(dtheta)*dtheta_dot, using the already captured complete form row and exact phase-rate numerators. These are source rates, not measured derivatives or rates throughout the uncertain window. Initial-center observation, initial-box identification and future-window identification have different evidence. same_pairing_as_initial_box is unavailable unless both boxes/windows have complete certified matchings; two absent pairings do not establish equality. Resolved nonmutual choices remain distinct from unresolved distance bounds. Agreement at preparation and on a later window does not by itself certify the intervening time gap; the fixed-source theorem proves that additional interval separately.

support_admission_status tests the same strict replica contract after observation. A certified joint matching can fail that support test. It then has no right to inherit the strict replica law; rejection does not prove that every possible collective description fails. The unchanged-source comparison establishes different joint partners already present in both original form-reversed boxes, while their phase-only verdicts differ. It does not assert that joint proximity defines a unique NFR or physical geometry.

The schema tnfr.relational-sine-joint-pairing-projection.v1 supports the generic SDK exporter and retains its nested phase-window evidence. Public dataclass construction does not authenticate provenance. No support birth, permanent identity, graph mutation or new trajectory is inferred.

The separate critical-boundary theorem uses a symbolically specified trigonometric form amplitude. Its exact ties are algebraic identities, not overlaps of numerical distance intervals. The theorem proves a local acquisition of support-compatible joint pairing from the complete form and phase rates, with no numerical lifetime. Existing graph readers still capture represented real coordinates; inserting a rounded critical amplitude changes that mathematical source. Neither a rounded graph nor the window projection automatically inherits the exact tie or local-time proof. Shared interval kernels check strict coefficient signs and bound the symbolic amplitude without installing a new runtime law or preparation type.

Joint form-phase identification during exchange

observe_joint_pairs(nodes=..., forms=..., phases=..., storage_scale=...) uses squared distance (x_i-x_j)^2+2*beta*(1-cos(theta_i-theta_j)). It accepts distinct ordered labels, signed exact/represented real scalars, circular phase representatives and a positive scale. It does not accept BEPI containers, project complex forms to magnitudes or infer the scale. The 64-node limit is a work budget. Strict mutual nearest partners are inferred from outward distance bounds without a supplied partition, label tiebreak or matching repair. Resolved nonmutual choices and unresolved comparisons have separate statuses; neither supplies candidate_pairs. This is an instantaneous observation, not a future-state or support claim.

assess_sine_joint_pairing_window(graph, reference_model=..., pairs=..., form_error_bounds=..., phase_error_bounds=..., window_end=..., phase_turns=None) instead captures one exact reference through the replica owner. It requires conservative normalized-sine dynamics, fixed unit K2,2 support, two internally synchronized pairs, equal initial forms, common positive held capacity and a certified interpair lift strictly below pi. The same law, support, capacity and structural clock hold for all compared trajectories. Positive window_end=T declares the interval [0,T]; both error vectors must supply four finite nonnegative initial radii. Independent errors in all eight coordinates need not retain synchronization.

The joint identity theorem derives a constant positive reference separation J* from inherited energy, including when the phases coincide. The full-field bound uses scaled real phase lifts, with r0^2=sum_i(form_error_i^2+beta*phase_error_i^2) and L=2*w*nu/(pi*sqrt(beta)). Circular chord errors are bounded by these lift errors; no global Lipschitz claim is made for an embedded field. identification_budget_margin is a conservative lower bound on J*-8*r0^2*exp(2*L*T). Certification requires both its strict positivity and resolved mutual nearest comparisons of the distance tubes. joint_distance_window_bounds contains squared distances, indexed by distance_indices; it is not an evolved snapshot.

reference_identity_certified records the exact reference theorem separately from whole_window_identity_certified. Zero amplitude makes the reference identity unavailable. A positive amplitude too small for interval resolution, excessive uncertainty or unrepresentable exponential amplification makes the finite certificate unavailable, without asserting actual identity loss. Zero error bypasses amplification arithmetic but retains the reference and distance checks. reference_period_bounds reuses the P2 period owner; window_covers_reference_period concerns that reference only, not perturbed periodicity or a recurrence deadline. The scale is inherited from the declared storage and coordinate chart, not selected as a unique physical metric.

No graph is advanced or mutated. These readers do not prove attraction, permanent perturbed identity, autonomous formation, operator occurrence or physical constituents. Source capture, errors, horizon, bounds and availability are retained by schemas tnfr.joint-pairs.v1 and tnfr.relational-sine-joint-pairing-window.v1, including generic SDK export.

Pair-state interchangeability on fixed support

assess_sine_pair_support_symmetry(graph, reference_model=..., pairs=..., witness_pair=None) checks the complete-law symmetry criterion on one captured simple connected unit graph. It requires zero form loss and a common strictly positive held capacity. The supplied ordered pairs must partition every captured node; unlike strict replica execution, the symmetry question admits a single pair and within-pair edges. The 64-node limit is a work budget. This reader neither discovers a partition nor changes the graph.

An independent member swap preserves the full sine field for all states if and only if the two members have the same external neighbors. Equivalently, every interpair adjacency block is empty or contains all four edges; internal pair edges may be present. Necessity follows from the exact normalized phase row; sufficiency also accounts for the nonlinear sine row. Equal neighbor counts without equal attachments do not establish this symmetry.

SinePairSupportSymmetryAssessment retains the captured comparison, declared pairs, their pair_indices, within_pair_edge_indices, complete cross_block_edge_counts and incomplete_cross_blocks. Each entry of external_neighbor_difference_indices names the captured indices that distinguish that pair's attachments. pair_swap_symmetry records each result. For each failed swap, phase_row_commutator_witnesses contains one exact (row, column, numerator) entry of G[P(i),P(j)]-G[i,j], where theta'=G*x/pi; successful swaps have None. This is an all-state algebraic witness, not a residual sampled at the source.

independent_pair_swaps_equivariant and unordered_pair_quotient_status distinguish certified_by_independent_swap_symmetry from obstructed_by_fixed_support. This concerns the full unordered member-state orbits, represented by the existing invariants only on their declared chart; it neither compresses them to pair means nor invents a scalar macro-node law. Separate strict_replica_admission_status and reasons retain the existing no-internal-edge contract. A graph can satisfy the symmetry theorem while its internal edges require different inherited rows and reject that older domain.

Optional witness_pair is an explicit nonboolean index in the declared pair list. No failing pair is selected automatically. The nested SinePairSwapWitness exchanges both captured form and phase within that pair, retaining fixed support and capacities. It records the permutation, swapped state, exact form gradient, shared phase-rate numerators, bounded full rates and equivariance residuals. It does not construct or commit a live graph.

The witness also retains source and swapped local pair-mean phase-rate numerators, their exact differences and bounds after division by pi. source_orbit_equal checks exact unordered raw member-state multisets, not overlap of trigonometric enclosures. A nonzero mean-rate difference then certifies collective_phase_rate_obstruction_certified. A zero difference does not repair a failed all-state symmetry test. The reported means refer to continuous local lifts; they do not define a global circular mean or identify a regional angle with primitive phase. Interval residuals containing zero are not proofs of exact field equality.

The fixed one-edge-deletion control yields receiver mean rates +1/(48*pi) and -1/(48*pi) from identical unordered pair states. It identifies missing state-to-attachment information, not a missing force or a support event law. The schema is tnfr.relational-sine-pair-support-symmetry.v1, supported by the generic SDK exporter. Graph/state relabeling, phase-only observation and autonomous support selection remain separate questions.

Sufficient pair state on asymmetric support

assess_sine_mixed_pair_state(graph, reference_model=..., pairs=..., phase_turns=None) shares the preceding support-symmetry admission and a single complete sine capture. Its mixed-state theorem retains signed member-to-attachment information exactly where an independent pair swap is not a graph symmetry. This removes redundant member labels; it does not remove internal continuous degrees of freedom.

The same explicit phase-lift convention and strict |delta|<pi/2 pair chart as the replica observer apply, independently of strict replica support. Missing turns mean zero supplied turns, not inferred unwrapping. A failed chart rejects this representation; it does not invalidate the smooth fine law. Within-pair edges and incomplete cross-blocks are admitted here without extending the older complete-replica formulas or persistence certificates.

SineMixedPairStateAssessment retains support_symmetry, phase_turns, phase_chart_margins, coordinates and rates. Its comparison and pairs properties expose the retained source. Each coordinate records form_mean and the exact radian/turn split of its lifted phase mean. Its mode is:

• ordered: keep form_half_difference and the radian/turn split of phase_half_difference, relative to the declared ordered attachments.
• unordered: keep resultant_magnitude_bounds for R=cos(delta), exact internal_form_squared=U=u**2, and form_phase_correlation_bounds for Q=u*sin(delta). The exact realized invariants obey Q**2=U*(1-R**2), 0<R<=1, U>=0.

Fields not retained in a mode are None, not zero. reconstruction_stratum distinguishes the ordered chart and the unordered reconstruction cases. The exact realized coordinates identify precisely the orbits generated by the individually permitted swaps. Synchronized tips are admitted; their quotient singularity does not make the fine field singular. No numerical inverse of arbitrary interval tuples or independent interval realizability test is provided. The retained ordered fine source is provenance, not an additional coordinate of the mathematical quotient.

Rate records retain form_mean_bounds and exact phase_mean_numerator; the phase rate is that numerator divided by mathematical pi. Ordered pairs also retain form_half_difference_bounds and phase_half_difference_numerator. Unordered pairs instead retain rates of R, U and Q, obtained by differentiating those invariants along the shared fine field. These are instantaneous bounds at the realized source, not a new coarse integrator or a future chart certificate. The proof gives reconstruction up to allowed symmetries and exact inherited rows, including the contributions of asymmetric neighbors and internal edges.

On the fixed nineteen-edge control, modes are (ordered, unordered, unordered, unordered, ordered). Swapping source members 0,1 now changes retained signed coordinates. The known receiver rates +1/(48*pi) and -1/(48*pi) therefore correspond to distinguishable states. This repairs the prior information loss without changing the law or repeating the frozen finite-window evaluation. A simultaneous graph/state relabeling changes reporting labels, not the mechanism. The schema is tnfr.relational-sine-mixed-pair-state.v1, supported by the generic SDK exporter.

Structural Emission on an unordered pair

assess_sine_pair_emission(graph, reference_model=..., pairs=..., pair_index=..., boost=..., phase_turns=None) reuses one mixed-pair source capture and its conservative normalized-sine admission. The selected pair must admit an independent swap; other pairs may retain ordered attachments. The SDK delegates through Network.relational_sine_pair_emission. This is the structural descent comparison, not a native Arg evolution step or a public AL execution.

The explicit positive finite boost uses the registered AL factor contract. The actual emission_epi_proposal owner supplies each candidate's represented addition, clipping and nondecrease check. Both member proposals must be admitted before any report is returned; rejection cannot become a no-op. The shared operator boundary policy retains its existing hard fallback for unrecognized modes and default soft steepness. It does not consume the integrator's configurable CLIP_SOFT_K. The report records the effective clip_policy=(minimum, maximum, mode, steepness) without changing those rules.

SinePairEmissionAssessment retains source_state, the selected index, represented boost and proposals, and three outcomes: first_member, second_member, whole_pair. Each outcome records target positions within the selected pair, ordered endpoint forms, effective increments and collective X,U,Q; phase mean and R remain those of the source. Ordered endpoints are fine provenance, not an extra coordinate of the unmarked collective state. Binary64 AL endpoints are stored as exact represented rationals; bounded trigonometric projections retain the source's strict phase chart and lifts.

single_member_source_orbit_equal compares exact member-state multisets. Its status distinguishes equal_at_source_synchronized_tip, equal_at_source_both_noops and obstructed_at_source. Interval overlap, equal means or equal storage costs cannot certify that equality. A source exception does not establish all-state single-target closure. In general, the missing information is a marked constituent relative to the pair.

whole_pair_structural_descent_certified concerns the common scalar map on both members and its invariant admission domain. Clipping can change internal form/correlation despite this symmetry. runtime_admission_certified and event_occurrence_derived remain false: grammar, configured public preconditions, history, lifecycle, pressure refresh and event timing are not assessed. No graph state, support or cache is modified; no subsequent flow, storage passivity, maintained identity or physical interpretation is inferred.

The owner schema is tnfr.relational-sine-pair-emission.v1. Generic SDK export uses the common tnfr.relational-report.v1 envelope and delegates the same payload, including source provenance and unavailable runtime claims. Its optional form_increment(outcome=...) uses the separate weighted-form ledger to test a necessary condition for being a closed-flow endpoint. Structural descent and that endpoint test have different obligations.

This dependent reader re-admits the source, pair declaration, phase lifts, boost and effective clip policy and rebuilds symmetry, target indices and the shared AL proposals before testing the increments. Saved increments and verdicts are not authoritative inputs. The actual binary64 clipping/rounding, including legitimate no-ops, remains part of the declared structural action; the reconstruction neither recaptures a graph nor executes AL.

Exhaustive fully acute equilibrium families

assess_sine_replica_equilibria(graph, reference_model=..., pairs=...) reuses one comparison capture and the shared complete-replica support admission. The ordered structural pairs must describe the full unit doubled C5; form loss is exactly zero and every fine capacity is held and strictly positive. Member capacities need not agree. No pair phase chart, target winding or phase lift is supplied: the classification derives circular pair synchrony only for fully acute equilibria.

SineReplicaEquilibriaAssessment.targets lists the integer solutions of 4*abs(k)<5 in the declared cycle orientation. Each target retains its exact target_phase_turns, target_edge_turns, shared target_geometry reconstruction, edge_cosine_bounds and storage_bounds. The reconstruction proves circular consistency and sine cancellation; its separate phase-law interpretations are not imported. Uniform form and arbitrary common form and phase origins complete each symbolic family. The shared target helper also serves the earlier trapping readers, whose public winding admission remains exactly -1 or 1.

acute_equilibrium_classification_certified concerns the complete family classification under this law/support, independently of the captured phase sector. It does not assert that the source is an equilibrium. The source keeps separate evidence:

• source_edge_cosine_bounds and source_acute_status distinguish fully_acute, outside_fully_acute_sector and unresolved. Strictly positive edge cosines certify acuteness; an interval crossing zero is not exclusion.
• source_form_uniform and source_raw_phase_uniform compare exact captured values, without a tolerance. Nonuniform form excludes full equilibrium by connectedness and positive capacity, even outside the acute sector.
• source_equilibrium_status is certified_consensus_equilibrium for uniform captured form and phase, certified_not_equilibrium for a proved exclusion, or unavailable when the admitted evidence does not decide it. source_equilibrium_reasons records the applicable argument; source_equilibrium_winding is zero only for certified source consensus, and otherwise None.

Captured radians are exact rational numbers after shared engine staging. Within the fully acute sector a nonzero equilibrium twist requires irrational phase differences 2*pi*k/5 modulo full turns, so nonidentical captured rational phases exclude exact equilibrium. This is a theorem about the captured representation, not an inference from a small rate residual or an uncertainty claim about a physical measurement. If the source has uniform form but nonacute or unresolved phases, this reader makes no global classification.

Consensus remains a valid full equilibrium even when the phase-only partner observer abstains because of ties. The structural pairing used to classify the supplied graph never replaces that observer's output. The two nonzero windings are exchanged by orientation reversal; the report selects neither. It introduces no graph mutation, attraction, new force, preferred spatial shape, formation event or physical identification. Its to_dict() schema is tnfr.relational-sine-replica-equilibria.v1, supported by the generic SDK exporter.

Exact prepared internal pulse

assess_sine_replica_pulse(reference_model=..., form_half_difference=..., phase_half_difference=..., capacity=...) in the replica pulse owner assesses an exact mathematical preparation, rather than capturing a graph. The support is the complete double-replica C5, with uniform mean form, exact mean phase turns i/5, identical signed internal coordinates u,delta in every pair, and one common strictly positive held capacity. Common form and phase origins are immaterial. The input phase half-difference is in radians and must have certified magnitude less than pi/2. Shared exact/represented real admission rejects Booleans and nonfinite values; rational inputs retain their exact value.

This owner also supplies the variation, splitting, stiffness-curve and work-response reports below. The established relational_sine_scale imports remain aliases to the same classes and functions; SDK export schemas are unchanged.

The reference model must declare phase_domain="regular" and zero EPI loss, as required by the shared sine comparison interface. Its coefficients and storage scale are reused for the sine comparison law, with no inputs, events or support changes. Neither the selected native phase chamber nor a small graph residual certifies this symbolic preparation. No live graph membership is inferred and no graph or runtime law is changed.

The proof derives the invariant family directly from the full retained rows. The report preserves the input scalars and exact target turns, enclosures of the internal storage H=u^2+beta*cos(2*pi/5)*sin(delta)^2, the elliptic parameter m=H/(beta*cos(2*pi/5)), rates and period bounds. A proved 0<m<1 certifies a nonstationary libration and permanent validity of the pair chart. For zero internal form and nonzero admitted phase, the chart proves this strict inequality even if the projected energy interval touches the barrier; numerical period availability remains a separate calculation. The full labeled state returns after T; the unordered state returns after T/2 because the constituents swap. A stationary state has no nonstationary period. Above-barrier states leave this chart; an unresolved boundary receives no libration certificate. This is not a theorem excluding all circular-state rotations.

All fine edges remain strictly acute throughout the orbit only when m<sin(pi/20)^2. This stronger test is separate from chart retention and the fixed collective winding. The small-amplitude limit and nonlinear period enclosures use the shared elliptic-integrand inequality, not an elliptic solver or a measured numerical return. Unresolved numerical bounds remain unavailable even when a qualitative theorem applies. SDK export preserves those distinctions. period_bounds describes labeled return; unordered_period_bounds is its half. nonlinear_periodic_exchange_certified and all_fine_edges_acute_status report different tests, while graph_membership_certified remains false. The report's to_dict() schema is tnfr.relational-sine-replica-pulse.v1. No exact period, internal locking or attraction is certified for transverse perturbations of this lower-dimensional preparation.

Full perturbations of the prepared pulse

assess_sine_replica_pulse_variation(...) accepts the same keyword arguments and delegates preparation, model and scalar admission to the pulse owner. Its SineReplicaPulseVariation report retains that assessment in pulse. It differentiates the complete fine law with fixed support, coefficients and capacity; it does not restrict disturbances to identical internal pairs. Changing those held parameters is a different perturbation problem.

mode_blocks contains three real four-by-four coefficient matrices, indexed by the C5 Fourier representatives mode_indices=(0,1,2). The convention is exp(+2*pi*i*k*j/5). For nonzero representatives, the coordinates are (X_hat,Theta_hat,i*u_hat,i*delta_hat); their real and imaginary parts each follow the displayed real block. The zero mode uses ordinary real collective and internal perturbations. Thus mode_multiplicities=(1,2,2) accounts for all twenty real fine-state directions. This is an invertible symmetry decomposition, not discarded state or a fitted low-dimensional model.

These matrices are evaluated at the supplied initial internal state. Their formulas apply along its exact reference evolution; they are not finite-time transition maps. dimensionless_mode_blocks uses form perturbations divided by sqrt(beta*cos(2*pi/5)) and the clock tau=Omega*t, with both scales enclosed separately. The instantaneous blocks depend on internal phase. Only after following a complete libration does the return spectrum depend on the dimensionless energy m and geometry alone, independently of initial orbit phase up to conjugacy. Holding raw form fixed while changing beta generally changes m.

variation_identity_certified concerns the exact differentiation and decomposition. periodic_reference_certified separately requires the pulse owner's nonstationary libration certificate. Equilibrium, unresolved energy and above-barrier preparations still have an instantaneous variation law; they do not receive that periodic-reference flag. The zero-amplitude frequency bounds describe the stationary target limit, not the finite-amplitude spectrum.

The constant symplectic_form and computed hamiltonian_structure_residuals check the inherited linear Hamiltonian identity. They introduce no additional physical law or energy. half_return_swap is diag(1,1,-1,-1): when the pulse is certified, half-period coefficients are conjugated by this swap. The unordered return derivative must include it; a raw half-period propagator cannot be treated as the return map. The proof owner derives these statements, the uniform amplitude/time shear and the remaining transverse stability criterion.

No propagator or Floquet multiplier is evaluated here, and orbital_stability_status remains "not_assessed". Static trapping, instantaneous eigenvalues and interval residuals alone cannot supply that verdict. No nonlinear finite-perturbation error bound is claimed. Exact export uses tnfr.relational-sine-replica-pulse-variation.v1, with the shared SDK dispatcher preserving the same scope and symbolic-source provenance.

Finite-time work response along the moving pulse

assess_sine_replica_pulse_work_response(reference_model=..., form_half_difference=..., phase_half_difference=..., capacity=..., scaled_duration=..., order=10) re-admits the primitive preparation through the pulse/variation owner. Its symbolic support is the same conservative doubled C5. No captured graph, cached verdict or externally supplied tangent is accepted as evidence. Exchange weight, beta and held capacity retain their admitted values; the duration is strictly positive in tau=t/pi, distinct from the variation report's Omega*t normalization.

The fixed real k=1 probes are distributed form masks: collective cosine and opposite-member sine. Both leave initial phase unchanged and subtract the same unperturbed moving background. Fine storage-work outputs have coefficients (10*lambda_1,20), accounting for each mask's squared norm five. The proof and port definition derive the normalization and distinguish reversal from relabeling.

SineReplicaPulseWorkResponse retains reference, initial_state_bounds, state_order, scaled_duration, taylor_order and work coefficients. The shared Picard/Taylor kernel encloses the joint ten-coordinate field: the moving background and both complete four-coordinate variation columns. The strict whole-time tube must retain the internal pair chart. A successful step gives response_bounds (output rows, input columns) and antisymmetric_response_bounds=H_CI-H_IC. response_sign is +1 or -1 only when this interval excludes zero; zero means unresolved direction, not exact reciprocity. directional_response_certified records that distinction.

An invalid primitive raises; a failed enclosure instead returns status="unavailable", reasons and available failure evidence, with response fields set to None. A successful enclosure has status="certified_finite_response", even when direction is unresolved. There is no automatic change of horizon, order or preparation to obtain a sign. The certificate concerns an infinitesimal probe at a finite delay, not a finite-amplitude disturbance, Floquet multiplier or orbital stability. The report installs no runtime law, edge, input or physical interpretation. Its direct schema is tnfr.relational-sine-replica-pulse-work-response.v1; the generic SDK wrapper preserves the same report and explicit availability.

Full nonlinear centered kicks under the fixed pulse protocol

assess_sine_replica_pulse_finite_work_response(probe_amplitude=..., order=10) is a fixed mathematical preparation, not a generic graph reader. The SineReplicaPulseFiniteWorkResponse report declares unit conservative sine exchange on doubled C5, capacity one, u_0=delta_0=1/32, the same two distributed form masks, and the fixed horizon tau=1/16. It admits only 0<probe_amplitude<=2^-20 through shared scalar admission and validates the Taylor order before propagation. These supplied experiment parameters do not become physical constants or a selected law.

The report freshly rebuilds its tangent from these primitives. Its initial_boxes, neighbors, edges, port_form_directions and preparation_order declare the four complete fine preparations. Shared sine rows evolve all twenty coordinates for each positive/negative kick, using interval-enclosed symbolic phases and masks. Multiplying every original-time row by outward-enclosed pi applies the clock change tau=t/pi. Interval boxes conservatively enclose the correlated symbolic states; they do not claim that every independent box point is exactly on the prepared family.

The uniform flow-error proof supplies flow_third_derivative_upper_bound, finite_probe_error_upper_bound, analytic_tangent_lower_bound, analytic_contrast_lower_bound and fine_edge_margin_lower_bound. These certify the exact symbolic preparation independently of numerical success. If a tangent enclosure is available, predicted_antisymmetric_response_bounds widens it by the nonlinear error.

Four independent shared Picard/Taylor steps retain steps, failed_tubes and failure_reasons in the declared order. They use the globally smooth sine field; acute identity is covered separately by the analytic margin. There is no early stopping or retry when one preparation fails. A complete evaluation gives response_bounds and antisymmetric_response_bounds from the centered full-node work readouts. response_sign=0 means unresolved direction, not exact zero. If a preparation is unavailable, numerical response fields are None, status="unavailable" and named reasons remain; the independent analytic evidence is retained. Accordingly analytic_directional_response_certified and numerical_directional_response_certified have separate meanings.

The raw four-reading contrast is twice the probe amplitude times the reported normalized contrast. No measurement precision, laboratory clock, physical pump, magnetic field or practical signal-to-noise ratio follows. Direct export uses tnfr.relational-sine-replica-pulse-finite-work-response.v1; the generic SDK wrapper retains all evidence and availability.

Necessary stiffness-family screen with observation uncertainty

assess_sine_replica_stiffness_trace_curve(trace_bounds=..., determinant_bounds=...) admits exactly three ordered real scalars or outward intervals per input through the shared scalar/interval boundary. The inputs must refer to tr(M^-1 K) and det(M^-1 K) under independently specified constant positive-definite mass M and a common clock. This reader does not identify mass, select a coordinate projection or reconstruct stiffness from the evaluated response.

The SineReplicaStiffnessTraceCurve report retains the primitive boxes, the conditional geometry coefficient, the trace difference product Q and cross-multiplied numerator N from the proof. template_residual_bounds=N-a_*Q tests a necessary template identity; affine_obstruction_bounds=(4*N-Q)*Q tests the symmetric affine one-control class. The calculation avoids division by small observed trace differences.

trace_separation_certified requires Q to exclude zero. Without it both verdicts are "unresolved_trace_separation". With separated traces, template_curve_status="excluded" requires a residual excluding zero and affine_family_status="excluded" requires strictly positive obstruction. Otherwise the corresponding verdict is "not_excluded", never a successful physical admission or exact equality. Wide input boxes may cause abstention. No threshold, phase fit, time fit, trajectory or runtime mutation is installed. The coefficient is specific to doubled C5/k1, not a new universal constant. Direct export uses tnfr.relational-sine-replica-stiffness-trace-curve.v1; the SDK wrapper retains the same primitive evidence and scope.

Small-amplitude return splitting

assess_sine_replica_pulse_splitting(reference_model=..., capacity=...) reuses the exact zero-amplitude variation template. It accepts no finite amplitude, observed graph or horizon. Its SineReplicaPulseSplitting report assesses the analytic family approaching that reference through m>0; the stationary reference itself receives no instability verdict.

The proof retains the changing-period clock and the induced collective response before extracting the internal resonant terms. p_exact_coefficients and q_exact_coefficients encode a+b*sqrt(5) pairs for the local harmonic coefficients P_k,Q_k, not nodal pressure or internal correlation. Their interval evaluations generate slow_generators in rotating amplitude order (A,B), with G_k=[[0,Q_k/2],[-P_k/2,0]] and squared exponents -P_k*Q_k/4. The effective logarithmic return generator is m*G_k+O(m^2) in an analytic internal basis, not an instantaneous constant-coefficient law for the nodes.

The exact signs give mode_classifications=("hyperbolic","elliptic") for representatives (1,2), each with two real spatial copies. Analytic remainder control establishes some strictly positive amplitude interval in which mode 1 destabilizes the orbit. Mode 2 is bounded in the transverse linear problem on a sufficiently small interval; it does not imply nonlinear stability. existential_amplitude_interval_certified and sufficiently_small_nonlinear_orbital_instability_certified express these conditional theorems. They do not evaluate a chosen amplitude.

labeled_log_multiplier_slope_magnitude_bounds encloses 2*pi*sqrt(abs(-P*Q/4)); the unordered slope is half that value. These are leading coefficients of m, real for the hyperbolic case and imaginary for the elliptic case. They are not finite log-multiplier bounds. amplitude_upper_bound and finite_amplitude_remainder_bound remain None; finite_preparation_assessed and return_multipliers_computed remain false. The API therefore cannot certify that a caller's small numerical amplitude lies within the instability interval.

Orbital departure is compatible with the separate static winding barrier and conserved full storage. This reader installs no drive, controller, changed law or numerical return solver. Its exact projection uses tnfr.relational-sine-replica-pulse-splitting.v1; the generic SDK dispatcher preserves the same absence of a finite-preparation verdict.