Conservative sine regional and collective dynamics

Regional storage and currents, conservative winding acquisition and retention, compatible collective families, actual contact feedback and forecast-derived organization.

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

Regional storage and two-channel boundary work

comparison.regional_storage_balance(region=...) returns SineRegionalStorageBalance. Its shared owner re-admits primitive support, model, phase, form and held capacity, then rebuilds the entire comparison. Cached rates, gradients, resultants, dissipation and storage cannot substitute for those premises. It admits nonnegative capacities without division by them.

The ordered nonempty region contains distinct source nodes. Internal region, internal complement and cross-boundary edge sets partition storage exactly once. *_form_storage are exact fractions; *_phase_storage enclose the unit cosine potential; *_storage includes the model's beta multiplier. Boundary edges are oriented (region,complement) in source-index coordinates. A full-support region has an empty complement and cut.

All rates use complete-network degrees, gradients and the original clock clock="structural_t". The regional identity is dE_R/dt=regional_boundary_work-regional_loss. Work retains both form and phase channels. The complement has its own identity; cross-edge storage changes by the negative sum of the two inward boundary powers. Consequently those powers need not be opposites. direct_*_storage_rate are computed from edge derivatives independently of the boundary-work sums. Balance and partition residual intervals expose arithmetic agreement, not a new theorem.

This storage ledger differs from the following weighted-form transfer. A positive instantaneous power is not accumulated work, monotonic regional storage, an external drive or evidence of pattern formation. The derivation and controls retain that distinction. Direct schema tnfr.relational-sine-regional-storage-balance.v1 and the generic SDK envelope preserve the rebuilt comparison, edge partition, exact fractions and intervals.

regional_channels and complement_channels add the separate storage rows under the same admitted general coefficients. Their internal conversion J is positive from phase to form. boundary_form_input and boundary_phase_input are the inputs to the respective internal stores; they are not the existing conjugate *_boundary_form_work and *_boundary_phase_work. signed_form_loss may be negative and is distinct from the nonnegative full-node loss. Phase rates of storage include beta. The direct rates satisfy F_dot=J+B_F-D_F and (beta*V)_dot=-J+B_V; independently reconstructed residuals retain interval uncertainty. Nested gradient/current/rate rows follow their region_indices order. flat_phase_storage_acceleration is available when each internal edge has an exactly zero represented phase gap. It encloses beta*sum_internal(theta_dot_j-theta_dot_i)^2 in original time; otherwise it is None. This conservative flatness admission does not infer equality modulo an unrepresented exact multiple of pi or discard uncertain angles. The curvature describes an initial response, not finite acquisition. The channel derivation and independent controls own the coefficient and sign checks. The direct schema remains v1 with additive nested fields; manually constructed older reports may retain None there and supply no channel evidence.

Regional form transfer and supplied increments

comparison.regional_transfer(region=...) reuses a captured SineExchangeComparison, without a graph reread. All held capacities must be strictly positive. The fixed-support weights are rho_i=d_i/nu_i; the report uses weighted form sums, not arithmetic means or physical mass. The boundary identity holds for both zero and positive form loss in this law. It does not transfer to native Arg pressure or state-dependent reciprocal mobility. Both readers use chained-report admission. Both readers rebuild their complete comparison from admitted primitive state before retaining it as nested evidence. An equivalent original comparison is kept only after calculation on normalized values; altered cached rates or storage cannot accompany a fresh ledger as if they were consistent evidence. Regional accounting consumes rebuilt rates and currents. The increment identity itself needs only forms, weights and supplied signed increments; the additional rebuild keeps its attached full-state report consistent.

SineRegionalTransfer retains the source, supplied region and its complement, weights, weighted form sums and the inward cut current. The diffusion and sine contributions are separate. Internal edges cancel algebraically; each cut edge contributes e*(x_out-x_in)+(w/pi)*sin(theta_out-theta_in) to the region. Direct weighted sums of the rebuilt nodal rates and their cut residuals are computed independently. global_weighted_form_conserved records the conditional identity; an interval residual containing zero is a numerical consistency check, not its proof. A regional current does not close the region's evolution without the consumed boundary state or certify persistent identity.

No pair midpoint or nonzero regional resultant is required by this cut reader. The zero-pair-resultant theorem shows why cancelled collective form transport can coexist with internal phase motion. Its exact antipodal preparation is symbolic; a graph storing represented radians remains its own source. The cut reader reports current, not a future activation time. With the theorem's zero loss, node-relative resultant_kinematics() bounds nodal form acceleration after the declared capacity/degree and phase-weight scaling; it does not report that pair's phasor derivative or supply a midpoint at cancellation. The separately admitted global pair state retains the complete unordered state through cancellation on the fixed unit doubled C5; it does not broaden this comparison reader's model or input format.

comparison.assess_form_increment(increments=...) accepts a complete ordered vector of finite signed increments in captured node order. It does not infer unobserved changes or silently fill missing nodes. SineFormIncrementAssessment retains exact weights, increments and before/after weighted form sums. closed_flow_endpoint_obstructed is true precisely when the weighted change is nonzero. Zero change leaves reachability unresolved; endpoint_reachability_certified remains false. No phase endpoint, storage budget, runtime event or trajectory is supplied by this necessary condition. The test concerns a full endpoint in the same form chart; it does not establish an obstruction on a quotient that discards common form origins.

The pair-Emission report's form_increment(outcome=...) accepts first_member, second_member or whole_pair and delegates those actual clipped/rounded proposals to the same increment owner. Other nodes have zero increment because these three declared actions leave them unchanged. A positive AL-only increment fails the global invariant; an increase in one region with compensating change elsewhere need not fail it. Saturated or rounded no-ops are not reported as positive injections.

Owner schemas are tnfr.relational-sine-regional-transfer.v1 and tnfr.relational-sine-form-increment.v1. Generic SDK export delegates their payloads through tnfr.relational-report.v1. Source provenance and label validation are retained. No observation mutates the graph, installs an activation rule or identifies form with an independently measured quantity.

Finite conservative regional winding acquisition

certify_sine_conservative_winding_entry(source, *, cycle, scaled_window, edge_turn_offsets, source_error_bound=0) in relational_sine_entry.py returns SineConservativeWindingEntry. This separate theorem requires an exact SineExchangeComparison as its nominal center, common nominal phase lifts, constant nominal form on the selected receiver cycle, unit capacities, zero loss, w=beta=1 and no forcing or events. All source nodes and edges remain present; the receiver's initial phase velocities follow from the environmental form and full degrees. source_error_bound=epsilon admits independent initial errors of at most that nonnegative size in every form coordinate and phase lift, including the environment. Phase errors use radians; the same numerical bound applies in the declared form units. Capacity and support remain fixed. A nonuniform nominal receiver is outside this contract; receiver contrasts within the supplied uncertainty box are admitted.

cycle is an ordered simple cycle of source labels. scaled_window=(a,b) contains admitted exact-or-represented real values with 0<a<b in tau=t/pi. original_time_window_bounds enclose pi*a and pi*b in the source clock. Each supplied edge offset is a nonboolean integer in cycle traversal order; it is a declared observation branch, not an executed event.

The full normalized law has abs(x_i')<=1 and abs(theta_i'')<=2. initial_phase_velocity is the freshly computed exact K L x(0). whole_window_form_bounds and whole_window_phase_bounds enclose the full nonlinear solution; cycle_raw_gap_bounds use the tighter direct edge expression with remainder 2*(b^2+epsilon*(1+2*b)). True lifted cycle gaps telescope to zero; independent points selected from these outer boxes need not do so.

After subtracting each declared 2*pi multiple, strict whole-window inclusion in (-pi,pi) certifies certified_winding=-sum(edge_turn_offsets). Initial winding is zero throughout the source box when 2*epsilon<pi is strictly certified; otherwise initial_winding=None and acquisition is unavailable. Only certified initial zero winding and a certified nonzero final period set acquisition_certified=True and status="certified_finite_acquisition". A certified zero period remains distinct from acquisition. Failed sufficient branch bounds yield status="unavailable" and explicit reasons; they do not prove that the actual trajectory fails. Malformed or unsupported premises reject before certification.

acute_margin_lower_bound additionally bounds the distance of every principal gap from +/-pi/2. acute_acquisition_certified requires both nonzero certified winding and a strictly positive whole-window acute margin. These additive fields do not strengthen a nonacute certificate or change the existing branch status. An unavailable bound is not a failed trajectory.

The same reader exposes an explicit full-state box at time a through entry_form_bounds and entry_phase_bounds. Its centers are x(0) and theta(0)+a*K*L*x(0); its analytic radii are a+epsilon and a^2+epsilon*(1+2*a), retained as analytic_entry_form_radius and analytic_entry_phase_radius. Interval materialization can round endpoints outward. entry_form_radius and entry_phase_radius therefore measure the actual stored box around its exact centers. entry_box_contains_source_flow certifies containment of every actual source member's endpoint. These centers are enclosure references, not substituted actual states or held-pressure dynamics. Every point of this independent box has continuation bounds for scaled_retention_duration=h=b-a under the same law. They use entry_box_form_remainder_bound=entry_form_radius+h and entry_box_phase_remainder_bound=entry_phase_radius+2*h*entry_form_radius+h^2. Its own entry_box_cycle_principal_gap_bounds, branch margin and acute margin retain that materialization allowance. entry_box_acute_retention_certified requires a nonzero period and strict margins from these bounds. The original whole_window_* and node_* fields still bound actual source trajectories; they must not silently substitute for this independently rounded target box. Continuation is also separate from proving that the source initially lacked the identity. The dyadic documented witness requires no entry-box rounding.

initial_total_storage retains the nominal-center value for compatibility; initial_storage_bounds encloses the entire uncertain source. It is not an energy bound for every member of the later rectangular entry box, which can include unreachable states. Each actual trajectory retains its own conserved origins and storage. No midpoint, shared-energy assumption or cached source rate replaces these premises.

The proof and frozen witness certify nonacute winding on one finite interval. This does not transfer positive-loss capture, acute equilibrium, attraction, indefinite maintenance or complete NFR identity. The robust short-passage result uses the same owner to certify acute identity with nonzero source and target widths. It does not establish entry into the separate invariant-family tube or retention beyond its declared short interval. The regional storage ledger keeps its own original-clock rates; a snapshot power is not the theorem's integrated work. Direct schema tnfr.sine-conservative-winding-entry.v1 and generic SDK export retain both clock conventions, branch evidence, source and scope. See admission tests, independent full-law controls and the usage.

Environmental source geometry and acute entry

analyze_sine_conservative_source_geometry(source, *, receiver) returns SineConservativeSourceGeometry from the same entry owner. It shares the conservative common-phase admission above and requires uniform initial form on the nonempty receiver. The receiver need not be a cycle. All environmental nodes remain present, including those without direct receiver contact.

The exact map T=-D_R^-1 A_RQ uses full degrees and maps environmental form contrasts relative to the receiver's common form to initial receiver phase velocities. The report retains the primitive source, receiver/environment order, map, relative map, exact ranks, reconstructed velocities, residuals and identical-row groups. Relative rows subtract the first receiver row. Empty environment and one-node receivers have explicit zero-dimensional maps. Cached source rates or storage are not input evidence.

Full relative rank suffices to prescribe every initial relative velocity; it is not necessary for one acute configuration. Identical rows establish equal initial velocities, not an invariant partition or a later obstruction. The source-image theorem combines a declared interior phase cell with the existing global nonlinear remainder to certify finite acute entry. It supplies no autonomous source preparation or indefinite maintenance. Direct schema tnfr.sine-conservative-source-geometry.v1 and the generic SDK exporter retain the exact map and its scope. See the map controls, analytic protocol and usage.

Acquired low storage and subsequent acute exit

assess_sine_conservative_handoff(source, *, cycle) returns SineConservativeHandoff from the shared entry owner. Its admitted source has common phase, uniform receiver form, the unchanged conservative unit-capacity law and an induced C5 receiver. The actual full-degree phase velocities on the ordered receiver must form an arithmetic progression with nonzero signed step. Its magnitude is omega and its sign is orientation. This is an exact restriction on a source class, not a fitted frequency or new phase law.

The reader bounds the actual acquired state at tau_star=(2*pi/5)/omega using the global nonlinear remainder. Correlated cycle errors telescope, giving the tighter phase-storage upper bound V5+10*tau_star^4, separately from the form bound 10*tau_star^2. It retains strict acute-entry and regional-subbarrier predicates separately. entry_relative_phase_rate_bounds enclose the actual five edge rates at entry in tau, using their complete initial velocities and the global acceleration bound. These rates can be large even with small internal storage. At tau_out=5/(3*omega), a lower bound on four oriented gaps can prove acute exit. handoff_obstruction_certified requires all these predicates: acquired low regional storage has not supplied retention through this deadline.

unavoidable_boundary_work_lower_bound, when available, bounds work reached by the first acute exit after entry. It is not net work at the declared endpoint; later return remains possible. The report retains both clocks, exact interval evidence, source and unavailable reasons. Failed sufficient inequalities do not establish failed formation or successful maintenance. Sources outside the declared law, cycle or ramp class reject. Re-admission rebuilds consumed state and rates instead of trusting cached source fields.

Direct schema tnfr.sine-conservative-handoff.v1 and generic SDK export preserve the conditional obstruction. The reader runs no trajectory, changes no support and selects no source. See the proof, independent and adversarial controls and usage.

Compatible phase-offset collective families

assess_sine_phase_offset_partition(source, *, blocks, phase_offset_turns) returns SinePhaseOffsetPartition from the shared partition owner. The source supplies an admitted full-support SineExchangeComparison under the unit-capacity conservative law e=0, w=beta=1, with no forcing or events. Its captured form and phase are a law/support anchor, not evidence that the source already belongs to the proposed family. blocks declares a partition of all fine nodes, with the shared software admission requiring 2 <= number of blocks < number of nodes. The mathematical theorem also allows trivial partitions, which this API does not admit. phase_offset_turns independently declares the fixed offsets in source node order using integers or Fraction values. Floating turns or rounded radians do not stand for these exact offsets.

The proposed family has block-constant form and phase equal to a free collective phase plus its supplied fine offset. Admission compares external normalized neighbor counts and complex phase sums within each block, and requires zero internal sine current. Equality of internal cosine sums is not required. These are joint full-law conditions for every collective state, not observed agreement of rates at one snapshot.

status is certified, excluded or unavailable. invariance_certified is true only for the first status. Exact recognized cancellations can prove equalities; an interval excluding zero can prove inequality. Neither a nonempty symbolic residual nor an interval containing zero proves that an equality fails. An undecided transcendental identity therefore stays unavailable. The report retains normalized counts, row sine/cosine expressions and their bounds. Quotient coefficients are available only for a certified family. Their form and phase couplings remain separate; phase cancellation need not preserve the bare coarse normalized-sine formula.

partition.evaluate(collective_form, collective_phase_turns) re-admits the primitive source, partition and offsets and rebuilds the required family evidence. A certified family then yields a detached SinePhaseOffsetState for the supplied collective coordinates. block_form_rates and full_form_rates retain rigorous enclosures; block_phase_rates and full_phase_rates retain exact represented-real coefficients. Rates use tau=t/pi; phase rates are radian derivatives, while collective phase inputs are integer or Fraction turns. Evaluation reconstructs the fine state, inherited collective response and storage; it does not advance a solver, mutate a network or trust cached certification fields. Zero-containing rate residuals do not replace the invariant-family proof. Direct exports use tnfr.sine-phase-offset-partition.v1 and tnfr.sine-phase-offset-state.v1; generic SDK export preserves the same premises and availability.

Internal regional storage and net internal regional power are constant and zero-rate, respectively, along the exact family. Collective motion and cross-edge interaction can still be nonzero. Invariance supplies neither autonomous formation into the family nor robustness to states outside it; smooth uniqueness excludes finite-time exact entry from outside under the same unchanged autonomous law. See the theorem and standalone usage.

Full-state uncertainty around a compatible moving pattern

assess_sine_moving_pattern_window(reference, *, form_error_bounds, phase_error_bounds, scaled_horizon, receiver_phase_radius, contact_phase_radius, reference_contact_bound) returns SineMovingPatternWindow from the partition owner. The input is a SinePhaseOffsetState; its primitive source, partition, offsets and collective coordinates are rebuilt before deriving a reference. Cached fields or verdicts cannot substitute for those inputs.

This reader admits the first block as an ordered induced C5 and the second as its five private leaves, with unit conservative law, uniform signed fifth-turn receiver gaps and one common contact phase. Every fine node continues to evolve. The supplied nodal form/phase error bounds are nonnegative and independently cover all ten nodes. Phase errors and the three phase bounds are radians; scaled_horizon is a strictly positive duration in tau=t/pi. The three phase bounds are also strictly positive. Reference coordinates retain the preceding API's exact turn convention.

Separate checks admit the initial phase-error chart and a reference contact libration envelope. The relative Bregman storage uses all fine form and phase edge errors. A strict first-exit inequality combines initial_relative_storage_upper_bound, relative_storage_barrier and growth_rate_upper_bound; it preserves receiver and contact margins separately. whole_window_retention_certified requires every premise. Successful reports retain propagated storage, edge error bounds, receiver storage excess and conserved weighted-mean error bounds. The edge estimates do not bound absolute coordinates without those retained origins.

status is certified or unavailable; an unavailable sufficient estimate does not exclude maintenance. Contact convexity is sufficient for this certificate, not necessary for identity or for the exact compatible family. The proof covers both forward and backward intervals of the declared duration. It therefore does not establish first formation at the central reference state or attraction toward that state.

initial_full_storage_bounds is independently rebuilt from primitive full-edge uncertainty boxes, even if retention is unavailable. When the acute unit-winding receiver chart is admitted, its lower bound also retains the exact cycle minimum V_R>=V_5, together with all edge-form and contact phase lower bounds. Independent edge intervals alone can forget that constraint and suggest a spurious budget overlap. If the receiver chart is admitted and that storage upper bound is below 7/2, phase_flat_acquisition_status="excluded" retains the existing phase-flat acquisition obstruction. Otherwise the value is not_excluded, which is not evidence of successful acquisition. A proposed earlier source must also have a conserved storage compatible with this interval; overlap alone does not provide entry. These checks concern full storage, not just a regional excess or an untracked environmental reservoir.

The direct schema is tnfr.sine-moving-pattern-window.v1; generic SDK export preserves its primitive reference, independent error budget, unavailable fields and scoped conclusions. No trajectory is run or modified. See the moving-reference proof and example.

Actual contact-pulse feedback outside an invariant family

observe_sine_collective_pulse(source, *, cycle, contact_turn_offsets) returns SineCollectivePulseBalance from the partition owner. It re-admits the actual comparison's full primitive state, conservative unit-capacity law and C5/private-leaf support. Membership in a phase-offset family is not assumed. cycle supplies the five receiver labels in traversal order; its private leaves are matched from the unchanged support.

The five contact_turn_offsets must be exact integers, excluding Booleans. In cycle order they declare the retained lifts phi_i=theta_leaf-theta_receiver-2*pi*k_i. No principal-branch wrapping or discarded history is inferred. The contact mean, its deviations and the collective storage depend on this declaration; the full nodal storage does not. Phase values are radians and all derivatives use tau=t/pi.

The report retains both block form means, their form_gap, full contact rates, contact phase/deviation bounds and the complex deviation resultant. collective_energy_bounds encloses h=u^2/2+1-cos(mean(phi)); collective_storage_bounds encloses 5h. collective_energy_rate_bounds and feedback_energy_rate_bounds evaluate the same exact mean-contact transfer identity in two forms. A zero-containing factorization residual records numerical consistency, not its proof. remainder_storage_bounds=H-5h is signed; its rate is -5h'. Nonnegativity requires extra contact-convexity premises and is not presumed by this reader.

The same report resolves receiver phase motion into receiver_internal_phase_rates=L*q/3 and receiver_contact_phase_rates=P*r/3, where q is centered receiver form, r=x_R-x_Q, P subtracts the receiver mean and L is the cycle Laplacian. Their exact sum is receiver_relative_phase_rates. The fields receiver_relative_phase_acceleration_bounds and receiver_relative_phase_jerk_bounds enclose its first and second derivatives from the full nodal rows. All five tuples use the declared cycle order; their derivative units are radians per corresponding power of tau. They are independent of the declared integer contact lifts and common origins. They consume the actual contact phases and velocities, not an imposed pulse. Cached source rates, gradients and storage do not supply these values.

These are instantaneous derivatives, not retention verdicts. Even exact zero relative velocity and acceleration can coexist with nonzero third phase derivative. Interval overlap with zero proves neither exact cancellation nor rigidity. The relative-feedback theorem classifies exactly rigid acute trajectories separately from finite identity whose internal gaps are allowed to change.

flat_phase_jet_available requires equal exact source phase lifts at every node and zero declared offsets. Only then are first_three_energy_derivatives=(0,0,0) and the exact fourth_energy_derivative available. The latter uses the actual contact-rate variance and third central moment. For another preparation these derivatives are None, not zero. A nonzero first available derivative fixes the initial direction of transfer, but supplies no certified numerical time window, later entry or retention.

The direct export schema is tnfr.sine-collective-pulse-balance.v1. The report observes an actual state without installing a law, advancing time or certifying formation. The derivation and standalone example keep the conserved-budget and phase-lift limitations explicit.

Conservative cycle-sector source admission

assess_sine_cycle_barrier(source, *, cycle) returns SineCycleBarrier from the regional owner. It re-admits a complete SineExchangeComparison, zero loss, unit exchange, unit storage coefficient and held unit capacities. Five distinct source labels must trace a C5. Additional nodes, contacts and chords remain in the full law and rebuilt storage; this check does not require an induced cycle.

The report independently assesses the two wider sectors with winding -1,+1 and all principal gaps strictly below 2*pi/3 in magnitude. sectors records each membership, exclusion and invariance verdict. If full_storage_bounds.hi <= energy_barrier, with the shared derived barrier 7/2, an outside source cannot reach any acute state of that orientation; an inside source cannot leave that wider sector in either time direction. The latter does not certify strict acuteness, a moving-reference error box or an attracting waveform. No initially flat phase is assumed.

finite_time_energy_bound_certified records this closed upper-bound test. The existing strict_energy_bound_certified retains its strict < meaning. At boundary equality, nonnegative full storage forces uniform form, aligned extra-edge phases and balanced cycle currents. The full field is therefore zero, and smooth uniqueness prevents a distinct orbit from reaching this boundary equilibrium in finite time. This does not assert that a source with energy 7/2 is itself stationary, nor exclude asymptotic approach to a boundary equilibrium.

Principal-branch ambiguity leaves initial_winding unavailable. A separately proved edge outside the wider chart can still establish nonmembership without assigning a winding. An upper storage endpoint equal to the barrier suffices for the finite-time test; an interval extending above it does not. Unresolved membership never supplies a sector verdict. Cached storage and rates are ignored. Source association and JSON projection do not authenticate a preparation. Direct schema: tnfr.sine-cycle-barrier.v1.

The moving-pattern window also exposes zero_winding_acquisition_status: a certified acute target box entirely below the barrier cannot be reached from any zero-winding source under the same law. Its existing phase_flat_acquisition_status retains the restricted compatible conclusion. not_excluded is not an entry claim. See the proof.

Finite acute retention from conserved full storage

assess_sine_cycle_retention(source, *, cycle, scaled_duration, source_error_bound=0) returns SineCycleRetention from the regional owner. Its complete law and ordered C5 admission are shared with the cycle barrier: zero loss, unit exchange/storage coefficients, held unit capacities and full simple unit support, including every environmental node and additional edge.

The positive duration uses tau=t/pi. The nonnegative error bound declares independent errors of at most that amount in every full-source form and continuous phase coordinate, in their declared model units. The mathematical set is defined by the exact admitted center and radius. Displayed gap intervals are outward bounds; they do not replace that set with an independently enlarged full-state box. Boolean, textual, negative or nonfinite budgets reject.

The reader rebuilds nominal storage and both a direct edge-box bound and a bound from the full storage gradients and the global Taylor remainder. It intersects their upper bounds in full_storage_bounds; no cached source storage, torque or phase rate supplies evidence. Nominal phase torques remain present even when a represented phase is close to an ideal critical twist.

initial_acute_unit_winding_certified requires every state in the declared box to have the same acute winding -1 or +1. Under that premise, the phase-storage floor is V5=5*(1-cos(2*pi/5)). Each oriented receiver edge has edge_cauchy_factors = 1/d_i + 1/d_j + 2/(d_i*d_j), using actual full degrees. Its phase speed is bounded by the square root of twice this factor times full_storage_bounds.hi - phase_storage_floor_bounds.lo. edge_phase_transport_upper_bounds multiplies these speed bounds by the duration. Subtracting them from the initial edge margins gives edge_retention_margin_lower_bounds.

Only a strictly positive minimum margin certifies whole_window_retention_certified, valid throughout both time directions up to the declared duration. Missing initial geometry or a failed sufficient budget yields unavailable, not a prediction of failure. The speed bounds come from a first-exit proof; the reader does not extrapolate an instantaneous velocity or assume future acuteness. It installs no law and runs no solver.

This is prepared maintenance. An above-barrier storage interval does not prove entry from zero winding, and a zero-winding source must precede the protected backward interval. No exact invariant reference, rigid receiver geometry or acute contact phases are required. Schema: tnfr.sine-cycle-retention.v1; shared SDK projection and atomic export preserve the primitive source, uncertainty and availability. See the proof and example.

Reversible preparation into a retained full-state region

assess_sine_reversible_preparation(forecast, checkpoint, *, cycle, target_error_bound, source_error_bound, scaled_retention_duration=1) returns SineReversiblePreparation from the same regional owner. Both radii and the scaled duration must be positive admitted real scalars. It re-admits the complete conservative unit-law forecast and checkpoint, including support, node order, unit capacities and every form/phase coordinate. The forecast starts at time zero without prior conditioning or a frozen hidden node; its initial enclosure must contain R(checkpoint), where R(x,theta)=(-x,theta).

The reader rebuilds the checkpoint's energy-speed retention certificate. At each retained grid endpoint at original time A, it reverses the exact rational midpoint to obtain a source center c, retaining the largest endpoint halfwidth eta. The independent source set is the exact coordinate ball B_delta(c), not the larger backward enclosure. The shared field has global sup-norm Lipschitz bound 2/pi in this clock. An outward exponential bound E >= exp(2*A/pi) therefore proves forward inclusion in the retained target of radius epsilon when E*(eta+delta) < epsilon strictly.

Certification additionally requires whole-source winding zero, overlapping source/target conserved-storage bounds, and A > pi*T, where T is the scaled retention duration. Storage overlap alone does not prove entry. Every endpoint retains its exact center, uncertainty, amplification, winding, energy and failed predicates. The first passing endpoint is selected. A certified prefix survives an unavailable suffix; without success, incomplete coverage yields unavailable, and complete coverage yields no_certificate_on_declared_grid. Neither means that formation is impossible.

The reader checks primitive admission, the forecast chain and Picard inclusions; it does not replay or authenticate the supplied Taylor endpoint proofs. Its conclusion is conditional on that numerical evidence. Schema tnfr.sine-reversible-preparation.v1 is supported by shared SDK projection and atomic export. This constructs mathematical preparations, without autonomous selection or physical identification. See the proof and workflow.

Finite exclusion from rapid contact motion

assess_sine_contact_averaging(source, *, cycle, scaled_horizon) returns SineContactAveraging from the partition owner. It uses the same complete unit conservative law but requires exactly the C5/private-leaf support, uniform initial form within each block and equal represented initial receiver phase lifts. Initial leaf phases are arbitrary. The finite positive horizon T uses tau=t/pi; Boolean, textual and nonfinite scalars reject through shared admission.

For the signed initial form gap u, the report bounds each contact's speed away from zero by M=4*abs(u)/3-4*T. When M>0, it computes the whole-window current-integral bound epsilon=2/M+4*T/M**2 and receiver phase-storage bound 640*T**2*epsilon**2/81. If that bound is strictly below 7/2, whole_window_acute_winding_excluded holds on the entire declared forward interval. This proof can apply when full storage exceeds the cycle barrier. It uses exact transformed full-law rows and integration by parts, not a truncated tangent model, sampled trajectory or an installed averaged law.

If speed separation or the final strict test fails, status is unavailable; there is no prediction of formation. The conclusion does not extend to later times or nonuniform block forms, including the separate distributed-ramp entry witness. Direct schema: tnfr.sine-contact-averaging.v1. Both source readers support the shared SDK report projection and atomic export. See the proof and usage.

Conservative phase transport from a flat preparation

analyze_sine_conservative_phase_transport(source, *, contrasts) returns SineConservativePhaseTransport from the shared entry owner. It requires an exact comparison, globally common initial phase lifts, zero loss, unit held capacities and w=beta=1. Its clock is tau=t/pi. Initial form may be nonuniform; unlike environmental winding entry, no receiver cycle is required. Primitive source admission and exact graph rows are shared with winding entry; cached source gradients, rates and storage are not derivative evidence.

Each supplied contrast is a nonzero row in source-node order, with exact zero sum. Coefficients use shared exact-or-represented real admission; Boolean and nonfinite values reject. These rows describe relative phase observations, not new coupling weights or additional dynamics.

Writing A=K L, the report retains exact initial derivatives initial_phase_velocity=A*x, initial_phase_acceleration=0, initial_form_acceleration=-A^2*x and initial_phase_jerk=-A^3*x. The jerk is a third derivative at the initial instant. It cannot be extrapolated into a finite-time trajectory without a remainder certificate.

For each contrast c, the reader also derives current weights c*A*K, oriented edge-current coefficients, and the global acceleration bound B=sum_edges abs((c*A*K)_i-(c*A*K)_j). Thus for every nonnegative tau, abs(c*theta(tau)-c*theta(0)-tau*c*A*x(0)) <= B*tau^2/2. contrast_quadratic_remainder_coefficients retains B/2 separately from contrast_initial_jerk. Correlated paths can cancel shared nodes before bounding; summing independent edge boxes would lose that information.

The regional derivation uses these quantities to identify delayed internal propagation and a necessary acute-entry condition. Neither a nonzero jerk nor an upper acceleration bound certifies acquisition, future work, capture or permanent identity. Direct schema tnfr.sine-conservative-phase-transport.v1 and generic SDK export retain the source, contrasts, derivatives, clock and scope. See the independent controls.

Continuous regional organization from a complete forecast

assess_sine_regional_organization(forecast, *, cycle_indices, minimum_duration, acute_margin) in relational_sine_regional.py returns SineRegionalOrganization. It observes a declared induced C5 inside the complete support of a SineForecast, under zero loss, unit held capacities and w=beta=1. Cycle entries are distinct source indices in traversal order. The strictly positive duration uses original structural t; the nonnegative requested acute margin uses radians. Both are observation policies, not evolution laws.

The reader re-admits the law, support, state intervals, time grid and every consumed enclosure. Consecutive steps must cover the actual validated prefix, retain their input and endpoint boxes, and preserve the augmented held capacity. Whole-time Picard inclusion is recomputed from full tube rates. Derived winding, storage, rates and work are rebuilt rather than copied from cached verdicts. This checks the consistency of supplied Taylor certificates; it does not authenticate their production or replay the full derivative proof. Retain the shared solver's actual evidence and source provenance.

Each whole-time tube is checked for fixed integer branches, nonzero cycle winding and a strict margin above the requested acute margin. A contiguous interval must keep the same offsets and meet minimum_duration before the outcome becomes finite_acute_retention_certified. Samples or an acute endpoint are insufficient. Observations retain internal form and phase storage, full-network phase and relative phase rates, and net boundary power computed from both conjugate channels. The separate channel reader resolves internal conversion and inputs to each regional store.

Conservative regional balance gives net work from the initial time as E_R(t)-E_R(initial). Subtracting interval enclosures bounds this quantity; it does not establish the instantaneous sign, monotonic supply or a fitted damping term. Internal regional storage includes form as well as phase. Acute phase retention alone does not certify the stronger low-storage protection theorem or indefinite maintenance.

acute_winding_excluded_on_horizon requires complete declared coverage and an independently recomputed exclusion on every tube. A nonacute edge, nonnegative cosine across a two-edge path, or certified zero winding can exclude an acute unit-winding C5 under their stated interval signs. Failure to certify the requested margin/duration is not itself such an exclusion. Otherwise the outcome is unresolved, retaining partial coverage and numerical reasons. A certified interval inside a validated prefix can remain valid even when a later step is unavailable; horizon_complete keeps that distinction explicit.

Direct schema tnfr.sine-regional-organization.v1 and generic SDK export retain the full forecast and observer evidence. The control freezes a particular source and budget separately from this reusable reader.

Cumulative regional channel observations and phase accessibility

assess_sine_regional_channels(forecast, cycle_indices=...) reuses the complete conservative forecast admission and the regional channel kernel. It admits the same induced C5, unit capacities and zero-loss w=beta=1 law. Every step retains the whole-time channel rates and cumulative interval integrals of internal conversion and both boundary inputs in original structural time. Positive conversion means phase-to-form transfer. Summing rate intervals times full step durations bounds the integrals; endpoint samples are not quadrature evidence. Endpoint storage changes must be compatible with both integrated balances or the reader rejects the supplied evidence.

phase_storage_upper_bound is the running bound from primitive whole-time boxes. running_phase_work_upper_bound independently bounds the running integral of -J+B_V, including intermediate times inside each step. initial_phase_flat_certified requires exactly equal point-valued receiver phase lifts; false means this premise was not certified. Under that premise, either a strict phase-storage bound below acute_accessibility_barrier=7/2 or the corresponding integrated-work bound excludes any fully acute unit winding entry on the validated prefix. Reaching or failing to resolve the barrier is not entry. A prepared acute state or arbitrary nonacute winding is outside this exclusion's conclusion.

horizon_complete separately reports coverage. Even an empty validated prefix cannot establish absence later in the requested horizon. The observer re-admits primitives and recomputes Picard inclusion; it does not replay Taylor endpoint proofs, authenticate a producer or infer new dynamics. Schema tnfr.sine-regional-channel-history.v1 and the generic SDK envelope retain normalized full forecast evidence and all cumulative bounds.

The benchmark's --analyze-channels SOURCE_RESPONSE --output PATH mode strictly decodes its retained response schema, binds it to the saved protocol and archived declaration, and checks original file/manifest hashes. It writes a separately identified retrospective analysis and current-source archive, leaving the source response intact. Its hashes establish file associations, not external provenance authentication; historical source need not equal current analysis source. The result is not a new reserved prediction. The proof and scope remain independent of the transport and serialization checks.