Sine response, constitutive comparisons and hidden memory

Declared input/output resonance, bridge-law and clock discrimination, return-path geometry, causal hidden memory, autonomous pulses and recurrence distinctions.

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

Conservative form/phase response at an aligned bridge

tnfr.physics.relational_sine_resonance.assess_sine_bridge_channels(source, *, target_phase_turns, bridge) returns SineBridgeChannelAssessment. It re-admits the primitive SineExchangeComparison, then admits an exact rational-turn critical target on its connected simple unit support. The target must be strictly acute, capacities held and positive, exchange/storage coefficients positive and loss zero. The selected edge must be an actual cut edge with aligned target phases. A positive-loss recovery certificate is not required and cannot supply attraction to this conservative calculation.

The observation is the bridge form contrast and sqrt(beta) times the bridge phase perturbation, in clock tau=w*t/pi. It retains the nodal cut-space constraints and uses transpose preparation in the edge energy chart. nodal_bridge_lift has unit oriented bridge contrast and zero weighted mean: use it for form preparation, and divide it by sqrt(beta) for phase preparation. Origins are retained separately; the homogeneous jets use zero hidden state. General hidden initialization remains a separate source.

The report keeps admitted source and target separately, exact mobility, storage scale, clock_rate_pi_numerator=w, bridge index/orientation, lift, incidence row and outward cosine bounds. first_jet_bounds and second_jet_diagonal_bounds enclose response derivatives, not Taylor coefficients. channel_difference_bounds encloses form minus phase second derivative. Its nonnegative exact sign is positive precisely when an edge incident to a bridge endpoint has a nonzero target gap; the independent channel_difference_is_positive preserves this fact even if interval resolution includes zero. A zero difference does not certify equality at later orders. Cached rates, storage and verdicts do not determine the result.

The proof, executable usage and full-row derivative controls share this domain. Direct to_dict() uses schema tnfr.relational-sine-bridge-channels.v1; relational_report_to_dict supports the generic SDK envelope. Both retain exact fractions and target provenance. No graph mutation, loss law, finite-amplitude approximation, autonomous connection or physical identification is derived.

Pattern and response under a declared phase-storage family

assess_bridge_storage_family(source, *, left_cycle, right_cycle, target_phase_turns, epsilon) in the resonance owner returns BridgeStorageFamilyAssessment. The two disjoint ordered C6 rings cover the full unit support, with their first nodes joined by the sole bridge. Held capacities, storage scale and normalized exchange equal one, loss is zero, and the clock is tau=t/pi. Each internal exact target gap has absolute turn 1/6; the bridge is aligned. Both winding signs are admitted. Shared source, target and support admission is also used by the sine bridge-memory reader.

The required nonnegative epsilon declares the static alternative U_epsilon=1-cos(delta)+epsilon*(2/3-cos(delta)+cos(delta)^3/3). Shared exact-or-represented scalar admission rejects Booleans and nonfinite values, retaining rational coefficients exactly. law identifies this reciprocal cubic-sine family. The normalized sine source retains primitive support, capacity and reference provenance; its cached sine rates and verdicts are not rates or certificates of the alternative. The supplied exact target is distinct from the captured source state. No executor or default changes.

The report retains full node order, target and bridge orientation, exact form_laplacian, phase_hessian, full_tangent_generator and full_energy_metric. Full matrices order form then phase perturbations. edge_phase_curvatures follows the retained target geometry's edge order. bridge_observation_rows and nodal_bridge_preparation retain the same raw bridge contrasts and nodal preparation for every coefficient. These maps are orthonormal in each law's tangent energy metric because the bridge curvature remains one. Equal finite perturbations need not have equal nonlinear storage.

first_jet, second_jet and channel_difference are exact derivatives in the declared clock; the difference is the form diagonal minus the phase diagonal of the second jet. It can be positive, zero or negative. A zero difference does not generally imply identical future responses. For the specific value epsilon=4/9, the complete phase Hessian equals the form Laplacian; the proof establishes tangent channel symmetry, separately from nonlinear dynamics.

target_storage is the potential at the declared target. The local theorem retains local_phase_radius_turns=1/24, an enclosure of m=cos(5*pi/12) in local_curvature_lower_bounds, and an enclosure of m*pi**2/288 in local_excess_storage_threshold_bounds. Protection requires the initial lifted edge perturbations to lie strictly inside that radius and the same law's full excess storage above the target to lie strictly below the threshold. Comparing a separately justified upper excess bound with the threshold enclosure's lower endpoint is sufficient; its upper endpoint is not a safe cutoff. Excess storage is not globally nonnegative. The reader does not evaluate these initial-state premises or certify that the captured source lies in the protected neighborhood. No global basin, attraction, formation or physical identification follows.

Direct to_dict() and the generic SDK envelope preserve exact fractions, target association and scope through the shared atomic JSON writer. See the usage, admission tests and independent complete-law checks.

Finite nonlinear bridge-law discrimination

assess_bridge_finite_law_discrimination(source, *, left_cycle, right_cycle, target_phase_turns, amplitude, duration, preparation_error, observation_error) in relational_bridge_discrimination.py returns BridgeFiniteLawDiscrimination. Shared two-C6 bridge admission rebuilds the complete support, unit capacities, target and common preparation/readout for the fixed comparison eta=0 versus eta=1/100. The captured source anchors support and reference-law metadata; the two declared finite preparations are separate states, with opposite form rows and the same exact rational-turn target.

Each actual preparation may differ by preparation_error in every one of its twenty-four form and lifted-phase coordinates. observation_error bounds each complete bridge-form reading, not each individual sensor. The statistic is C=(2*a-u_plus(h)+u_minus(h))/(a*h**2) from two positive-time observations. Amplitude and duration are declared in the normalized model and tau=t/pi; clock uncertainty is not included in either error parameter.

The proof uses full-law reversibility and the exact finite-amplitude initial curvature. It separately bounds finite-time remainder, preparation propagation and reading error. curvature_prediction_bounds retain the resulting outward intervals; separation_margin_lower_bound>0 and phase_chamber_certified are both needed for discrimination_certified. An unavailable margin does not imply identical laws. The reader computes no trajectory and consumes no measured response.

Require positive amplitude and duration, nonnegative error budgets, and 2*duration<=1 for the shared exponential enclosure. Invalid inputs reject; failed sufficient inequalities return explicit reasons. All derivatives and uncertainty bounds retain the full nonlinear state. Symbolic target angles are not converted into supposedly exact binary64 phases. The projected initial boxes can include additional outward rounding width; this does not enlarge the declared admissible preparation error. Any implemented angle approximation must independently fit within that error.

The direct schema is tnfr.bridge-finite-law-discrimination.v1; generic SDK exports use the shared report wrapper. This is a conditional analytical prediction under a declared clock, not physical law selection, formation, clock identification or an alternative runtime installation. See the finite-budget example.

Finite bridge-law discrimination with a bounded common clock

assess_bridge_clock_law_discrimination in the same shared owner returns BridgeClockLawDiscrimination. It retains the preceding support, exact target, two complete laws and opposite form preparations, and adds a third preparation with zero form and phase theta_target + amplitude*p. This uses the existing independent phase column; none of the twenty-four coordinates is discarded.

All three observations occur at the same declared sampling_duration=s. Their actual structural duration is h=k*s, where one unknown constant positive k is shared by every evolution row and every experiment and lies within independently declared clock_scale_bounds. This is not a bound on arbitrary clock drift or independently reset clocks.

The observable is the raw ratio Q=(2*a-u_plus(h)+u_minus(h))/(2*(a-v(h))). Here v is the signed bridge phase gap relative to its exact aligned target lift: any integer target-turn offset must be removed. The certified chamber keeps this branch near zero. An unsigned phase distance is not this observable. Each of the three initial states has its own full-coordinate preparation error cube; the common reading error bounds each complete gap in its declared normalized coordinate units. An actual experiment must independently admit both form and phase readouts.

The proof divides both raw responses by the same unknown a*h**2 as an algebraic device, then cancels it in the quotient. These normalized responses are proof coordinates, not separately measured values. The exact finite-amplitude phase coefficient contains sin(a)/a; replacing it by its tangent is unsupported. Uniform errors retain the dependence on the actual time: with fixed growth and derivative bounds at the longest duration, each complete error budget is A*h**2+B/h**2. Convexity in h**2 makes its maximum the larger complete endpoint total. Taking separate maxima of its components is unnecessarily weaker. Clock uncertainty does not disappear from these remainders.

Both normalized phase prediction intervals must have strictly positive lower bounds before quotient formation. An unresolved denominator produces no ratio bounds or separation margin. Certification additionally requires the phase chamber and strict separation of the resulting ratio intervals. Overlap or a failed sufficient bound is unavailable evidence, not equivalence of the laws. No observed response is consumed and no selected law is installed.

The numerical domain is 0<amplitude<=1, positive sampling duration, ordered positive clock bounds, nonnegative preparation/reading errors, and 2*sampling_duration*clock_scale_bounds[1]<=1. Invalid inputs reject. Outward initial boxes do not enlarge the admitted preparation cubes. The direct schema is tnfr.bridge-clock-law-discrimination.v1; generic SDK export uses the shared wrapper. The example keeps the previous amplitude, sampling time and errors and adds ten-percent clock uncertainty. This is conditional discrimination between two supplied models, not clock identification, a unique pressure law or physical validation.

Exact two-cycle bridge memory and the local-loss boundary

tnfr.physics.relational_sine_bridge_memory.assess_sine_bridge_memory(source, *, left_cycle, right_cycle, target_phase_turns) returns the immutable SineBridgeMemoryAssessment. It reuses the preceding bridge admission and restricts its scope to two ordered, disjoint C6 cycles covering the full support, with their first nodes joined by the sole bridge. Every internal target edge has absolute turn 1/6. Capacities, storage scale and effective exchange are one, and loss is zero. Either winding sign, cycle reversal, common target-phase shifts and node relabeling preserve the calculation. Unsupported geometry or primitive state raises; cached source verdicts do not provide the tangent law.

projection_rows maps a supplied full nodal tangent vector, ordered as all form perturbations then all phase perturbations, to eight rational shell coordinates. The coordinate order is four form increments followed by four phase increments. nodal_lift is an invariant section, not reconstruction of the complete nodal state. Exact intertwining and energy orthogonality authenticate the reduction; the discarded symmetry sectors are unobservable at this bridge in this tangent law. They need not be absent from the network.

The report retains full_tangent_generator, coordinate_generator and energy_metric. Shared derive_linear_observation checks that all eight coordinates are required for this all-state bridge observation. Shared derive_coordinate_memory selects visible indices (0,4) and retains the other six coordinates, in original complementary order. Its blocks A,B,C,D give the exact causal identity, in clock tau=t/pi,

y' = A*y + B*exp(D*tau)*h0 + integral B*exp(D*(tau-s))*C*y(s) ds.

hidden_initial_projection_rows maps a separately declared nodal tangent initial state to h0; initial_source_rows gives its source at time zero. The captured source state is neither a target perturbation nor a zero-hidden preparation. No initial perturbation is invented or exponential evaluated. The report also retains the zero-lag kernel through coordinate_memory, its first two derivatives and the exact matrices static_visible_generator=A-B*D^-1*C and low_frequency_rate_matrix=I+B*D^-2*C.

Those last matrices are resolvent coefficients. The static generator is (2/13)*R, with R=((0,-1),(1,0)); the rate matrix is diag(309/169,449/169). They do not replace the causal convolution or supply a memoryless executor. The proof establishes a nondecaying three-frequency kernel, a visible slow hidden mode and the absence of symmetric damping in the zero-frequency coefficient. Time-domain approximation needs separate preparation, time/frequency and error premises. No nonlinear formation, physical mass or irreversible-loss claim follows from the retained rate matrix.

Direct export uses schema tnfr.relational-sine-bridge-memory.v1; the generic SDK envelope and atomic JSON writer also support the report. Exact fractions, target provenance, matrix order and scope are preserved. See the guide, admission controls and independent full-law kernel checks.

Constitutive dependence of an implicit return-path equilibrium

assess_return_path_storage_geometry(source, *, left_cycle, right_cycle, mediator, epsilon, refinements=40) in the phase-cycle owner returns ReturnPathStorageGeometryAssessment. The ordered disjoint C5 rings and the distinct mediator cover the full eleven-node support. Their first nodes are linked through the mediator, and their second nodes have a direct return edge. No additional edge is admitted. Shared source admission retains unit conductance support, unit held capacities, zero loss, w=beta=1 and clock tau=t/pi; the sine comparison is a primitive reference association, not an observation of the alternative law.

The required nonnegative epsilon declares the same cubic-sine potential as the bridge comparison. Exact-or-represented scalar admission rejects Boolean and nonfinite inputs. refinements is an integer from 1 through 64; this is a numerical work budget. The same strict-sign bisection kernel also serves the retained native return-geometry instrument, whose original sine bracket and default preparation remain distinct.

The target has constant zero form and prescribed named periods (1,-1,0). Its special edge angle t is defined by the unique root of F_epsilon(t)=j_epsilon(pi/2-t/4)-j_epsilon(t)-j_epsilon(2*t/3). root_turn_bracket encloses s=t/(2*pi) between strict certified endpoint signs, starting in (1/12,1/5). Its residuals enclose F_epsilon/(1+epsilon); this positive normalization improves arithmetic and changes neither roots nor the declared dynamics or clock. Unresolved interval signs reject a refinement instead of selecting a half by rounded values or a tolerance.

nodal_turn_affine_coefficients and edge_turn_affine_coefficients store (intercept,slope) in the same unknown root s, preserving correlations. Node lifts follow source.nodes; oriented edge gaps follow geometry.edges. edge_integer_offsets retain the circular reconstruction, and named_cycle_periods are exact affine identities. The corresponding bounds enclose one implicitly defined equilibrium; arbitrary choices from their Cartesian product need not satisfy its periods or current balance. A root midpoint is an approximation and is not certified as an exact critical state.

The proof establishes existence, uniqueness throughout the acute period cell and constitutive motion of the geometry. nodal_root_residual_multipliers keep the complete nodal current identities, so criticality does not follow merely from small nodal_current_residual_bounds. edge_current_bounds, edge_curvature_bounds and target_phase_storage_bounds refer to the original unscaled potential. minimum_acute_margin_turns_bounds retains the distance from every target gap to a quarter turn. The positive Hessian supports conditional local stability under this conservative law; it supplies no capture radius, attraction, trajectory or spontaneous winding formation.

Direct schema tnfr.return-path-storage-geometry.v1 and generic SDK projections preserve the law, exact affine data, implicit root, source association and unavailable claims. Export is detached and does not authenticate provenance. See the usage, admission controls and complete-nodal checks.

Infer a constitutive interval and a separate nodal response

assess_return_path_geometry_response(source, *, left_cycle, right_cycle, mediator, special_turn_bounds, form_direction, observation_origin) returns ReturnPathGeometryResponseAssessment from the same phase-cycle owner. It shares the preceding reader's complete support, coefficient-independent geometry admission and unit-capacity, zero-loss, w=beta=1, tau=t/pi premises. It does not run a forward coefficient root to construct its own observation.

special_turn_bounds contains two ordered exact-or-represented real endpoints for the special angle divided by mathematical 2*pi. form_direction contains eleven admitted signed real values in source-node order. Boolean, nonfinite and malformed inputs reject before the calculation. observation_origin is explicitly supplied_mathematical_interval or measured_interval; the latter declaration authenticates neither measurement nor physical correspondence. Stored source phases, cached rates and storage are not observations of the inferred equilibrium.

The inverse theorem identifies the admissible half-open branch from the sine root through, but excluding, the pure-cubic limiting root. The coefficient is -F0/F3 on that branch. Strict outward endpoint signs yield the following statuses:

coefficient_status	Meaning and availability
bounded	A compatible finite coefficient enclosure; both bounds and the full initial acceleration enclosure are available
unbounded_above	The compatible angle interval reaches the limiting geometry; coefficient_upper and finite response bounds are None
incompatible	The interval misses every finite nonnegative member; coefficient and response bounds are None
unresolved_interval_arithmetic	Numerical signs cannot certify classification; bounds remain unavailable rather than implying incompatibility

The reader retains original observation bounds, endpoint residual enclosures, geometric_outer_turn_bounds, exact nodal/edge affine correlations and named periods. Outer angle/coefficient boxes enclose candidates; their Cartesian product is not a set of independent equilibria. Geometric error can make coefficient error unbounded and does not identify the dynamic rate scale.

For a bounded case, phase_velocity_direction=K L h and response_acceleration_bounds encloses -K H_epsilon K L h for all compatible equilibria. H_epsilon is the full phase Hessian, evaluated with outward trigonometric intervals. This uses a prescribed initial form direction and zero initial phase perturbation. response_status is bounded_initial_tangent_acceleration or unavailable; the known phase velocity direction remains available even when the acceleration is not. This conservative availability rule also withholds a response for unbounded coefficients when a specially chosen null direction could have zero response. No positive-time trajectory, event, runtime installation or formation claim follows from an initial derivative.

Direct schema tnfr.return-path-geometry-response.v1 and the generic SDK envelope preserve exact fractions, unavailable values and declared origins. The guide, admission tests and independent full-row controls exercise this boundary. The staged producer consumes a fixed known-source protocol, saves its prediction before evaluation and checks saved bounds against rebuilt geometric premises before using them.

Sine-law resonance and declared input/output response

relational_sine_resonance.py owns static tangent assessments for the complete normalized-sine law. assess_sine_cycle_resonance(model=..., node_count=..., mode_index=..., capacity=..., winding=0) declares a complete unit cycle with uniform form, exact target phase turns winding*j/node_count, common held positive capacity and an explicit regular reference model with positive EPI loss weight. The reference model supplies normalized coefficients, not native Arg rates.

Counts, winding and mode must be exact nonboolean integers, with n>=3, 4*abs(winding)<n and 1<=mode_index<n. Shared represented-real admission retains exact positive rational capacity. The analytic template allocates no graph and performs no mode search or floating eigensolve. Its modal rows differentiate the shared sine field; bounds use mathematical pi and outward interval arithmetic. This is not admission of a live observed cycle.

SineCycleResonance retains the template, coefficient ratio beta*(e/w)^2, modal matrix, damping ratio and separate pole/phase-peak classifications. Its input is a unit Euclidean real Fourier mode added to the form-rate row. Form amplitude has transfer s/D(s), phase amplitude has b*r/D(s), with D=s^2+e*r*s+a*b*cos(alpha)*r^2 and r=capacity*lambda/2. The work-conjugate output is lambda*X. Natural frequency equals the form/work gain peak; it is not necessarily an observable free-oscillation frequency. The exact work peak gain is 2/(e*capacity). Positive-frequency phase gain and complex poles require different inequalities.

report.gain(angular_frequency) admits a finite nonnegative exact or represented real frequency in the model clock and encloses all three gains separately. Under chained-report admission, it rebuilds and retains the mode from its law, model, cycle size, winding, mode index and capacity. Unresolved denominators or classification signs retain unavailable reasons; an interval containing zero does not establish a zero gain, critical damping or absent susceptibility. At zero frequency the form/work transfer is zero. Invalid domains reject rather than being coerced into an admissible mode.

certify_sine_recovery_resonance(recovery, port=(i,j)) reuses a shared full-support sine pattern/cycle recovery report through chained-report admission. The target must meet that owner's sufficient acute criticality, positive capacity, support and recovery hypotheses. With K=diag(nu_i/d_i) and p=e_i-e_j, the declared probe is f=K*p, the input term is f*u(t) and the conjugate output is y=f^T*L*delta_x. The theorem establishes an interior gain maximum and the bound (K_ii+K_jj)/e, without selecting its frequency. All original environmental coordinates remain in the target; the statement concerns its tangent, not an observed transient's transfer. Uncertain held capacities describe a family of systems. If a selected port's capacity is uncertain, the corresponding probes/readouts also form a family, rather than one calibrated fixed physical port.

Reports use exact shared JSON projection through to_dict() and the SDK atomic writer for export. relational_report_to_dict also accepts the resonance report types and delegates projection to this owner while preserving the generic SDK envelope. Export does not authenticate a manually constructed dataclass. No report installs forcing, executes RA, adds an edge, advances a solver or demonstrates nonlinear formation, autonomous excitation or physical resonance. The foundation owns the full-network proof, gain ceiling, counterexamples and clock scope.

Remote sine response through a retained intermediary

assess_sine_mediated_response(recovery, donor_cycle=..., receiver_cycle=..., mediator=...) in the same resonance owner uses chained-report admission for its supplied recovery and requires exactly two disjoint ordered five-node cycles and one intermediary. Each cycle's first node is its sole contact with the intermediary. All eleven nodes and twelve edges must be present, with no additional edges. Capacities must be exact held positive scalars; uncertain capacity families are outside this adapter.

Shared recovery admits the exact acute critical target, independently of the nominal observed perturbation. Both bridge target gaps must be zero modulo a turn. Those hypotheses force uniform acute cycle twists; winding -1, 0 and 1 are permitted separately on each cycle. No graph mutation or target fitting occurs. The declared clock and normalized coefficients are retained.

The donor direction is f=K*p_D, where p_D is one at the donor port, minus one quarter at its other four nodes, and zero elsewhere. The receiver contrast r_R has the same weights on the receiver. It observes phase or form relative to the other receiver nodes, using target-relative phase lifts. It is not a structural-work-conjugate readout. The report concerns the exact target's tangent, not a time-invariant approximation inferred from the observed transient.

SineMediatedResponse retains form_markov_bounds and phase_markov_bounds for derivative orders 0, 1 and 2, and the corresponding full-state derivative vectors. These are C*J**j*(f,0), without Taylor factorials. They describe both the autonomous tangent response to initial form epsilon*f and the impulse kernel for hypothetical form-rate input f*u(t). A constant input switched on from zero has its corresponding first phase effect at order 3, not order 2. No impulse or other forcing is installed.

Static bridge balance gives exact zero DC contrasts. The phase second derivative is phase_order_two_pi_numerator/pi, which is strictly negative by exact positive-capacity/coefficient admission. This sign remains distinct from an outward enclosure that touches zero. The form coefficient can cancel. positive_frequency_phase_peak_certified and phase_impulse_sign_reversal_certified apply the remote-response proof: there is a finite-frequency phase-gain maximum and the autonomous tangent phase contrast becomes positive after its initial negative response. Peak frequency, reversal time and a quantitative nonlinear amplitude bound remain unavailable. Neither positive-realness nor the collocated gain ceiling is assigned to this remote phase output.

The retained 2-by-2 contact blocks use form-then-phase order. They define the intermediary generator and the donor-to-intermediary and intermediary-to-receiver couplings. memory_kernel_at_zero_bounds encloses their product. The complete kernel also depends on the intermediary exponential, hidden initial state and visible history; no instantaneous direct edge replaces it.

The two *_silent flags certify sufficient tangent reflection symmetries by checking support, exact target cosine geometry and equal capacities on reflected pairs. A false flag means this proof is unavailable, not that transmission has been demonstrated. Finitely many zero Markov entries never prove permanent silence. Invalid support, incompatible recovery or inexact capacities reject. to_dict() uses schema tnfr.relational-sine-mediated-response.v1; the generic SDK export delegates to it and retains the generic envelope. This is a static theorem report, not a trajectory, formation event or physical observation.

Autonomous nonlinear pulse on a complete sine pair

assess_sine_pair_pulse(graph, reference_model=...) in the shared resonance owner reuses sine state admission, rates and work. It requires the complete two-node, single-unit-edge support, held nonnegative capacities and absent inputs/events. The model supplies the coefficients, storage scale and clock. The reader never changes a positive loss coefficient to zero. It does not execute the native argument-pressure runtime or install a new pressure law.

In captured node order, u=x0-x1, delta=theta1-theta0 and sigma=nu0+nu1. Storage is E=u**2/2+beta*(1-cos(delta)); normalized_energy_bounds encloses E/beta. Equal captured phase values also give exact_energy, preserving exact rational decisions at zero energy and the separatrix. Other energies use outward enclosures; unresolved signs remain unavailable rather than being rounded into a certificate.

SinePairPulseAssessment separates the following cases:

• libration_certified: supplied e=0, sigma>0 and certified 0<E<2*beta. The full nonlinear continuous state is periodic, including a pair with exactly one zero capacity.
• dissipative_recurrence_excluded: e>0. Nonstationary recurrence is excluded even if the initial loss happens to be zero.
• frozen or equilibrium: respectively both capacities vanish, or the captured forms and phases coincide. These checks certify stationary cases; they do not enumerate every circular representation of an equilibrium.
• separatrix_out_of_scope, rotation_out_of_scope or energy_classification_unresolved: no nonstationary libration certificate.

small_amplitude_period_bounds encloses T0=2*pi**2*sqrt(beta)/(w*sigma) when e=0 and sigma>0. For certified libration, period_bounds uses the exact elliptic-integral inequalities T0<T<=T0/sqrt(1-E/(2*beta)). The period refers to the model clock, not an identified laboratory unit. A certificate of period existence can remain true while its numerical bound is unavailable near the separatrix; period_unavailable_reasons retains that distinction. Relative-rate bounds and initial_continuous_loss come from the shared complete-law reader, not a fitted oscillation.

to_dict() uses schema tnfr.relational-sine-pair-pulse.v1; the generic SDK report export delegates to the same projection. The detached assessment does not advance time, modify a graph, certify an Euler trajectory, select an attracting amplitude or generate motion from equilibrium. The proof also establishes the broader finite closed positive-loss sine obstruction. Zero loss and a nonzero initial preparation remain explicit premises, not a derived fundamental selection or physical identification.

Conservative path memory and nonlinear regional transfer

assess_sine_path_memory(graph, reference_model=..., mediator=..., consensus_form=0, consensus_phase=0) uses the same resonance owner. It requires a complete unit P3 path, the correct middle node, common positive held capacity and an explicitly zero EPI loss weight. Shared capture owns scalar/alias, support and absent-source admission. Positive loss, unequal or zero capacity and extra support reject; this adapter never changes the law.

The SinePathMemoryAssessment report separates two uses of the captured state:

• nonlinear_edge_storage_bounds and nonlinear_edge_rate_bounds describe actual incident-edge storage and its instantaneous work in the original structural clock. Endpoint order follows graph capture; path and path_indices retain left, middle, right. nonlinear_transfer_bounds encloses the left edge's rate, whose exact negative is the right edge's rate. The two nonlinear_edge_rate_residual_bounds compare these formulas with the independent chain-rule work. Total continuous loss is exactly zero.
• coordinate_memory and linear_observation concern the separately declared uniform-form/uniform-phase reference tangent. The supplied origins are admitted through shared exact-or-represented-real admission. The report does not infer this target from the graph or linearize about a moving state. initial_tangent_state contains all form differences from the reference, followed by all chosen real phase-lift differences, in captured node order. No error estimate identifies its linear evolution with the nonlinear future.

The exact clock is tau=w*t/pi; clock_rate_pi_numerator and clock_rate_bounds retain that conversion. In this clock, the consensus generator is [[0,-M],[M/beta,0]], where M=nu*D^-1*L uses the complete path degrees. Its entries are exact rationals for admitted captured values; mathematical pi has not been replaced by a floating-point coefficient. The existing coordinate-memory owner partitions endpoint forms then endpoint phases as visible, retaining middle form and phase as hidden.

With omega_squared=hidden_frequency_squared=nu**2/beta, the exact kernel is cos(omega*tau)*K0+sin(omega*tau)*K1/omega. K0 belongs to coordinate_memory.kernel_at_zero; K1=kernel_derivative_at_zero is its normalized-time derivative. Hidden initialization contributes the analogous combination of hidden_initial_forcing=B*h0 and hidden_initial_forcing_derivative=B*D*h0. Omitting that term changes the preparation. No kernel, exponential or memory convolution is numerically propagated by this static report.

The shared minimal-observation construction requires all six tangent coordinates to close the four endpoint observations. This is an all-state linear statement, not a physical count of hidden constituents. The full tangent has a return period enclosed by full_tangent_period_bounds, 2*pi**2*sqrt(beta)/(w*nu) in structural time. A particular state can be stationary or have a shorter fundamental period; the exact tangent_motion_nonstationary flag checks its generator action. Neither that flag nor the period bound certifies a nonlinear pulse.

to_dict() uses schema tnfr.relational-sine-path-memory.v1; generic SDK export delegates to this same projection. The proof derives regional transfer, the periodic memory and the obstruction to exact irreversible finite tangent reduction. A finite-window damping approximation, microscopic zero-loss selection and universal fractality remain separate questions. No graph mutation, external drive or runtime-default change occurs.

Full nonlinear recurrence: family admission and individual limits

assess_sine_recurrence(graph, reference_model=..., energy_ceiling=..., form_mean_bounds=(lower, upper)) uses the shared resonance owner. It admits complete finite connected simple unit support with at least two nodes, strictly positive held capacities, explicit zero EPI loss weight and the shared absent-input/event premises. It applies to the full nonlinear sine law, with real form and circular phase, without an equilibrium target or small-amplitude assumption.

The caller supplies a strictly positive energy ceiling and a strictly positive-width mean interval. Shared exact-or-represented-real admission rejects Booleans, nonfinite values and unordered/invalid bounds. These values select the analyzed family; they do not add parameters to the dynamics. The family contains every state with E<=energy_ceiling and weighted form mean in that interval, at the captured support, capacity and law coefficients. It is not a Cartesian product of independently bounded phase and form data.

SineRecurrenceAssessment retains exact weights rho_i=d_i/nu_i, their normalization and weighted_form_mean. Its weighted_mean_rate_residual_bounds is computed from the shared form rates, and comparison.storage_rate retains the computed work enclosure. The exact divergence identity is -e*sum(nu_i), hence zero in the admitted law. Interval residuals checking zero do not by themselves prove either invariant.

The report uses path_length_upper_bound=n-1 to enclose R=sqrt(2*(n-1)*energy_ceiling) in form_radius_bounds and supplies the family coordinate box [lower-R,upper+R] in form_coordinate_bounds. This conservative connected-path bound needs no spectral computation or all-pairs distance search. Compactness includes the phase torus, not arbitrary unwrapped real phase lifts. Tiny exact positive capacities and slab widths remain positive in admission even if an outward projection touches zero.

The report keeps three conclusions separate:

• almost_everywhere_recurrence_certified and finite_positive_family_volume_certified apply to the declared family in ambient form Lebesgue measure times phase-circle Haar measure.
• snapshot_family_membership is inside, outside or unresolved, with explicit reasons. Exact mean comparison and, for identical captured phases, captured_energy_exact can resolve boundaries that projected energy cannot. Otherwise overlapping energy bounds remain unresolved. An outside snapshot does not invalidate the separately admitted family theorem or inherit its form box.
• snapshot_motion_status is stationary, nonstationary or unresolved. A nonzero exact form Laplacian or a sign-resolved sine current proves motion. Uniform captured phases and zero form Laplacian certify a stationary case; these are sufficient checks, not an exhaustive equilibrium classifier. individual_recurrence_status is trivial_stationary_recurrence for that stationary case and unavailable_for_chosen_state otherwise.

In particular, confirmed family membership and known motion do not imply an individual recurrence certificate. The proof and exact separatrix counterexample explain why this distinction is necessary. No return deadline, exact period, attracting amplitude or preserved pattern identity is reported. A finite numeric grid or seed distribution does not automatically inherit a theorem for absolutely continuous ambient preparations; fixed-energy and fixed-mean surfaces need their own measure argument.

The joint-episode theorem combines this family result with a separately proved open acquisition region on the same doubled-C5 law. It establishes repeated finite matching episodes for almost every preparation there. This reader alone does not admit that region, infer an identity predicate, or certify any captured state's episodes. Leaving a collective-coordinate chart is not itself loss of joint matching.

to_dict() uses schema tnfr.relational-sine-recurrence.v1; generic SDK export uses the same projection. The reader executes no trajectory, changes no runtime law and does not reinterpret a dissipative recovery report.