Conservative sine saddle preparations and certificates

Signed-involution preparation, the C5/private-leaf saddle, directed passage, same-orbit retention, sensitivity, retained-metric forecasts and constitutive controls.

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

Directed nonlinear passage through the principal phase seam

assess_sine_saddle_corridor(source, *, cycle, lower_phase, upper_phase) in physics/relational_sine_corridor.py returns SineSaddleCorridor. It reads the complete conservative unit-capacity C5/private-leaf source and reuses exact signed-involution admission. The ordered source must belong to the real-lift family (a,b,0,-b,-a) and (u,v,0,-v,-u), with the matched leaf rows retained. The proof uses the scaled clock tau=t/pi.

The declared strip has -2*pi/3 < lower_phase < u < -pi/2 < upper_phase < 0. A source with winding +1 and four nonclosing receiver gaps strictly inside their principal branches can receive a directed-exit certificate when the rebuilt full storage is below four, the energy-controlled conjugate force is strictly positive, and its initial momentum exceeds the lower-face exit allowance. The owner uses exact represented-real admission and outward rational intervals; a failed sufficient comparison remains explicitly unavailable.

The momentum 6*a+3*b+2*c+d includes the leaf forms. It is not just the closing phase rate. The shared full storage bounds both this momentum and the transverse motion, giving a finite residence-time upper bound. The lower-face condition rules out the wrong exit; the remaining exit is beyond the principal seam with winding zero. The proof does not assume monotone closing phase, aligned contacts throughout the motion, or a tangent law.

The source, phase coordinates, energy, momentum, force bound, exit margin and time bounds remain in the report. Cached input bounds are rebuilt from primitive state and law. The direct schema is tnfr.sine-saddle-corridor.v1; the SDK delegates projection and atomic export to the shared owners.

This is an exact-family finite passage theorem, with no trajectory evaluated and no independent full-state uncertainty box certified. Its initial unit-winding state is outside the wider protected region and is nonacute. Reversing the motion supplies a zero-to-unit passage with the same limitation; it does not alone establish acute retention. The same-orbit formation theorem adds the required inner passage and retained band. See the proof and workflow.

Same-orbit formation and finite retention

assess_sine_saddle_retention_band(source, *, cycle) and assess_sine_saddle_formation(source, *, cycle, epsilon=Fraction(1, 2**32)) in physics/relational_sine_corridor.py return SineSaddleRetentionBand and SineSaddleFormation. Both re-admit the complete conservative unit law and C5/private-leaf support. Here the supplied comparison anchors law and support only: its captured state is not asserted to be the mathematical preparation, a hitting state or a target.

The band reader verifies a sufficient implication from an actual deep hit u=-5/2 on the admitted signed family, with conserved storage at most 35001/10000 and retained path branches. The band [-13/5,-12/5] gives a uniform acute margin greater than 1/12000 and a phase-coordinate speed below 1/7. From the deep hit, either band exit takes more than 7/10 scaled time units in either direction. The target checkpoint is one half-unit before the deep hit; its next full unit remains acute. Independent form and phase errors of radius 2^-18 at every fine node use the complete-field Lipschitz bound, including perturbations that break symmetry.

The formation reader admits 0 < epsilon <= 2^-32 through shared exact scalar admission. Epsilon specifies a favorable preparation, not a new law parameter. It rebuilds the saddle and band and checks the correlated recipe z_saddle + epsilon*(3*X_lambda, -v_lambda). Its exact eigenvector components share one algebraic root; interval components cannot be selected independently. Bounds at scaled times +3 and -3 place actual states of this one orbit in both nonlinear corridors. Reversal connects the outer zero-winding endpoint to the inner band under the same complete law.

formation_existence_certified and independent_source_ball_existence_certified concern this analytical family. The report retains energy and nonlinear error bounds, both directional gates, connection-time ceiling 64/epsilon, target radius and duration. The source radius is represented exactly by source_radius_prefactor and source_radius_exponent, meaning prefactor * exp(-exponent), namely 2^-19 * exp(-128/epsilon). It is positive but generally far too small to materialize in a floating scalar. Do not replace it with zero, a rounded positive floor or a practical tolerance.

The nominal source is the reversed outer hitting state; the checkpoint is the same reversed orbit's deep-hit state evaluated one half-unit earlier. Their numerical coordinates and hitting times are not returned. Accordingly, numerical_source_center_available, captured_source_formation_certified and captured_source_retention_certified remain false. A certified theorem report is not a concrete preparation, source checkpoint or numerical forecast. No trajectory is run and no frozen response is modified.

Direct schemas are tnfr.sine-saddle-retention-band.v1 and tnfr.sine-saddle-formation.v1. Shared SDK projection retains the exact radius expression and validates nested labels; export does not authenticate or turn the analytical recipe into observed state. The proof closes conditional finite formation/retention existence under supplied law and support, with autonomous preparation, physical identification and an operational source still separate. See the workflow.

Rational preparation and conditional full-state sensitivity

prepare_sine_saddle_state(source, *, cycle, epsilon=Fraction(1, 2**32)) in physics/relational_sine_corridor.py returns SineSaddlePreparation. The source anchors the admitted unit conservative law and C5/private-leaf support. prepared_state contains a new exact rational form/phase state, with every fine node retained and rates in original structural time. The correlated algebraic recipe is enclosed, materialized with exact signed symmetry, and compared to the actual rational center. The representation error is propagated into both local corridor gates. Actual full-edge energy and initial phase branches are recomputed; cached certificates cannot substitute for these checks. Full and regional sine storage share the 128-bit interval cosine kernel, resolving the nearcritical energy excess without changing the pressure or evolution law.

preparation_certified means the actual rational intermediate state passes these sufficient conditions. It has winding one; the zero-winding source coordinates and hitting times remain unavailable. No trajectory is evaluated. An unresolved energy or local margin returns unavailable, even when the ideal analytical family remains certified. Computational representation error is not a physical preparation tolerance.

assess_sine_saddle_sensitivity(source, *, cycle, phase_radius=Fraction(1, 1000)) in physics/relational_sine_sensitivity.py returns SineSaddleSensitivity. It rebuilds the exact saddle and a positive rational metric on all twenty form/phase coordinates, including both conserved origins and perturbations outside the signed family. Exact matrix inequalities certify tangent growth at most 1/3 in both time directions. The nonlinear bound is 1/3 + 28*phase_radius in scaled time tau=t/pi, conditional on both complete trajectories remaining within that primitive-phase radius of the exact saddle throughout the compared interval. Radius admission is shared, finite and nonnegative, excluding Booleans. This radius is a proof domain, not a constitutive parameter.

The source again anchors law/support only. In particular, captured_source_flow_bound_certified remains false: static metric admission does not establish whole-time containment, handle a numerical residual or advance a validated forecast. Norm conversions are supplied, but repeated conversion to independent coordinate boxes loses the retained correlations. The metric is an analysis norm, not a new physical storage law.

Direct schemas are tnfr.sine-saddle-preparation.v1 and tnfr.sine-saddle-sensitivity.v1. Shared SDK projection preserves exact values and validates nested labels without authenticating a checkpoint. The proof and workflow separate rational preparation, conditional error bounds and an actual forecast.

Retained-metric full-state saddle forecast

forecast_sine_saddle_metric(source, *, cycle, duration, time_step, initial_coordinate_radius=0, phase_radius=Fraction(1, 1000), growth_rate_bounds=None, growth_bisections=8, direction=1, order=12, max_steps=256) in physics/relational_sine_metric_forecast.py returns SineSaddleMetricForecast. Here the re-admitted source is the actual initial state, not just a support anchor. The law remains conservative unit sine exchange on the complete C5/private-leaf support, with held unit capacities and no forcing or events. The sensitivity metric and all theorem premises are rebuilt from primitive state and law.

The source's twenty exact form/phase coordinates center an independent coordinate-error cube of the supplied nonnegative radius. The cube is enclosed in a positive-metric ball once. The shared numerical owner mathematics/_validated_metric.py carries that ball's center and radius between steps. Its coordinate projection serves strict Picard inclusion, whole-time phase admission, Taylor remainders and output observations. An output box never becomes the next metric radius. Complete center-solution rounding and truncation error are added in the same metric, using a scaled norm calculation to avoid losing small squared errors to the interval grid.

Both duration and time_step use original structural time. Exact integer direction=+1 or -1 selects the forward or reversed autonomous field, with positive elapsed duration. Every phase in every Picard tube must remain strictly within phase_radius of the exact saddle phase lift; neither wrapping nor a rounded saddle midpoint replaces that comparison. The original-clock growth bound is (1/3+28*phase_radius)/pi_lower.

With phase_radius=None, the alternative declared numerical mode instead rebuilds the full interval Jacobian on every admitted Picard tube. It retains the same metric and complete sine law, which is smooth on all real phase lifts. For S=J.T*W+W*J, the midpoint plus the diagonal row sums of interval radii dominates every actual symmetric Jacobian. An exact positive-semidefinite test of 2*gamma*W-majorant certifies the whole-tube rate. The rational bracket defaults to (0, 1); growth_bisections admits 0--16 fixed refinements. An insufficient upper endpoint returns unavailable, without bracket expansion. These rates already use the original clock and receive no further pi scaling. Supplying a growth bracket together with a finite phase radius is rejected.

growth_mode identifies the chosen method. Computed-mode steps retain their Jacobian enclosures, rational majorant, declared bracket, selected rate and matrix certificate. Its forecast-level growth_rate_upper_bound is the maximum certified step rate, or unavailable if no step passed. This replaces the small-phase-domain growth premise with a fresh whole-tube proof; it does not select or alter the evolution law.

Positive finite time arguments, nonnegative coordinate error, Taylor order 1--16 and a positive integer step budget no larger than 256 are admitted before execution. The shared scalar exponential additionally requires abs(growth_rate*time_step) <= 1 for each actual step. Numerical work limits are not physical restrictions. A failed tube returns unavailable with the last certified center/radius, validated_duration, failed enclosure and reason. There is no automatic retry, radius enlargement or changed law.

Each step retains its Picard/domain evidence, propagated initial radius, center polynomial and remainder, local metric error and endpoint ball. status="admitted" certifies enclosure through the requested horizon; it does not itself certify formation, winding change or retention. The distinct schema tnfr.sine-saddle-metric-forecast.v1 and SDK report type preserve this scope. Existing consumers of the coordinate-box SineForecast do not accept this report without a separate admission contract. Export is not checkpoint authentication. See the step proof and manufactured example.

Two-direction same-orbit connection evaluation

forecast_sine_metric_connection(source, *, cycle, duration, time_step, initial_coordinate_radius, growth_rate_bounds=(0, 1), growth_bisections=8, order=16, max_steps=256) in physics/relational_sine_metric_connection.py re-admits one complete source and runs two fresh metric forecasts from it, with directions +1 and -1. It uses whole-tube Jacobian growth admission, not a phase cutoff. The normalized bracket and all preparation, clock and work parameters are shared by both calls. A caller-supplied forecast or favorable cached verdict cannot replace either computation.

The reader selects the first forward endpoint whose complete enclosure has unique principal branches and winding zero. On the backward leg it selects the first contiguous sequence of complete Picard tubes with strict acute receiver gaps and fixed winding one, spanning at least pi_upper in original time. Endpoint samples do not establish that continuous interval. Reversibility then links the reflected forward state to the reflected backward window on the same orbit; it changes forms' signs and leaves phases unchanged.

SineMetricConnection keeps both complete or partial forecasts, the selected times, phase margins and independent horizon-completion flags. A valid prefix can certify same_orbit_connection_certified even if a later step fails; declared_horizons_complete still reports that failure. Missing observations on complete horizons mean no_certificate_on_declared_horizons, not a global formation obstruction. Incomplete coverage without the observations remains unavailable.

The forward endpoint enclosure contains the image of the initial family, but need not itself lie inside that image. Consequently, independent_zero_winding_source_ball_certified remains false. Neither a new independent source preparation nor an infinite lifetime follows from these two forecasts. Direct schema tnfr.sine-metric-connection.v1 and the shared SDK retain these distinctions. The frozen-trial instrument separates preparation from evaluation and additionally requires both complete horizons for its study-level pass. See the proof and source boundary.

Local agreement and a different global phase barrier

assess_saddle_storage_discriminator(source, *, cycle, initial_coordinate_radius, epsilon=Fraction(1, 2**32)) in physics/relational_phase_storage.py returns SaddleStorageDiscriminator. The source anchors the admitted conservative unit C5/private-leaf support. The reader freshly rebuilds the named rational saddle preparation; it does not advance the captured source or accept a cached preparation verdict.

The report compares sine with one explicitly supplied smooth cutoff storage law. Its pressure and phase storage change together; its distinct law tag never installs that law in the native or sine runtime. The complete-law proof owns its potential, companion rows, conserved storage and regularity. The cutoff law is infinitely differentiable but not real analytic.

The nonnegative independent radius applies to every form and phase coordinate. Shared circular admission bounds every initial support edge, and shared sine storage bounds apply to the alternative only when the whole initial box lies in the region where both potentials coincide. Outside that certified region, alternative initial storage is unavailable. Invalid primitive declarations reject before calculation; unresolved sufficient inequalities remain explicit.

The whole local window uses the bound b=729*(3*epsilon+preparation_error+initial_coordinate_radius) and the strict edge margin pi_lower/12-2*b. A positive margin proves that the two complete flows agree for |tau|<=3, including every admitted perturbation. The two directional gate flags concern the nominal signed preparation; arbitrary perturbations need not preserve that symmetry or inherit its gate verdicts.

If the complete initial storage is strictly below the alternative seam cost 4, and the initial cycle branches are unique, conservation excludes a winding change for both time directions. This is a conditional all-time topological obstruction, not a trajectory, an all-time acute-identity theorem or a statement that the alternative admits no formation from other sources. The corresponding sine passage belongs to its unchanged retained evidence; this reader neither reruns nor authenticates that response.

Exact report projection uses tnfr.saddle-storage-discriminator.v1 through to_dict() and the shared SDK. sine_full_cube_passage_certified remains false: the nominal sine gates do not establish passage for every perturbed member, and this reader does not consume the separate frozen sine response. Source, prepared state, uncertainty, margins and separate claim flags remain visible. An export is not a checkpoint or evidence authentication. See the static example.

Same-source cubic-law acquisition obstruction

assess_sine_constitutive_robustness(evidence_directory) in research/sine_constitutive_robustness.py reads the fixed two-direction evidence bundle. It verifies retained file identity, re-admits the consumed complete-law/support/state premises and rebuilds the source winding, reference retention and storage observations. It does not run a producer, reconstruct an alternative source or integrate the changed law. File hashes establish byte identity, not mathematical truth or authentication of the historical execution; validated-reference soundness remains an explicit premise.

The comparison coefficient is fixed at eta=1/100 for j_eta(delta)=sin(delta)+eta*sin(delta)^3, with its own reciprocal storage, unchanged phase row, held unit capacities and all twenty coordinates. The source is exactly R Phi_sine(237,z), with R reversing form. Its enclosing endpoint box is used only to bound observations of that mapped family; it is not an independently prepared rectangular source set.

The original initial enclosure bounds conserved sine storage H0(z). The retained source phases bound the added potential A on all ten edges. Their sum H0+eta*A bounds the changed law's storage at the same source. The conditional barrier theorem gives B_eta=7/2+47*eta/24: strict H_eta<B_eta excludes entry into either acute unit-winding sector at any time from the source's zero winding. This is not permanent zero winding, loss of an already formed identity, nonformation from every source, or a physical selection of either law.

SineConstitutiveRobustness.to_dict() uses tnfr.sine-constitutive-robustness.v1; the generic SDK exporter preserves its exact rational evidence. Missing or inconsistent evidence cannot become a passing zero observation. The original frozen passed=false is preserved independently of the new obstruction. See the read-only example.

Exact signed involution reduction

assess_sine_involution_reduction(source, *, permutation_indices, sign=-1) in relational_sine_symmetry.py returns SineInvolutionReduction. It shares complete source, support, capacity and permutation admission with the cycle-symmetry reader below. The permutation must additionally be an involution and sign the exact integer -1 or +1, excluding Booleans. No group search is performed.

The fixed family obeys x[p(i)]=sign*x[i] and theta[p(i)]=sign*theta[i] in the declared real phase lifts. The negative sign reverses both state rows together; capacities keep their original sign. Every two-node orbit retains one coordinate per row. For sign -1, fixed nodes have zero form and phase; for sign +1, their coordinates remain free. The report exposes representatives, the exact reconstruction matrix and captured-source residuals. It does not implicitly recenter or wrap phases.

family_invariance_certified requires a full-support automorphism and preserved held capacities. source_membership_certified independently checks the actual captured form and phase. Both are needed for source_trajectory_reduction_certified; status="certified" alone describes the family, not membership. Nonpreserved support or capacity makes the family unavailable. Malformed primitives, signs or noninvolutive permutations reject.

report.evaluate(form_coordinates, phase_coordinates) rebuilds the report's primitive law and symmetry before reconstructing a complete detached state. It returns SineInvolutionState, retaining the shared full comparison, storage and representative rates in original structural time. It does not evolve or project the captured source. Cached favorable flags, matrices and residuals cannot replace admission. Reported residual intervals check arithmetic; exact row equality follows from equivariance, not interval overlap.

An arbitrary independent uncertainty box around a symmetric center need not lie in this family. No transverse stability, attraction or formation is implied. In particular, combined sign/reflection preserves winding and does not inherit the pure-reflection zero-winding obstruction below. This symmetry is distinct from conservative time reversal, which reverses form alone. Direct schemas are tnfr.sine-involution-reduction.v1 and tnfr.sine-involution-state.v1; shared SDK projection and atomic export apply. See the proof and workflow.

Conservative C5 saddle and its full tangent

assess_sine_c5_leaf_saddle(source, *, cycle) in relational_sine_resonance.py returns SineC5LeafSaddle. It reuses complete private-leaf admission: an ordered C5, exactly one private leaf per receiver, unit capacities, zero loss and unit exchange/storage coefficients. Extra edges, a different law or omitted environment reject. The captured source anchors support and law; it is not asserted to occupy the hypothetical saddle.

The target has zero form and receiver/leaf phases (i-2)/6 exact turns. Exact circular reconstruction checks its sine balance. Four cycle curvatures are 1/2, the closing curvature is -1/2, and contacts have curvature one. The full Laplacian, phase Hessian, tangent and quadratic energy use the same shared assembly as existing bridge assessments. Their clock is tau=t/pi.

The signed involution supplies an invariant odd reconstruction. This adapter orders its four coordinates positively at cycle positions 0 and 1 and their leaves, independently of source node order. It derives the reduced blocks theta'=A*x, x'=-B*theta from full matrices and verifies their intertwining identities. Full relative phase inertia is (8 positive, 1 negative, 0 zero); the conservative tangent has one growing/decaying real pair, eight oscillatory pairs and two full-state origin modes. Positive-loss stability verdicts are not reused at zero loss.

The characteristic polynomial is recomputed from the exact reduced matrix. Its sign variation and exact rational endpoint signs isolate the unique positive root; growth_rate_bounds bounds its square root. No floating eigensolver supplies a certificate. unstable_direction_bounds and its cycle-gap image enclose one correlated algebraic eigenvector defined by that root, not every independently selected member of the displayed intervals. The target remains at winding +1: crossing its 2*pi/3 protected-region boundary is distinct from reaching winding zero or an acute target.

The report is a static conditional assessment, not a solver, event or formation certificate. Schema tnfr.sine-c5-leaf-saddle.v1 supports shared SDK projection and atomic export. The proof and nonlinear remainder state how far a tangent comparison can legitimately be used.