Uniform relaxation accuracy of composed sine ports¶
Prospective all-time claim and frozen protocol¶
The degree-aware composition theorem bounds full-state approximation on a short window and protects actual identity for all subsequent times. The identity result alone does not bound the difference between the full and reduced trajectories. This gate asks for uniform tracking through the entire uninterrupted relaxation, with separate form and phase resolution tests.
Freeze the following before the first reserved assessment:
- Retain the three simple unit C9 components with classes \((1,2,1)\), central contacts \((0,1),(1,2)\), and original phase origins \((0,1/1000,0)\). Their port degrees after attachment are \((3,4,3)\). Retain unit capacities, beta one, \(e=1023/1024\), \(w=1/1024\), \(\tau=et\), and \(\gamma=1/(1023\pi)\). On the actual support, \(x'=-KLx+\gamma KS(\theta)\), \(\theta'=\gamma KLx\).
- Retain the original class-specific form sources \(k_i(2046/9)(355/113)^2(j-4)\), phase-flat sources at the stated origins, and independent per-coordinate form and continuous phase-lift errors \(10^{-10}\). Each component/channel error row has exact zero sum. Retain all original relative coordinates and each source storage ceiling \(10^9\); do not equate unequal class preparation costs.
- Freshly admit formation at \(\tau=100\), radius \(r=1/12\), and uninterrupted unprobed relaxation for \(D=10^{13}\). Require each component's actual form norm and target-phase error norm to be at most \(\epsilon=10^{-32}\), using exact \(2^{-512}\) only if \(512\le\kappa D\le4096\). Add both central unit contacts at \(\tau_c=100+D\), with no reset and total work allowance \(2\,10^{-6}\). Hold the support and complete law afterward.
- Use exactly the existing degree-aware thirty-coordinate surrogate: class tangent internal currents, exact nonlinear bridge currents, and nominal initial zero forms/deviations about the limiting twists and supplied origins. Compare with the actual fifty-four-coordinate fine trajectories. Retain both initial odd errors and subsequently generated odd modes. The observable is trajectory error on all \(0\le\tau-\tau_c<\infty\), not a new endpoint response.
- Prove the normalized even spectral gap candidate \(q=3/100\) by exact positive-semidefinite admission of the actual reduced degree metric and joined Laplacian. Use the independent C9 odd gap \(7/30\). Use the acute energy bootstrap, fixed curvature floor \(c=1/6\), parity-separated defects and mixed-coordinate invariant rectangles below. Preserve global weighted mean differences as explicit constant error floors. An isolated target-decay certificate cannot substitute for this trajectory comparison.
- Let \(\varphi=\max o_i-\min o_i=1/1000\). Require separately a uniform phase-coordinate error strictly below \(\varphi/2\) and a uniform form-coordinate error strictly below \(\gamma\varphi/2\). Use the outward lower endpoint of the latter allowance and strict outward lower margins. Do not combine the two channels into a norm that hides a poor form bound.
- Use the same proved mixed-coordinate invariant-rectangle method for both parity sectors and both output channels. No unsupported cancellation, fitted gain or enlarged allowance is admitted to force a joint pass.
- Preserve protocol, derivation and producing source before reserved evaluation. Keep exact rational primitives and fixed outward interval arithmetic. No trajectory solver, later-time sampling, horizon search, changed source/dwell, or precision scan is admitted. Report a joint certificate only when both channel tests and every prerequisite pass. A phase-only certificate remains explicitly phase-only, with its coarser form enclosure retained. Failure of a sufficient bound is not a counterexample to an actual trajectory or to the reduction.
This gate does not extend the previous \(2\,10^{-32}\) finite-window allowance to infinite time. It freezes different channel-specific resolution tests for a new time domain. The structural dwell, source accuracy and supplied law retain their previous nonempirical status.
Same complete trajectories, weighted means and two invariant charts¶
Use the fixed finite connected simple unit central-contact graph of the composition owner, with \(m\ge2\) components and classes in \(\{1,2\}\). Write \(\mathsf D=\operatorname{diag}(d_{\rm fine})\), \(K=\mathsf D^{-1}\), \(A=KL\), and \(\|v\|_{\mathsf D}^2=v^{\mathsf T}\mathsf Dv\). Then \(A\) is self-adjoint and nonnegative in this norm and \(\|A\|_{\mathsf D}\le2\). All orthogonal projections below use this metric. The structural clock and coordinate units are fixed.
Let \(\Theta\) be the aligned collection of centered class twists. For the full trajectory write \(u=\theta-\Theta\); for the lifted surrogate write \(\widetilde u=o+T\bar y\), with the block lift \(T\) and the supplied component origins restored. Its form is \(\widetilde x=T\bar x\). The surrogate begins at zero form and at these origins; the actual trajectories begin at their uninterrupted formation images. Their difference, rather than two separate distances to equilibrium, is the object to be bounded.
The actual degree-weighted common means of each flow are constant, but may differ between the flows. Put
The exact source zero sums and endpoint norm bounds imply
Subtract only these global constant differences when applying spectral estimates. Individual component means remain dynamical. Because common form and phase shifts leave both fields unchanged, this centering changes neither the difference dynamics nor their forcing. Restore (1) in each final raw-coordinate error bound.
Require the original full-law contact radius and storage tests to pass. Let \(E_H\) be the retained upper bound on actual initial full storage excess; let \(c_r=\cos(4\pi/9+\sqrt2r)>0\) be its acute-chart curvature lower bound. The ideal surrogate has initial internal storage zero and bridge storage \(\sum_E[1-\cos(o_j-o_i)]\le E_H\). Its quotient radius is no greater than the actual-family radius bound. In the same chart its internal quadratic curvature is \(c_i=\cos(2\pi k_i/9)\ge c_r\), and its bridge curvature is at least \(c_r\). The composition owner's full-graph gap and first-exit proof therefore also trap the surrogate, using its own exact storage balance. This is an additional argument about that flow, not a transfer of full-law identity based solely on proximity at one time.
Both storages remain at most \(E_H\) above their respective critical values. Their nonnegative form terms and phase coercivity give, for every future time,
Here edge differences are deviations from the aligned twist, with its fixed closing-edge integers. Taylor's linear terms cancel because the reference is critical for the full joined graph. The bridge reference gaps are zero. Both charts use continuous lifts; no wrapping of real deviations is performed during this argument.
Set \(\beta_*=(2E_H/c_r)^{1/2}\). Admit the sharpened fixed floor
Every edge deviation is bounded by \(\beta_*\), so (3) bounds the curvature along the reference-to-state segments for both flows and along each segment between their bridge gaps. Reapplying the phase energy estimate yields the useful edge budget
This is not a node-phase norm bound. In particular, it does not lose another factor of the slow intercomponent spectral gap.
Parity and independently admitted spectral gaps¶
Reflect each C9 by \(j\mapsto8-j\), fixing its central port. Let \(P_+,P_-\) be the blockwise even and odd projections. They are orthogonal in \(\mathsf D\), commute with \(A\), and preserve the global mean-free leaf. Every odd port value is zero. Contact incidence therefore annihilates odd vectors; the changed port degree has no effect on the odd block.
For each component the normalized odd diffusion block, in the four paired amplitudes, is the symmetric tridiagonal matrix with diagonal \((1,1,1,3/2)\) and off-diagonal entries \(-1/2\). Subtracting \(7I/30\) gives leading principal minors
all positive. Hence the uniform odd gap is \(\lambda_o=7/30\), independently of contacts and class.
The even gap must be admitted on the actual contact graph. Let \(\widehat L=T^{\mathsf T}LT\), \(M=T^{\mathsf T}\mathsf DT\), \(v=M\mathbf1\). A positive rational candidate \(q\le7/30\) is accepted only after the exact matrix inequality
It implies \(A\ge\lambda_e=q\) on the even global mean-free subspace. An unnormalized Laplacian gap cannot be inserted here unchanged. The frozen path uses \(q=3/100\); its finite rational PSD check is part of admission, not a fitted relaxation rate. An unsupported candidate must leave the corresponding all-time certificate unavailable.
Difference dynamics keep the nonlinear coupling in explicit defects¶
Extend the surrogate to the fine space only as a proof device, retaining the class-linear internal currents and exact bridge sine currents. Its even subspace is invariant. For any full state \(z\), split the difference as
The first bracket uses the exact normalized averaged phase Hessian \(B_t\), namely \(K\) times the averaged Euclidean phase Hessian. Its internal weights are the constants \(c_i\), while each bridge weight is the integral of cosine along the segment between its two phase gaps. This operator is self-adjoint in the degree metric. By (3),
All these operators commute with both parity projections. On each odd component \(B_t=c_iA\), although no commuting assumption is needed for the even estimate. Parity mixing remains in the second bracket of (6); the actual nonlinear sine field is not declared parity-invariant.
Let \(e=P_+u\), \(h=P_-u\). Orient the internal cycle edges consistently and use their fixed reference angle \(a=k_i2\pi/9\). Writing \(E_b=\Delta e\), \(H_b=\Delta h\) on an edge, reflection pairs its phase gap with the same \(H_b\) and opposite \(E_b\). The exact even and odd internal currents, after subtracting the constant reference current where necessary, are
The surrogate currents are \(\cos(a)E_b\) and \(\cos(a)H_b\). Their residuals satisfy globally
The odd inequality splits the sine remainder in \(H_b\) from \(\sin(a+H_b)(\cos E_b-1)\). The even inequality splits the cosine change in \(H_b\) from the cubic sine remainder in \(E_b\). Thus it retains both hidden odd initialization and generated odd motion.
The internal incidence map obeys \(\|K\partial_{\rm int}^{\mathsf T}r\|_{\mathsf D} \le\sqrt2\|r\|_2\), because its squared norm is the largest normalized internal Laplacian eigenvalue, at most two. Even and odd internal edge vectors are orthogonal, so (4) bounds the sum of their squared norms. Equations (8) imply the time-independent odd forcing bound
Once \(\|h(t)\|_{\mathsf D}\le O\) is established, use \(\|\Delta h\|_2\le\sqrt2O\), \(\max|E_b|\le B\), and \(\|\Delta e\|_2\le B\) to obtain
There is no circular closure: (9) follows from energy alone; it first gives \(O\), which then supplies (10). Bridge sine is exact and contributes no omitted-current defect.
A forced mixed-coordinate estimate with initial-state exchange¶
For either centered parity difference, equations (6)--(7) are
Set \(\eta=\gamma^2\), \(z=\gamma\delta x\), \(y=\delta u+z\). These are comparison coordinates, not new primitive state or a pressure reconstructed from response. They give
Suppose this parity sector has gap \(\lambda\), and forcing norm at most \(D\). With \(Y=\|y\|_{\mathsf D}\), \(Z=\|z\|_{\mathsf D}\), (7) yields the upper right-derivative inequalities
They hold also at zero norms by their integral or Dini-derivative form. This is an estimate of trajectory differences under forcing; the earlier Lyapunov decay toward a fixed target was not applied to their difference.
Use positive interval endpoints \(\gamma_-\le\gamma\le\gamma_+\). In slow comparison time \(s=\eta(\tau-\tau_c)\), set
Require \(\Delta_\lambda>0\). Then \(-C_\lambda\) has a positive inverse. If both initial centered form and phase norms are at most \(N\), define
The following rectangle retains possible initial exchange between channels:
Indeed, its vector is \(r=(y_0,z_0)^{\mathsf T}+(-C_\lambda)^{-1} [D\mathbf1+(C_\lambda(y_0,z_0)^{\mathsf T})_+]\). It dominates the initial vector and satisfies \(C_\lambda r+D\mathbf1 =C_\lambda(y_0,z_0)^{\mathsf T} -(C_\lambda(y_0,z_0)^{\mathsf T})_+\le0\). The positive comparison system therefore cannot exit this rectangle. Neither initial channel norm is assumed individually monotone.
In particular, for all subsequent times,
The second estimate may be conservative because its norm comparison does not retain every cancellation in the form response. It remains a valid enclosure; failure of a form-resolution test using it is not proof of a large actual form error.
All-time parity bounds and separately resolved output channels¶
Let \(d_{\max}\) be the maximum contact degree. The admitted endpoint norms imply the conservative initial bounds
Odd port values vanish, so the odd degree metric is exactly two times the fine Euclidean metric. For even errors the largest fine degree is \(2+d_{\max}\). Parity projection and removal of the global weighted mean are orthogonal contractions, so they need no additional initial error inflation.
Apply (13) with \((\lambda_o,D_o,N_o)\), writing its outputs \((Y_o,Z_o)\), and put \(O=Y_o+Z_o\). The lifted surrogate is even, so this bounds the full odd phase itself. Insert that \(O\) in (10) and apply (13) with \((q,D_e,N_e)\), giving \((Y_e,Z_e)\). Both determinant conditions and all earlier basin, curvature, spectral and source premises must be admitted.
Since parity is orthogonal in the degree metric and each fine degree is at least two, the raw-coordinate bounds are
They hold uniformly on \([\tau_c,\infty)\) for every actual admitted source member and every fine coordinate, relative to the same lifted surrogate. The mean floors in (1) have been restored exactly once. The argument does not compare independently chosen states on different mean leaves while silently setting their conserved differences to zero.
For fixed contact geometry and coefficients, \(E_H=O(\varphi^2+ \epsilon\varphi+\epsilon^2)\), \(D_o=O(E_H)\), and (13) yields \(O=O(E_H+\epsilon)\). Equation (10) is then \(O(E_H^{3/2}+\epsilon\sqrt{E_H})\). Thus the dominant all-time tracking error is of order \(\epsilon+\varphi^2\), rather than the order-\(\varphi\) sum of two unrelated distances to a target. The constants remain dependent on the admitted graph gaps and complete law. This order statement does not by itself certify either frozen channel allowance.
Reporting and stopping boundary¶
For positive origin span \(\varphi\), the report admits separate fractions \(f_\theta,f_x\in(0,1]\), with allowances \(f_\theta\varphi\) and \(f_x\gamma\varphi\). The frozen control sets both fractions to \(1/2\). The form comparison uses the outward lower endpoint of its allowance. Each strict margin must itself have positive outward lower endpoint. A zero origin span supplies no positive resolution allowance; it does not turn a missing comparison into a passing flag.
Use exact rational gap admission and conservative outward upper bounds through (1)--(16). In particular, \(B^2=12E_H\) remains exact when forming \(D_o\) and \(B^3=B\,B^2\); an upper square-root bound for \(B\) is sufficient. Use upper \(\sqrt2\) for defect factors and lower \(\sqrt2\), \(\gamma\) for output denominators. The API's finite component cap is an implementation budget, not a theorem about a maximal number of components.
The all-time theorem requires both invariant charts, the energy bootstrap, the correct mean leaf and the two positive comparison determinants. Its certificates distinguish unavailable premises, proved uniform bounds, phase resolution and form resolution. Passing only the phase test cannot be described as all-channel predictive accuracy or as resolving a small contact-form signature. A failed sufficient bound is not an observed failure of either trajectory. The finite-window composition and earlier contact-response protocols remain unchanged.
Retained qualified result of the unchanged protocol¶
The first reserved assessment was performed after preserving the
protocol, proof and producing source. It returned phase_only:
the all-time envelopes are admitted and the phase-resolution test passes,
while the form-resolution test and the frozen joint stopping rule are
false. No parameter, allowance, source or method was changed to turn
this qualified outcome into a joint certificate.
Fresh source handoff, full and surrogate trapping, supplied work,
exact normalized-gap admission, the refined curvature floor and both
comparison rectangles passed their prerequisites. The saved report has
no unavailable-premise reasons and retains the resolution limitation
form_resolution_not_certified.
| Channel | Uniform coordinate-error upper bound, approximately | Frozen allowance, approximately | Resolution |
|---|---|---|---|
| Phase | \(2.652351894767271\,10^{-4}\) | \(5\,10^{-4}\) | Certified for all subsequent uninterrupted times |
| Form | \(4.505933179946952\,10^{-6}\) | \(1.555766794642183\,10^{-7}\) | Not certified by this bound |
The exact outward phase-margin endpoints are
which are strictly positive. The form-margin endpoints are
which are both negative. The latter compares a conservative form-error upper bound with the allowance. It does not show that any actual trajectory exceeds the allowance, and it does not refute the surrogate. No trajectories were integrated or observed in this assessment.
The retained form_mean_error_floor and phase_mean_error_floor fields
equal \(\mu_*=4\epsilon/58\). They are permanent uncertainty
allowances for possible conserved actual-minus-nominal offsets, not
positive lower bounds on actual errors. Some admitted members have zero
offset. These terms were retained once in the raw-coordinate bounds.
The result establishes uniform phase tracking on the declared natural phase scale and an explicit, coarser all-time form enclosure. It does not certify the requested natural form scale, resolve an arbitrarily small contact-form signature, or extend the earlier short-window \(2\,10^{-32}\) allowance to infinite time. The source accuracy, uninterrupted dwell \(10^{13}\), complete law and support remain the same supplied mathematical premises, without an empirical clock or measurement bridge.
Retain the frozen protocol,
complete report,
producing source archive
and evidence manifest.
The source archive SHA-256 is
5ed129494818ad786d92d4e21fc53fd148646f98287e06aa0fb492db78066d48.
It preserves the pre-evaluation protocol and derivation; this retained-result
section was appended afterward. An exact read-only audit of the saved
report verified primitive inputs, parity coefficients and forcings,
invariant-rectangle inequalities, restored mean allowances, margin
endpoints and the separate stopping outcomes, without rerunning the
assessor.
The shared owner exposes assess_sine_port_relaxation and
SinePortRelaxation in
relational_sine_port_relaxation.py.
The contract
and guide
retain admission, availability and separate channel criteria. The
independent controls
cover the fine-state parity identities, spectral and energy premises,
initial-state exchange, mean restoration and strict reporting boundaries;
they do not substitute sampled times for the uniform proof.