Passive cycle relaxation with canonical delayed memory

Status: Exact conditional cycle relaxation derived from the existing REMESH history theorem; finite runtime correspondence is checked separately. Research links: B2.d.4/O3.a, S3, S8, S9 and S16.

1. Retained positive capacity removes the finite-clock obstruction

Use a fixed simple unit-conductance cycle with n>=3 nodes, scalar EPI x, uniform capacity nu>0 and a regular phase twist. Its adjacent circular gap is strictly below both the effective U3 gate and pi/2. Target-only all-node Coupling without functional links preserves this regular twist and uniform capacity. The topology, capacity and phase pressure gradients vanish; with the effective fully normalized EPI coefficient e>0, the refreshed canonical pressure and nodal flow are

\[ \Delta\mathrm{NFR}=-eLx,\qquad \dot x=-e\nu Lx, \qquad L=I-W/2. \]

Thus the flow uses the same canonical channels as joint support dynamics. No Silence attenuation or additional restoring law is inserted. Capacity persists, so repeating one fixed partition satisfying the exact Euler restriction 0 < e*nu*h_j <= 1/2 from section 2 supplies a common strict contraction on every delay circuit. Positive duration alone is insufficient for Euler stability; the continuous-flow result has its separate spectral bound. This removes the finite-clock explanation for EPI contrast in the preceding block; it does not itself prove a localized restoring mechanism.

The graph, coefficients, phase chart and capacities must remain fixed over the declared family. Grammar admission of a complete event word is a separate requirement. In particular, a registered REMESH word token is advisory and must not be identified with the separately executed delayed EPI map. The admitted bridge uses Coupling, Coherence and this advisory Recursivity token. Coherence has its own phase-lock action and is not generally inert in the phase sector: the regular strict twist is a fixed point of its exact neighbor-midpoint update. In this restricted preparation the bridge leaves EPI, capacity and phase unchanged before the flow; its direct pressure changes are replaced by the canonical refresh needed to identify the flow above. Runtime phase drift and pressure-realization residuals remain separate measured quantities.

2. A rational full-state flow gain comes from the cycle spectrum

Let bar(x) be the spatial arithmetic mean and E(x)=sum_i(x_i-bar(x))^2/2. A constant positive multiple of this metric gives the same gain. The cycle eigenvalues are lambda_j=1-cos(2*pi*j/n). In particular,

\[ \lambda_2=2\sin^2(\pi/n)\geq\ell_n:=\frac8{n^2}>0, \qquad \lambda_{\max}\leq2. \]

Here the subscript 2 denotes the first nonconstant eigenvalue. The inequality uses sin(t)>=2t/pi on [0,pi/2]; ell_n is a derived rational bound, not a fitted parameter or an asserted exact cycle eigenvalue.

For an exact refreshed Euler partition with positive segment durations h_j, put kappa_j=e*nu*h_j and require kappa_j<=1/2. Every modal factor lies in [0,1], and for each nonconstant mode it is at most 1-kappa_j*ell_n. Therefore the complete partition map S_E preserves the spatial mean and obeys

\[ E(S_E x)\leq q_E E(x),\qquad q_E=\prod_j(1-\kappa_j\ell_n)^2<1. \]

This bounds all spatial modes, including a local EPI bump; it is not a single-eigenmode fit. For the exact continuous flow over h=sum_j h_j, a separate rational upper bound is

\[ E(S_Cx)\leq e^{-2e\nu h\lambda_2}E(x) \leq q_CE(x),\qquad q_C=\frac1{1+2e\nu h\ell_n}<1. \]

The last step uses exp(-u)<=1/(1+u) for u>=0. The continuous and Euler maps have separate bounds; this comparison is not an Euler accuracy or binary64 convergence theorem. The restricted half-step condition avoids the nondecaying checkerboard boundary at kappa=1 on an even cycle.

3. The memory state contains pre-REMESH samples

The actual event/REMESH executor first completes its schedule and physical flow, appends the resulting EPI field to _epi_hist, and then applies the delayed map. A delay tau reads history[-(tau+1)] after that append. The post-REMESH output is not appended again.

Write y_k for this stored pre-REMESH EPI and x_k for its same-cycle post-REMESH result. Here y is a history label, not the shifted support coordinate of the preceding note. For one uniform alpha in [0,1],

\[ \begin{aligned} \beta&=(1-\alpha)^2,& \gamma&=\alpha(1-\alpha),&\delta&=\alpha,\\ x_k&=\beta y_k+\gamma y_{k-\tau_l}+\delta y_{k-\tau_g},\\ y_{k+1}&=Sx_k. \end{aligned} \]

The endpoint alpha=0 is a formal identity-memory control of the exact certificate. The actual delayed REMESH configuration requires positive alpha. A runtime memory-free control instead executes the same admitted word and flow while omitting the separate delayed-map call.

Coincident delays combine. Let m be the greatest delay with a positive coefficient and set H=m+1. Then Y_k=(y_k,...,y_{k-m}) is the active history. At alpha=0, m=0; at alpha=1, only tau_global is active. Older retained rows lie outside this active state. A startup buffer is declared initial data and is not automatically a past solution of the nodal equation. Insufficient-history no-op behavior lies outside the applied recurrence until the required samples exist.

This indexing matters. The schedule acts on the mixed head to produce the next stored pre-REMESH row. Storing post-REMESH states instead changes the history recurrence and cannot be substituted without a separate proof.

Nor can unchanged delayed rows be repeatedly reused as rolling memory. If their combined contribution is held at b, the pre-map recurrence is y_(k+1)=S*(beta*y_k+b). For a spatial mode with schedule factor 0<r<1, its stationary amplitude is r*b_mode/(1-beta*r), which can be nonzero. This is sustained by held input, not the advancing companion proved below. History advancement and its causal provenance distinguish these experiments.

4. Reuse of the sealed common-gain history theorem

Let P and its positive stationary row pi be the temporal companion and stationary distribution already constructed by remesh_history_stability.py. For the chronological energies in newest-first order,

\[ v_k=(E(y_k),\ldots,E(y_{k-m}))^\top,\qquad V_k=\pi^\top v_k. \]

Convexity of the centered quadratic energy and either gain q=q_E or q=q_C give the componentwise envelope

\[ v_{k+1}\leq D_qPv_k,\qquad D_q=\operatorname{diag}(q,1,\ldots,1). \]

This is exactly the order and convention of the existing policy certificate, whose proof is in the REMESH derivation. That certificate proves prefix gain at most one and

\[ V_{k+N}\leq q^{\lfloor N/H\rfloor}V_k. \]

No new temporal stability theorem is required: the cycle calculation supplies the previously conditional spatial schedule gain. Since q<1 and every pi_j>0, all active rows lose their nonuniform spatial component. The post-REMESH field also loses it, by the same convexity inequality. The physical pressure therefore tends to zero in this exact fixed family.

With a prepared row y_{-1}=x_initial and unit delays, the first executed flow creates y_0. The initial augmented energy is then formed from (y_0,y_{-1}); two successive executed cycles provide one transition from V_0 to V_1. Counting the initial preparation as an additional verified schedule transition would change this finite bound's provenance.

At alpha=1, the recurrence becomes y_{k+1}=S*y_{k-tau_global}: diffusion contracts each delay circuit once per tau_global+1 stored transitions. Pure-delay temporal permutation therefore does not protect nonuniform spatial structure when every circuit includes the positive-capacity flow. At alpha=0, H=1 and the certificate reduces to the ordinary memory-free schedule estimate.

The common strict gain cannot be replaced by total elapsed time alone. For alpha=1, tau_global=1, each mode obeys a_k=exp(-e*nu*lambda*h_k)*a_{k-2}. Taking h_{2j}=1 and h_{2j-1}=2^{-j} gives infinite total time, but the odd delay lineage receives only finite exposure and can retain nonzero contrast. The fixed partition assumed here excludes that control; a broader varying-duration theorem would need sufficient exposure along every active delay lineage.

5. Mean history, transients and the localization boundary

The result concerns spatial disagreement. The schedule preserves means, so historical means obey the scalar REMESH companion. If every initial active row has the same mean, every subsequent pre- and post-REMESH field has it, and spatial relaxation gives full consensus to that constant. Otherwise 0<alpha<1 mixes these means to their stationary history barycenter, while alpha=1 can retain a period dividing tau_global+1 among spatially uniform fields. That temporal effect is not localized structure. Zero pressure during a physical flow also does not forbid a subsequent named REMESH event from changing a uniform EPI level.

Even in the decaying regime, the current EPI energy need not decrease at each delayed map: older high-contrast rows can raise the current contrast. The nonincrease theorem applies to the augmented energy at stored-row boundaries. It does not justify multiplying a gain assigned only to the current REMESH input across an arbitrary history.

A simple comparison shows why memory can prolong a passive transient. Let a nonconstant mode have a positive schedule factor 0<r<1, and prepare every active startup amplitude equal to a_0>0. The exact scalar recurrence has nonnegative coefficients summing to one:

\[ a_{k+1}=r\left(\beta a_k+ \gamma a_{k-\tau_l}+\delta a_{k-\tau_g}\right). \]

Induction gives a_{k+1}<=a_k and a_{k+1}>=r*a_k; hence a_k>=r^k*a_0, the memory-free amplitude. The common-gain theorem still forces a_k->0. This comparison requires the specified positive mode and repeated startup preparation; arbitrary signed histories can interfere differently. It is not a claim that memory always slows every trajectory.

Thus uniform delayed memory plus retained positive-capacity diffusion does not sustain nonuniform EPI in this canonical cycle class. A persistent twisted phase pattern can coexist with spatially uniform EPI, but delayed echoes alone do not turn the passive EPI transient into a self-restoring localized entity. Moving structural support, nonlinear branch changes, clipping and more general graph/operator families require their own analysis rather than a transfer of this result.

6. Exact adapter and finite executor evidence

cycle_memory_relaxation.py derives ell_n, the complete Euler-partition gain and the continuous gain. It passes each gain to the existing sealed policy certificate while retaining their common sealed REMESH history model. Node count, uniform capacity, fully normalized EPI coefficient, positive physical partition, alpha and active delays remain explicit inputs. Exact rationals are preserved, and other supported real values use the shared materialized-real reader. The adapter introduces no second history recurrence or new transport kernel. Its declared physical partition repeats unchanged in the exact family; the history delays count stored cycle samples, not units of physical time.

Independent exact tests check local-bump gains across cycle sizes, the true pre-REMESH recurrence, finite augmented envelopes, coincident and inactive delays, common means, pure-delay damping, uniform temporal cycles and current-energy transients. They also verify that altered nested policy payloads lose their seal.

Runtime tests and the benchmark compare the same admitted Coupling/Coherence/advisory-Recursivity word and physical partition with and without the separately invoked delayed map. The memory case uses explicit unit delays and a prepared initial history row. Four executed cycles expose three adjacent pre-REMESH history transitions. The records separately retain actual channel states, applied delayed proposals, the ideal exact mixture, canonical pressure refresh and Euler arithmetic residuals. The regular phase twist, uniform positive capacity and fixed support must be checked in these records; they are not inferred from the exact companion's input vectors.

These finite observations connect the implementation to the restricted reference. They do not prove that future binary64 histories remain inside a uniform relative-defect class, that rounding preserves a positive contraction margin indefinitely, or that a generic adaptive operator schedule satisfies the same assumptions. Those remain distinct from the exact passive-relaxation theorem established here.

In the bounded default C8 run at physical T=2, all capacities stay at one and every accumulated nodal clock is exactly two. The initial centered energy 0.109375 becomes 0.07105253529907486 with delayed memory and 0.05179949153244585 in the advisory-only control. Winding remains one. The four observed augmented energies decrease, and both complete two-step block slacks against the conservative q_E≈0.9773016233624776 are positive. Clipping is inactive; the maximum measured REMESH rounding defect is 8.326672684688674e-17. These recorded finite values demonstrate delayed relaxation in this preparation; the asymptotic statement belongs to the separate exact model and its hypotheses.