REMESH Fixed-Delay Models and Runtime-Limit Boundary¶

Status: CORRECTED N15 HISTORICAL RECORD — restricted finite cyclic, companion, conditional policy, robust-defect and represented-number boundary results, including transactional per-invocation P2 EN/REMESH execution and finite certification; runtime limit and catalog completeness open Date: May 26, 2026 — corrected September 2026 Owner: theory/REMESH_INFINITY_DERIVATION.md Source implementations: src/tnfr/operators/remesh.py::apply_network_remesh, src/tnfr/physics/remesh_history_stability.py, src/tnfr/physics/remesh_schedule_policy_stability.py, src/tnfr/physics/remesh_schedule_policy_stability.pyi, src/tnfr/physics/remesh_schedule_relative_defect_stability.py, src/tnfr/physics/remesh_schedule_relative_defect_stability.pyi, src/tnfr/physics/binary64_remesh_relative_defect.py, src/tnfr/physics/binary64_remesh_relative_defect.pyi, src/tnfr/physics/binary64_p2_reception_stability.py, src/tnfr/physics/binary64_p2_reception_stability.pyi, src/tnfr/physics/runtime_p2_reception_stage.py, src/tnfr/physics/runtime_p2_reception_stage.pyi, src/tnfr/physics/runtime_p2_reception_remesh_sequence.py, src/tnfr/physics/runtime_p2_reception_remesh_sequence.pyi, src/tnfr/physics/runtime_p2_reception_remesh_policy.py, src/tnfr/physics/runtime_p2_reception_remesh_policy.pyi, src/tnfr/physics/runtime_remesh_history_stability.py, src/tnfr/physics/remesh_schedule_stability.py, src/tnfr/physics/runtime_remesh_schedule_stability.py, src/tnfr/operators/event_remesh_causal_runtime.py, src/tnfr/physics/runtime_remesh_schedule_block_margin.py, and src/tnfr/physics/runtime_remesh_schedule_relative_defect.py, with their typed interfaces where present, plus the finite reference-family certificate and typed interface in src/tnfr/physics/event_remesh_reference.py and src/tnfr/physics/event_remesh_reference.pyi.

Abstract¶

The historical N15 programme used the name \(\mathcal R_\infty\) for several different objects: a delay parameter limit, iterates of a history-advance map, and a Fourier projection. Those objects are not interchangeable.

Two exact finite results survive after separating them. On a finite cyclic history window, with fixed delays, fixed \(0<\alpha<1\), and no clipping, the filter

\[F=\beta I+\gamma S^{\tau_l}+\delta S^{\tau_g}\]

is a normal contraction because \(S\) is a unitary cyclic shift and \(\beta,\gamma,\delta>0\) sum to one. Its Cesàro averages converge to the orthogonal projection onto \(\ker(I-F)\). The fixed modes satisfy both delay conditions, so the fixed subspace consists of sequences whose periods divide \(\gcd(\tau_l,\tau_g)\).

For the distinct finite companion history recurrence, one fixed uniform \(\alpha\in[0,1]\), fixed delays, fixed ordered spatial support, one fixed positive diagonal metric and no clipping yield an exact stationary-weighted Jensen Lyapunov functional. It is nonincreasing for every history transition. When \(0<\alpha<1\), the companion is primitive and every spatial coordinate converges to its stationary history barycenter. At \(\alpha=1\), the companion is a permutation: the functional is conserved and periodic histories remain possible.

For a further conditional exact REMESH/schedule policy class, assume every possibly varying schedule preserves spatial consensus and has one common fixed-metric disagreement gain bound \(q\). The post-schedule history energy has prefix gain at most one and block gain at most \(q\) over the sufficient horizon \(L=\texttt{active\_max\_delay}+1\). Thus \(q<1\) gives uniform normalized margin \(1-q\) and repeated geometric decay of spatial disagreement. This theorem does not verify the schedule family or identify it with the binary64 runtime.

A robust conditional extension admits a bounded signed pre-schedule head defect. If \(J_k\) is the Jensen energy envelope of the active REMESH history, \(y_k\) is the ideal head, \(z_k\) is the bounded head and \(E_H(z_k)-E_H(y_k)\leq\eta J_k\), then the combined head gain is \(q_{\mathrm{eff}}=q(1+\eta)\). The same companion proof gives nonincrease for \(q_{\mathrm{eff}}\leq1\) and geometric spatial-disagreement decay for \(q_{\mathrm{eff}}<1\). A finite causal observer checks these inequalities on a recorded execution block. It does not prove that one uniform \(\eta\) is forward invariant for a runtime class.

The represented-number boundary is now exact at two complementary points. A pairwise reduction computes the optimal nonnegative relative defect for one admissible pair of binary64 temporal tuples. A normal-valued \(\alpha=1/2\) witness requires \(\eta=2^{210}-1/4\), so a bounded hard-clipped box alone is far too broad for the existing \(q=9/16\) policy. At \(\alpha=1\), every sufficient finite represented history inside a fixed hard-clipping interval instead has \(\eta=0\) and is forward invariant under REMESH alone. This boundary class does not by itself certify a schedule or its execution.

Between those two boundaries, the first useful restricted \(0<\alpha<1\) class is now exact. On ordered \(P_2\) support, fix any positive diagonal metric, positive delays, \(\alpha=1/2\), a symmetric hard interval \([-B,B]\) with \(B\geq4\,2^{-1074}\), and sufficient histories whose rows are \((a,-a)\). IEEE round-to-nearest-even and symmetric hard clipping preserve numeric antisymmetry. An analytic binary64 error bound for \(\sqrt D\geq11\,2^{-1074}\) plus exact enumeration of 6,615 integer-core triples, 3,890 admissible, below that boundary proves the sharp uniform bound \(\eta^*=135/124\). The maximizer is \((c,l,g)=(-3,-2,-3)2^{-1074}\). Hence strict robust composition requires \(q<124/259\); \(q=4/9\) gives \(q_{\mathrm{eff}}=259/279\) and margin \(20/279\), while equality at \(q=124/259\) gives only nonincrease. This forward-invariance claim concerns the REMESH map alone and excludes general metric-centered rows, fixed lattices, verified schedule families and complete runtime execution.

A first restricted global represented numeric EPI kernel closes the composition on \(P_2\). With mutual singleton neighbor sets, an immutable all-target Reception snapshot, the exact binary64 mix 0.5 and one common hard clamp, both target proposals are the same reversed sum of two half-scaled operands. They are numerically equal for every finite represented input pair in the source interval, so the global kernel gain is \(q=0\). Composed with the \(\alpha=1\), \(\eta=0\) REMESH class, the active post-schedule history has zero spatial disagreement after \(L=\tau_g+1\) cycles. This proves arbitrary finite repetition of the restricted numeric EPI kernels. One sealed adapter binds a single executed, grammar-admitted two-phase EN EPI stage and its captured endpoint to the first kernel. A second adapter binds each such stage and its same-cycle \(\alpha=1\) REMESH transition inside one completed graph-owned causal sequence. For an observed sequence of \(N\geq L\) cycles, it certifies extinction only on the active history suffix of length \(L\). It does not prove future or unobserved complete runtime repetition.

Neither base finite model is the complete clipped, history-gated runtime operation. They do not establish a literal \(\tau_g\to\infty\) limit, conserve the TNFR structural charge, make the full tetrad energy monotone, imply spatial consensus or \(\Delta\mathrm{NFR}=0\), or prove that the 13 registered operators exhaust all admissible TNFR transformations.

A separate effective-\(P_2\) reference family now connects three executed pressure-refreshed Euler meshes to the exact continuous nonuniform mode and to one unit-delay REMESH step. It supplies a rational finite-mesh error enclosure, strict improvement under the two declared proper subdivisions, exact ideal REMESH error scaling and a runtime residual bound. This finite family is not a general or binary64 asymptotic convergence theorem.

The sections below retain the N15 programme structure and commit anchors while recording the corrected statements.

§1. Runtime REMESH Semantics¶

§1.1 Exact raw update¶

For every node with sufficient history, apply_network_remesh computes

\[ x_{\rm raw} =(1-\alpha)^2x_0+\alpha(1-\alpha)x_l+\alpha x_g, \]

where \(x_0\) is the current EPI and \(x_l,x_g\) are snapshots at the configured local and global delays. With

\[ \beta=(1-\alpha)^2, \qquad \gamma=\alpha(1-\alpha), \qquad \delta=\alpha, \]

the coefficient identity is exact:

\[\beta+\gamma+\delta=1.\]

The runtime factor contract requires 0 < alpha <= 1. It rejects values outside that interval instead of clamping them. The exact-real coefficients represented by a validated alpha are therefore nonnegative and sum to one. The implementation evaluates the equivalent nested binary64 expression; the optional evidence reports its rounding residual from the exact-real affine value.

§1.2 Guards, shape and clipping¶

Observation transport. Uniform scalar head mixing commutes with a fixed linear spatial projection in exact arithmetic. On unit-weight P5, the reflection quotient ((0,4),(1,3),(2,)) also intertwines pure-EPI diffusion, so the two declared operations share an exact three-node quotient. Its stationary-history energy and the discarded fine energy give a Pythagorean split using the same companion weights. This is compatible composition, not an identification of temporal echo with eliminated-state memory: genuine past samples of a decaying nodal mode make the echo larger than the current amplitude, while the forward nodal amplitude is smaller. The proof, minimal-state scope and production rounding/clipping controls are centralized in Derived EPI memory, section 9.

Both delays are strict positive integers. The outer EPI history must be replayable and support indexed access. If it contains fewer than max(tau_l, tau_g) + 1 snapshots, the planner returns an immutable insufficient-history no-op and the executor changes no graph state. Empty live support returns the distinct empty_support no-op and emits no metadata, callback, history event or cooldown update.

Once the length guard passes, both selected lag entries must be mappings with support exactly equal to the live node support, and every selected EPI must lie in the real-scalar chart. There is no fallback from a missing historical node to its present EPI. The local and global labels denote two temporal lags for each node; the global-lag snapshot is not a network mean.

The public committed effect is

\[x_{\rm new}=\operatorname{structural\_clip}(x_{\rm raw}).\]

Every immutable node proposal retains both raw and bounded EPI and records whether clipping intervened. Hard and soft clipping can therefore be inspected without treating either as part of the raw affine recurrence. The operation reads the history but does not append or shift it; advancement belongs to the surrounding runtime.

The executor validates the complete proposal before writing and wraps EPI, topology, graph metadata, history, caches and capturable graph-owned callback state in one rollback boundary. A propagated commit, telemetry or strict callback failure restores that graph state. An external effect already emitted by a callback cannot be undone by a graph snapshot.

§1.3 Mean and disagreement statements¶

If all three scalar inputs are the same constant and clipping leaves that constant in range, the value is fixed:

\[x_0=x_l=x_g=c \quad\Longrightarrow\quad x_{\rm new}=c.\]

For network vectors and any declared positive diagonal weights h, the exact-real raw weighted mean obeys

\[ \mu_h(x_{\rm raw}) =\beta\mu_h(x_0)+\gamma\mu_h(x_l)+\delta\mu_h(x_g). \]

This is a mean-combination identity, not automatic preservation of the current mean. Let V_h be weighted disagreement from that mean. Convexity gives

\[ V_h(x_{\rm raw}) \leq \beta V_h(x_0)+\gamma V_h(x_l)+\delta V_h(x_g). \]

The right-hand side contains all three snapshots. If delayed histories remain free, it supplies no finite multiplicative gain relative to V_h(x_0) alone.

For a one-step map with the delayed vectors fixed, write b = gamma x_l + delta x_g. The raw map of x_0 preserves the consensus subspace exactly when b is uniform. Under that condition its disagreement gain is at most beta squared. If b is nonuniform, a consensus x_0 can be sent to positive disagreement and no finite global multiplicative gain exists. Hard clipping after a uniform-offset raw map is nonexpansive and retains the beta-squared bound in the exact-real scalar model. The soft knee is globally 4/3-Lipschitz, giving the sufficient bound (4 beta / 3) squared. A degenerate clipping interval is a constant map with zero gain.

The returned opt-in evidence keeps these universal fixed-history bounds apart from observed pre/post mean and disagreement values. It also reports binary64 rounding and clipping interventions separately. Exact rational diagnostics outside the finite binary64 reporting range produce an explicit domain error before the graph commit rather than an infinite or partial result. None of these one-step facts alone proves stability of the history-advance recurrence, pressure closure, structural-charge conservation, or U2 convergence. Section 2.4 derives a separate exact result only after the finite history shift, fixed coefficients and fixed metric are declared explicitly.

§2. Three Operators That Must Be Distinguished¶

§2.1 Runtime map \(M_{G,h}\)¶

The runtime map acts on a graph and a stored history, applies the guard and clipping, and uses configuration-dependent parameters. It is the canonical engine operation.

§2.2 Companion history advance \(T\)¶

A mathematical recurrence can insert the new value at the head of a history vector and shift all prior entries. This companion-style map is useful for studying an isolated recurrence, but apply_network_remesh does not perform that shift itself. Section 2.4 analyzes its finite, uniform, unclipped form without replacing it by a convolution.

§2.3 Finite cyclic filter \(F\)¶

On \(\mathbb C^n\), let \(S\) be the unitary cyclic shift and define

\[F=\beta I+\gamma S^{\tau_l}+\delta S^{\tau_g}.\]

This is a convolution filter on a periodic sample window. It is the model used by the corrected Fourier projector in src/tnfr/riemann/remesh_infinity_residue_split.py.

The spectrum of \(F\) does not describe the companion map \(T\) merely because both contain the same coefficients. The historical N15 derivation conflated these two constructions.

§2.4 Exact finite companion-history stability¶

Let \(m\) be the largest delay with a positive coefficient and define the newest-first augmented history

\[X_k=(x_k,x_{k-1},\ldots,x_{k-m}).\]

After combining coincident delays, write \(c_0=\beta\), \(c_{\tau_l}\mathrel{+}=\gamma\) and \(c_{\tau_g}\mathrel{+}=\delta\). The recurrence and shift are

\[ x_{k+1}=\sum_{r=0}^{m}c_r x_{k-r}, \qquad X_{k+1}=P X_k, \]

where the first row of \(P\) is \((c_0,\ldots,c_m)\) and its remaining rows shift the history. The coefficients are nonnegative and sum to one, so \(P\) is row stochastic. Its exact stationary distribution is

\[ D=1+\gamma\tau_l+\delta\tau_g, \qquad \pi_0=D^{-1}, \qquad \pi_j=\frac{\gamma\mathbf 1_{j\leq\tau_l} +\delta\mathbf 1_{j\leq\tau_g}}{D} \quad(1\leq j\leq m). \]

Fix a positive diagonal spatial metric \(H\) and let \(Q_H\) remove the \(H\)-weighted spatial mean. With \(E_H(x)=\tfrac12\lVert Q_Hx\rVert_H^2\), define

\[V(X_k)=\sum_{j=0}^{m}\pi_j E_H(x_{k-j}).\]

Stationarity of \(\pi\) and the quadratic Jensen identity give the exact balance

\[ V(X_k)-V(X_{k+1}) =\frac{\pi_0}{2} \sum_{a<b}c_a c_b \lVert Q_Hx_{k-a}-Q_Hx_{k-b}\rVert_H^2 \geq0. \]

Equality holds exactly when all active centered input fields agree pairwise. This is an augmented temporal disagreement functional. It does not control differences between spatially uniform history rows and therefore is not a strict Lyapunov function for the complete augmented state.

For \(0<\alpha<1\), \(c_0=\beta>0\) and the maximum-delay coefficient is positive. The finite companion is irreducible and aperiodic, hence primitive. It follows independently of strict decrease of \(V\) that every coordinate converges to the preserved stationary history barycenter \(\sum_j\pi_jx_{k-j}\). This is temporal pointwise convergence; it does not force different nodes to share one value.

At \(\alpha=0\), \(m=0\) and the map is the identity. At \(\alpha=1\), only the global-delay coefficient remains: \(P\) is a cyclic permutation of order \(\tau_g+1\), \(V\) is conserved and an individual orbit can have any period dividing that order. The alternating-history witness is therefore retained.

certify_uniform_remesh_history_stability materializes the exact companion and stationary measure. observe_uniform_remesh_history_transition checks one rational transition, its barycenter and the dissipation identity. Both APIs exclude binary64 runtime identification, clipping, changing coefficients, metric or support, and schedule/REMESH gain composition.

§2.5 One executed binary64 transition¶

observe_runtime_remesh_history_bridge closes the first runtime boundary for one applied, executor-sealed EventRemeshCycleResult. It reverses exactly the retained oldest-first runtime window required by the newest-first companion, rebuilds the exact theorem from the represented alpha and delays, and replays both the nested binary64 affine formula and the canonical clipping call bit for bit.

Let y be the exact companion head, b the rational value represented by the binary64 raw result, and z the rational value represented by the committed bounded result. The bridge retains the signed decomposition

\[ b=y+r_{\rm round},\qquad z=b+r_{\rm clip},\qquad z=y+r_{\rm round}+r_{\rm clip}. \]

Only the new history head changes relative to the ideal companion step. If \(\pi_0\) is the first stationary temporal weight, the exact lifted balances are

\[ \begin{aligned} V(X)-V(b,x_0,\ldots,x_{m-1}) &=D_J-\pi_0\bigl(E_H(b)-E_H(y)\bigr),\\ V(X)-V(z,x_0,\ldots,x_{m-1}) &=D_J-\pi_0\bigl(E_H(z)-E_H(y)\bigr). \end{aligned} \]

For a signed error \(e\), the bridge also evaluates the exact a posteriori bound

\[ \left|E_H(y+e)-E_H(y)\right| \leq \sum_i h_i\left|(Q_Hy)_i(Q_He)_i\right|+E_H(e). \]

Multiplying this bound by \(\pi_0\) gives a sufficient one-step Lyapunov margin. The signed defect remains separate because it can establish decrease when the absolute bound is conservative. Rounding and clipping are staged: the clipping defect is based at b, so the cross term between the two residuals is not lost.

Hard clipping on one common scalar interval is nonexpansive for every positive diagonal spatial metric. This follows from the pairwise identity

\[ E_H(x)=\frac{1}{2\sum_i h_i} \sum_{i<j}h_i h_j(x_i-x_j)^2 \]

and the 1-Lipschitz property of the scalar clamp. Soft clipping has no such nonincrease theorem: points in its Hermite knee can expand pairwise separation by almost 4/3, and disagreement by almost 16/9. The runtime bridge therefore records its exact observed defect rather than promoting the analytic ideal-map Lipschitz bound to the binary64 evaluator.

The result is deliberately a lifted companion observation. The current runtime appends the pre-REMESH state and does not immediately insert z into _epi_hist. Consequently, this bridge alone proves neither live history advance nor repeated runtime stability. It also records the exact stationary history-barycenter drift \(\pi_0r_{\rm round}\) or \(\pi_0(r_{\rm round}+r_{\rm clip})\); disagreement nonincrease does not imply preservation of that barycenter.

Exact diagnostics used by delayed-REMESH evidence now reject a nonzero rational that would underflow to displayed binary64 zero. This prevents a real rounding residual from being published as 0.0.

§2.6 Exact REMESH-head/schedule-head balance¶

The next algebraic step inserts a scheduled head s after the ideal, raw and bounded REMESH heads y, b and z. In one fixed positive diagonal metric, define the stationary-weighted defects

\[ \varepsilon_r=\pi_0(E_H(b)-E_H(y)),\qquad \varepsilon_c=\pi_0(E_H(z)-E_H(b)),\qquad \varepsilon_s=\pi_0(E_H(s)-E_H(z)). \]

Replacing only the newest history row gives the exact telescoping identity

\[ V(X)-V(s,x_0,\ldots,x_{m-1}) =D_J-\varepsilon_r-\varepsilon_c-\varepsilon_s. \]

Suppose a separately established represented schedule bound satisfies E_H(s) <= q E_H(z) with q >= 0. Its nonnegative slack is sigma = q E_H(z) - E_H(s), and therefore

\[ \begin{aligned} L &=D_J-\varepsilon_r-\varepsilon_c +\pi_0(1-q)E_H(z),\\ V(X)-V(s,x_0,\ldots,x_{m-1}) &=L+\pi_0\sigma \geq L. \end{aligned} \]

Thus L >= 0 is a sufficient finite-step nonincrease condition. A schedule gain q <= 1 is not sufficient by itself when binary64 rounding or a disagreement-expanding clipping map consumes the Jensen margin. Conversely, q > 1 need not force increase when the ideal Jensen dissipation pays for the scheduled expansion. The stationary-history barycenter changes exactly by pi[0] * (s - y).

observe_remesh_schedule_history_transition implements this identity as a sealed exact-rational observation. Its inputs are caller-supplied algebraic heads and a declared nonnegative gain bound that is verified against those heads; the result does not identify them with one executor trace, prove repetition or constrain future schedules. The stricter recorded-artifact binding is handled in §2.7.

§2.7 Adjacent recorded-cycle binding¶

observe_runtime_remesh_schedule_sequence binds the preceding pure identity to an ObservedEventRemeshCycleSequence. For every adjacent supplied pair it starts from the applied REMESH bridge of cycle i, then requires the represented EPI composition inside cycle i+1 to start at the bounded REMESH head z_i, end at its exact pre-REMESH head s_i, and expose a gain in the sequence's common normalized metric.

The decisive runtime-history check is

\[ X_{i+1}=(s_i,x_i,\ldots,x_{i-m+1}), \]

where the right side must equal the newest-first window reconstructed from the next cycle's sealed outgoing _epi_hist. Equal REMESH configuration fixes the same coefficients, delay width and clipping policy across every paired row. This certifies the recorded append identity between the supplied artifacts, which the isolated runtime bridge could only lift hypothetically.

For r adjacent boundaries in the same metric, exact history equality makes the intermediate augmented energies cancel:

\[ \sum_{i=0}^{r-1}\bigl(V(X_i)-V(X_{i+1})\bigr) =V(X_0)-V(X_r). \]

Each summand retains its own Jensen, raw, clipping and schedule defects. The finite total drop is also the sum of the gain-based lower bounds and the nonnegative augmented schedule-gain slacks \(\pi_0\sigma_i\). This is an additive balance; no fixed-history REMESH gain is multiplied across cycles.

The adapter identifies exact represented values in the supplied sealed cycle records. The underlying sequence remains a caller-ordered observation of individually atomic executions: it does not prove that the records came from one graph in causal succession, make the calls jointly atomic, identify a global binary64 map, or establish repeated or future stability.

§2.8 Effective-P2 three-mesh reference family¶

The generic three-mesh observer becomes runtime-linked on one deliberately small domain. Its diffusion component is exactly the two-node, homogeneous-capacity specialization of the reversible single-eigenmode Euler theorem, where the complete exponential, Euler, norm-bound, subdivision and conditional exact-real convergence proofs are centralized.

For the effective path \(P_2\), fixed positive conductance and homogeneous capacity \(\nu>0\) give

\[ L_{\rm rw}=\begin{pmatrix}1&-1\\-1&1\end{pmatrix}, \qquad u=(1,-1)^\mathsf T, \qquad A u=\lambda u,\quad \lambda=2\nu. \]

Thus a nonuniform initial field has the form

\[ x_0=m\mathbf 1+d u,\qquad d\ne0, \]

and the general theorem supplies

\[ x(T)=m\mathbf 1+d e^{-\lambda T}u, \qquad x_h(T)=m\mathbf 1+d p_h u, \qquad p_h=\prod_j(1-\lambda h_j). \]

The adapter requires each exact represented partition to have positive durations summing to the same \(T\), every segment to expose an intact pressure-refreshed exact-affine Euler identification, and \(0<\lambda h_j<1\). Its inherited finite bounds are

\[ 0\le e^{-\lambda T}-p_h \le \frac{\lambda^2}{2}\sum_j h_j^2 \le \frac{\lambda^2}{2}T h_{\max}, \qquad \lVert x(T)-x_h(T)\rVert_\infty =|d|\bigl(e^{-\lambda T}-p_h\bigr). \]

The implementation stores rational lower and upper enclosures of both errors and requires two successive proper positive subdivisions. It submits the three executor-owned physical partitions to the general runtime eigenmode observer once, then strengthens that result by requiring exact-affine identification, zero pressure-realization and held-input residuals, and zero local and endpoint defects in every row. The strict finite improvements follow from the cited general theorem. Its rational exponential routine restricts \(\lambda T\le4096\) solely to cap the integer-power exponent used by the enclosure. That restriction is not a dynamical threshold and does not bound the total bit size of arbitrary rational inputs. Although the pure theorem proves convergence for fixed-data admissible exact-real partition families with \(h_{\max}\to0\), this runtime adapter observes only three finite meshes and does not promote binary64 asymptotic convergence or solver order.

The REMESH part fixes \(\tau_l=\tau_g=1\) and requires the delayed row to be the exact initial field \(x_0\). With

\[ \beta=(1-\alpha)^2, \]

the ideal Euler-driven and continuous-reference REMESH heads are

\[ y_h=\beta x_h(T)+(1-\beta)x_0, \qquad y_*=\beta x(T)+(1-\beta)x_0. \]

Their error therefore scales exactly as

\[ \lVert y_*-y_h\rVert_\infty =\beta |d|\bigl(e^{-\lambda T}-p_h\bigr). \]

The strict subdivision improvement survives this ideal REMESH step when \(\beta>0\); at \(\alpha=1\), \(\beta=0\) and the ideal error is already zero. For the committed binary64 head, the executor-linked bridge supplies

\[ z_h=y_h+r_{\rm round}+r_{\rm clip}, \]

and hence the certified finite runtime bound is

\[ \lVert z_h-y_*\rVert_\infty \le \beta |d|\bigl(U_{\exp}-p_h\bigr) +\lVert r_{\rm round}+r_{\rm clip}\rVert_\infty, \]

where \(U_{\exp}\) is the rational upper enclosure of \(e^{-\lambda T}\). The runtime family requires hard clipping on one common nonempty scalar interval; soft clipping is rejected.

observe_p2_event_remesh_reference_family constructs the sealed family from three already executed cycles. Its mesh and family records preserve every exact premise, factor, enclosure, ideal scaling and runtime residual. It certifies neither solver order nor generic mesh or binary64 asymptotic convergence, arbitrary glyphs or mixed channels, changing support or metric, repetition, or future behavior. The executable 166_event_remesh_reference_family.py instantiates the scope with two, four and eight equal segments.

The separate pure 167_reversible_eigenmode_reference.py exercises both exact nonuniform modes of nonregular P3 without claiming a runtime or REMESH binding.

§2.9 Causal finite cycle-sequence execution¶

The algebraic telescope in §2.7 becomes causally identified for one finite invocation through execute_event_remesh_cycle_sequence. It accepts ordered EventRemeshCycleExecutionSpec values and executes every cycle on one graph inside one outer graph transaction. Each sealed CausalEventRemeshCycleReceipt retains its zero-based ordinal, exact submitted spec and EventRemeshCycleResult; the executed schedule is the schedule object carried by that spec. The sealed ExecutedEventRemeshCycleSequence always retains the receipts and the ordinary offline ObservedEventRemeshCycleSequence. By default, require_runtime_telescope=True also requires and retains the compatible RuntimeRemeshScheduleSequenceObservation. The explicit False branch always stores runtime_telescope=None, thereby admitting a valid word without one common affine schedule metric; telescope-specific properties then remain false.

Both branches of the outer wrapper prove same-invocation causal order, one graph identity and finite graph-owned atomicity. It does not alter the contracts of the nested offline observers when they are used separately. The finite additive energy identity is still not a global schedule-times-REMESH gain and has no uniform positive normalized block margin over a forward-invariant runtime class or runtime intrablock prefix-amplification bound. Solver accuracy/order, mesh convergence, repeated or future stability and rollback of emitted I/O, warnings, external resources or external-only aliases remain outside the result. In particular, the lag-one alpha=1 alternating history remains a valid causal counterexample to inferring convergence from provenance alone.

§2.10 Exact normalized margin on one causal finite block¶

Let \(B=(j_0,\ldots,j_0+r-1)\) be a nonempty contiguous block of boundaries in one intact ExecutedEventRemeshCycleSequence. Every boundary already satisfies

\[ D_j=V_j-V_{j+1}=K_j+S_j, \qquad S_j\geq0, \]

where \(K_j\) is its gain-based lower bound and \(S_j\) is its exact augmented schedule-gain slack. Exact intermediate-energy continuity gives

\[ D_B=V_{j_0}-V_{j_0+r} =K_B+S_B, \quad K_B=\sum_{j\in B}K_j, \quad S_B=\sum_{j\in B}S_j. \]

When \(V_{j_0}>0\), define the block-specific normalized lower margin

\[ \kappa_B=\frac{K_B}{V_{j_0}}. \]

Then the observed endpoint obeys the exact bound

\[ \frac{V_{j_0+r}}{V_{j_0}}\leq1-\kappa_B. \]

observe_executed_event_remesh_block_margin revalidates the causal source, runtime telescope and selected boundary objects by identity, and seals every term above. Its normalized fields are undefined when \(V_{j_0}=0\). The executable 170_runtime_remesh_block_margin.py has one finite block with \(\kappa_B=139/256\) and endpoint gain upper bound \(117/256\). Its separate lag-one alpha=1 orbit has \(\kappa_B=0\), so causal provenance alone supplies no positive margin.

There cannot be a positive uniform absolute drop over a class containing equilibrium or amplitude-scaled copies approaching it: this disagreement energy and every term in the balance scale quadratically, while equilibrium has zero drop. A repeated result would instead need a declared forward-invariant class with one uniform \(\kappa_*>0\) for every positive-energy block, together with a finite constant bounding energy amplification at each intrablock prefix. The finite observer proves neither condition.

§2.11 Uniform exact REMESH/schedule policy stability¶

The missing uniform margin and prefix bound can be proved for a restricted exact-model class. Fix the REMESH companion \(P\), its positive stationary row vector \(\pi\), the ordered spatial support and one positive spatial metric \(H\). Let every schedule map \(S_k\) preserve the spatial-consensus subspace and obey the same disagreement-energy estimate

\[ E_H(S_kx)\leq qE_H(x), \qquad 0\leq q\leq1. \]

The maps may vary with \(k\). The hypotheses, fixed metric and common \(q\) must hold for every member of the declared family. Index the augmented history immediately after each schedule. The next operation mixes those stored heads by REMESH, after which the next schedule acts on the new mixture. If

\[ e_k=(E_H(x_k),E_H(x_{k-1}),\ldots,E_H(x_{k-m}))^\mathsf T, \]

Jensen's inequality and the schedule bound give the componentwise envelope

\[ e_{k+1}\leq B_qe_k, \qquad B_q=D_qP, \qquad D_q=\operatorname{diag}(q,1,\ldots,1). \]

The order \(D_qP\) is essential: the schedule contracts the newly mixed head. Since \(0\leq B_q\leq P\) entrywise and both matrices are nonnegative, \(B_q^s\leq P^s\) for every \(s\geq0\). With

\[ V_k=\pi^\mathsf Te_k, \qquad \pi^\mathsf TP=\pi^\mathsf T, \]

every finite prefix is nonexpansive:

\[V_{k+s}\leq V_k.\]

Set \(L=m+1\). A length-\(L\) path through a companion matrix cannot remain in the shift rows: every such path visits the head at least once. Equivalently, the head-avoiding matrix \(B_0=D_0P\) satisfies \(B_0^L=0\). Every monomial contributing to \(B_q^L\) therefore contains at least one factor \(q\), so

\[ B_q^L\leq qP^L, \qquad \pi^\mathsf TB_q^L\leq q\pi^\mathsf T, \qquad V_{k+L}\leq qV_k. \]

Repeated blocks and their remaining prefix yield

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

Thus this exact class has intrablock amplification bound one and uniform normalized block-margin lower bound \(1-q\). If \(q<1\), the augmented spatial disagreement converges geometrically to zero. Positivity of every \(\pi_j\) then forces the spatial disagreement of every retained history row to vanish. This does not control differences among their spatially uniform temporal means.

The horizon \(L\) is a sufficient universal path bound. It is sharp for the worst-case pure-delay companion, but it need not be the smallest contractive horizon for a particular interior coefficient choice. The endpoint cases are also exact:

at \(\alpha=0\), \(L=1\) and the bound is the ordinary schedule gain \(q\);
at \(\alpha=1\) and \(q<1\), each spatial-disagreement component is damped once per pure-delay circuit, so the former periodic disagreement obstruction is removed;
at \(q=1\), the margin supplied by this theorem is zero. Identity schedules and a nonzero constant centered history retain the earlier counterexample, but the loose bound \(q=1\) alone does not prove that another concrete family fails to converge.

certify_uniform_remesh_schedule_policy_stability materializes \(P\), \(B_q\), the head-avoiding matrix and every rational power needed at the \(L\)-cycle block boundary, then rechecks the one-step induction hypotheses and all block and stationary-weight inequalities in its sealed result. Intermediate prefix powers are not retained. The API receives no schedule map, metric or runtime record; the common-gain and consensus-preservation statements are explicit hypotheses. It is therefore a repeated theorem for the declared exact family, not a promotion of the causal binary64 observer in §2.10. Such a promotion still needs a forward-invariant runtime class, fixed support and metric, and a derived uniform centered relative defect bound. Section §2.12 shows how such a bound enters the exact margin; absolute additive defect bounds can at most imply convergence to a neighborhood.

§2.12 Relative signed-defect envelope and finite causal verification¶

The common-\(q\) theorem can absorb a bounded discrepancy between the ideal REMESH head and the bounded head actually presented to the next schedule. For one transition, write

\[ J_k=\sum_d c_d E_H(x_{k-d}), \qquad y_k=\sum_d c_d x_{k-d}, \]

where the nonnegative active coefficients sum to one. Let \(z_k\) be the bounded binary64 head represented as an exact rational vector. The relevant quantity is the signed centered-energy defect

\[ \delta_k=E_H(z_k)-E_H(y_k). \]

It is not \(E_H(z_k-y_k)\) and it is not a norm of the state residual: the energy cross term prevents either substitution. Assume one finite \(\eta\geq0\) satisfies

\[ \delta_k\leq\eta J_k \]

for every transition in the declared fixed-support, fixed-metric family. Jensen gives \(E_H(y_k)\leq J_k\), hence

\[ E_H(z_k)\leq(1+\eta)J_k. \]

If the following consensus-preserving schedule has disagreement-energy gain at most \(q\), including on \(z_k\), then

\[ E_H(S_k z_k)\leq q(1+\eta)J_k =q_{\mathrm{eff}}J_k, \qquad q_{\mathrm{eff}}=q(1+\eta). \]

Therefore the proof in §2.11 applies verbatim with \(q_{\mathrm{eff}}\) in the head row:

\[ B_{q_{\mathrm{eff}}} =\operatorname{diag}(q_{\mathrm{eff}},1,\ldots,1)P, \qquad V_{k+n}\leq q_{\mathrm{eff}}^{\lfloor n/L\rfloor}V_k. \]

The exact certificate rejects \(q_{\mathrm{eff}}>1\). At \(q_{\mathrm{eff}}=1\) it proves prefix and block nonexpansion with zero margin. At \(q_{\mathrm{eff}}<1\) it proves normalized block-margin lower bound \(1-q_{\mathrm{eff}}\) and geometric decay of spatial disagreement in every retained history row. If \(q=0\), every finite \(\eta\) gives \(q_{\mathrm{eff}}=0\). These conclusions still leave spatially uniform temporal means uncontrolled.

The normalization by \(J_k\) is essential. Opposing active centered fields can cancel in \(y_k\), making \(E_H(y_k)=0\) while \(J_k>0\); a ratio against the ideal energy would then be singular even though a finite robust budget exists. If \(J_k=0\), Jensen forces \(E_H(y_k)=0\), and the declared inequality directly requires \(E_H(z_k)=0\). Implementations must check that algebraic condition and must not divide by zero.

certify_uniform_remesh_schedule_relative_defect_stability materializes this conditional theorem by reusing the sealed policy envelope, rather than maintaining a second matrix-power implementation. It keeps the schedule-only \(q\) separate from \(q_{\mathrm{eff}}\) and exact-rationalizes the caller-supplied \(\eta\).

observe_executed_event_remesh_relative_defect_block provides the finite causal bridge. For every selected adjacent boundary of one intact ExecutedEventRemeshCycleSequence that retained the compatible runtime telescope, it reconstructs \(J_k\), \(E_H(y_k)\), \(E_H(z_k)\) and \(\delta_k\); checks \(\delta_k\leq\eta J_k\) and the represented schedule gain \(q_k\leq q\); and verifies the complete post-schedule history-energy vector against \(B_{q_{\mathrm{eff}}}\) times its input vector. For \(N\) selected boundaries it checks the finite endpoint factor \(q_{\mathrm{eff}}^{\lfloor N/L\rfloor}\). A prefix shorter than \(L\) therefore retains factor one.

This observer establishes the inequalities only on the recorded causal block. It does not show that future heads remain in a class satisfying the same \(\eta\), or that support, metric, REMESH coefficients and schedule policy stay fixed. A repeated binary64 theorem still requires that forward-invariance argument. Solver accuracy and order, adaptive grammar, and full multichannel TNFR stability also remain outside scope.

§2.13 Pairwise binary64 boundary and two uniform classes¶

An EPI interval and hard clipping do not by themselves provide a useful uniform \(\eta\). This follows from an exact scalar reduction. For the temporal coefficients \(c=(\beta,\gamma,\delta)\), let \(a\) and \(b\) contain the current, local-delay and global-delay values at two spatial coordinates, and define

\[ D_c(a,b)=\sum_d c_d(a_d-b_d)^2. \]

Fix the represented \(\alpha\), the production evaluation order and one common hard clamp. Let \(g\) be the exact-rational convex combination and let \(\widehat g\) be that actual nested binary64 evaluation followed by the clamp. For a finite binary64 alphabet \(K=[\mathrm{EPI}_{\min},\mathrm{EPI}_{\max}]\cap\mathbb F_{64}\) and any finite set \(A\subseteq K^3\) of admissible nodewise temporal tuples, including the equality of local and global entries when the two delays coincide, define

\[ \eta_A^\star= \max_{a,b\in A,\,D_c(a,b)>0} \frac{\left[(\widehat g(a)-\widehat g(b))^2 -(g(a)-g(b))^2\right]_+}{D_c(a,b)}. \]

The weighted centered energy has the pairwise identity

\[ E_H(v)=\frac{1}{2\sum_i h_i} \sum_{i<j}h_i h_j(v_i-v_j)^2. \]

Applying the scalar inequality to every pair proves \(E_H(z)-E_H(y)\leq\eta_A^\star J\) for every finite support and positive diagonal metric whose nodewise tuples belong to \(A\). The constant is optimal for that tuple class because a two-node support realizes each admissible pair. If no pair has positive \(D_c\), set the maximum to zero. A common hard clamp cannot increase the pairwise difference. If \(D_c=0\), the two input tuples agree on every active coefficient; the deterministic evaluator gives equal outputs as numeric values, although an IEEE signed-zero bit may differ, and the branch closes without division. This characterizes a finite tuple class exactly, but it does not imply that its maximum is small enough for \(q(1+\eta_A^\star)<1\).

The obstruction already occurs with normal values and inactive clipping. Set \(\alpha=1/2\), use two equally weighted nodes, and take

\[ \begin{aligned} x_{0,1}=x_{0,2}&=2^{-52},\\ x_{g,1}=x_{g,2}&=1,\\ x_{l,1}&=2^{-105},\\ x_{l,2}&=2^{-105}+2^{-157}. \end{aligned} \]

All inputs lie in \([0,1]\). The exact head separation is \(2^{-159}\), whereas the nested runtime evaluation crosses two rounding-cell boundaries and returns a separation \(2^{-53}\). With the repository energy convention,

\[ J=2^{-318},\qquad E_H(y)=2^{-320},\qquad E_H(z)=2^{-108}, \]

so this one witness requires

\[ \eta\geq 2^{210}-\frac14. \]

Consequently, strict contraction would require \(q<4/(2^{212}+3)\approx2^{-210}\); the finite witness policy \(q=9/16\) cannot be promoted over the whole bounded binary64 box. The cause is structural: \(J\) removes spatially uniform modes, while floating-point evaluation is not translation equivariant. Large common temporal components can therefore move nearby coordinates across different rounding cells. Replacing \(J\) by the centered augmented energy \(V\) does not see those common components either.

There is one immediate positive class. Fix a nonempty ordered support, one positive spatial metric, two positive delays, and a sufficient represented history of at least \(\max(\tau_l,\tau_g)+1\) rows. At \(\alpha=1\), the exact recurrence is the pure global delay and the nested runtime expression returns the global delayed value numerically. If every retained value is finite and lies inside one fixed hard-clipping interval, the clamp is the identity, hence

\[ E_H(z)=E_H(y),\qquad \eta=0,\qquad q_{\mathrm{eff}}=q. \]

This REMESH-only step preserves that interval and the fixed support. IEEE signed zero may be canonicalized by the zero multiplications and addition, so the claim is equality as a real/binary64 numeric value and in exact-rational EPI, not preservation of the zero sign bit. This REMESH-only class supplies no schedule. Section §2.14 supplies one restricted global P2 numeric EPI-kernel family that preserves the interval and has common gain \(q=0\). Promoting that composition to repeated complete runtime execution still requires real-graph, stage, grammar and event provenance plus preservation of support, metric and configuration. A gain measured on a finite trace is insufficient.

observe_binary64_remesh_pair_relative_defect replays the shared production scalar kernel and records \(D_c\), both exact squared separations, the signed rounding and clipping defects, and the local \(\eta\) witness without promoting it to a uniform class bound. certify_alpha_one_hard_clip_remesh_class seals the fixed-support, fixed-metric, bounded-history REMESH-only class above. It checks represented-history membership and explicitly leaves schedule-family, repeated-execution and future-runtime claims false.

The antisymmetric \(P_2\) branch closes the first useful uniform class with \(0<\alpha<1\). Fix \(\alpha=1/2\), two ordered nodes, any positive diagonal metric, two positive delays and the symmetric hard interval \([-B,B]\), where \(B\geq4s\) and \(s=2^{-1074}\) is the least positive binary64 subnormal. Require every row of a sufficient represented history to have the form \((a,-a)\). Write \(H(x)=\operatorname{RN}(x/2)\) for binary64 halving and evaluate the production scalar kernel in its actual order:

\[ \begin{aligned} a&=H(c), & b&=H(l), & m&=\operatorname{RN}(a+b),\\ d&=H(m), & e&=H(g), & r&=\operatorname{RN}(d+e). \end{aligned} \]

The ideal amplitude is \(y=(c+l)/4+g/2\). Round-to-nearest-even is odd as a numeric map, up to irrelevant signed-zero representation, and the symmetric clamp is odd. The runtime output therefore remains \((r,-r)\) inside \([-B,B]\). This proves REMESH-only forward invariance of the represented history class.

The uniform defect bound is global over binary64 amplitudes, rather than an empirical sample. Put

\[ D=c^2+l^2+2g^2,\qquad x=\sqrt D,\qquad u=2^{-53}. \]

The pairwise Jensen denominator for \((c,-c)\), \((l,-l)\) and \((g,-g)\) is exactly \(D\), while the output signed defect is \(4(r^2-y^2)\). Binary64 halving has absolute error at most \(s/2\), and every finite rounded sum has error at most \(u|z|+s/2\). Propagating those six operations gives

\[ |r-y|\leq Ax+Cs,\qquad A=\frac{3u}{2}+\frac{u^2}{2},\qquad C=\frac94+\frac{9u}{4}+\frac{u^2}{2}. \]

The sums cannot overflow because each combines halved finite operands. Also \(|y|\leq x/2\) by the weighted Cauchy inequality. For \(x\geq11s\), let \(z=A+C/11\). Then \(z<21/100\) and

\[ \frac{4(r^2-y^2)}{D} \leq4(z+z^2) <\frac{2541}{2500} <\frac{135}{124}. \]

For \(0<x<11s\), all three inputs are integer multiples of \(s\), with \(|c/s|,|l/s|\leq10\) and \(|g/s|\leq7\). The executable proof enumerates all \(21\cdot21\cdot15=6615\) integer triples exactly. Of these, 3,890 satisfy \(0<D/s^2<121\). Applying the exact integer ties-to-even map \(h(n)=\operatorname{RN}(n/2)\) finds

\[ \max\frac{4(r^2-y^2)}{D}=\frac{135}{124} \]

at \((c,l,g)=(-3,-2,-3)s\). Here \(y=-11s/4\) and \(r=-4s\); the centered Jensen energy and defect are respectively \(31s^2/4\) and \(135s^2/16\). Since \(B\geq4s\), the witness and its runtime output belong to every declared class. The tail and finite core therefore prove that the optimal uniform constant is \(\eta^*=135/124\).

Composition with §2.12 is strict exactly when

\[ q<\frac{1}{1+\eta^*}=\frac{124}{259}. \]

For \(q=4/9\), \(q_{\mathrm{eff}}=259/279\) and the normalized block margin is \(20/279\). At \(q=124/259\), \(q_{\mathrm{eff}}=1\) and the theorem gives only nonincrease with zero margin. The prior \(q=9/16\) policy gives \(q_{\mathrm{eff}}=2331/1984>1\) and is rejected even on this class.

The antisymmetric hypothesis is essential. General centering in the positive metric \((1,3)\) permits rows \((3a,-a)\) whose exact centers vanish, yet one binary64 triple acquires runtime center \(-2^{-84}\). A fixed lattice is also not automatically closed: scalar inputs \((h,0,0)\) produce \(h/4\). These falsifiers exclude broader claims without excluding future classes with additional structure or quantization.

certify_half_alpha_antisymmetric_hard_clip_remesh_class stores the analytic constants, exact 6,615/3,890 core result, maximizer and sharp witness; checks strict represented-history membership; and composes its bound through the existing robust theorem. Its class-forward-invariance claim is limited to REMESH. Schedule verification, graph/event execution, repeated complete-runtime behavior, solver properties and future stability remain false. Example 178_half_alpha_antisymmetric_remesh_class.py reports both proof regions, the strict and zero-margin schedule boundaries and the two excluded generalizations.

For broader \(0<\alpha<1\) classes, viable later branches include a quantization-aware invariant alphabet, other symmetry representations with explicit guards, or an augmented defect budget. If only the lifted positive defect obeys \(U_k\leq\rho V_k\), the correct envelope is a rank-one update of the companion matrix and must be analyzed with its induced block factors; it cannot be replaced by the scalar \(q(1+\eta)\) formula.

§2.14 A global binary64 \(P_2\) half-Reception kernel family¶

The first restricted global numeric EPI-kernel family closes on the smallest connected support. Fix two ordered nodes with mutual singleton neighbor sets, the exact binary64 factor EN_mix=0.5, one immutable all-target Jacobi snapshot and one common hard-clipping interval. For an input pair \((x_1,x_2)\), the shared Reception kernel evaluates

\[ \begin{aligned} \widehat S_1(x)&= \operatorname{clip}\!\left(\operatorname{fl} (\operatorname{fl}(x_1/2)+\operatorname{fl}(x_2/2))\right),\\ \widehat S_2(x)&= \operatorname{clip}\!\left(\operatorname{fl} (\operatorname{fl}(x_2/2)+\operatorname{fl}(x_1/2))\right). \end{aligned} \]

A singleton fmean reproduces its nonzero operand; for zero it may normalize the sign without changing the numeric value. Multiplication by one half leaves both terms finite, IEEE addition is commutative, and the two raw outputs are therefore numerically equal. A common hard clamp preserves that equality and returns a finite value in the interval. Thus, for every positive diagonal spatial metric,

\[ E_H(\widehat S(x))=0, \qquad q=0. \]

This is a global statement over every finite represented pair in the declared interval; it is not a gain fitted to a finite trace. It remains true when underflow makes the numeric consensus value differ from the exact-real average. Consequently the binary64 kernel is not promoted to a global affine map. Signed-zero bits are also outside the theorem.

Compose this numeric EPI kernel with the \(\alpha=1\) class of §2.13. REMESH has \(\eta=0\), so \(q_{\mathrm{eff}}=0\). Each cycle inserts one consensus row into the post-schedule history. Since the only active delay is \(\tau_g\), the common policy theorem has

\[ L=\tau_g+1, \qquad V_{k+n}=0\quad\text{for every }n\geq L. \]

This proves arbitrary finite repetition and exact spatial-disagreement extinction for the restricted pair of numeric EPI kernels. It does not certify the complete Reception stage: operator preconditions, grammar replacement, semantic EPI-kind and history writes, callbacks, graph transactions and event ownership still require a causal execution binding. It also says nothing about solver accuracy, temporal consensus values or full TNFR stability.

The selected half mix is essential to this interval-independent commutative argument. Narrow or degenerate intervals may admit other factors. The canonical default represented factor is 0x1.ee7eea04ddca0p-3. For the adjacent normal values

x1 = 0x1.d2775ff0a1eccp-1021
x2 = 0x1.d2775ff0a1ecdp-1021,

the two default-factor outputs are respectively 0x1.d2775ff0a1eccp-1021 and 0x1.d2775ff0a1ecep-1021. Their separation is twice the input separation, so the centered-energy gain is exactly four. Any broad interval containing this inactive-clipping witness therefore admits no global default-factor bound \(q<1\).

certify_p2_half_reception_remesh_stability seals the fixed-support, fixed-metric composition by reusing the \(\alpha=1\) REMESH class, the common-\(q\) theorem and its relative-defect extension. Its pair evaluator calls the shared production mean, blend and clipping kernels. The result exposes the restricted global and finite-repetition claims while keeping complete-stage, grammar, live-execution and solver properties false.

§2.15 Finite causal binding of one executed \(P_2\) Reception stage¶

The abstract kernel can now be identified with one completed executor event. certify_executed_p2_half_reception_stage accepts one intact kernel certificate and one intact OperatorEventExecutionResult. It selects an executor-owned EN stage and requires same-invocation event identity, zero duration, exact two-phase Jacobi scheduling, two captured EPI endpoints and a deeply revalidated one-step neighbor-stage certificate. The observed targets must equal the ordered P2 support; runtime neighbor sets must be the two mutual singletons; the binary64 mix must be exactly one half; and the hard interval, fixed effective conductance, capacity and normalized diffusion-metric ray must match the source class.

The adapter then replays both coordinates through the shared production mean, blend and clipping kernel and requires bit identity with the accepted stage state and captured right endpoint. This proves for that completed EPI jump

\[ E_H(x^+)=0, \qquad q=0, \]

including underflow cases where the generic represented-affine stage bridge correctly abstains. Opposite signed-zero outputs remain valid numeric consensus. The result carries the finite grammar admission and graph-owned schedule atomicity of its source execution.

The underlying ExecutedGlyphStage now also carries one sealed ReceptionStageObservation for each ordered target. These observations bind the EPI and semantic-kind endpoints, neighbour inputs and optional tracked source record to the same owner-bound pre-EN snapshot, and verify their final commit after fallible callbacks. The P2 energy certificate does not yet consume that richer evidence as an auxiliary-state invariant.

This binding stops at one observed EPI stage. By itself it does not identify the source \(\alpha=1\) REMESH history or configuration with that graph, certify all Reception auxiliary state, preserve raw topology or current live-graph identity, or prove repeated/future runtime stability, solver accuracy or full TNFR stability.

§2.16 Finite causal binding of executed \(P_2\) Reception/REMESH cycles¶

certify_executed_p2_half_reception_remesh_sequence closes the finite same-graph boundary left open by §2.15. It accepts the global kernel certificate and one intact ExecutedEventRemeshCycleSequence. For each cycle it selects one executor-owned EN event, constructs the stage certificate of §2.15 and requires its two EPI endpoints to equal the complete schedule EPI endpoints. Thus the observed schedule EPI transition is the represented half-Reception kernel with \(q=0\), even when the schedule has no common affine metric and the causal executor was invoked with require_runtime_telescope=False.

The adapter also constructs a RuntimeRemeshHistoryBridgeObservation for every cycle, including the last. Each bridge must be identity-bound to that cycle and must identify an applied hard-clipped binary64 \(\alpha=1\) REMESH as the exact numeric copy of its selected global-delay row. The signed rounding and clipping residuals vanish, realizing the source class's \(\eta=0\) boundary. Ordered P2 support, the positive metric ray, interval, delays, history capacity and the runtime alpha-source provenance are fixed across the finite invocation.

Let \(L=\tau_g+1\) and write the outgoing runtime history chronologically. Only its last \(L\) rows are active in this pure-global-delay recurrence. Each cycle appends its consensus post-schedule row before REMESH, so after \(N\geq L\) observed cycles the final active suffix has zero spatial-disagreement energy in every row. The same-cycle bridges identify zero-energy post-REMESH fields from the horizon onward and at the final committed endpoint. Retained rows before the suffix may be stale or outside the source interval; no claim uses them.

This is a sealed certificate of one completed finite trace. It does not bind the current live graph after observation, audit auxiliary Reception writes, prove future or unobserved complete-runtime repetition, establish solver accuracy or extend the result to full TNFR stability.

§2.17 Transactional policy for each finite \(P_2\) invocation¶

execute_p2_half_reception_remesh_policy_invocation closes the operational gap between preparing a live graph and constructing the certificate of §2.16. Every call takes an outer graph snapshot before materializing caller-owned specifications or metric weights. Its read-only preflight requires the intact abstract kernel; the same ordered undirected mutual-singleton two-node support and positive diffusion-metric ray; positive capacities and symmetric conductance; exact binary64 EN_mix=0.5; matching hard-clipped \(\alpha=1\) controls, delays, interval and history capacity; an in-interval live EPI pair that is not identically zero; active global-delay history; and at least \(L=\tau_g+1\) distinct, clock-chained, one-cycle zero-flow EN -> IL -> REMESH specifications. This initial policy excludes physical flow partitions.

The policy then invokes the graph-owned causal executor with a live grammar context that rederives the U1a premise at every cycle start. A cycle-start pair that is identically zero rejects the next Reception. The policy requests no affine telescope and constructs, revalidates and identity-binds the existing finite certificate before the outer transaction can commit. Post-certification must leave the graph unchanged. Any preflight, execution, callback, observation, certificate or post-certification-read failure restores graph-owned state to the call boundary. Emitted I/O, warnings, external resources and aliases that exist only outside the graph remain subject to the executor's existing rollback boundary.

This mechanism permits successive calls, but each call is validated from its own live entry state and produces a distinct finite trace. It does not prove that the hypotheses will hold before a later unobserved call. Reception also writes auxiliary kind, source, operator-history and pattern telemetry; those fields are transactionally restored on failure, but their stability is not part of the centered-EPI theorem. The forward-invariant complete-runtime class, solver properties and full TNFR stability therefore remain open.

§3. The Literal \(\tau_g\to\infty\) Question¶

For a fixed finite _epi_hist, increasing \(\tau_g\) eventually triggers the insufficient-history guard, so the runtime call becomes a no-op. If the stored history grows with \(\tau_g\), then the domain, selected snapshot, deque length, and possibly the graph state change together. A nontrivial limit requires a declared common state space, an embedding of histories, parameter bounds, and a mode of convergence.

N15 supplied none of those data. Consequently:

the fixed-history pointwise behavior is eventually the guard-induced no-op;
a growing-history runtime limit remains undefined until an embedding and trajectory are specified;
neither behavior is the finite cyclic Cesàro projection of §7.

§4. Corrected Finite Cyclic Surrogate¶

Fix integers \(n,\tau_l,\tau_g>0\) with \(n\) divisible by both delays, and fix \(0<\alpha<1\). Let \(S\) be the cyclic shift on \(\mathbb C^n\). The DFT basis \(v_k(j)=n^{-1/2}e^{2\pi i kj/n}\) diagonalizes \(S\) and therefore \(F\).

For \(\omega_k=2\pi k/n\), the eigenvalue is

\[ \mu_k =\beta+\gamma e^{-i\omega_k\tau_l} +\delta e^{-i\omega_k\tau_g}. \]

This statement is exact for the finite cyclic filter. It is not a transfer function for the companion history-advance map.

§5. Contractivity and Power Boundedness¶

Because all three coefficients are positive and sum to one,

\[|\mu_k|\leq\beta+\gamma+\delta=1.\]

The matrix \(F\) is a polynomial in the unitary matrix \(S\), hence is normal. It follows that

\[\|F\|_2=\max_k|\mu_k|\leq1, \qquad \|F^m\|_2\leq1. \]

Power boundedness is therefore proved for this finite cyclic model. It was not proved for the historical infinite companion operator. In particular, for the old weight \(w(k)=\rho^{-k}\), the stated right-shift norm \(\sqrt\rho\) had the direction reversed; the norm is \(\rho^{-1/2}\) with that convention.

§6. Fixed Modes: GCD, Not LCM¶

For \(0<\alpha<1\), equality \(\mu_k=1\) in the convex combination requires

\[ e^{-i\omega_k\tau_l}=1, \qquad e^{-i\omega_k\tau_g}=1. \]

Let \(d=\gcd(\tau_l,\tau_g)\). On a compatible window, the common solutions are

\[ \omega_m=\frac{2\pi m}{d}, \qquad m=0,\ldots,d-1, \]

or DFT-bin indices \(k=mn/d\). The fixed subspace has dimension \(d\).

The historical use of \(\operatorname{lcm}(\tau_l,\tau_g)\) as the fixed-mode period was incorrect. The LCM can still be used as a convenient sample-window alignment, but it does not determine the common fixed modes. For the defaults \((\tau_l,\tau_g)=(4,8)\), \(d=4\), so the fixed frequencies are \(0,\pi/2,\pi,3\pi/2\) modulo \(2\pi\).

§7. Cesàro Projection Theorem for the Surrogate¶

Define

\[ A_M=\frac1M\sum_{j=0}^{M-1}F^j. \]

In the DFT basis, the multiplier of \(A_M\) is \(1\) when \(\mu_k=1\) and

\[ \frac{1-\mu_k^M}{M(1-\mu_k)} \]

otherwise. Since the model is finite-dimensional, these nonfixed multipliers converge to zero. Therefore

\[ A_M\xrightarrow[M\to\infty]{\|\cdot\|_2}P_d, \]

where \(P_d\) is the orthogonal projector onto \(\ker(I-F)\).

For a fixed window and fixed delays, an \(O(1/M)\) operator-norm bound follows with a constant depending on \(\min_{\mu_k\ne1}|1-\mu_k|\). No uniform constant follows as the window, delays, or coefficients vary. No rate for the nonlinear structural candidate energy follows from this state-space estimate.

§8. Projection Algebra¶

The finite surrogate projection satisfies

\[P_d^*=P_d, \qquad P_d^2=P_d, \qquad \|P_d\|_2=1\]

when its range is nonzero. Its spectrum is contained in \(\{0,1\}\); both values occur when the selected subspace is proper and nontrivial.

These are properties of \(P_d\) in the Euclidean norm on the declared periodic history window. They do not transfer automatically to graph observables or to the runtime map.

§9. Why the Historical \(H^2\) Proof Does Not Apply¶

The historical note represented the companion head-insertion plus shift as if it were multiplication by \(\beta+\gamma z^{\tau_l}+\delta z^{\tau_g}\). That symbol belongs to a translation-invariant convolution or cyclic filter, not to the companion map whose first row is updated while the remaining rows shift.

Additional problems were:

constant infinite histories do not belong to unweighted \(\ell^2\) or \(H^2\) coefficient space;
the claimed shift norm used the inverse geometric factor;
a spectral-radius bound was asserted without a valid spectrum formula;
power boundedness of the companion map was assumed rather than proved;
symmetrizing an operator does not preserve the spectra of all its powers.

The mean ergodic theorem is valid when its hypotheses hold. Those hypotheses were not established for the operator that the historical proof defined. The finite cyclic theorem in §7 supplies a valid restricted replacement.

§10. Relation to the Nodal Equation¶

REMESH changes EPI through a canonical operator path. The finite filter can be applied to a sampled EPI history as an auxiliary diagnostic. An identity such as

\[\partial_t(P_d\mathrm{EPI})=P_d(\partial_t\mathrm{EPI})\]

requires a common linear function space and sufficient regularity. Even when that commutation is valid, substituting the nodal equation only gives

\[\partial_t(P_d\mathrm{EPI})=P_d(\nu_f\Delta\mathrm{NFR}).\]

It does not show that the projected trajectory is produced by the runtime REMESH operator or that pressure and capacity close on the projected state.

§11. Structural Charge¶

The structural charge diagnostic is

\[Q=\sum_i(\Phi_s(i)+K_\phi(i)).\]

It depends on pressure, phase, topology, and the corresponding field extractions. An EPI-history projector does not act on all of these arguments. Exact preservation would require a specified lifted state map and an invariance relation such as \(Q\circ P_d=Q\) along a separately conserved evolution. Neither follows from \(P_d^2=P_d\) or \(P_d^*=P_d\).

Thus the historical projected-Noether conservation claim is superseded. Charge before and after REMESH remains trajectory telemetry unless a model-specific proof supplies the missing commutation and conservation laws.

§12. Energy and Isometry Boundary¶

The structural functional

\[ E=\frac12\sum_i (\Phi_s^2+|\nabla\phi|^2+K_\phi^2+J_\phi^2+J_{\Delta\mathrm{NFR}}^2)_i \]

is a nonnegative candidate energy. EPI is not an explicit term. Therefore an EPI-only write leaves a same-snapshot evaluation unchanged when every derived field is held fixed. This dependency fact is not an isometry and not a conservation theorem after pressure, currents, phase, or fields are updated.

For the finite surrogate, orthogonality gives

\[\|P_dx\|_2\leq\|x\|_2.\]

This contracts the declared history norm, not the structural functional \(E\). For \(E[P_dx]\leq E[x]\) one would need an \(E\)-compatible state space and contractivity in that energy metric. Monotonic decay along a trajectory needs an evolution law with a verified nonpositive derivative.

§13. Registry Reuse¶

P_d is a read-only numerical projection derived from a selected surrogate. It can be computed by a utility function without adding a class to the engine's operator registry. This establishes only implementation reuse:

\[\text{auxiliary computation} \not\Rightarrow \text{new registered state transformation}.\]

The legacy phrase “no fourteenth operator is required” is valid only in this narrow engineering sense.

§14. Catalog Completeness Remains Open¶

The current registry has 13 declared operators and coherent metadata. To prove that these operators exhaust all admissible TNFR transformations would require:

a transformation space defined independently of the existing names;
admissibility axioms derived from the nodal equation and invariants;
a representation or generation theorem for every admissible map;
an irreducibility or equivalence criterion for proposed new maps.

Registry size, reload idempotence, metadata alignment, and reuse of P_d do not supply those ingredients. This is the open S10 boundary in the theory catalog.

§15. Spectral Comparisons¶

For fixed \(n\) the projector has finitely many selected DFT modes. Calling their spacing a continuum spectral density adds an unsupported limit. Comparisons with Riemann-zero density, Kolmogorov spatial spectra, or random-matrix spacing laws can be posed only after specifying a scaling family and an intertwining map between observables.

The corrected fixed-mode calculation is useful as a finite periodicity test. It proves no spectral universality and no mismatch theorem about the full runtime operation.

§16. P50 Finite Fourier Diagnostic¶

build_resonant_bin_mask retains the historical requirement that the sample count be divisible by \(\operatorname{lcm}(\tau_l,\tau_g)\). Within that window, it now selects the bins fixed by both delays, using \(d=\gcd(\tau_l,\tau_g)\).

split_residue_by_remesh_infinity is a legacy API name. It returns \(P_dx\) and \((I-P_d)x\) using an orthogonal DFT mask. Parseval gives, up to roundoff,

\[ \frac{\|P_dx\|_2^2}{\|x\|_2^2} +\frac{\|(I-P_d)x\|_2^2}{\|x\|_2^2}=1. \]

This identity validates the numerical split. It does not identify the analytic support of an off-grid signal: a finite rectangular window creates spectral leakage.

§17. TNFR-Riemann Boundary¶

The P31 prime-ladder signal can be projected onto the finite periodic subspace and its complement. A small selected fraction means only that the declared finite sample has little DFT energy in those bins. No equality relates this split to the smooth and oscillatory parts of the Riemann counting formula.

Consequently N15 and P50 do not advance T-HP or RH. Historical B1/B2 language is retained only as programme history, not as a theorem implied by the Fourier certificate.

§18. Navier–Stokes Boundary¶

The finite surrogate acts on a one-dimensional sample index. A temporal projection does not by itself produce, preserve, or rule out a spatial \(k^{-5/3}\) spectrum. Any statement about a Navier–Stokes field requires a declared space-time tensor product and proof that the temporal map acts as the identity on the spatial factor.

No conclusion about vortex stretching, regularity, or a cascade follows from the REMESH surrogate.

§19. Historical Claim Ledger¶

Historical N15 claim	Current status
\(\tau_g\to\infty\) equals a Cesàro projector	Superseded: these are different limits
Companion history map has the polynomial Fourier symbol	Superseded: the symbol belongs to the cyclic/convolution filter
Fixed modes use \(\operatorname{lcm}(\tau_l,\tau_g)\)	Corrected to \(\gcd(\tau_l,\tau_g)\) for \(0<\alpha<1\)
The history operator is power bounded on the stated \(H^2\) model	Unproved for that companion map
No finite companion-history Lyapunov functional is available	Corrected in the fixed uniform, unclipped, finite-dimensional scope of §2.4
Projected structural charge is conserved	Conditional and unproved
Projected structural energy is monotone with universal \(O(1/n)\) decay	Superseded
The surrogate closes the 13-operator catalog	Superseded; registry reuse only
Direct Riemann/K41/RMT verdicts apply to runtime REMESH	Superseded; finite-surrogate comparisons only

The original commits remain useful provenance:

a1f298fd: historical W1 operator-existence claim;
badac156: historical W2 conservation and Lyapunov claim;
48b0574a: historical W3 spectral and branch verdict.

§20. Corrected Branch Status¶

Branch A: established only for the finite cyclic fixed-delay surrogate: its Cesàro projector exists.
Finite companion branch: the stationary-weighted disagreement functional of §2.4 is nonincreasing; strict temporal mixing holds only for \(0<\alpha<1\), while \(\alpha=1\) retains periodic orbits.
Exact policy branch: under the common fixed-metric schedule-gain hypothesis of §2.11, prefix gain is at most one and q<1 gives uniform geometric spatial-disagreement decay, including at \(\alpha=1\).
Relative-defect branch: under the uniform signed bound of §2.12, the effective factor is q_eff=q*(1+eta); a causal adapter verifies that bound and its vector envelope on finite executed blocks, but not future invariance.
Represented-number boundary: §2.13 gives the exact pairwise reduction, a bounded normal-valued obstruction at \(\alpha=1/2\), and the forward-invariant hard-clipped REMESH-only class at \(\alpha=1\) with \(\eta=0\). It also proves the first useful intermediate class: antisymmetric \(P_2\) histories at \(\alpha=1/2\) have sharp uniform \(\eta=135/124\) and strict robust threshold \(q<124/259\) under REMESH alone.
Binary64 \(P_2\) kernel branch: the configured half-Reception Jacobi kernel has global gain \(q=0\) and composes with that \(\alpha=1\) class to extinguish active-history disagreement after \(\tau_g+1\) restricted cycles. One sealed executor event binds this kernel to a real two-phase EN EPI stage and its captured endpoint. A finite sequence adapter additionally binds every selected EN and same-cycle REMESH, verifies the active suffix of length \(\tau_g+1\), and records post-horizon extinction on one executor-owned graph. A transactional policy now revalidates and executes this restricted protocol per call and post-certifies before commit. Future or unobserved repeated complete-runtime composition remains open.
Branch B1: no universality conclusion follows without a scaling family and an intertwining map.
Branch B2: no extra registry entry is needed to compute this projection; admissible-transformation completeness remains open.
Branch B3: nonexistence is ruled out for the finite cyclic surrogate, not for the literal runtime \(\tau_g\to\infty\) problem or the historical infinite companion map.

§21. Scope Across TNFR Programs¶

The corrected results are internal and limited:

it supplies a finite periodic-history diagnostic;
it supplies an exact augmented-history disagreement balance and temporal limit for one fixed uniform unclipped companion recurrence;
it supplies a conditional exact common-\(q\) schedule-family theorem with a uniform normalized margin and prefix bound;
it supplies a conditional signed relative-defect extension and a finite causal verifier for its complete energy-vector envelope;
it characterizes pairwise represented REMESH defects and certifies the sufficient-history \(\alpha=1\) hard-clip class under REMESH alone;
it proves REMESH-only forward invariance and sharp uniform \(\eta=135/124\) for the \(\alpha=1/2\) antisymmetric \(P_2\) hard-clip class by an analytic IEEE tail bound and an exact 6,615/3,890 finite-core enumeration;
it composes that class with the global \(P_2\) half-Reception numeric kernel and obtains exact finite-time spatial-disagreement extinction for the restricted repeated binary64 recurrence;
it binds one completed graph-owned EN event and its captured EPI endpoints to that global \(P_2\) kernel by exact binary64 replay;
it binds every selected EN and \(\alpha=1\) REMESH in one finite graph-owned P2 sequence and verifies extinction on the active suffix, without promoting the observation to future repetition;
it supplies a reusable outer transaction that preflights, executes and post-certifies every finite P2 invocation independently;
it separates an auxiliary linear model from the canonical clipped runtime;
it corrects the fixed-mode arithmetic from LCM to GCD;
it leaves all classical open problems unchanged;
it leaves the S10 catalog-completeness problem open.

No “structural/operational universality” follows across arbitrary graphs, initial states, clipping policies, or time-varying parameters.

§22. Reproducibility and Direct Checks¶

The corrected numerical surface is src/tnfr/riemann/remesh_infinity_residue_split.py. For a compatible sample count \(n\):

the selected bin count is \(d=\gcd(\tau_l,\tau_g)\);
the selected indices are multiples of \(n/d\);
applying the split twice leaves each part unchanged;
the two parts reconstruct the input up to FFT roundoff;
their inner product vanishes up to FFT roundoff;
their squared-norm fractions sum to one for nonzero input.

Canonical catalog checks belong to operator_contracts.verify_contract_consistency() and its tests. They check declared metadata consistency, not completeness of the possible dynamics.

The finite companion theorem and its one-transition identity are exercised by tests/physics/test_remesh_history_stability.py. The executed residual bridge, pure schedule balance and adjacent runtime telescope are checked by tests/physics/test_runtime_remesh_history_stability.py, tests/physics/test_remesh_schedule_stability.py, and tests/physics/test_runtime_remesh_schedule_stability.py. Together they verify the exact stationary distribution, Jensen dissipation, equality case, preserved ideal-history barycenter, signed runtime defects, recorded history advance, finite additive telescoping and the distinct \(\alpha=0\), \(\alpha=1\) and \(0<\alpha<1\) regimes.

The exact common-\(q\) policy theorem and its public example are checked by test_remesh_schedule_policy_stability.py and test_remesh_schedule_policy_stability_example.py. They verify the D_q P order, exact block powers, varying rational schedule witnesses, prefix and repeated bounds, endpoint regimes, public facade and fail-closed seals. A separate consensus-amplification witness confirms that the result controls spatial disagreement rather than uniform temporal means.

The relative-defect theorem, causal adapter and public example are checked by test_remesh_schedule_relative_defect_stability.py, test_runtime_remesh_schedule_relative_defect.py and test_runtime_remesh_schedule_relative_defect_example.py. They verify exact q_eff, cancellation-safe normalization by J, the J=0 case without division, schedule-gain and componentwise envelope inequalities, complete-block endpoint factors, a positive binary64 defect at its exact minimum eta, causal identity and fail-closed seals.

The represented-number boundary and its public example are checked by test_binary64_remesh_relative_defect.py, test_delayed_remesh_contract.py and test_binary64_remesh_relative_defect_example.py. They verify shared-kernel replay, the exact \(\alpha=1/2\) obstruction, the zero-denominator branch, hard-clamp nonexpansiveness, signed-zero scope, the \(\alpha=1\) class at endpoints and subnormals, public type boundaries and fail-closed proof seals.

The restricted half-alpha class and example are checked by test_half_alpha_antisymmetric_remesh_class.py and test_half_alpha_antisymmetric_remesh_class_example.py. They verify the exact analytic-tail constants, exhaustive 6,615-candidate and 3,890-admissible core, sharp subnormal witness, normal/subnormal/signed-zero history membership, REMESH forward invariance, strict and zero-margin schedule boundaries, q=9/16 rejection, excluded metric-centered and fixed-lattice generalizations, public facade and fail-closed proof fields. Example 178_half_alpha_antisymmetric_remesh_class.py records the same proof boundary.

The global P2 half-Reception composition and its public example are checked by test_binary64_p2_reception_stability.py and test_binary64_p2_reception_stability_example.py. They cover the exact q=eta=q_eff=0 composition, finite extinction horizon, maximum finite values, subnormals, noncentered and degenerate intervals, numeric-but-not-bitwise signed-zero consensus, hostile reseals and the canonical default-factor energy-gain-four counterexample.

The finite executor binding and its public example are checked by test_runtime_p2_reception_stage.py and test_runtime_p2_reception_stage_example.py. They verify same-invocation event/stage identity, grammar-aware two-phase EN, runtime P2 neighbors, exact half mix, hard interval, capacity/conductance and metric continuity, bit-exact endpoint replay, underflow and signed-zero scope, nested proof revalidation and explicit negative future-runtime claims.

The finite causal P2 EN/REMESH sequence is checked by test_runtime_p2_reception_remesh_sequence.py. It covers same-cycle stage/REMESH identity, fixed support, metric and configuration provenance, the active suffix of length \(\tau_g+1\), observed post-horizon extinction and fail-closed broader claims. Its public facade, stub and example are checked by test_runtime_p2_reception_remesh_sequence_example.py. Example 176_runtime_p2_reception_remesh_sequence.py records the finite same-invocation witness.

The transactional per-invocation policy is checked by test_runtime_p2_reception_remesh_policy.py. It covers successful and successive calls, exact static preconditions, active history scope, live U1a derivation, directed preflight rejection and complete rollback when execution or post-certification fails or mutates the graph. Its public facade, stub and example are checked by test_runtime_p2_reception_remesh_policy_example.py. Example 177_runtime_p2_reception_remesh_policy.py records two independently validated finite invocations.

The graph-owned finite causal wrapper and public example are checked by test_event_remesh_causal_runtime.py and test_event_remesh_causal_runtime_example.py. They verify receipt/spec/schedule identity, same-invocation graph provenance, outer rollback and the explicit negative repeated-stability scope.

The finite causal block-margin observer and its public example are checked by test_runtime_remesh_schedule_block_margin.py and test_runtime_remesh_schedule_block_margin_example.py. They verify exact contiguous telescoping, normalized fields, identity binding, zero-energy abstention, kappa=139/256, the alpha=1 zero-margin boundary and the explicit false uniform-class scope.

The effective-P2 family and its public example are checked by test_event_remesh_reference.py and test_event_remesh_reference_example.py. They verify the exact Euler recurrence, rational enclosures, strict proper-subdivision inequalities, ideal REMESH scaling, runtime residual bound, public stub and explicit negative scope properties. The general pure theorem and nonregular-P3 example are checked separately by test_reversible_eigenmode_reference.py and test_reversible_eigenmode_reference_example.py. They verify the reversible metric, exact eigenmode identity, exponential and Euler enclosures, both norm bounds, proper subdivision, public stub and the conditional exact-real convergence scope without promoting a REMESH bridge.

§23. Open Research Questions¶

The following problems remain open:

Define a common state space and convergence mode for a nontrivial runtime \(\tau_g\to\infty\) limit.
Extend the \(\alpha=1/2\) antisymmetric \(P_2\) class to broader represented states, support or other \(0<\alpha<1\) coefficients while retaining a useful uniform eta and q_eff<1; retain the common-mode, canonical-factor, zero-margin, scaling and soft-knee obstructions.
Promote the finite same-graph \(P_2\) EN/REMESH observation to a forward-invariant class under future complete grammar-aware event execution, including the auxiliary Reception state; or extend the numeric result to a broader verified kernel family without replacing class-wide proof by finite traces.
Determine the extra hypotheses needed to compose the implemented general finite executor/eigenmode binding with REMESH beyond effective \(P_2\); its pressure, held-input and full-matrix endpoint defects currently stop at the pre-REMESH boundary.
Extend the companion result to changing \(\alpha\), metric, delays or node support, or produce counterexamples.
Determine when a lifted REMESH map preserves a declared structural charge or dissipates the full tetrad energy.
Establish any bridge from the finite periodic projection to the Riemann, Navier–Stokes, or other programme observables.
Define the admissible TNFR transformation space independently and resolve catalog generation and irreducibility within it.

The exact current conclusion is therefore narrow: a fixed finite cyclic REMESH filter has a computable Cesàro fixed-mode projection, and a distinct fixed finite companion recurrence has a nonincreasing augmented disagreement functional with sharply separated mixing and pure-delay regimes. One applied binary64 transition is decomposed into exact ideal, rounding and clipping terms, and compatible adjacent recorded cycles have an exact finite additive schedule/history telescope. One graph-owned outer executor supplies a compatible requested telescope with same-invocation causal provenance and finite atomicity; its explicit no-telescope branch retains the finite causal trace when the schedule has no common affine metric. A sealed observer extracts exact normalized lower margins from its contiguous finite blocks; the implemented witnesses give 139/256 and zero without establishing uniform runtime-class coercivity. A separate conditional exact common-\(q\) policy theorem gives prefix gain upper bound one, uniform normalized margin \(1-q\) and repeated geometric spatial-disagreement decay on the fixed companion. Its conditional relative-defect extension replaces \(q\) by \(q_{\mathrm{eff}}=q(1+\eta)\), and a finite causal observer verifies every signed defect, represented gain, history-energy envelope and complete-block endpoint bound, including a positive binary64-defect witness. The exact pairwise boundary then shows that a bounded hard-clipped box can require \(\eta=2^{210}-1/4\), while the \(\alpha=1\) class has uniform \(\eta=0\) and is forward invariant under REMESH alone. The restricted antisymmetric \(P_2\) hard-clip class at \(\alpha=1/2\) separately has sharp uniform \(\eta=135/124\), proved by an analytic IEEE tail and exact finite-core enumeration, and is REMESH-forward-invariant. Its strict schedule threshold is \(q<124/259\); \(q=4/9\) gives margin \(20/279\), while equality gives zero margin. A global binary64 \(P_2\) half-Reception EPI kernel has \(q=0\) and composes with that class to extinguish active-history spatial disagreement after \(\tau_g+1\) restricted cycles. This repeated result belongs to the numeric EPI kernels. One completed graph-owned event now binds the P2 kernel to a grammar-admitted two-phase Reception EPI stage and its captured endpoints. A second finite adapter binds every selected stage and same-cycle \(\alpha=1\) REMESH on one graph, verifies only the active suffix of length \(\tau_g+1\), and records post-horizon extinction. It does not establish future or unobserved repetition. A reusable transactional policy now revalidates the current restricted conditions on every call, runs the causal sequence and constructs that finite certificate before commit; successive accepted calls remain separate observations, and auxiliary Reception-state stability is not certified. One event-free effective-\(P_2\) family additionally has a rational continuous/Euler error enclosure, strict improvement across its two declared proper subdivisions, exact ideal REMESH scaling and an explicit runtime residual bound. A separate general reversible exact-mode runtime adapter binds finite executor-owned pressure-refreshed partitions and propagates represented defects through complete Euler matrices, but it contains no REMESH claim. Extension of the half-alpha antisymmetric class to broader represented states, support or coefficients, a priori preservation of the P2 EN/REMESH hypotheses for future unobserved complete event execution, generic or binary64 asymptotic convergence, repeated complete-runtime stability, the clipped binary64 runtime limit, full structural invariants and global operator completeness remain unresolved.