Memory derived by observing groups of EPI nodes¶

Status: Exact fixed-generator elimination, captured-affine closure and minimal linear realizations, P5 finite-history error bounds, and nested P5 orbit closure with an obstruction to direct REMESH substitution; exact controls and finite numerical/runtime observations. General multichannel closure and broader memory-to-operator realizations remain open. Research links: B2/O4.a, S3, S8, S9 and S16. Updated: 2026-09-18. Implementation: epi_memory.py, using the reversible partition geometry in structural_morphism.py.

For the selected nonlinear phase/form model, the joint pattern memory reuses the elimination and error-propagation methods below. Its hidden generator and nonlinear sources have their own admission; reversible kernel positivity, P5 truncation formulas and REMESH equivalence do not transfer automatically.

1. What is derived, and from what¶

The microscopic dynamics is the existing pure-EPI channel of the TNFR nodal equation. It contains no imposed delay or additional memory parameter:

\[ \dot x_i=\nu_i\Delta\mathrm{NFR}_{\mathrm{epi},i},\qquad \Delta\mathrm{NFR}_{\mathrm{epi},i} =\frac{\sum_j W_{ij}x_j}{d_i}-x_i, \qquad d_i=\sum_jW_{ij}. \]

Here \(x_i\) is the real scalar EPI coordinate. Assume finite fixed effective conductance \(W=W^\top\geq0\), positive row strengths \(d_i>0\), and fixed finite positive capacities \(\nu_i\). These are the hypotheses of this restricted channel, not conclusions about all TNFR executions. Set

\[ D=\operatorname{diag}(d_i),\quad B=D-W,\quad H=\operatorname{diag}(d_i/\nu_i),\quad A=\operatorname{diag}(\nu_i)(I-D^{-1}W)=H^{-1}B. \]

The nodal law is \(\dot x=-Ax\). In particular, \(A^\top H=HA=B\), and \(B\) is positive semidefinite. Algebra in this note is over the exact real coefficients of the declared model. Floating-point construction and evaluation of the associated matrices have separate numerical residuals.

An observer retains the weighted EPI means of a partition and discards the remaining within-block coordinates. The partition specifies which TNFR nodes are observed together; it is not an additional evolution law or a proof that nature selects those particular blocks. Eliminating the discarded coordinates derives the memory below from the same nodal law. It does not add a fundamental memory mechanism to that law.

2. The reversible observation geometry¶

Let \(P\in\mathbb R^{n\times m}\) lift one macro value to every node of its nonempty partition block: \(P_{i\alpha}=1\) when node \(i\) belongs to block \(\alpha\), and zero otherwise. Define

\[ \bar H=P^\top HP,\qquad R=\bar H^{-1}P^\top H,\qquad Q=I-PR. \]

Thus \(R\) takes the \(H\)-weighted block means, \(RP=I\), \(Q^2=Q\), \(RQ=0\), \(QP=0\), and \(Q^\top H=HQ\). The macro and hidden coordinates are

\[ y=Rx,\qquad z=Qx,\qquad x=Py+z. \]

The hidden subspace is \(\mathcal Z=\ker R=\operatorname{im}Q\). It is the \(H\)-orthogonal complement of \(\operatorname{im}P\). The instantaneous macro generator is fixed by the same graph and capacities:

\[ \bar A=RAP=\bar H^{-1}P^\top BP. \]

This is the existing reversible quotient construction. Its off-diagonal macro conductances aggregate the microscopic conductances between blocks. When a macro node has positive external strength \(\bar d_\alpha\), its corresponding capacity is \(\bar\nu_\alpha=\bar d_\alpha/\bar H_{\alpha\alpha}\); it is not a separately fitted parameter. The shared implementation additionally requires a strict reduction to at least two blocks and connected macro support.

3. Exact elimination, including the initial hidden state¶

Applying \(R\) and \(Q\) to the nodal equation gives

\[ \dot y=-\bar A y-RAz,\qquad \dot z=-QAPy-QAQz. \]

Let \(T=QAQ\) and \(C=QAP\). The subspace \(\mathcal Z\) is invariant under \(T\). Since \(z_0=Qx_0\) and \(Cy(s)\) lie in that subspace, variation of constants gives

\[ z(t)=e^{-Tt}Qx_0-\int_0^t e^{-T(t-s)}Cy(s)\,ds. \]

Substitution into the first equation proves the exact Volterra equation

\[ \boxed{ \dot y(t)=-\bar A y(t) +\int_0^t K(t-s)y(s)\,ds+f(t) } \]

with

\[ \boxed{K(t)=RAe^{-Tt}C},\qquad \boxed{f(t)=-RAe^{-Tt}Qx_0}. \]

The plus sign on the convolution follows from substituting the negative hidden forcing into \(-RAz\). The source \(f\) is the contribution of the initial hidden state, not an external drive or random noise. Omitting it generally changes the solution. Conversely, \(Qx_0=0\) removes this source but does not generally remove the convolution generated by later within-block evolution.

Writing \(RAQ\) instead of \(RA\) is equivalent in these two expressions: the inputs to the exponential are hidden and the exponential preserves \(\mathcal Z\). Without that subspace fact, dropping projection factors would be unjustified.

This equation reproduces \(y(t)=Re^{-At}x_0\) for all \(t\geq0\) and every initial state under the declared fixed-coefficient hypotheses. The converse follows by recovering \(z\) from its formula: the reconstructed \(x=Py+z\) solves the same linear initial-value problem. No empirical fitting or extra constitutive law enters this equivalence.

4. Kernel positivity and the exact closure criterion¶

The hidden generator is self-adjoint and nonnegative in the \(H\) metric:

\[ T^\top H=HT,\qquad v^\top HTv=(Qv)^\top B(Qv)\geq0. \]

Consequently \(H^{1/2}TH^{-1/2}\) is symmetric positive semidefinite, and its exponential is symmetric positive definite at each finite \(t\geq0\). Using \(Q^\top H=HQ\) and \(A^\top H=HA\) gives

\[ \boxed{\bar H K(t)=C^\top H e^{-Tt}C}. \]

Indeed, \(C^\top H=P^\top HAQ\) and \(Qe^{-Tt}C=e^{-Tt}C\). For every macro vector \(v\), this identity yields

\[ v^\top\bar H K(t)v =(H^{1/2}Cv)^\top e^{-H^{1/2}TH^{-1/2}t}(H^{1/2}Cv)\geq0. \]

Thus \(K(t)\) is self-adjoint and positive semidefinite in the macro metric. This does not assert that all its individual entries are nonnegative. At zero lag, \(\bar H K(0)=C^\top HC\), so positive definiteness of \(H\) proves

\[ \begin{aligned} K(0)=0 &\iff QAP=0\\ &\iff AP=P\bar A\\ &\iff RA=\bar A R. \end{aligned} \]

For the final equivalence, \(QAP=0\) says that the lifted block-constant subspace is invariant under \(A\). Self-adjointness in \(H\) makes its orthogonal complement invariant too, hence \(RAQ=0\) and \(RA=\bar A R\). Conversely, \(RAQ=0\) has weighted adjoint \(QAP=0\). Therefore the kernel vanishes identically exactly when every microstate admits the autonomous macro law \(\dot y=-\bar A y\). In that case the initial hidden source also vanishes for every \(x_0\).

Zero hidden source for one state or agreement of one initial derivative is a strictly weaker condition. A numerical residual below a tolerance is likewise not an exact zero-kernel proof.

If microscopic positive-conductance support is connected, \(B\) has only the constant null direction. Because \(\mathcal Z\) is orthogonal to that direction, \(T\) restricted to \(\mathcal Z\) is positive definite. The ambient matrix \(T\) still has the lifted macro subspace in its kernel. Connected macro support alone does not establish connected microscopic support, so it cannot justify the stronger hidden-space assertion for every accepted graph.

The positive convolution sign does not create energy. The original nodal balance remains

\[ \frac12 x^\top Hx =\frac12 y^\top\bar Hy+\frac12 z^\top Hz,\qquad \frac{d}{dt}\frac12 x^\top Hx=-x^\top Bx\leq0. \]

The macro equation retains the influence of hidden coordinates of this same dissipative system. It does not establish a new source of sustained vibration.

5. Exact zero-memory control: P4¶

Take the unweighted path \(0-1-2-3\), unit capacities and partition \(((0,3),(1,2))\). Here

\[ a=(x_0+x_3)/2,\qquad b=(x_1+x_2)/2,\qquad \bar H=\operatorname{diag}(2,4). \]

For every microscopic initial state, direct application of the nodal equation gives

\[ \dot a=-a+b,\qquad \dot b=(a-b)/2, \qquad \bar A=\begin{pmatrix}1&-1\\-1/2&1/2\end{pmatrix}. \]

Thus \(QAP=0\), \(K(t)=0\), and \(f(t)=0\). Reflection-antisymmetric hidden states can evolve, but they cannot affect these macro observations. Grouping nodes does not universally produce nonzero memory.

6. Exact nonzero-memory witness: P5¶

Take the unweighted path \(0-1-2-3-4\), unit capacities and partition \(((0,4),(1,2,3))\). The reversible macro coordinates and one relevant hidden coordinate are

\[ a=(x_0+x_4)/2,\quad b=(x_1+x_2+x_3)/3,\quad u=x_2-(x_1+x_3)/2. \]

The full hidden subspace has dimension three, but its other two coordinates do not couple to these macro observations. No reflection-symmetry assumption on the initial state is required. Direct substitution proves

\[ \begin{aligned} \dot a&=-a+b-u/3,\\ \dot b&=(a-b)/3+u/9,\\ \dot u&=(b-a)/2-5u/3. \end{aligned} \]

Therefore

\[ \bar H=\operatorname{diag}(2,6),\qquad \bar A=\begin{pmatrix}1&-1\\-1/3&1/3\end{pmatrix}, \]

\[ K(t)=e^{-5t/3} \begin{pmatrix}1/6&-1/6\\-1/18&1/18\end{pmatrix},\qquad f(t)=u_0e^{-5t/3}\begin{pmatrix}-1/3\\1/9\end{pmatrix}. \]

In particular,

\[ \bar H K(t)=\frac{e^{-5t/3}}3 \begin{pmatrix}1&-1\\-1&1\end{pmatrix}\succeq0. \]

Three controls distinguish the claims:

Same macrostate, different initial rates. The nonnegative integer states \(x^{(A)}=(0,1,1,1,0)\) and \(x^{(B)}=(0,0,3,0,0)\) both have \((a,b)=(0,1)\). Their hidden coordinates are \(u_A=0\) and \(u_B=3\), and their initial macro rates are \((1,-1/3)\) and \((0,0)\). No autonomous first-order right-hand side depending only on \((a,b)\) can represent both.
Zero source, nonzero generated memory. For \(x_0=P(1,0)\), \(u_0=0\) and \(f\) vanishes. Nevertheless \(\dot u(0)=-1/2\). The second derivative differs from the source-free autonomous quotient prediction by \(K(0)y_0=(1/6,-1/18)^\top\). Thus removing the convolution is already wrong at second order in time.
Relaxation, not an emergent harmonic wave. The contrast \(d=a-b\) satisfies \(\ddot d+3\dot d+2d=0\). Its two decay rates are 1 and 2. The added time derivative is a reformulation of the eliminated hidden coordinate, not a separately postulated inertia or proof of sustained oscillation.

All these coefficients follow from the path, the partition and the existing unit-capacity nodal law. For a common fixed capacity \(\nu>0\), the kernel is \(\nu^2 e^{-5\nu t/3}K_0\), with \(K_0\) the displayed constant matrix, and the source is \(\nu u_0e^{-5\nu t/3}(-1/3,1/9)^\top\). The contrast decay rates are \(\nu\) and \(2\nu\); no new relaxation parameter is fitted.

7. Implementation and the evidence boundary¶

observe_epi_memory in epi_memory.py reads one fixed graph, partition and initial EPI state and samples the corresponding linear-model quantities. It reuses the existing reversible partition builder; it does not evolve the live graph or add an operator. Its binary64 matrices, matrix-exponential samples and reported residuals are finite numerical observations of the exact model derived here. They are not exact symbolic certificates, a validated matrix-exponential error bound, or a proof of a particular engine solver trajectory's accuracy.

The P4/P5 formulas supply independent analytical controls. Tests check both the initial hidden source and the driven hidden contribution, weighted kernel symmetry/positivity, shared-macrostate counterexamples, zero-memory closure, and failure to justify closure from a single block-constant initial state. A finite tolerance must remain visible when interpreting numerical agreement. Observed finite agreement does not establish asymptotic binary64 convergence or every future execution.

The observer keeps closure tolerance separate from numerical_tolerance. Before sampling, it checks relative constant-field and weighted-total defects in the represented fine generator. Each fine and Markov propagator is then checked for row-sum, nonnegativity and normalized stationary-weight defects. Samples outside this declared numerical policy are rejected. These are necessary diffusion identities, not a sufficient accuracy test: even a stochastic matrix with the right stationary measure can be the wrong flow. The checks catch an extreme-time P4 exponential drift that leaves the rate residual zero, and severe self-loop cancellation in generator construction, including when the only requested time is zero. Neither tolerance is a new physical parameter, and neither causes small memory terms to be discarded.

derived_epi_memory.py reproduces four deterministic P4/P5 controls at times 0, 0.125, 0.5, 1, 2. It writes a local JSON artifact with the source revision, dirty-source digest, dependency versions, matrices, samples and omission errors through the existing CoreExperimentManifest. Its status is MEASURED, separate from the exact theorem above. It uses no random seed, runtime timestep or physical dataset.

The older reduction_certificates.py contains a related Kron reduction control for \(\dot x=-Bx\), including the initial kernel norm. That supplied-conductance model equals the nodal dynamics \(\dot x=-H^{-1}Bx\) only under unit mobility \(H=I\), for example \(\nu_i=d_i\). It supplies a useful static-versus-dynamic reduction comparison, but does not replace the heterogeneous-capacity derivation above.

The result is restricted to fixed-coefficient pure-EPI diffusion and a fixed partition. It does not prove closure for the nonlinear phasor channel, changing capacities, support, topology or observation partitions, arbitrary canonical operator sequences, grammar or complete runtime histories. It does not identify this continuous memory kernel with REMESH's configured temporal echo.

No external physical mechanism, astronomical data or empirical outcome is used in the derivation. A future laboratory interpretation still needs a specified observation map, derived TNFR dynamics and a discriminating measurement. This result establishes an exact mathematical form of memory in grouped TNFR observations; it does not establish that physical reality or quantum behavior emerges from it.

8. Finite-history approximation on the derived P5 model¶

The exact P5 kernel in section 6 decays with a rate already fixed by the nodal law. This permits a separate approximation question: how much older observed history can be omitted, and how much can that omission change the resulting macro trajectory? A bound on the omitted forcing alone does not answer the second question, because the approximate trajectory subsequently feeds back through its retained history. The following theorem controls that feedback.

8.1 Full and finite-history equations¶

Keep the unweighted P5 graph, fixed partition and common positive capacity \(\nu\) of section 6. Define the macro contrast and conserved mean by

\[ d=a-b,\qquad m=(a+3b)/4, \]

and set

\[ \alpha=\frac{4\nu}{3},\qquad \beta=\frac{5\nu}{3},\qquad \gamma=\frac{2\nu^2}{9},\qquad k(t)=\gamma e^{-\beta t},\qquad g(t)=-\frac{4\nu u_0}{9}e^{-\beta t}. \]

Here \(\alpha\), \(\beta\) and \(\gamma\) abbreviate derived coefficients; they are not configurable laws or the REMESH mixing coefficient. Taking the contrast of the exact macro equation gives

\[ \dot d(t)=-\alpha d(t)+g(t)+\int_0^t k(t-s)d(s)\,ds. \]

For a selected retention window \(\ell\geq0\), define \(d_\ell\) by the same initial contrast and the same initial-hidden-state source:

\[ \boxed{ \dot d_\ell(t)=-\alpha d_\ell(t)+g(t) +\int_{\max(0,t-\ell)}^t k(t-s)d_\ell(s)\,ds, \qquad d_\ell(0)=d_0. } \]

Thus only the driven memory integral is shortened. The source \(g\) is retained at every time; discarding it would define a different approximation requiring an additional error term. The window \(\ell\) is an approximation resource chosen against an error target, not a new physical timescale. For each finite window the continuous Volterra equation has a unique continuous trajectory with the displayed derivative. No timestep or live-graph update is needed to define it.

Both macro equations conserve the same \(m\): the row vector \((1,3)\) annihilates the instantaneous generator, memory kernel and source. Consequently

\[ y_\ell(t)-y(t)= \begin{pmatrix}3/4\\-1/4\end{pmatrix} \bigl(d_\ell(t)-d(t)\bigr),\qquad \|y_\ell(t)-y(t)\|_\infty=\frac34|d_\ell(t)-d(t)|. \]

8.2 Exact reference and omitted forcing¶

The full contrast has the closed form

\[ d(t)=c_1e^{-\nu t}+c_2e^{-2\nu t},\qquad c_1=\frac{2d_0}{3}-\frac{4u_0}{9},\quad c_2=\frac{d_0}{3}+\frac{4u_0}{9}. \]

Its exact amplitude \(D=\sup_{t\geq0}|d(t)|\) can be calculated without sampling. Under \(z=e^{-\nu t}\), the contrast is \(c_1z+c_2z^2\) on the closed interval \([0,1]\). Therefore \(D\) is the maximum of \(|d_0|\) and, only when \(c_2\ne0\) and \(z_*=-c_1/(2c_2)\in(0,1)\), the additional candidate \(|-c_1^2/(4c_2)|\). The omitted endpoint \(z=0\) has value zero. The simpler \(D\leq |c_1|+|c_2|\) supplies a conservative alternative. On a finite horizon \([0,T]\), the corresponding \(D_T\) follows from the endpoints and any interior vertex of the same quadratic on \([e^{-\nu T},1]\).

Write \(\delta=e^{-\beta\ell}\) and \(q=t-\ell\). For \(t\leq\ell\), the omitted forcing \(r_\ell(t)\) is zero. For \(t>\ell\), its value on the full reference is

\[ \begin{aligned} r_\ell(t) &=\int_0^{t-\ell} k(t-s)d(s)\,ds\\ &=\delta\left[ \frac{\nu c_1}{3}\bigl(e^{-\nu q}-e^{-5\nu q/3}\bigr) +\frac{2\nu c_2}{3}\bigl(e^{-5\nu q/3}-e^{-2\nu q}\bigr) \right]. \end{aligned} \]

Direct integration also gives the sign-independent bound

\[ |r_\ell(t)|\leq \frac{2\nu}{15}D\delta\bigl(1-e^{-\beta q}\bigr) \leq\frac{2\nu}{15}D\delta. \]

These are statements about a forcing evaluated on the exact full trajectory. The error between the two actual solutions requires the next step.

8.3 Positive resolvent and the uniform trajectory bound¶

Let \(k_\ell(t)=k(t)\mathbf1_{0\leq t\leq\ell}\), with causal convolution on \([0,\infty)\). The value at the window endpoint does not change its integral. Subtracting the two equations, \(e=d_\ell-d\) satisfies

\[ \dot e=-\alpha e+k_\ell*e-r_\ell,\qquad e(0)=0. \]

Its impulse resolvent \(h_\ell\) is nonnegative. To see this, set \(p(t)=e^{-\alpha t}\) and \(j_\ell=p*k_\ell\). Variation of constants gives the nonnegative series

\[ h_\ell=p+\sum_{n=1}^\infty j_\ell^{*n}*p. \]

The series converges in \(L^1\) because

\[ \|j_\ell\|_1= \frac{\|k_\ell\|_1}{\alpha}<1, \qquad \|k_\ell\|_1=\frac{2\nu}{15}(1-\delta). \]

Nonnegativity permits integration term by term, yielding

\[ \|h_\ell\|_1 =\frac1{\alpha-\|k_\ell\|_1} =\frac{15}{2\nu(9+\delta)}. \]

As \(e=-h_\ell*r_\ell\), the convolution inequality now proves

\[ \boxed{ \sup_{t\geq0}|d_\ell(t)-d(t)| \leq\frac{D\delta}{9+\delta},\qquad \sup_{t\geq0}\|y_\ell(t)-y(t)\|_\infty \leq\frac{3D\delta}{4(9+\delta)}. } \]

This is a bound on the solution of the truncated equation, including its feedback through the retained history. It assumes no unproved bound on that approximate solution. For a finite horizon, \(D_T\) can replace \(D\) and the suprema can be restricted to \([0,T]\).

8.4 Causal and decaying error bounds¶

The full-kernel resolvent is explicit:

\[ h(t)=\frac23e^{-\nu t}+\frac13e^{-2\nu t}. \]

Its Laplace transform is \((p+\beta)/((p+\nu)(p+2\nu))\), as follows directly from \(p+\alpha-\gamma/(p+\beta)\). Since \(0\leq k_\ell\leq k\), the positive resolvent series proves \(0\leq h_\ell\leq h\). For \(q=(t-\ell)_+\), integration gives

\[ \int_0^q h(q-v)\bigl(1-e^{-\beta v}\bigr)\,dv =\frac{5}{6\nu}\bigl(1-e^{-\nu q}\bigr)^2. \]

Combining this identity with the forcing estimate and the uniform bound gives

\[ \boxed{ |d_\ell(t)-d(t)|\leq D\delta\min\left\{ \frac1{9+\delta},\; \frac{(1-e^{-\nu q})^2}{9} \right\}. } \]

The macro infinity-norm bound is again three quarters of this expression. The error vanishes through time \(\ell\) and its bound starts quadratically after that time. Thus \(\ell\geq T\) gives exact agreement over \([0,T]\).

A bound that also resolves late-time relaxation follows by convolving the absolute values of the two explicit forcing terms with \(h\). With \(s=\nu(t-\ell)_+\), both bracketed expressions below are nonnegative, and

\[ |d_\ell(t)-d(t)|\leq\frac{2\delta}{9}\left[ |c_1|e^{-s}(s-1+e^{-s}) +|c_2|\bigl(e^{-s}-(1+s)e^{-2s}\bigr) \right]. \]

This expression tends to zero as \(O((1+s)e^{-s})\). The truncated solution therefore converges to the same conserved macro mean, although its transient trajectory generally differs. This conclusion concerns this derived scalar P5 reduction with its retained source, not arbitrary memory truncations.

8.5 Window choice, limiting controls and scope¶

For a prescribed positive contrast-error target \(\varepsilon\), a sufficient uniform window is

\[ \ell_{\mathrm{req}}= \begin{cases} 0,&\varepsilon\geq D/10,\\[1mm] \dfrac{3}{5\nu}\log\dfrac{D-\varepsilon}{9\varepsilon}, &0<\varepsilon<D/10. \end{cases} \]

This is obtained by solving \(D\delta/(9+\delta)\leq\varepsilon\); it is a sufficient budget, not a proof of a smallest possible window. For a macro infinity-norm target \(\varepsilon_y\), substitute \(4\varepsilon_y/3\) for \(\varepsilon\). On a finite horizon, retaining \(\min(T,\ell_{\mathrm{req}})\) suffices, since the \(T\) branch is exact there. If \(D=0\), both \(d_0\) and \(u_0\) vanish, and every window is exact for these macro observations. A zero error target for a nonzero trajectory is not covered by the positive-target logarithmic formula.

At the other boundary, \(\ell=0\) removes the convolution but keeps the initial hidden source. Its solution is

\[ d_{\ell=0}(t)= \left(d_0-\frac{4u_0}{3}\right)e^{-4\nu t/3} +\frac{4u_0}{3}e^{-5\nu t/3}. \]

This supplies an independent approximation control. In particular, it must not be confused with the source-free instantaneous quotient, which would give only \(d_0e^{-4\nu t/3}\). Changing the common capacity replaces time by \(\nu t\) and the retention window by \(\nu\ell\) throughout: the retention budget scales as \(1/\nu\), without an independently selected physical rate.

The theorem is over exact real model coefficients and applies to every finite real initial P5 state, including states without reflection symmetry. Binary64 evaluation of its exponentials or bounds is not automatically a certified enclosure. In particular, exponential underflow to a displayed zero cannot prove exact zero omitted history. Exact zero for \(t\leq\ell\) instead follows from the integration domain itself. Any executable rational reference must separately enclose the exponential evaluations and its finite-history solution.

This approximation result neither evolves a live graph nor certifies a canonical operator sequence, REMESH history, numerical solver timestep, changing topology or empirical correspondence. Its memory and decay rates remain consequences of the fixed P5 nodal dynamics already derived above.

8.6 Executable rational finite-history reference¶

bound_p5_memory_truncation implements this fixed P5 reference from five explicit initial coordinates, the common capacity, a window and observation times. It accepts integers and Fractions exactly; other real inputs retain the shared materialized binary64 rational value. It does not infer a P5 model from an arbitrary live graph.

The finite-history solution itself has an exact finite formula. Set sigma=nu*t, lambda=nu*ell. Differentiating the derived convolution equation and retaining its source gives

\[ d_\lambda''+3d_\lambda'+2d_\lambda =-\frac29e^{-5\lambda/3} d_\lambda(\sigma-\lambda)\mathbf1_{\sigma>\lambda}, \qquad d_\lambda'(0)=-4d_0/3-4u_0/9, \]

where primes now use scaled time. The source remains in the initial derivative even though it cancels on differentiating the equation. This second derivative is a reformulation of projected diffusion, without an independently added inertia. The value of the right-hand side at its single switching time does not change the continuous solution.

For lambda>0, define polynomials for positive integers p,q:

\[ P_{pq}(r)=\sum_{k=1}^{p}(-1)^{p-k} {p+q-k-1\choose p-k}\frac{r^{k-1}}{(k-1)!},\qquad Q_{pq}(r)=\sum_{k=1}^{q}(-1)^p {p+q-k-1\choose q-k}\frac{r^{k-1}}{(k-1)!}. \]

Partial fractions show that P_pq(r)*exp(-r)+Q_pq(r)*exp(-2r) is the inverse Laplace transform of 1/((s+1)^p*(s+2)^q). The boundary cases are P_10=1,Q_10=0 and P_01=0,Q_01=1. Expanding the delayed transfer equation gives

\[ d_\lambda(\sigma)= \sum_{j=0}^{\lfloor\sigma/\lambda\rfloor} \left(-\frac29\right)^j \left[ (c_1P_{j+1,j}(r_j)+c_2P_{j,j+1}(r_j)) e^{-(\sigma+2j\lambda/3)} +(c_1Q_{j+1,j}(r_j)+c_2Q_{j,j+1}(r_j)) e^{-(2\sigma-j\lambda/3)} \right],\quad r_j=\sigma-j\lambda. \]

Equivalently, each term is a delayed convolution power of the second-order resolvent with the full reference. Terms with j>=1,r_j=0 vanish exactly. The implementation handles the zero-window formula in section 8.5 separately and uses the full reference unchanged when t<=ell.

Every coefficient and exponential argument in this formula is rational for rational inputs. The implementation reuses the rational negative-exponential enclosure from reversible_eigenmode_reference.py and its shared Taylor bound in _exact_time.py. Signed interval sums preserve rigorous endpoints and exact cancellation of equal exponents. Unlike the approximate matrix-exponential observer in section 7, these results explicitly enclose evaluation uncertainty. No integration timestep or sampled-history interpolation error occurs in this formula.

The output keeps three quantities separate: the omitted reference forcing, the error between full and truncated solutions, and the width of the numerical enclosure. The theorem's macro error bound is rounded upward with rational exponential bounds; the raw solution-error enclosure can be wider because interval evaluation is conservative. Integer/Fraction interval endpoints remain authoritative. The benchmark uses directed outward rounding for any displayed binary64 error bounds, rather than ordinary float casts.

Resource admission currently allows at most 32 nonzero delayed corrections per sample and nu*t<=2048, which keeps all required exponential arguments within the existing limit of 4096. These are workload and representation limits; neither is a physical scale or a bound on rational bit size. The theorem itself has no such finite-time cutoff. In particular, an extremely large window is handled by the exact t<=ell branch without exponentiating the window.

Tests compare the enclosures with independent high-precision full, zero-window and first-delay formulas, and compare later windows with a separate block-matrix method of steps. They also check the source, amplitude maximum, bounds, capacity scaling and resource boundaries. The existing benchmark now records three P5 initial states at windows 0,1/4,1/2,2 and times 0,1/8,1/2,1,2. Its separate B2.b manifest labels the rational reference DERIVED, preserves exact endpoints and records the working-source digest. Physical correspondence remains untested.

9. Nested nodal closure and the REMESH comparison¶

The derived history integral and the configured REMESH echo both involve past structure, but that description alone establishes no dynamical equivalence. The following comparison retains the same fixed unweighted P5, common positive capacity nu, and real scalar EPI channel. It separates a failure of direct substitution from an exact way to compose the two declared dynamics.

9.1 A causal-history obstruction to direct substitution¶

Write the microscopic generator as A5=nu*L_rw. The reflection-even vector

\[ v=(1,0,-1,0,1)^\top,\qquad A_5v=\nu v \]

gives the exact nodal trajectory x(t)=c0*1+epsilon*exp(-nu*t)*v. Let h>0 be an observation interval and take the current state at zero together with its actual nodal past at -tau_local*h and -tau_global*h. For positive integer delays and 0<alpha<=1, uniform unclipped REMESH uses

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

Its output is c0*1+epsilon*r*v, where

\[ r=\beta+\gamma e^{\nu\tau_{\rm local}h} +\delta e^{\nu\tau_{\rm global}h}>1. \]

The exact forward nodal state at time h instead has amplitude exp(-nu*h)<1. Thus no coefficient in this class makes the complete REMESH map a forward nodal step on this same causal history. The original two-group projection also sees v, so projection does not remove the contradiction. Equivalently, no real decay factor z in (0,1) satisfies

\[ z=\beta+\gamma z^{-\tau_{\rm local}} +\delta z^{-\tau_{\rm global}}. \]

This is not an argument based on fabricated constant past data. For any finite set of these samples, a finite baseline and amplitude can keep every coordinate nonnegative and inside one interval where clipping is inactive. Neither the observation interval nor that preparation is a new physical law.

An exact dyadic control uses nu=1, h=log(2), c0=4, epsilon=1, alpha=1/2, and delays 1,2. Newest first, its three rows are 4*1+v, 4*1+2*v, 4*1+4*v; every value lies in [0,8]. REMESH gives r=11/4, whereas forward diffusion gives 1/2. The current microscopic disagreement energy rises from 2 to 121/8. This does not contradict the existing augmented-history theorem: with stationary weights (4/9,1/3,2/9), that energy decreases from 32/3 to 55/6, an exact drop of 3/2.

The obstruction concerns direct substitution of the complete echo map for a forward nodal or projected evolution step. It does not rule out every possible derived embedding, discretization of the memory term, or larger drift-plus-echo construction. Such a construction would need its own exact observation map, history initialization and dynamical correspondence; none follows from this comparison, and fitting a coefficient to one trajectory would not prove it.

9.2 The memory is removed by a genuine nested nodal quotient¶

Retain the reflection-orbit coordinates

\[ q=(a,p,c)^\top,\qquad a=(x_0+x_4)/2,\quad p=(x_1+x_3)/2,\quad c=x_2. \]

The associated projection and lift are

\[ R_3=\begin{pmatrix} 1/2&0&0&0&1/2\\ 0&1/2&0&1/2&0\\ 0&0&1&0&0 \end{pmatrix},\qquad J=\begin{pmatrix} 1&0&0\\0&1&0\\0&0&1\\0&1&0\\1&0&0 \end{pmatrix}. \]

This is the existing reversible quotient construction for the equitable partition ((0,4),(1,3),(2,)). The quotient is a three-node path with conductance 2 on each edge, strengths (2,4,2) and capacity nu at every node. Direct application of the nodal law gives

\[ A_3=\nu\begin{pmatrix} 1&-1&0\\-1/2&1&-1/2\\0&-1&1 \end{pmatrix},\qquad \dot q=-A_3q,\qquad R_3J=I,\quad R_3A_5=A_3R_3,\quad A_5J=JA_3. \]

These identities hold for every microscopic initial state; reflection symmetry of that state is unnecessary. The original two-group observer is the further weighted merge

\[ \begin{pmatrix}a\\b\end{pmatrix} =\begin{pmatrix}1&0&0\\0&2/3&1/3\end{pmatrix}q,\qquad b=(2p+c)/3,\quad u=c-p. \]

Its projection is exactly this merge composed with R3. The first reduction closes without memory. The second merges two quotient nodes with distinct neighbor profiles, discards their contrast u, and produces the kernel of section 6. Thus the memory-producing step is identified inside the same hierarchy of TNFR node observations.

The chart (a,b,u) is related to genuine quotient EPI values by q=(a,b-u/3,b+2*u/3). Its generator is

\[ A_{abu}=\nu\begin{pmatrix} 1&-1&1/3\\-1/3&1/3&-1/9\\1/2&-1/2&5/3 \end{pmatrix}. \]

This is a coordinate change, not a proposal to treat u as a new TNFR node. In particular, its consensus direction is (1,1,0), not (1,1,1).

9.3 Minimality of the retained linear state¶

Let R2 denote the original two-group projection. Any autonomous linear state containing both its outputs for every microscopic initial state must have a row space containing R2 and invariant under right multiplication by A5. It must therefore contain the two rows of R2 and the first row of R2*A5. Their minor in columns 0,1,2 is

\[ \det\begin{pmatrix} 1/2&0&0\\0&1/3&1/3\\\nu/2&-\nu/2&0 \end{pmatrix}=\frac{\nu}{12}>0. \]

At least three independent coordinates are required. The invariant three-coordinate quotient above attains that bound. This proves minimality within autonomous linear realizations containing (a,b); it does not prove full microscopic observability, nonlinear state completeness, or minimality of every possible history representation.

9.4 One metric and an exact visible/discarded energy split¶

The metrics are inherited from the same nodal capacity and conductance:

\[ H_5=\frac1\nu\operatorname{diag}(1,2,2,2,1),\qquad H_3=J^\top H_5J=\frac1\nu\operatorname{diag}(2,4,2). \]

Set rho=(x0-x4)/2, sigma=(x1-x3)/2, and w=x-J*q=(rho,sigma,0,-sigma,-rho). The lift and discarded component are H5-orthogonal. The latter has zero weighted mean, while both x and q have mean m=(a+2*p+c)/4=(a+3*b)/4. Consequently the centered disagreement energy E_H(x)=1/2*(x-m*1)^T*H*(x-m*1) satisfies

\[ E_{H_5}(x)=E_{H_3}(q)+\frac{\rho^2+2\sigma^2}{\nu},\qquad E_{H_3}(q)=\frac{3(a-b)^2}{4\nu}+\frac{2u^2}{3\nu}. \]

In the (a,b,u) chart the induced diagonal metric is diag(2,6,4/3)/nu, but centering must remove the direction (1,1,0). Applying an all-ones-centered node-energy routine directly to that chart would change the meaning of consensus. Existing REMESH certificates can instead read the genuine three-node q and H3 without changing their semantics.

The reduced nodal balance is

\[ \frac{d}{dt}E_{H_3}(q) =-2(a-p)^2-2(p-c)^2\leq0. \]

The two-coordinate contribution alone need not decrease. For example, x0=(2,1,4,1,2) has a=b=2,u=3: its visible contrast starts at zero and then becomes (4/3)*(exp(-2*nu*t)-exp(-nu*t)), through the hidden source. The full three-coordinate energy still decreases. Conversely, quotient consensus can hide nonzero microscopic disagreement in w.

The distinction also affects rates. Quotient diffusion has eigenvalues 0,nu,2*nu and centered energy gain exp(-2*nu*h) over an interval h. The discarded coordinates satisfy

\[ \begin{pmatrix}\dot\rho\\\dot\sigma\end{pmatrix} =-\nu\begin{pmatrix}1&-1\\-1/2&1\end{pmatrix} \begin{pmatrix}\rho\\\sigma\end{pmatrix}, \]

with rates nu*(1-1/sqrt(2)) and nu*(1+1/sqrt(2)). Therefore the faster quotient gain cannot be promoted to all microscopic disagreement. Both rates and the energy partition follow from the fixed P5 nodal generator.

9.5 Exact composition with a declared temporal echo¶

Uniform scalar REMESH mixing commutes with R3:

\[ R_3(\beta x_0+\gamma x_l+\delta x_g) =\beta R_3x_0+\gamma R_3x_l+\delta R_3x_g. \]

History shifts likewise commute with applying R3 to every retained row. The generator identities imply the same transport for exact diffusion segments and for a declared Euler map I-h*A5. Hence any finite composition of those fixed-support linear operations closes on 3*(m_delay+1) state coordinates when m_delay is the largest active delay and corresponding fine/coarse history rows are supplied. This is a sufficient finite state dimension, not a new minimality theorem for temporal companions. On each diffusion segment, eliminating u recovers the derived memory with that segment's actual initial hidden coordinate.

Adding a REMESH operation changes the trajectory; commutation proves that its effect can be represented by the same quotient, not that it was already present in the eliminated continuous dynamics. These claims require uniform mixing and matching delays across nodes. General node-dependent coefficients, clipping, changing support/capacity or other channels are outside this proof.

For fixed REMESH coefficients and delays, let pi be the existing companion's stationary temporal weights. Summing the spatial identity in section 9.4 over retained rows gives the exact augmented-history split

\[ V_5=V_3+\sum_j\pi_j\frac{\rho_j^2+2\sigma_j^2}{\nu}. \]

The before/after values and Jensen drops therefore split under one exact REMESH transition. This reuses the uniform history theorem. The schedule policy theorem can separately use the quotient diffusion gain when its hypotheses and operation order are satisfied: it contracts a newly mixed head. It does not automatically apply that gain to a diffusion-before-mixing cycle or to the discarded microscopic modes. Neither theorem controls spatially uniform temporal means merely from a spatial-disagreement bound.

9.6 Implementation, finite runtime defects and evidence boundary¶

p5_reduction.py materializes the exact fixed P5 geometry, verifies the intertwining identities and rank, and observes one supplied newest-first REMESH history. Its fine, quotient and discarded observations reuse the existing sealed history certificates. The shared reduce_p5_state also supplies the coordinate, contrast-coefficient and exact amplitude calculation used by p5_memory_truncation.py. Integers and Fractions remain exact; other accepted real inputs use the shared represented-binary64 rational rule. These detached observations do not bind input histories to a live executor or make u an additional operator.

Exact tests compare the quotient with the existing graph coarse-graining builder, verify the rank and energy split, and reproduce the causal decay-mode obstruction. Production tests exercise the actual delayed REMESH kernel on paired five- and three-node graphs. A dyadic witness gives exact commutation, including one declared refreshed nodal Euler segment followed by REMESH. A separate non-dyadic witness with represented alpha=0.3 has a first-orbit commutation defect 2^-57, entirely accounted for by the projected signed binary64 rounding defects. Exact-real intertwining therefore does not promise bitwise runtime equality.

Clipping supplies an independent obstruction. At alpha=1, let an older history row have endpoint values (4,-2) and zero elsewhere, with hard target bounds [-1,1]. Clipping the microscopic endpoints gives mean zero; projecting first gives the endpoint mean one, which the quotient clip leaves unchanged. The first-orbit defect is -1, with zero rounding defect. The selected older row is deliberately outside the target interval. This example does not contradict the fact that an exact convex mixture of inputs all inside one common hard interval remains inside it.

The causal decay fixture is also exercised by the production kernel with inactive [0,8] clipping. Its input values are analytically compatible with one exact nodal past, but direct caller-supplied history is not a sealed executor-owned causal-history certificate. These finite production checks establish neither repeated complete-runtime stability, solver convergence, generic grammar/event admission nor future binary64 commutation.

The reproducible benchmark records the fixed-model reduction and causal-history comparison alongside the elimination and finite-window controls. The result identifies an exact synergy between nodal coarse-graining, retained structural information and declared temporal echoes. It supplies no physical dataset, laboratory confirmation or new fundamental parameter, and it leaves broader derived memory-to-operator realizations open.

10. Exact affine closure on a captured forced support¶

The same projection criterion applies to the held canonical channels in FORCED_SUPPORT_BALANCE.md. With that reference's independently captured forcing F, positive EPI coefficient e, positive capacities and connected symmetric conductance, write

x' = -A*x+b, A=e*diag(nu)*D^-1*B, b=diag(nu)*F, H=diag(d/nu).

All coefficients are exact rationals on the declared materialized inputs; H*A=e*B. In particular, constructing A does not require writing a rounded e*nu into a new graph. The partition geometry P,R,Q is that of Section 2, with Hbar=P^T*H*P. No forcing is inferred from a measured derivative.

10.1 Projected autonomy and affine lift invariance differ¶

For y=R*x and eta=Q*x, direct projection gives

\[ \dot y=-RAPy-RA\eta+Rb,\qquad \dot\eta=-QAPy-QAQ\eta+Qb. \]

Therefore an affine law for y valid for every microscopic EPI state is available precisely when R*A*Q=0. The projected law is then y'=-R*A*P*y+R*b. A known constant R*b is allowed, and Q*b need not vanish: hidden coordinates may be driven without feeding back into the observed ones. By reversibility, R*A*Q=0 is equivalent to Q*A*P=0, as in Section 4. In contrast, invariance of the entire lifted affine state space additionally requires Q*b=0. These are separate claims.

If column j of R*A*Q is nonzero, choose v=Q*e_j, where e_j is a coordinate basis vector. The two detached states x and x+v satisfy

\[ R(x+v)=Rx,\qquad R[-A(x+v)+b]-R[-Ax+b]=-RAQe_j\ne0. \]

This gives an exact same-observation/different-rate obstruction. The witness belongs to the declared unrestricted scalar model; it is not a fabricated runtime history or a certificate of operator, clipping or chart admission.

10.2 Derived memory and the current relative profile¶

Let C=Q*A*P and retain the zero-lag kernel K(0)=R*A*Q*A*P. The existing Section 4 identity becomes

\[ \bar H K(0)=C^\top HC\succeq0. \]

Positive H implies K(0)=0 exactly when C=0. Thus a nonzero closure defect already establishes a nonzero derived memory kernel at zero lag, without a matrix exponential, fitted delay or extra constitutive term. It does not specify a finite-memory approximation or certify its error.

For the current held reference, let z be its centered relative profile, m_H(x) its weighted mean, and vbar its mean drift. Its established balance -A*z+b=vbar*1 implies

u=x-m_H(x)*1-z, u'=-A*u.

Section 3's homogeneous elimination can therefore be used on u while keeping the captured forcing visible in the change of coordinates. Applying a pure-EPI memory formula directly to forced raw x would omit a real source. An older reference used to score retention is not substituted for the current model's centering: after a profile-changing event, that older error has its own source.

10.3 Observer changes at a structural event¶

Across a same-EPI event that changes the metric, the family averages satisfy

y1-y0=(R1-R0)*x.

This is a change of observation weights, not EPI motion. Current-model centered observations additionally change by the mean and profile reset:

R1*u1-R0*u0=(R1-R0)*x-(m1-m0)*1-(R1*z1-R0*z0).

These identities require the same ordered nodes and partition and the stated unchanged EPI. They do not certify an operator event or replace its retained execution evidence. The original z0/H0 retention score remains a separate fixed observable throughout.

The closure claim concerns this fixed affine EPI model. A closed macro NFR for phase, capacity, changing support and operator policy needs their corresponding closure results too. In particular, actual parent-child fibers have internal edges and can have unequal capacities; the sufficient phase-quotient hypotheses cannot be assumed merely because an EPI projection is defined.

10.4 Implementation and evidence boundary¶

observe_forced_support_closure in epi_memory.py rebuilds the reference through the existing forced-support owner and materializes the shared reversible geometry over exact fractions. It verifies the projection, weighted adjoint, Gram, affine-rate and current-profile identities and returns a deterministic same-observation/different-rate witness when closure fails. Its proper partition domain matches the existing EPI geometry: at least two nonempty blocks, strictly fewer blocks than nodes, and each ordered node used once.

The observer reuses the existing rational matrix product and profile arithmetic. It neither changes the previous numerical memory API nor makes its approximate exponential a certified propagator. The floating-point projection/lift checks retain the existing structural-morphism residual and tolerance policy; their flags remain distinct from exact zero tests. No new delay, force, operator or runtime certificate is introduced.

Exact controls include both closed and nonclosed partitions, retained nonzero forcing, invalid domains and stale cached reference fields. An affine control with R*A*Q=0 but Q*b!=0 verifies that projected closure is not confused with lifted affine invariance. The actual generated-family application is owned by THOL_BIRTH_AND_TRANSPORT.md.

11. Minimal linear state retaining a declared observation¶

The failure of instantaneous closure does not specify how much additional information is needed. For the same fixed affine nodal model x'=-A*x+b and declared affine observation y=O*x+o0, define the increasing row spaces

\[ \mathcal O_k=\operatorname{rowspan}(O,OA,\ldots,OA^k). \]

Every coefficient comes from the retained nodal generator and observation; these are derived observations, not added physical channels. The forcing b and output offset o0 are known and fixed. They affect the reduced source and decoder, respectively, but not these spaces. Partition means are the special case O=R,o0=0. Stacking means and linear field observations uses this same construction; repeated, dependent and zero output rows are allowed.

11.1 Complete-level stabilization and minimality¶

If O_k=O_(k-1), right multiplication by A maps every generating row of O_(k-1) into O_k, so the stabilized space is invariant. No later power adds a direction. The check must process an entire level O*A^k; a dependent first row does not imply that the remaining rows are dependent. Finite dimension ensures stabilization, with rank at most the microscopic dimension n. For initial rank r0>0, a complete no-growth level occurs by power n-r0+1. A rank-one observation can therefore require all n+1 levels, including power n; partition means have at least two independent rows and retain their earlier bound. Rank zero is the constant-output case and is checked through power one without an inverse.

Choose independent rows of the stabilized space to form C, with rank r. Consider any competing all-state linear observation s_tilde=L*x that retains the output and admits closed affine dynamics. Its identities must satisfy O=D_tilde*L and L*A=G_tilde*L; its constant source is L*b. Consequently its row space contains O and all O*A^k. Thus rank(L)>=r. Constructing a closed observation of dimension r attains this bound. This extends the P5 argument to the admitted held references without a single-seed or scalar-moment proxy.

11.2 Exact affine realization¶

The invariant-row calculation now delegates to the shared mathematics/linear_observation.py owner for supplied matrices. That owner uses z'=Jz; this diffusion-specific wrapper retains its original x'=-Ax+b, deterministic O*A^k basis and source/reference admission. The extraction preserves reports and resource counts. The relational composition study reuses only this algebra, not the diffusion law, reversible metric or memory positivity. Rational matrix identities cannot certify exact irrational coefficients without a separate argument.

Select r independent columns J of C. Let I_J embed the corresponding coordinate vectors into the microscopic state space and define

\[ T=I_J(C_J)^{-1},\qquad G=CAT,\qquad D=OT. \]

Then C*T=I. Row-space invariance and output inclusion give the full identities C*A=G*C and O=D*C. Therefore

\[ s=Cx,\qquad \dot s=-Gs+Cb,\qquad y=Ds+o0. \]

This realization is exact for all scalar initial states in the held model. The known source and offset need no extra fitted coefficient or added constant state. For r=0, s is empty and y=o0 for every fine trajectory; the displayed inverse construction is unnecessary. If r equals the microscopic dimension, C is invertible: the realization is a change of coordinates, with no state compression. Otherwise, different microscopic states can share s while having identical output evolution.

Rows selected from O*A^k can contain signed, nonlocal coefficients and different powers of the existing structural rate. They must not be read as ordinary new NFRs with an independently established phase/capacity triad. In particular, the pivot right inverse T is an algebraic section, not an automatically admissible graph state or an H-orthogonal nodal lift.

11.3 Scope and implementation¶

observe_affine_nodal_realization in epi_memory.py accepts finite exact or represented observation rows and known offsets for a rebuilt held nodal reference. It selects independent rows in deterministic power/row order, tests complete-level stabilization, selects pivot columns, and verifies the full realization identities. observe_forced_support_realization reuses the partition closure owner and delegates its means to the same algorithm; its public result fields, row ordering and resource accounting are preserved. The held generator/source assembly also has one owner. The shared exact_rank, rectangular matrix product and rational inverse supply the arithmetic. Numerical rank is not a fallback.

The rank-call limit is an operational work guard, not a TNFR scale. Exhaustion must report incomplete computation and return no successful realization. Rank-call and matrix-product counts, processed levels and retained coefficient bit sizes describe the calculation; they do not bound every intermediate allocation or the cost inside rational elimination.

Full rank excludes proper all-state linear compression preserving this exact O on this fixed model. It does not exclude approximate observations, restricted reachable-state descriptions, nonlinear reductions or a history representation. Nor does exact invertibility certify a well-conditioned numerical reconstruction. Changes of support, capacity or source need new model admission and separate event accounting. Autonomous maintenance and physical identification remain outside this theorem.

Independent affine-output controls cover offsets with nonzero forcing, redundant/zero rows, exact tiny independent rows, rank-one stabilization and the partition API. The joint geometry adapter derives its rows and offsets from the nodal pressure and inherited distance kernel; the scale bridge owns that physical observation and its numerical boundary. The generic API does not make a caller-declared observation canonical by itself.

11.4 Full-EPI Euler prediction in the sufficient coordinates¶

When C has full microscopic rank, all EPI coordinates are available without an approximation. For declared durations h_k, the shared Euler arithmetic gives a detached forecast

\[ s_{k+1}=s_k+h_k(-Gs_k+Cb),\qquad x_{k+1}=Ts_{k+1}=(I-h_kA)x_k+h_kb. \]

predict_forced_support_realization_euler rebuilds the held reference and realization, requires full rank, and checks the encoding, rates, outputs and fine-coordinate recurrence at every frame. It reuses euler_update with rational operands; it neither writes a graph nor replaces the canonical runtime integrator. Known forcing remains explicit and no observed future endpoint is accepted as input. Durations are nonnegative, with zero giving an identity. A step outside the convex range still defines an exact Euler formula, with a false convex-step flag rather than a claim of stability. Bounded step materialization and the rank-call guard return no partial successful forecast on exhaustion.

The forecast is exact for this declared discretization. It is not the exact continuous solution, a binary64 trajectory or a causal guarantee that a caller retained it before execution. A runtime comparison must separately establish chronology, model constancy and its actual numerical defects.

For a matched held-model runtime segment, the existing observe_forced_support_step separates the stored-pressure defect epsilon_k from the endpoint integration defect delta_k. With M_k=I-h_k*A, the actual endpoint obeys

\[ x_{k+1}^{\rm run}=M_kx_k^{\rm run}+h_kb+r_k,\qquad r_k=h_k\operatorname{diag}(\nu)\epsilon_k+\delta_k. \]

If the forecast starts at the same EPI, its discrepancy satisfies e_0=0, e_(k+1)=M_k*e_k+r_k. In the sufficient chart it is exactly C*e_k. This vector propagation is stronger information than an energy telescope; it does not erase a nonzero measured discrepancy or certify continuous-time solver accuracy. Graph-owned execution evidence supplies the runtime link.

For two branches with the same held A,b, the source cancels in their difference d=x_perturbed-x_control:

d_(k+1)=M_k*d_k+r_k_perturbed-r_k_control.

The H-weighted mean offset is conserved in the ideal recurrence because (H*1)^T*M_k=(H*1)^T. The centered spatial difference can contract while this offset persists. The original fixed-target error, full EPI separation, mean offset and centered energy are therefore distinct observables. A return in EPI also would not erase an operator's history or lineage metadata. Such calculations concern all EPI coordinates of a held model, not autonomous closure of the full phase/capacity/support/operator dynamics.

Prediction tests use independent fine-coordinate rational updates, positive and zero steps, explicit forcing, incomplete observations and work-limit failures. The causal application is owned by THOL full-state response.

11.5 A held-model forecast versus a composite runtime step¶

A full runtime step may apply glyphs and change phase, capacity or support. The old held forecast then remains a counterfactual, not a prediction of the changed dynamics. Its discrepancy must not be classified entirely as numerical error.

On one unchanged ordered node space, let x0 be the pre-step EPI, xg the actual integrator input, xi its output and xf the final runtime state. The frozen model supplies f0(x)=-A0*x+b0, M0=I-h*A0 and xhat=x0+h*f0(x0). Write j=xg-x0, let rg=diag(nu_g)*p_stored be the stored nodal base at integrator entry, and define delta=xi-xg-h*rg and a=xf-xi. Gamma or extended dynamics can contribute beyond that base. Direct substitution gives the exact vector identity

\[ x_f-\widehat x=M_0j+h\{f_g(x_g)-f_0(x_g)\} +h\{r_g-f_g(x_g)\}+\delta+a. \]

Here fg=diag(nu_g)*(e_g*grad_EPI+F_g) is the current rate realization when the shared forcing observer admits its actual pre-integrator support, capacity and independently captured non-EPI channels. It needs no equilibrium/profile solve; a current relative profile can be unavailable while this rate is still defined. If p_kernel is the separately read fresh binary64 pressure and p_model its exact represented-coefficient model, the third term separates further into

h*diag(nu_g)*(p_stored-p_kernel) and h*diag(nu_g)*(p_kernel-p_model).

The first difference includes intentional glyph pressure writes and stale stored pressure; it is not merely rounding. The term delta is an integration residual until the actual solver, held inputs, clipping, Gamma and extended dynamics justify a narrower interpretation. The post-integrator term includes any later EPI writes. Later phase/capacity changes can affect future evolution even when this step's post-integrator EPI term is zero.

If the current affine model is inadmissible but the node space remains the same, the coarser identity xf-xhat=j+h*(rg-f0(x0))+delta+a still holds. If node identity or order changes, neither vector comparison is silently extended by padding or an invented projection. These are finite algebraic accounts of observed boundaries, not causal ablation experiments, future stability bounds or a proof that runtime policies follow from the bare nodal equation. Multiplication by the original C merely expresses the same identity in its retained coordinates.

11.6 Conditional paired evolution under common Coherence¶

For two branches on the same fixed support/capacity, suppose the actual pressure-generation boundary has the same EPI coefficient and non-EPI source, and each node subsequently receives exactly one IL with common retention a, followed by one Euler step of duration h. Between pressure generation and integration, no write changes EPI, capacity or support, and there is no intervening pressure refresh or pressure writer other than the declared IL. The existing IL pressure owner computes p_after=RN(a*p_before); a is the configured operator factor, not a newly fitted coefficient or an actual capacity update.

Let p_before=-e*L*x+F+epsilon_p, let epsilon_IL be the signed difference between stored post-IL pressure and exact a*p_before, and keep the actual integration remainder and post-integration EPI writes separate. Subtracting the two nodal updates gives

\[ d_+=(I-haA)d+h\operatorname{diag}(\nu) (a\Delta\epsilon_p+\Delta\epsilon_{\rm IL}) +\Delta\delta_{\rm int}+\Delta j_{\rm post}, \qquad A=e\operatorname{diag}(\nu)L. \]

The common forcing cancels at its actual generation boundary. Phase changes after generation do not retroactively refresh the stored pressure. A new step needs new admission of its generated source, policy, support and capacity; common endpoint phases alone do not authenticate those intermediate conditions. Unseparated Gamma, clipping or extended dynamics remain in the integration remainder, not silently in a rounding term.

For the single native witness, IL also leaves phase unchanged and the common captured source can be bound at integrator entry. The represented retention is exactly 854047988748571/1125899906842624 (about 0.7585469929947761). Independent arithmetic checks the paired formula with maximum absolute residual 3.531e-17, fully separated into its observed components. This conditional finite identity does not prove future class invariance, a uniform numerical error bound or maintenance of either individual target profile.

When these conditions hold over a finite sequence, the same identity can be composed step by step, re-admitting each generated source and consumed state. No common source need be constant in time for it to cancel between branches. That cancellation says nothing about preservation of an individual fixed profile: its compatibility residual P_H*(b_k-A_k*z0) can change even when the two branches share b_k exactly. The target-compatibility observer in forced_support.py owns that separate test. IL scales the entire stored pressure in this restricted route, including both the EPI transport contribution and the common source. It is not an additional EPI reset, nor does its presence alone prove faster relaxation.

11.7 EPI resets with pre-generated pressure¶

The same nodal accounting identifies a distinct conditional model when a glyph changes EPI but preserves pressure already generated at x. Suppose a declared ideal affine reset is J(x)=S*x+c, the ordered node space and capacity remain fixed, and the following Euler call consumes the original pressure for f(x)=-A*x+b. Before numerical and clipping defects,

x_next=S*x+c+h*(-A*x+b).

This follows by adding the reset displacement and the nodal increment. It is not Euler with refreshed pressure at J(x). The pre-generated result minus the refreshed result is exactly h*A*(J(x)-x). For two branches with the same admitted S,c,A,b,

d_next=(S-h*A)*d.

For sequential Reception, S is the ordered product of the individual neighbor-blend rows, because later targets read earlier EPI writes. It is not the simultaneous immutable-snapshot stage matrix. For a common unclipped Emission boost, S=I and c cancels between branches, giving d_next=(I-h*A)*d. Neither expression is the pressure-scaled common-IL map.

A runtime link still requires the actual per-node order, scalar chart, mix, neighbor readouts, clipping domain, unchanged pressure and integration boundaries. The observed reset defect and its propagation through later Reception rows, pressure-generation defect, integration remainder and later EPI writes remain separate. These formulas neither make the complete binary64 reset globally affine nor supply a global gain. The retained five-step window provides candidate records for this next admission task; its current common-IL observer correctly declines the EN and AL steps.