Inherited form dynamics and observation

Fine-to-coarse form, inherited pressure and metric, tetrad reconstruction and changing support.

Section numbers are stable locators across this document family. The parameter reference owns the reading map; the execution plan alone owns active tasks. Each result retains its stated model and scope.

12. Intrinsic response from a closed fine nodal model

The constitutive-origin review has a positive, restricted candidate: differential relaxation of internal EPI modes induces an autonomous normalized-shape response and a changing relaxation rate. Both follow from an already closed fine model. Neither requires selecting a new potential, reinjecting a diagnostic or fitting pressure from an observed derivative. This is a derived effective response; identifying it with primitive capacity, synchronization phase or a persistent macro-NFR is a separate obligation.

12.1 Mechanism inventory and reuse decision

The review separates actual dynamical constructions from their names and read-outs. It covers the following owners, not every historical repository claim or every possible TNFR completion.

Route and existing owner	Reusable mechanism	What it does not derive
Reversible EPI transport: structural_diffusion.py, forced_support.py	Closed fine linear evolution, fixed metric, modal decay and exact removal of held forcing/profile drift. These supply the candidate below.	Initial support, a variable primitive capacity law or a sustaining source.
Projection and memory: epi_memory.py, structural_morphism.py	Exact hidden-mode feedback, projectability, inherited coarse coefficients; the reciprocal kernel obeys Hbar*K(0)=C^T*H*C.	Laws for the fine model's supplied capacity/forcing, autonomous selection of a partition or a new energy source.
Capacity feedback/localization: capacity_feedback.py, capacity_localization.py	Exact consequences of configured Coupling/Euler maps and held capacity contrasts.	The origin of their gain, event timing or prepared contrast.
Phase and adaptation: phase_evolution.py, adaptation.py	Shared implementations and explicit admission/parameter contracts.	The identification of capacity with angular speed, or a fundamental law selected by pressure/Si thresholds.
Auxiliary geometry: symplectic_substrate.py, variational scope	A specified harmonic model and explicit graph-field realizability tests.	A bridge from its ambient flow or snapshot bilinear contractions to the missing nodal laws. Existing P2 obstructions remain applicable.
Gauge, coherence geometry and transitions: gauge.py, coherence_geometry.py, phase_transition.py	Conditional geometric identities, level sets and transition diagnostics.	A new restoring force merely from their geometric or criticality labels.
Birth and changing support: birth/transport, remesh.py	Executable canonical transformations, finite causal birth/transport and state-dependent reconstruction policies.	A uniquely derived occurrence/selection law for those transformations.
Rich EPI and ontology: epi.py, EMERGENT_ONTOLOGY.md	Richer representations, spectral observations and scoped comparisons.	Autonomous closure from storage dimensionality or identification of those observations with physical particles.

An important distinction is that an emergent observable can have a derived law while the primitive law remains an input. The present candidate takes the already-studied pure EPI channel with fixed positive fine capacities and fixed reciprocal conductance as that input. It does not derive those fine premises or claim to complete the full multichannel engine.

12.2 Shape and rate are induced by the nodal generator

Use the held model and metric already owned by forced_support:

\[ \dot x=-Ax+b,\qquad A=e\,\operatorname{diag}(\nu_i/d_i)B,\quad B=D-W,\quad H=\operatorname{diag}(d_i/\nu_i). \]

Assume connected symmetric nonnegative conductance with positive strengths, positive fixed capacities and e>0. Then HA=eB is symmetric positive semidefinite. With zero held forcing, subtract the conserved H mean. With nonzero held forcing, additionally subtract the exact relative profile from the shared owner; its existing drift identity still gives

\[ y=x-\operatorname{mean}_H(x)\mathbf1-z,\qquad \dot y=-Ay, \qquad \langle\mathbf1,y\rangle_H=0. \]

The intrinsic-origin control uses zero forcing. Allowing a held nonzero source extends the accounting; it does not explain that source's origin. For S=<y,y>_H>0, define

\[ R=\sqrt S,\qquad q=y/R,\qquad \kappa=\frac{\langle y,Ay\rangle_H}{S}. \]

These quantities are computed from current fine state and generator. Direct differentiation, using self-adjointness in H, gives

\[ \dot S=-2\kappa S,\qquad \dot R=-\kappa R,\qquad \dot q=-Aq+\kappa q, \]

\[ \dot\kappa =-2\left(\frac{\langle Ay,Ay\rangle_H}{S}-\kappa^2\right) =-2\frac{\|Ay-\kappa y\|_H^2}{S}\le0. \]

For example, differentiate N=<y,Ay>_H: Ndot=-2<Ay,Ay>_H; the quotient rule for N/S gives the last identity. Also <y,Ay>_H=2*e*E_D(y) and <q,qdot>_H=0. The term +kappa*q arises from differentiating the normalization; it is not a force fed back into the graph. The normalized-shape law closes without amplitude because kappa=<q,Aq>_H on the unit sphere. No primitive capacity has been changed.

The shape is an oriented unit vector, retaining the sign of EPI. Quotienting by positive amplitude does not identify y with -y. A real projective identification would discard that additional distinction. At y=0, shape, rate and their ratio-based derivatives are undefined, not measured zeros. A constant rescaling of the metric rescales S and R, but not kappa; amplitude units inherit the declared fixed metric normalization.

In an H-orthonormal eigenbasis, differential decay selects the lowest-rate eigenspace that has a nonzero initial component. A degenerate eigenspace retains the initial direction within it, so no unique orientation is selected there. A pure eigenmode has fixed normalized shape and constant kappa. The shape selection has a geometric restoring interpretation on this unit sphere, while the original amplitude continues to decay. Connected pure diffusion does not thereby maintain a nonuniform finite-amplitude NFR.

12.3 An exact induced angular response and closed rate law

On an invariant plane with H-orthonormal eigenvectors of rates r_1<r_2, write y=u*v_1+v*v_2, u=R*cos(alpha), v=R*sin(alpha). The fine equations udot=-r_1*u, vdot=-r_2*v imply, away from the zero vector,

\[ \dot\alpha=(r_1-r_2)\sin\alpha\cos\alpha,\qquad \kappa=r_1\cos^2\alpha+r_2\sin^2\alpha, \]

\[ \dot\kappa=-2(\kappa-r_1)(r_2-\kappa). \]

All coefficients are inherited decay rates. This is an induced angular response and a genuinely closed rate equation on the specified plane; it is not the assumption theta_dot=nu, and alpha is not automatically the engine's neighbor-synchronization phase. Basis orientation affects the angle. The strict rate decrease off eigenmodes also excludes nonstationary periodic normalized-shape motion in this fixed reversible model.

The scalar rate law does not generalize by retaining only kappa on every graph. If three distinct rates r_1<r_2<r_3 are present, a pure r_2 mode and a unit-norm mixture of the endpoint modes with energy fractions (r_3-r_2)/(r_3-r_1) and (r_2-r_1)/(r_3-r_1) have the same S and kappa=r_2. Their rate derivatives are respectively zero and -2*(r_2-r_1)*(r_3-r_2). A spectral variance or further state is required. This reuses the existing fiber/projectability criterion rather than treating a scalar summary as a complete macro state.

Exact P3 control. On the unit three-node path with fine capacity one, e=1, zero other source and x=(1,1/4,1/2), the actual neighbor-mean generator gives

H=diag(1,2,1), mean_H(x)=1/2, y=(1/2,-1/4,0),
Ay=(3/4,-1/2,1/4), S=3/8, E_D(y)=5/16,
kappa=5/3, -Ay+kappa*y=(1/12,1/12,-1/4),
spectral variance=2/9, kappa_dot=-4/9, S_dot=-5/4.

Its orthogonal modes (1,0,-1) and (1,-1,1) have rates one and two. Writing zeta=exp(-t), the exact solution is

\[ y(t)=\frac{\zeta(1,0,-1)+\zeta^2(1,-1,1)}4, \quad S=\frac{\zeta^2+2\zeta^4}{8}, \quad\kappa=\frac{1+4\zeta^2}{1+2\zeta^2}. \]

Differentiation using zeta_dot=-zeta yields kappa_dot=-4*zeta^2/(1+2*zeta^2)^2, exactly the two-rate law. This proves an interval identity; rational evaluations are regression controls, not a numerical trajectory campaign. As t increases, shape approaches the first mode, kappa approaches one and R approaches zero. The changing rate did not require a changing fine capacity or an imposed phase schedule.

Finite identity has both shape and retained amplitude. In this same P3 solution, the H-energy fraction in the slower mode and total retained squared amplitude are

\[ P=\frac1{1+2\zeta^2},\qquad F=\frac{S(t)}{S(0)}=\frac{\zeta^2+2\zeta^4}{3} =\frac{1-P}{6P^2}. \]

As P increases from 1/3 towards one, F decreases from one towards zero. Thus increasingly recognizable normalized form and loss of signal can occur together. A finite identity criterion must retain both quantities instead of promoting a normalized shape to finite-amplitude maintenance. This does not make a finite-lived coherent pattern inadmissible; it separates its lifetime from indefinite recurrence.

More generally, two positive decay rates r1<r2, initial modal energy ratio R=E2(0)/E1(0)>0 and Delta=r2-r1 give

\[ P(t)=\frac1{1+R e^{-2\Delta t}},\qquad F(P)=\frac1{(1+R)P} \left(\frac{1-P}{RP}\right)^{r_1/\Delta}. \]

For prospectively declared observation thresholds P_* in [P(0),1) and eta in (0,1], a time satisfying both P>=P_* and F>=eta exists exactly when eta<=F(P_*); strict inequality gives a positive time window. The thresholds define the tested observation, not dynamical coefficients or TNFR constants. At P_*=1 there is no such finite time. This analytic criterion reuses the derived two-mode law; no new graph, trajectory or maintaining force is selected. Controls: finite shape/retention frontier.

12.4 Canonical identification and observation scope

Writing Rdot=kappa*(-R) does not uniquely factor a nodal equation into capacity and pressure. For any positive scale c, the pair (c*kappa,-R/c) gives the same product. The current isolated canonical scalar node has zero neighbor pressure, so assigning it capacity kappa does not reproduce this internal contraction. An inherited internal pressure or explicit coupled environment must be derived through an observation map; it cannot be concealed by relabeling kappa or alpha.

The full tetrad remains tied to the fine primitives. In the P3 control, primitive phases may stay at consensus: phase gradient and curvature then stay zero even while modal orientation changes. Structural potential reads the evolving actual pressure. Coherence length retains its pressure-product and graph-distance provenance (on this small graph, its spectral fallback). Amplitude and the retained source/state are needed to reconstruct these observations; the normalized shape alone does not close the full tetrad.

observe_forced_support_shape in forced_support.py centralizes the exact rational identities, reusing reference rebuilding, relative profiles, the graph Laplacian and fixed metric. It returns the scaled tangent without evaluating a square root or choosing an eigenbasis. The source-state pressure defect stays visible; modeled derivatives are not promoted to derivatives of stale stored pressure, numerical execution or future schedules. The tests in test_forced_support_shape.py cover the exact P3, single-mode and zero-shape cases, covariance, held forcing, cache reconstruction and the isolated-macro-node obstruction.

This supplies a concrete endogenous geometric-response mechanism within a closed TNFR restriction. It does not provide the missing potential Psi of section 13.5 in the variational note, a sustained particle, or a universal closure for primitive phase/capacity. The following exact internal-mode construction tests a stronger, neighbor-coupled response. Other candidate mechanisms above remain references rather than parallel research queues.

12.5 Neighbor-coupled phase and internal pressure from scalar EPI

A fixed Cartesian product P2 square C3 supplies a bounded constructive test. Its six fine nodes have unit conductance, unit positive capacity and pure EPI pressure (e=1); primitive phases may stay equal and all other pressure coefficients are zero. These are declared fine-model premises, not a proof that this graph or configuration forms spontaneously. Each node has three neighbors, so the exact fine generator is A=B/3 and H=3*I. No angular equation is added to that model.

Write the three scalar EPI coordinates in fiber a as

\[ x_a=m_a\mathbf1+u_a p+v_a q,\qquad p=(1,-1,0),\quad q=(1,1,-2). \]

The projections are m_a=sum(x_a)/3, u_a=(x_a0-x_a1)/2 and v_a=(x_a0+x_a1-2*x_a2)/6. The two internal columns are orthogonal with Gram diag(2,6); both have internal combinatorial Laplacian eigenvalue three. Applying the actual fine neighbor-mean law gives, with b=1-a,

\[ \dot m_a=(m_b-m_a)/3,\qquad \dot u_a=(u_b-4u_a)/3,\qquad \dot v_a=(v_b-4v_a)/3. \]

Consequently the internal projection closes for every fine EPI state, independently of fiber means; its zero-mean lift is an invariant subspace. This is both a projection intertwining and an invariant-lift statement, not an inference from one trajectory. In the Euclidean-orthonormal internal chart z_a=sqrt(2)*u_a+i*sqrt(6)*v_a, the induced equation is

\[ \dot z_a=\frac{z_b-z_a}{3}-z_a. \]

The complex coordinate abbreviates two real internal form coordinates. It is not the lossy product of signed scalar EPI with an independently supplied primitive phase studied in section 9. A common internal basis across both fibers is part of this observation map. Independent local basis rotations would introduce corresponding edge connection matrices; plain phase differences could not be retained unchanged.

Where both amplitudes r_a=|z_a| are positive, set z_a=r_a*exp(i*psi_a). The chain rule then derives

\[ \dot r_a=\frac{r_b\cos(\psi_b-\psi_a)-r_a}{3}-r_a, \qquad \dot\psi_a=\frac{r_b}{3r_a}\sin(\psi_b-\psi_a). \]

For two nonzero amplitudes, delta=psi_1-psi_0 obeys delta_dot=-(r_1/r_0+r_0/r_1)*sin(delta)/3. Thus neighbor-coupled alignment is a derived response of internal scalar EPI structure, rather than a prescribed oscillator law or a feedback controller using telemetry. At zero amplitude the linear coordinates remain regular but the corresponding polar angle is unavailable. If only the neighboring amplitude vanishes, the local rate remains defined by rdot=Re(exp(-i*psi_a)*z_b)/3-4*r_a/3 and psi_dot=Im(exp(-i*psi_a)*z_b)/(3*r_a) without assigning that neighbor an angle. The induced psi is not automatically the original fine phase, and the ratio in its equation is not a new primitive capacity.

The internal restoring term is also inherited. Restricting the fine Dirichlet energy to the internal sector yields

\[ E_{\rm int}=\frac12\left(|z_0-z_1|^2+3|z_0|^2+3|z_1|^2\right). \]

Its gradient with mobility 1/3 gives exactly the induced equation. In the rational (u,v) chart the norm uses G=diag(2,6), the inherited metric is 3*G in each fiber, and 3*G*dot c_a=-partial E_int/partial c_a. The local term is the energy of internal edges, not a selected polynomial potential. On general fiber means, the full Dirichlet energy additionally contains 3*(m_0-m_1)^2/2, which decouples from this internal sector.

This derives a restoring contribution for macro form, not the missing primitive-capacity potential Psi of the prior P2 test. At equal nonzero z_0=z_1, internal relaxation still gives rdot=-r. The existing canonical macro-P2 formula instead has zero pressure when scalar form, phase, capacity and degree are uniform. Therefore the current scalar pressure formula is not preserved unchanged by this internal-mode reduction. The pressure (z_b-z_a)-3*z_a and inherited mobility 1/3 describe the derived vector-form model; they are not installed as a replacement engine law. Multiplying observations by exp(t) to erase the loss would conceal physical relaxation, not derive maintenance.

Indeed, S=|z_0|^2+|z_1|^2 satisfies

\[ \dot S=-2S-\frac23|z_0-z_1|^2. \]

The sum and difference modes decay at rates one and 5/3. Nonzero internal structure is not sustained: S(t)<=exp(-2*t)*S(0). The alignment response does not imply a self-maintaining NFR, sustained oscillation or a particle. Fine support, conductance, capacity and clock remain supplied premises.

Exact controls in test_internal_mode_pushforward.py reuse the existing rational transport and energy owners. Rational 1/3 belongs to this exact-real diffusion model, not an assertion that a binary64 mean, finite solver step or live event realizes it without residual.

12.6 What this changes in the research question

Three identifications remain distinct: a closed internal observation, a macro nodal realization with inherited pressure and metric, and a pattern maintaining nonzero internal amplitude indefinitely. The construction establishes the first and an explicit vector-form realization of the second. It disproves unchanged scalar pressure inheritance here, and its decay identity excludes the third within this fixed passive model. It does not exclude finite-lived coherent identity or settle persistence of phase/topological structure. Reconstructing the full fine tetrad also needs any discarded means and primitive phase data on which its fields depend. Internal polar coordinates alone are not a complete canonical triad or diagnostic replacement. Section 13 completes that finite identification gate; the single execution plan owns the remaining maintenance question.

13. Faithful macro state and tetrad inheritance on the retained prism

Keep exactly the fine model of section 12.5: unit P2 square C3, unit capacity and EPI coefficient, fixed conductance and unit path lengths, with other pressure channels disabled. The present statements concern the exact real diffusion law and its stated observation maps. They do not replace stored pressure with model pressure or promote binary64 capture to exact arithmetic. Primitive phases are separately retained diagnostic inputs; no new evolution law for them is assumed.

13.1 Four internal coordinates close, but omit a pressure direction

Let c=(u_0,v_0,u_1,v_1) be the four internal coordinates of section 12.5, P their six-by-four lift, R_int their left-inverse projection, and define

\[ \mu=(m_0+m_1)/2,\qquad \delta=m_0-m_1,\qquad s=(1,1,1,-1,-1,-1)^\top. \]

Every fine form has the exact decomposition

\[ x=\mu\mathbf1+\frac{\delta}{2}s+Pc. \]

The dynamics gives mu_dot=0, delta_dot=-2*delta/3, and the already derived four-dimensional law for c. Consequently (c,delta) is a closed five-coordinate state. Its lift with mu=0 reconstructs centered EPI and the full exact model pressure. The pressure map -A has rank five and kernel span(1); c omits the independent direction s. Thus one added coordinate is necessary and sufficient for all-state linear pressure reconstruction. The common mean is needed to recover absolute EPI but not this pressure. This result concerns the entire nodewise pressure vector, not merely its mean.

There is a simple bounded witness with identical c=(1/4,0,1/4,0) and mu=1/2:

x_A=(3/4,1/4,1/2,3/4,1/4,1/2), delta_A=0,
x_B=(15/16,7/16,11/16,9/16,1/16,5/16), delta_B=3/8,
p_B-p_A=-(1/8)*s.

Both states lie inside (0,1). All four internal coordinates and their derived angular/radial response agree, but the complete pressure differs. This does not contradict their autonomous internal law: pressure sufficiency is a stronger requirement than closure of those four observations.

The existing exact partition/memory owner independently gives closed triangle means, Hbar=9*I, and RAQ=QAP=K(0)=0. Its supplied mean coordinates cannot observe the four internal directions, and those directions cannot drive the means in this fixed model. A zero memory kernel is therefore not evidence that internal structure has disappeared or is reconstructed. The current generic realization API already serves partition outputs; this finite signed-coordinate proof needs no second realization engine.

13.2 Inherited metric, restoring pressure and coordinate dependence

In the full chart (mu,u_0,v_0,u_1,v_1,delta), the pullback of H=3*I is

\[ H_{\rm chart}=\operatorname{diag}(18,6,18,6,18,9/2),\qquad E_D=\frac32\delta^2+E_{\rm int}. \]

Its inverse metric acting on the negative energy gradient recovers the complete chart dynamics. The common mean has zero energy gradient. This derives the coordinate mobilities once the fine metric, energy and chart are specified; the nodal product identity alone still does not choose a unique pressure/capacity factorization.

For the internal normalized complex coordinate z_a=r_a*exp(i*psi_a), the inherited metric where r_a>0 is

\[ 3\,|dz_a|^2=3\,dr_a^2+3r_a^2\,d\psi_a^2. \]

Hence rdot_a=-(1/3)*partial E_int/partial r_a and psi_dot_a=-(1/(3*r_a^2))*partial E_int/partial psi_a. These yield the radial and angular equations already derived in section 12.5. Their state-dependent angular coefficient has a geometric origin in the change of coordinates; it is not an independently evolving primitive nu_f. The singular polar metric at zero amplitude requires the regular Cartesian internal coordinates there, not an invented angular value.

Retaining (m_a,u_a,v_a) in each triangle gives two vector-form units and an invertible six-coordinate chart. It is an exact realization of the fine law, without a reduction of its scalar state dimension. Retaining only (c,delta) discards the conserved common mean and therefore is not a faithful representation of absolute EPI or every canonical operator contract.

13.3 The potential kernel must be inherited too

On the unit prism, every node has three neighbors at distance one and two other nodes at distance two. Let W be its unit adjacency and J the all-ones matrix. The actual inverse-square potential kernel is

\[ K=\frac{3W+J-I}{4},\qquad \Phi_s=Kp. \]

Its eigenvalues are 7/2 on the common mean, 1/2 on s, -1/4 on the two common internal directions and -7/4 on the two opposed internal directions. None vanishes. In particular K*(-A) has rank five, so the same five linear EPI coordinates are necessary and sufficient to reconstruct the entire model potential on this fixed metric. The bounded witness above gives Phi_B-Phi_A=-s/16:

Phi_A=(1/16,-1/16,0,1/16,-1/16,0),
Phi_B=(0,-1/8,-1/16,1/8,0,1/16).

These displayed fields use exact model pressure. On the second fixture the actual fresh binary64 pressure has residual epsilon*(1,0,0,0,-1,0), epsilon=2^-54; its stored fresh-pressure potential consequently differs from Phi_B by (epsilon/4)*(-1,0,3,0,1,-3). The portable controls retain these measured defects. They do not present exact model identities as zero-defect runtime closure.

The stronger issue is that exact nodal-flow reduction does not imply that recomputing the same field formula on the coarse graph preserves its meaning. Let R average each triangle and T lift two constants. Exact multiplication gives

\[ \bar K=RKT= \begin{pmatrix}2&3/2\\3/2&2\end{pmatrix},\qquad RK=\bar K R. \]

The diagonal two is inherited from two distinct fine neighbors within the same triangle. It is not a self-interaction inserted into the fine graph. For pure EPI, Rp is antisymmetric and R*Phi_s=(1/2)*Rp. The mean quotient has inherited capacity 1/3 and canonical scalar pressure p_macro=(m_1-m_0,m_0-m_1)=3*Rp, so

\[ R\Phi_s=\frac16p_{\rm macro}. \]

By contrast, computing the usual self-excluded potential directly on a two-node macro graph with positive distance ell gives Phi_macro=-p_macro/ell^2. Its sign is opposite for nonzero contrast; no positive length fixes the mismatch. Using Rp instead of p_macro still gives the wrong sign. To preserve the averaged fine observation with canonical macro pressure units, the inherited source kernel is Kbar/3. This proves a failure of unchanged scalar potential inheritance while supplying the correct observation map. It does not change the canonical fine-graph potential definition.

13.4 Full tetrad dependency and representation boundary

Fine observation	Required information on the declared prism	Result of the identification gate
Phi_s	Actual pressure and the fine path-length kernel; model pressure is reconstructible from (c,delta)	Four internal coordinates are insufficient; the inherited aggregation kernel differs from a new scalar macro graph.
Phase gradient	Primitive fine phases and support neighbors	Internal EPI orientation does not supply these data. Even fiber-constant phases have different fine and macro normalization.
Phase curvature	Primitive fine phases, circular resultants and availability	Nonlinear neighbor resultants must remain defined; a modal angle does not replace primitive phase.
xi_C	Static coherence products derived from pressure, path distances and fit policy, or a separately identified spectral fallback	This unit prism has only two positive distance bins; the fit requires at least three. The read-out uses spectral_gap, with ideal scale sqrt(3/2), not a fitted coherence-correlation length.

For a direct phase witness, hold full EPI and unit capacity fixed and set primitive phases to zero in one triangle and pi/3 in the other. Pure-EPI pressure, potential and all internal modal coordinates stay unchanged. Every edge satisfies strict U3. The fine phase gradient is pi/9 at every node and curvature is -atan(sqrt(3)/5) in the first triangle and its opposite in the second; all resultants are nonzero. With consensus primitive phase both fields are zero. A scalar macro P2 with those two phases instead has gradient pi/3 and curvature (-pi/3,pi/3). Internal neighbors matter to the inherited diagnostics even though the mean EPI law closes exactly. This pressure independence uses the declared zero phase-channel coefficient.

The spectral fallback for xi_C is identical across these pressure/phase witnesses because the held graph is unchanged. It supplies no evidence that a fitted correlation length is generally pressure-independent. Retaining all primitive phase data is sufficient for the phase-sector read-outs; no minimal phase reconstruction theorem is claimed. When those data are fixed by preparation, that restriction must accompany the reduced state.

Pressure freshness and arithmetic remain separate. Model pressure can be reconstructed from five coordinates in exact arithmetic; an arbitrary stored pressure cannot. Changing stored pressure alone changes the corresponding potential without changing those coordinates. Actual fresh binary64 pressure can also differ from the exact-real model and must retain its measured realization residual. Exact-model sufficiency is not a proof of runtime state compression, finite-step intertwining or the future tetrad.

The portable controls share the fixture in _internal_mode_fixture.py, and reuse the existing transport, exact linear algebra, partition/memory, pressure-capture and field owners: macro-state proof and actual tetrad observations. The preceding nine internal-mode controls retain their behavior after fixture centralization. No new evolution law or certificate API is introduced.

The finite gate is complete: a closed internal response, a minimally extended exact pressure state and inherited diagnostic maps are distinguished. The construction still supplies neither a dynamically selected partition nor autonomous nonzero maintenance. Restoring the missing coordinate or correct potential kernel cannot overcome the passive loss proved in section 12.5.

14. Causal support versus changing geometry

The target of this retained branch is nonzero internal nonuniform form, measured by S=|z_0|^2+|z_1|^2 in the fixed coordinates of section 12.5. A conserved common EPI mean is not that target; neither is this target a definition of every NFR. This section separates three questions: whether changing conductance can feed this form, whether canonical source pressure can compensate its loss, and whether the source itself has a closed endogenous evolution. The first two admit scoped answers; the third remains open. The existing transport, regional-balance and forcing owners suffice.

14.1 Positive geometry alone cannot supply internal amplitude in this family

Keep the same six-node prism, partition and fixed internal basis. Give every triangle edge a common conductance a(t)>0, every matching cross edge a common conductance b(t)>0, and every node the same capacity nu(t)>0. Keep a fixed pure-EPI pressure coefficient e>0, with no other channels. Explicit edge lengths remain one; conductance is not a changing path metric. With d=2a+b, projection of the fine nodal equation gives exactly

\[ \dot z_0=\frac{e\nu}{2a+b}\{bz_1-(3a+b)z_0\},\qquad \dot z_1=\frac{e\nu}{2a+b}\{bz_0-(3a+b)z_1\}, \]

\[ \dot S=-\frac{2e\nu}{2a+b} \left(3aS+b|z_0-z_1|^2\right)<0\quad(S>0). \]

There is no a_dot or b_dot in this fixed-coordinate norm. The identity holds along any admitted coefficient history, including state-dependent ones that retain these symmetries. It supplies no law selecting that history. Arbitrary edgewise changes, heterogeneous capacity, changing partitions, support events and additional pressure channels are outside this result.

Pointwise positivity alone does not prove that the form vanishes at infinite time. In the fixed orthonormal sectors z_+=(z_0+z_1)/sqrt(2) and z_-=(z_0-z_1)/sqrt(2), the positive rates are

\[ k_+=\frac{3e\nu a}{2a+b},\qquad k_-=\frac{e\nu(3a+2b)}{2a+b},\qquad z_\pm(t)=z_\pm(0)\exp[-I_\pm(t)],\quad I_\pm(t)=\int_0^t k_\pm(\tau)\,d\tau. \]

For locally integrable rates, each occupied sector vanishes exactly when its own cumulative rate diverges. For one fixed supplied coefficient history, all internal initial vectors decay if and only if I_+(infinity)=infinity, because k_->k_+. For state-dependent coefficients this criterion must be applied along each solution, not transferred from one history to all others.

For example, the supplied counter-history a=exp(-t), b=nu=e=1 has I_+(infinity)=(3/2)*log(3) and I_-(infinity)=infinity. Its common internal sector retains amplitude factor 3^(-3/2) and squared amplitude factor 1/27. It loses effective internal transport asymptotically; this passive remnant is not derived active maintenance or recovery. No conductance law or numerical trajectory is proposed by this counterexample.

14.2 Geometry work can increase energy while the form shrinks

The existing moving-conductance identity is

\[ \dot E_D=(Bx)^\top\dot x+\frac14\sum_{ij}\dot W_{ij}(x_i-x_j)^2. \]

Its second term changes the energy assigned to a given form. It need not feed the internal coordinates. On the unit prism, take c=(1/4,0,1/4,0), both means 1/2, and e=nu=1. At that state S=1/4, S_dot=-1/2 and E_D=3/8. A declared coefficient jet a_dot=3, b_dot=0 gives

nodal_work       = -3/4,
conductance_work =  9/8,
E_D_dot         =  3/8 > 0,
S_dot           = -1/2 < 0.

observe_support_transport_derivative supplies this exact balance on a detached model-pressure snapshot. It does not execute or justify the supplied coefficient jet. The sign contrast can also persist over a bounded interval: under the declared history a=1+3t, b=nu=e=1, 0<=t<=1/12, the same common internal sector has S_dot/S=-6a/(2a+1)<=-2, whereas E_D_dot/E_D=3/a-6a/(2a+1)>=9/35>0. Thus a positive geometry-energy budget alone cannot establish maintenance of the actual form, even while coefficients stay positive and bounded.

14.3 Canonical phase pressure can compensate the loss instantaneously

Return to unit conductance and unit common capacity. For each triangle let y_i=x_i-m_a, so S=sum_i(y_i^2). Capture canonical non-EPI forcing F before evaluating any rate; never reconstruct it from a desired answer. Write the stored pressure as p=e*g_epi+F+epsilon, where epsilon is the sum of the captured pressure-assembly defect and stored-pressure residual. The exact represented-state budget is

\[ \dot S=-2eS-\frac{2e}{3}|z_0-z_1|^2 +2\sum_i y_i F_i+2\sum_i y_i\epsilon_i. \]

This is the nodal rate evaluated at the captured state, not a completed numerical step. The two existing regional observers give sum(E_region)=3*S/2; multiplying their summed variance-rate balance by 2/3 reproduces all three terms. Both defect sources remain visible. The degree-three topology gradient vanishes, and uniform capacity has zero capacity gradient. If primitive phase and capacity are constant inside each triangle, every non-EPI source channel is fiberwise constant and its internal projection vanishes. Multiplying by fiberwise-constant capacity preserves that statement, although the displayed simple S budget requires unit common capacity. A single collective phase per triangle cannot supply this internal drive. The zero mean/internal memory kernel of section 13.1 does not hide a sustaining source either.

A nonuniform internal primitive phase gives a positive witness. Prepare

e=1/2, w_phase=1/4, w_vf=1/4, w_topo=0,
nu_i=1, m_0=m_1=1/2, c=(1/16,0,1/16,0),
theta=(0,pi/3,pi/6) in each triangle.

Every support edge satisfies strict U3. Each node sees one copy of each phase, with nonzero phasor resultant and ideal direction pi/6, so the ideal phase channel is (1/6,-1/6,0) in each triangle. Consequently

\[ S=1/64,\qquad \dot S_{\rm passive}=-1/64,\qquad \dot S_{\rm source}=1/48,\qquad \dot S=1/192>0. \]

The actual pressure-capture owner retains represented source coefficients and their exact projected work. Regression controls evaluate the represented phasor arithmetic independently and distinguish it from the ideal 1/48 source work and 1/192 total rate; a historical binary64 fraction is not a portable model constant. The retained witness has positive source work above passive loss. Pressure assembly and stored-pressure defect work are separate from the phase-arithmetic difference. The ideal source sums to zero: internal redistribution can increase internal amplitude without increasing the common mean; the represented mean contribution must be retained independently. This is evidence for instantaneous source compensation under declared coefficients, not a law sustaining that phase arrangement or a self-maintaining NFR trajectory.

14.4 Capacity and causal closure remain explicit dependencies

Within-triangle capacity variation breaks the four-coordinate autonomous law. For capacities (1,2,1,1,1,1), adding an EPI constant delta only to the first triangle leaves c unchanged but changes its exact model rate by (e*delta/6,-e*delta/18,0,0). The hidden mean enters through diag(nu). At e=1/2, delta=3/16, this is (1/64,-1/192,0,0). A capacity extension must therefore retain enough state and derive its law; it cannot inherit the uniform-capacity closure by assumption.

The source-compensation result localizes the missing mechanism: a closed evolution must generate or preserve the required internal primitive source structure and account for its response as EPI changes. The internal modal angle derived from passive EPI is still not primitive phase. Existing phase steppers declare additional angular-rate and coupling premises; their availability does not derive those premises from the EPI equation. Neither holding the prepared phase forever nor choosing a restoring gain would resolve that origin question. No complete autonomous candidate or maintenance trajectory is admitted here. This is not a general nonexistence result for multichannel or hybrid TNFR dynamics.

Portable controls reuse the shared fixture and existing physics owners: geometry and exposure, source and capacity budgets. No production evolution rule, feedback controller, energy implementation or certificate API is added. The single execution plan owns the next gate.