Child Coupling and changes of the structural reference

Status: Exact event/target balances, restricted capacity/profile and local phase-response identities, bounded canonical response controls, and a self-consistent spatial envelope, finite pressure first-passage result and observed finite mean-deficit repayment for fixed-source carried C6 evolution; complete-carry-cell trapping obstruction, finite seven-cell escape proof, and a finite graph-owned bridge from the original preparation to its phase tail. Recovery of an earlier relative pattern requires a fixed comparison target. Research links: B2.d.8-B2.d.45/Q3/O3.a, S1, S2, S3, S8, S9, S10 and S16.

Research status: The regional-response and C6 continuations recorded here are parked. Their hypotheses, obstructions and historical follow-ups remain reusable evidence; they are not active instructions. Scientific resumption belongs only to the current G3 gate.

1. Closing an existing structural feedback path

THOL birth and transport established an actual child node followed by an admitted parent-child connection. Forced support balance then separated the retained graph's relative profile from its mean drift during physical EPI evolution. A child-target Coupling event now tests another existing path: the evolved EPI enters the compatibility score, actual links change the pressure neighborhoods, and refreshed nodal flow changes EPI.

The child must already have a U3-compatible neighbor. Equal phase alone does not admit Coupling on an isolate. The functional candidate sample, phase gate, EPI/sense compatibility and actual edge commit must be captured. The child-target selector is a declared policy; the study does not derive autonomous target selection from the resulting profile.

The canonical Coupling kernel preserves EPI during its event. Its capacity synchronization uses the old compatible neighbors, before functional edges are added. Default bidirectional phase updates also use that old neighborhood. New links can subsequently change the phase pressure even when the stored phases are unchanged, because the neighbor phasor sets have changed.

2. What singleton capacity synchronization predicts

Suppose the child's only old compatible neighbor is its parent, with parent capacity one and child capacity nu_c=1-delta, 0<delta<1. The existing default synchronization coefficient is s=COUPLING_GENTLE, whose ideal expression is 1/(4*pi). The canonical capacity proposal is

\[ \nu_c^+=\nu_c+s(1-\nu_c),\qquad \delta^+=(1-s)\delta. \]

This describes the exact companion using the represented coefficient; the actual binary64 write has its own observable arithmetic residual. The initial production capacity 0.95 likewise denotes its represented value, rather than the exact rational 19/20.

If the original cycle and single child edge of weight omega remain unchanged, all other capacities remain one, and no phase contribution changes, the weighted capacity forcing is

\[ b_{\nu}=\frac23w_{\nu}(\omega-1)\delta, \qquad b_{\nu}^+=(1-s)b_{\nu}. \]

The total forcing numerator scales this way only when the other weighted channel contributions vanish. Even in that case the mean drift changes with the metric mass:

\[ \frac{\bar v^+}{\bar v^-} =(1-s)\frac{Z^-}{Z^+},\qquad Z=16+\omega+\frac{\omega}{\nu_c} \]

for the C8 with one child edge. A smaller capacity mismatch therefore does not by itself establish the same reduction factor for mean drift.

The default functional-link branch can leave this singleton geometry. If the child ends with k edges into otherwise unit-capacity cycle vertices, with positive weights omega_j and no other added edges, then the exact unweighted capacity gradients give

\[ g_{\nu,c}=\delta,\qquad g_{\nu,j}=-\delta/3\ \text{at each adjacent cycle vertex}, \]

\[ \sum_i d_i g_{\nu,i} =\frac{2\delta}{3}\left(\sum_{j=1}^k\omega_j-k\right). \]

Adding links can increase the magnitude of this negative contribution despite a smaller deficit. Unequal link weights at reflected cycle vertices can also remove weighted phase cancellation while preserving the same stored phases. The actual refreshed channel decomposition is needed to interpret a changed total drift.

The reflected preparation gives a concrete nonlinear phase identity. Let eta=UM_theta_push, with ideal existing phases theta1=(1-eta)*pi/4, theta7=-theta1 and child/parent phase zero. After child edges to vertices 1 and 7 are added, vertex 1 has neighbor phasor sum 2+i, hence

\[ g_{\phi,1}=\frac{\arctan(1/2)}{\pi}-\frac{1-\eta}{4}, \qquad g_{\phi,7}=-g_{\phi,1}. \]

The default coefficient makes g_phi,1<0. The other equal-strength reflected pairs cancel in the weighted sum, while parent and child phase pressure vanish. Thus

\[ b_{\phi}=w_{\phi}(\omega_1-\omega_7)g_{\phi,1}. \]

For omega1<omega7, this term is positive and opposes the negative capacity contribution. The cause is a changed neighbor-phasor geometry; no independent phase evolution has been introduced. These expressions are the ideal chart identities. Actual stored phases and represented phasor arithmetic must be checked against the captured production kernel before attributing the measured cancellation to them.

Direct UM pressure attenuation is a separate operator write. The postevent held reference uses independently captured canonical forcing and a refreshed full pressure; an attenuated stored pressure must not be substituted for that reference.

3. Same-EPI events change both the model and its coordinates

Let two valid forced-support references describe the states before and after one event, on the same ordered node set. Each has connected symmetric positive conductance, positive capacity and the profile construction of the preceding note. Index their conductances, Laplacians, metric matrices, forcing, drift and centered relative profiles by 0 and 1.

At a canonical UM event the actual EPI field x is unchanged. Define

\[ m_j=\frac{h_j^\top x}{Z_j},\qquad u_j=x-m_j\mathbf1-z_j,\qquad j\in\{0,1\}. \]

The reference-induced coordinate change is exactly

\[ \delta u=u_1-u_0=(m_0-m_1)\mathbf1+z_0-z_1. \]

The change m1-m0 is a reweighting of the same EPI values. It is not an accumulated physical change of EPI during the event. Likewise the profile shift z1-z0 changes the comparison target. Neither is an additional term in the nodal evolution law.

The old reference may have been derived before a preceding physical flow interval. Its source EPI is not automatically the current event EPI. Reset observations therefore require explicit actual before/after snapshots, validate each under its own reference and enforce identical node order and EPI across the event.

4. Exact variance and Dirichlet error resets

For V_j=u_j^T H_j u_j/2, direct expansion gives

\[ \begin{aligned} V_1-V_0 ={}&\tfrac12u_0^\top(H_1-H_0)u_0\\ &+u_0^\top H_1\delta u +\tfrac12\delta u^\top H_1\delta u. \end{aligned} \]

These are the metric change, the signed interaction with the changed reference, and its quadratic term. A decrease of V need not describe any improved physical EPI field: that field is identical at the two event endpoints.

For the relative Dirichlet error E_j=u_j^T B_j u_j/2, the corresponding identity is

\[ \begin{aligned} E_1-E_0 ={}&\tfrac12u_0^\top(B_1-B_0)u_0\\ &+u_0^\top B_1\delta u +\tfrac12\delta u^\top B_1\delta u. \end{aligned} \]

The metric term reuses the existing conductance-reset identity by holding the error field u0 fixed across the old and new supports. The same helper applied to the actual EPI field instead gives

\[ \tfrac12x^\top(B_1-B_0)x, \]

the raw Dirichlet reset. Raw structural energy and error relative to a profile are distinct observables. The error coordinates are detached data; they are not written into live nodes or treated as new nodes.

Each subsequent held-support flow interval has the separate mean and Euler defect budgets from the preceding block. A finite event/flow sequence combines those interval budgets with the explicit resets above. Its changing weighted mean additionally includes each m1-m0 reweighting. Telescoping such terms is exact finite accounting; it does not prove nonincrease under arbitrary later events.

5. A fixed old-profile observable prevents a moving-target conclusion

Retain the original metric H0, projection P0=I-1*h0^T/Z0, and original relative profile z0. Define

\[ R_0(x)=\tfrac12(P_0x-z_0)^\top H_0(P_0x-z_0). \]

The target here is specifically the old derived relative profile, not the finite observed EPI field at an earlier time. It also ignores uniform EPI shifts by its declared projection. This fixed observable can be evaluated after capacity or conductance changes without asserting that the old dynamics still govern the graph.

Because the UM event preserves x, it preserves R0(x) exactly. Meanwhile its new-model error V1 can rise or fall due to the reset terms. Improvement of a changing-reference score alone therefore cannot establish recovery toward the old target.

Even exact future relaxation under the new held model has a distinct limit for this comparison. If its coefficients remain fixed and its unrestricted-chart error decays, then

\[ R_0(x(t))\longrightarrow \tfrac12(P_0z_1-z_0)^\top H_0(P_0z_1-z_0). \]

This value is generally nonzero even though the new-model error tends to zero. Actual finite trajectories must report both observables and their mean/pressure defects. Clipping and later structural events invalidate a direct promotion to that asymptotic reference.

6. Shared observations and the bounded comparison

The existing forced_support.py now centralizes the profile coordinates in both old state observations and the new observe_forced_support_pattern read-out. That read-out accepts only the ordered node identifiers and EPI alongside the retained reference. It keeps the original profile and metrics fixed without certifying a current pressure law.

observe_forced_support_reset takes both references and explicit event snapshots. It rebuilds their public cached fields, validates each snapshot under its own held model, and reuses support_transport.py for the raw and error-coordinate conductance resets. Its signed metric-first energy decomposition carries a checked exact residual. These detached observations do not establish operator admission, pressure refresh or a causal link between arbitrary supplied records.

The 19 independent controls in test_forced_support_reset.py include references predating the event, separate metric/profile changes, fixed old-profile comparisons after dynamics changes, and a direct counterexample to a moving-target recovery claim. In that example, unchanged EPI has positive old-profile error and exactly zero error under a newly chosen reference; the reset terms fully account for the apparent improvement.

The executed comparison in child_coupling_feedback.py retains the actual born and connected graph after the preceding 24 physical segments, at time 6.5. One case applies public child-target UM; the matched case takes no extra event. Each then executes 12 additional refreshed Euler intervals. The comparison retains both the old fixed-profile observable and each case's current-model error. Its independent runtime checks are in the runtime test suite.

The actual child-target event preserves every EPI and phase value. Its child capacity changes from 0.95 to 0.9539788735772974 and it adds child edges to cycle vertices 1 and 7, with respective weights 0.8791579029688148 and 0.8981657798702134. The old parent-child edge remains. The captured vertex-1 phase matches (1-eta)*pi/4; the wrapped vertex-7 reflection residual is approximately -2.45e-16. This finite check supports the ideal chart calculation above without certifying exact transcendental arithmetic.

The materialized weighted forcing contributions are approximately:

Channel	Before child UM	After child UM
Phase	0	+0.0006063375069147582
Capacity	-0.0003191519727040398	-0.0006920500301710027

The new phase contribution opposes a larger negative capacity term. Including the new metric mass, the reference mean drift changes from -1.8015381161585726e-5 to -4.014215906537859e-6. This is a combined structural effect, rather than the singleton capacity prediction alone.

At unchanged EPI the current-model variance jumps from 0.06603591731695654 to 0.9294668175027274. Its reset terms are approximately +0.03686666543873662 from the metric, -0.3938392738714086 from the reference interaction, and +1.2204035086184428 from the quadratic reference shift. Their exact rational identity residual is zero. The raw EPI Dirichlet reset is approximately +0.029393575677005273; the relative Dirichlet error reset is +0.46220825274344396. They also have exact zero budget residuals. The weighted mean reweights by approximately -0.018281219168014524 despite no EPI change during the event.

Both cases then execute 12 intervals of duration 1/4, reaching time 9.5. They share the initial fixed old-profile variance R0=0.06603591731695654 and give:

Continuation	Fixed old-profile variance at 9.5
Child UM	0.14196498883949968
No extra event	0.021532984486475837

In the child-UM case, the new-regime variance nevertheless decreases from 0.9294668175027274 to 0.5219712930713153. Thus a reduced reference drift and relaxation toward the new profile coexist with increased distance from the old target over this measured horizon. The finite comparison does not establish recovery toward that target.

The full represented states, reset terms and later interval budgets are generated as artifacts/research/child_coupling_feedback.json by

.venv313\Scripts\python.exe -X utf8 benchmarks/child_coupling_feedback.py

This tests one finite structural action and its subsequent evolution. The conditional singleton calculation is not silently substituted for the default functional-link branch or treated as an additional run. The operator policy and all measured intervals remain explicit.

7. Exact full-event accounting on the same ordered nodes

The new observe_forced_support_event in the shared forced_support.py separates an actual EPI jump from accompanying reference and metric changes. Each endpoint must satisfy its own reference's fixed-model domain: symmetric nonnegative conductance with connected positive-conductance support, positive weighted degrees, positive capacities and positive EPI coefficient. Node count and order agree across the event. The existing same-EPI observer retains its stricter EPI-equality contract.

For actual endpoint fields x0, x1, let

\[ \delta=x_1-x_0,\qquad a=\frac{h_0^\top\delta}{Z_0},\qquad v=\delta-a\mathbf1=P_0\delta. \]

In the original reference, u0=P0*x0-z0 and the algebraic midpoint error is u_mid=P0*x1-z0=u0+v. Direct expansion gives the EPI-jump contributions

\[ J_H=u_0^\top H_0v+\tfrac12v^\top H_0v,\qquad J_B=(B_0u_0)^\top v+\tfrac12v^\top B_0v. \]

The first term in each is signed; positive quadratic terms do not imply a positive total. Apply the same-EPI reset of section 4 at field x1, changing from the old to the new reference. With its reset differences R_H, R_B and mean reweighting r_m, the complete endpoint identities are

\[ V_1(x_1)-V_0(x_0)=J_H+R_H,\qquad E_1(x_1)-E_0(x_0)=J_B+R_B, \]

\[ m_1(x_1)-m_0(x_0)=a+r_m. \]

The complete relative-error change likewise equals the centered EPI jump v plus the reset's reference-error shift. Totals telescope along compatible endpoint/reference chains. The decomposition's separate cross, quadratic and metric terms need not cancel under reversal.

The midpoint is an algebraic evaluation of post-event EPI in old coordinates, not an observed intermediate graph or an internal operator write order. The public ForcedSupportEvent exposes it only as a ForcedSupportPattern; the copied pressure used internally to validate the reused reset is not exposed as observed midpoint pressure. Both actual endpoints and references are rebuilt and validated; reference source EPI may predate the event. The observer neither evolves EPI nor certifies elapsed flow, operator admission, pressure refresh, causal provenance or nonincrease.

Independent exact controls are in test_forced_support_event.py. For P2 with old w=1, nu=(1,2), e=1/2, F=(1/4,1/2) and x0=(1,0), the original metric is H0=diag(1,1/2) and profile z0=(-1/6,1/3). The pure jump to x1=(0,1) has H cross/quadratic terms -1,2/3 and B terms -3,2. Simultaneously changing to w=2,nu=(1,1) with the same forcing adds reset changes 23/96,7/16; complete changes are -3/32,-9/16. These hand-solvable balances do not require a canonical runtime campaign.

A separate exact control sets x1=(2,0) and a new reference with w=1,nu=(1,1),F=(1,-1). Its new-profile error is zero while the original H-error increases from 3/8 to 25/24. Even with a real EPI jump, a new zero-error reference does not establish recovery of an older target. The event's pre-reference and a study's fixed target are separate inputs to different existing observers; the event API does not silently choose a target.

8. Matched perturbation and response comparison

The B2.d.9b campaign independently reaches the actual child-UM graph at t=9.5 in each branch, before either pending SHA closure. The target remains the original B2.d.8 H0,z0 derived at t=6.5, not the current post-UM reference. The three branches use unchanged defaults and seed 17:

• U: no additional event.
• P: child VAL followed by IL, then canonical pressure refresh.
• F: the identical VAL/IL prefix, then child UM and pressure refresh.

No pressure refresh is inserted between VAL and IL. Both word validators, actual retained per-target histories and live operator gates admit the declared words. Materialized initial states, controls and RNG provenance agree across all branches. P/F additionally have identical VAL/IL records and complete materialized states at the pre-feedback boundary. Every branch then runs twelve shared Euler intervals of 0.25 to t=12.5; child and parent SHA closures execute separately after measurement. All closures are admitted, and no clipping occurs in these retained runs.

Default VAL changes child EPI from 1.140013776631149 to 1.2307331905030023 and capacity from 0.9539788735772974 to 1.0298941002448299. IL changes stored pressure but preserves EPI and phase in this particular symmetric preparation; its general phase-update path is not absent from the operator. The following refresh is captured separately. Feedback UM changes child capacity to 1.0275152033332051 and neighboring phases at nodes 1,7 from approximately (0.595761,5.687424) to (0.451913,5.831272). It adds no edges: all branches retain eleven edges. Neither a capacity-only perturbation nor an edge-addition explanation matches these actual writes.

The common initial original-profile error is 0.14196498883949968. The matched results and the conditional exact-model limits derived in section 9 are:

Branch	Original-profile error at 12.5	Held exact-model limiting error
U	0.2564525415209221	0.9461915126089225
P	0.2530099369965697	0.9461612626392493
F	0.17618454861095278	0.5998144725644323

With D=R_P-R_U, B=R_P-R_F and G=R_F-R_U, exact arithmetic on the represented endpoints gives

D = -0.0034426045243524145
B = +0.07682538838561694
G = -0.08026799290996935
D - B - G = 0 exactly (displayed decimal values are rounded).

Since D<0, the predeclared classification is no damage in the fixed observable. UM has a positive finite response benefit relative to P and U; this is not a damage/repair result, and B/D is deliberately undefined. All three final errors remain above their common initial error. A lower matched endpoint error does not establish attraction or full-state recovery.

VAL alone changes R0 by -0.0047656984792713; IL and UM have zero immediate effect on R0 because they leave EPI fixed. Subsequent flow accounts for the remaining change. The original-target and current-regime H-variance event/flow telescopes both close exactly. Current-regime Dirichlet budgets remain separate from H variance. The largest captured pressure and Euler endpoint defects are approximately 5.47e-18 and 1.11e-16; these measurements are not solver-error bounds or evidence about an infinite trajectory.

The campaign reuses prepare_child_feedback_endpoint and the shared interval helper in the earlier benchmark; its two old case payloads remain exactly unchanged. Reproduction:

.venv313\Scripts\python.exe -X utf8 benchmarks/structural_perturbation_response.py

The script writes the local ignored artifact artifacts/research/structural_perturbation_response.json, with actual source provenance, full endpoint captures and budgets. The runtime controls cover target continuity, matched preparation, actual writes, independent energy identities, executor/history bindings, refusal and classification boundaries. This finite result closes B2.d.9b; it does not justify a factor or horizon search until recovery appears.

9. Frozen-model compatibility with the original target

The existing references also give a mechanism discriminator without another trajectory or a new observable. Let j denote a post-event held model within the positive-capacity, connected reversible domain of the forced-support balance. Its exact continuous solution has the form

\[ x_j(t)=(m_j(0)+b_jt)\mathbf1+z_j+\exp(-A_jt)u_j(0), \qquad u_j(0)=P_jx_j(0)-z_j. \]

Here t is elapsed time in the held post-event regime, b_j is its mean_drift, A_j=e_j diag(nu_j) L_rw,j, and u_j(0) is centered in the current metric H_j. Reversible connected diffusion with positive capacity makes this centered term tend to zero. The original projector P0 annihilates the uniform drift, so the previously fixed observable has the limit

\[ R_{*,j}=\lim_{t\to\infty}R_0(x_j(t)) =\tfrac12\|P_0z_j-z_0\|_{H_0}^2. \]

Compute this quantity by passing the current reference's relative_profile to observe_forced_support_pattern(original_target, nodes=..., epi=...). The profile need not have zero mean in H0; the shared pattern observer performs the original projection. No second profile solver is needed.

A positive R_*,j excludes asymptotic restoration of the original profile under that unchanged exact model. It does not exclude transient passage through the target or later recovery under a different admitted model. A zero value establishes target compatibility only. Neither value proves the asymptotics of clipped binary64 execution, future operator admission, full-state restoration or autonomous policy selection.

Finite increments require a separate reading. For an actual represented EPI jump d=x_after-x_before, put u=P0*x_before-z0 and v=P0*d. Then

\[ R_0(x_{\rm after})-R_0(x_{\rm before}) =\langle u,v\rangle_{H_0}+\tfrac12\|v\|_{H_0}^2. \]

This same-target identity attributes an immediate change to the actual EPI write. Phase, capacity, conductance or pressure changes that leave EPI fixed have zero immediate contribution; they affect the following evolution. Read those following flow defects under each interval's current model using the existing step observer. Substituting the older recovery target as the current flow model would confuse two different contracts.

If only child c changes EPI by delta, the same identity simplifies to

\[ \Delta R_0=h_cu_c\delta+ \tfrac12h_c\left(1-\frac{h_c}{Z_0}\right)\delta^2, \qquad Z_0=\sum_i h_i. \]

The actual jump's alignment with the pre-event error determines its sign. An operator classified as a grammar destabilizer need not increase this particular target error. The named perturbation must therefore be measured before it can be classified as damage. This exact sign criterion is an observation, not a rule permitting ad hoc EPI writes or selection of a new operator factor after seeing the outcome.

For the retained VAL event, u_c=-0.10281937232096414 and delta=0.09071941387185345. Its exact H cross and quadratic terms are approximately -0.008206125390920804 and +0.003440426911649504, summing to the recorded negative jump. With this fixed pre-event error and metric, positive single-child jumps below -2*u_c/(1-h_c/Z0), approximately 0.21638445016881427, reduce the target error. The actual default VAL jump lies inside that interval. This explains its immediate direction without choosing a new factor or claiming that the interval is a policy invariant.

The calculation continued in section 10 concerns compatibility of the original target with a held post-event model. Equivalently to P0*z_j=z0, require

\[ P_0\operatorname{diag}(\nu_j) \left[-e_jL_{{\rm rw},j}z_0+F_j\right]=0. \]

Connected positive-capacity diffusion then identifies the unique relative profile modulo a uniform field. Every observed R_*,j above is positive: UM reduces the mismatch but does not remove it. A candidate restoring rule must explain how the canonical channels could satisfy this compatibility condition and control the old-target error. A repeated-policy theorem further requires an admitted full-state domain, its preservation and controlled execution defects. None follows from extending the observed word or horizon.

10. Compatibility, channel cancellation and the old metric

The shared observe_forced_support_target now evaluates that compatibility condition and the instantaneous signed balance at an explicit actual snapshot. Both references are rebuilt, the snapshot must match the current model, and all node orders must agree. Let

\[ A=e\operatorname{diag}(\nu)D^{-1}B,\quad b=\operatorname{diag}(\nu)F,\quad r=P_0(b-Az_0),\quad m=P_0z-z_0. \]

The current profile obeys -A*z+b=mean_drift*1. Since A*1=0,

\[ r=P_0Am. \]

This identity gives r=0 if and only if m=0: if P0*A*m=0, then A*m=c*1; multiplying by the current metric weights h^T gives c=0 because h^T*A=0. Connectivity makes m uniform, and its old-metric centering makes it zero. This proves relative-shape compatibility. It does not require zero mean drift. For example, P2 with nu=(1,2), e=1/2 and F=(1,1/2) preserves the zero relative profile while every EPI coordinate drifts at rate one. The existing scalar zero-pressure compatibility test and this vector target test answer different questions.

For an actual old-target error u=P0*x-z0, the held model gives

\[ \dot u=-P_0Au+r,\qquad \dot R_0=-u^\top H_0Au+u^\top H_0r. \]

The first term is signed; an old metric need not make it nonpositive. A hand-solvable P3 counterexample has unit edge conductance, old capacities (1,2,1), hence H0=I, current capacities (1/8,1/8,4), e=1, and held F=0 in both models. Both profiles are zero. At actual x=(1/8,3/4,5/8), u=(-3/8,1/4,1/8), compatibility holds but dR0/dt=11/512>0. This is a detached coefficient control: its zero forcing is not the canonical default forcing for heterogeneous capacities. If instead H0=c*H_current, the same identity reduces to dR0/dt=-c*e*u^T*B*u+u^T*H0*r. With r=0 it is strictly negative away from the target in this connected centered domain. Neither sufficient condition establishes future runtime admission or controls clipping.

decompose_non_epi_forcing reuses the capture owner's exact phase/vf/topology products. It rebuilds support-gradient caches and checks that their sum equals the supplied forcing; it does not rerun or authenticate the nonlinear phase kernel. Project each target-pressure component through P0*diag(nu) to obtain r_a, including the EPI component. Then

\[ r=\sum_a r_a,\quad G_{ab}=r_a^\top H_0r_b,\quad J_r=\tfrac12\|r\|_{H_0}^2 =\tfrac12\sum_a G_{aa}+\sum_{a<b}G_{ab}. \]

Cross terms can cancel. Nonzero channel magnitudes alone do not establish incompatibility, and summing only their squares is incorrect. J_r is a derived diagnostic, not a new physical energy. In the hypothetical exact model initialized at u=0, R0(t)=J_r*t^2+O(t^3), so zero instantaneous dR0/dt there does not imply a compatible target. No graph is assigned that hypothetical state by this observer.

At the supplied actual EPI, the observer also separates the stored-pressure contribution u^T*H0*diag(nu)*(p_stored-p_model). A kernel defect captured at another EPI cannot be reused as the defect at z0. Independent exact checks, including signed cancellation and stale-reference controls, are in test_forced_support_target.py.

11. What the retained U/P/F channels explain

The offline analysis in structural_target_compatibility.py reads the B2.d.9b captures without new trajectories or operator choices. It binds each current reference's EPI, capacity and stored pressure to the recorded actual post-event endpoint. U's first later forcing capture supplies only its checked held phase/capacity/support coefficients; its later EPI and kernel defect do not replace the earlier state. Parent evidence bytes and producer metadata are retained separately from the new analysis manifest.

At the three post-event states at t=9.5, the exact-model read-outs are:

Case	J_r	Homogeneous term	Target-source term	Total dR0/dt
U	0.0120728383	-0.0396520771	0.0761747684	0.0365226913
P	0.0120906598	-0.0389605486	0.0759599729	0.0369994243
F	0.0043113892	-0.0389606073	0.0468645978	0.0079039905

All targets remain incompatible, and all three instantaneous old-target rates remain positive. UM reduces the outward rate in this comparison; it does not reverse it. The stored-pressure contributions are approximately 2.25e-18, 1.84e-18 and 9.55e-19, recorded separately rather than silently included in the exact-model terms.

The phase channel's own squared magnitude actually grows from about 0.0030834147 in P to 0.0032578705 in F. Nevertheless its cross term with the EPI channel changes from +0.0037467097 to -0.0042059349. Better cancellation, not a decrease of every individual channel, explains the smaller combined incompatibility. These are projected target-rate channels, not an identity for every phase-gradient or tetrad diagnostic.

For a symmetric accounting of the P-to-F change, define

\[ \Delta J_a=\tfrac12(r_P+r_F)^\top H_0(r_{F,a}-r_{P,a}),\qquad J_F-J_P=\sum_a\Delta J_a. \]

The phase, capacity and EPI allocations are approximately -0.00777818875, -0.000001046589 and -0.0000000352605; topology is zero. Their exact sum equals the observed residual-energy change. This is a signed algebraic allocation of simultaneous writes, not a set of independently executed channel ablations or a universal causal ranking.

Reproduce the detached analysis after generating or retaining its input:

.venv313\Scripts\python.exe -X utf8 benchmarks/structural_target_compatibility.py

The output artifacts/research/structural_target_compatibility.json is local and ignored. It records the input SHA-256, historical producer manifest and current analysis-source digest; it does not relabel the old trajectory as new execution. Its schema controls reject stale endpoint sources, unmatched coefficients and overwritten input.

12. A canonical capacity-profile family and its obstruction

A useful exact family requires connected simple undirected support with one common positive edge conductance, positive capacities, all represented phases equal to zero, and fixed channel coefficients e>0, w_vf>0, w_topo=0. These are declared restrictions; the zero topology weight and channel ratio are read from the existing defaults in the implementation controls. Equal conductance makes the unweighted capacity neighborhood agree with L_rw:

\[ F=-w_{\nu}L_{\rm rw}\nu,\quad k=\frac{w_\nu}{e},\quad \dot x=-e\operatorname{diag}(\nu)L_{\rm rw}(x+k\nu). \]

Here k is the ratio of existing channel coefficients, not a fitted factor. The common phase has zero phase forcing, and d^T*F=0. Therefore the held profile and mean drift are exactly

\[ z=-kP_H\nu,\qquad \bar v=0. \]

For original/current states inside this family with the same ratio,

\[ P_0z-z_0=-kP_0(\nu-\nu_0). \]

The original relative target is compatible exactly when the capacity increment is spatially uniform. Nonuniform capacities can support a nonuniform profile, but this identity alone does not explain how the capacity pattern is generated or maintained. Heterogeneous conductance or nonzero phase invalidates the simple formula; explicit counterexamples are retained in the controls below. The U/P/F preparation has nonuniform phases and unequal conductances, so this theorem is not substituted for section 11.

The same formula exposes a reuse path: the detached coordinate y=x+k*nu has centered value P_H*y=P_H*x-z, exactly the existing forced-profile error. While capacity is held, y follows the same pure EPI diffusion. Across a capacity event, Delta y=Delta x+k*Delta nu. This is a change of analysis coordinates, not an extra engine state or an authorized EPI write. The coupled capacity/EPI model below reuses this coordinate and the existing event, metric and diffusion budgets; no parallel relaxation solver is needed.

On P2, let the original capacity be (c,c), and the actual new capacity be (c+a,c), a>0. With unit edge conductance,

\[ m=(-ka/2,ka/2),\quad r=\frac{w_\nu a(2c+a)}2(-1,1),\quad R_* = \frac{k^2a^2}{4c}. \]

A single default VAL with a positive capacity increment breaks compatibility. Under zero phase, IL preserves capacity/support and cannot remove that incompatibility after pressure refresh; its immediate stored-pressure contraction is a separate operator effect. Target-only UM has no new edge available on P2. Its ideal affine capacity rule reduces the gap by a'=(1-gamma)*a; binary64 execution must use its actually observed a'. For those represented endpoints, the exact-model limiting-error ratio is (a'/a)^2. A smaller positive gap gives a smaller positive mismatch, not complete restoration.

The capacity-family controls include exact P2/P3 preparations and one actual default VAL -> IL -> UM -> SHA word, with both word validators, live gates, histories and pressure refreshes. VAL breaks compatibility, IL leaves it broken, and UM reduces but does not eliminate the observed capacity gap. SHA is a separately checked closure; its EPI preservation is not recovery. No physical flow interval, repeated word, infinite-time binary64 theorem or invariant full-runtime domain is asserted by this finite control.

B2.d.10 closes the target-residual decomposition and this restricted mechanism/obstruction. The coupled model below uses the same coordinate and event/flow owners. Admission, history, phase, support, clipping and numerical defects remain separate obligations for a complete runtime policy.

13. Joint capacity and form evolution on synchronized P2

The conditional exact model in capacity_feedback.py retains the ordered target/neighbor support of section 12. Let their capacities be (c+a,c), with c>0, 0<=a<=A, and set

\[ k=w_\nu/e,\quad \rho=1-\gamma,\quad s=he,\quad d=x_0-x_1,\quad q=d+ka. \]

Here e>0, w_nu>=0 and 0<gamma<1 are declared pressure/operator coefficients; the runtime study below reads their existing defaults. The duration h>=0 and bounds A,ell,U specify numerical and state domains, not extra physical parameters. The zero-capacity case, changing support and nonzero phase forcing are outside this model.

Target-only UM preserves EPI and changes a to a'=rho*a. Refreshed nodal pressure is then p=(-e*(d+ka'), e*(d+ka')). A held-input Euler interval gives

\[ a_{n+1}=\rho a_n,\qquad q_{n+1}=[1-s(2c+\rho a_n)](q_n-k\gamma a_n),\qquad d_{n+1}=q_{n+1}-ka_{n+1}. \]

This is a coupled recurrence: treating the new capacity as merely a faster clock would omit its change to the forcing profile. The API evaluates detached rational data only; graph EPI still advances through the shared nodal integrator. Multiple exact internal Euler substeps with the same held pressure/capacity have this same total-duration endpoint. Their binary64 rounding and any pressure refresh inside the interval require separate checks.

A sufficient invariant domain is

\[ 0\le a\le A,\quad \ell\le x_0\le U-ka,\quad \ell+ka\le x_1\le U, \quad 0<\ell\le U,\quad kA\le U-\ell,\quad s(c+\rho A)\le1. \]

In lifted coordinates both components lie in [ell+k(c+a), U+kc]. The UM event subtracts k*gamma*a from y0 and reduces the required lower bound by that amount; y1 stays fixed. During flow the lifted update is row-stochastic, with mixing coefficients s(c+a') and s*c, so both coordinates remain in the interval. Converting back gives the same domain at a'. This proves exact-model positivity and bounded EPI over arbitrary finite repetitions. If the interval lies inside the configured hard clip range, exact clipping is inactive. Fitting this positive domain above the configured UM EPI/frequency thresholds is a sufficient scalar precondition, not a proof of full grammar/history admission.

The box can be constructed from initial data without searching: A=a0, ell=min(x0,x1-k*a0), U=max(x0+k*a0,x1), provided the resulting lower bound is positive and the timestep/clip conditions hold. Mere convexity of the lifted flow is insufficient for arbitrary positive EPI: with c=a=1, k=gamma=s=1/2, initial x=(3/16,3/16) gives x'=(0,5/16). These detached rational coefficients are a domain counterexample, not the default runtime fixture or a parameter fit.

Define the uniform multiplier bound

\[ \beta=\max\{|1-2sc|,\ |1-s(2c+\rho A)|\}\le1. \]

The executable finite envelope is

\[ |q_n|\le\beta^n|q_0|+ k\gamma a_0\beta\sum_{j=0}^{n-1}\beta^{n-1-j}\rho^j, \qquad |d_n|\le |q_n|+ka_0\rho^n. \]

The sum is (beta^n-rho^n)/(beta-rho) when the factors differ and n*beta^(n-1) when equal; zero cycles are handled separately. beta<1 implies decay of both capacity gap and spatial EPI disagreement. At h=0 or the equal-capacity full-swap boundary the uniform bound can be one; neither boundary supplies that contraction conclusion.

The mean is also accounted for. For M=(x0+x1)/2,

\[ M_{n+1}-M_n=-\frac{s a_{n+1}}2(d_n+ka_{n+1}),\qquad |M_\infty-M_n|\le \frac{s(U-\ell)a_0\rho^{n+1}}{2\gamma}. \]

The tail follows by summing the box bound on the lifted difference. Thus the arithmetic mean has a limit, but need not equal its initial value. For the current metric mean mu=[c*x0+(c+a)*x1]/(2c+a), the same cycle gives

\[ \mu_{n+1}-\mu_n= \frac{c\gamma a_n d_n}{(2c+a_n)(2c+\rho a_n)}. \]

This is exactly a same-EPI metric reset followed by zero H-mean model drift, already owned by observe_forced_support_reset and observe_forced_support_step. With beta<1, both EPI coordinates tend to the common arithmetic-mean limit; their endpoint error is bounded by half the disagreement envelope plus the mean-tail bound. This is an exact conditional relaxation theorem, not a formation mechanism for a persistent differentiated pattern. The exact controls retain mean identities, boundary cases, the domain obstruction and invalid-input checks.

14. Finite admission and the represented capacity obstruction

The supplied word VAL IL UM UM ... SHA fails the string-based canonical transition validator at UM -> UM; the separate ValidatedSequence check alone accepts it. Both results are retained. No transition policy is changed to obtain a run. The declared finite study instead uses VAL IL, then sixteen UM IL / refresh / Euler blocks, and finally SHA after measurement. Within this zero-phase P2 fixture, each separator IL leaves EPI, capacity and support unchanged. Its raw pressure reduction is recorded, and fresh canonical pressure matches the pre-IL pressure exactly. This verified identity explains why the same coupled recurrence can be used as the block's exact reference.

benchmarks/capacity_feedback.py independently prepares two unit P2 graphs with EPI (0.5,0.5), capacity (1,1), zero phase and seed 17. All operator factors are the defaults. One branch applies those sixteen blocks; the other keeps the post-VAL/IL capacity during sixteen flow intervals. Both end at engine time 4, with h=0.25, before their separate admitted SHA closures. This is a new restricted comparison, not another continuation of the nonuniform-phase U/P/F graph. No factor or horizon search is performed, and the original uniform profile and metric stay fixed.

The data-derived positive box is approximately [0.4746697041,0.5651190317], with A=0.07957747155; its additional capacity-dependent margins are checked at every recorded boundary. The exact reference has beta=0.9084232738 and rho=0.9204225285. Both branches start after preparation at R0=0.0003957858736. Their pre-closure read-outs are:

Branch	Actual capacity gap	Actual R0	Conditional held-profile R_*
UM/IL blocks	0.02111475158	1.397878163e-7	1.129307508e-5
Held capacity	0.07957747155	3.718826587e-5	1.604059727e-4

The lower final error is a finite response benefit, not monotonic recovery: the coupled branch's sixteenth interval increases R0 from approximately 4.802676566e-9 to 1.397878163e-7. In both branches the final EPI lies closer to the original uniform target than the current conditional frozen profile does. A transient crossing must not be reported as a maintained equilibrium. The arithmetic mean changes from 0.5198943679 to 0.5193772453 in the coupled branch and 0.5188997311 in the control.

Each flow uses the existing executor with four default internal Euler substeps and held pressure/capacity. The report binds actual endpoints to that executor's evidence, retains full materialized event records, checks the P2 family and domain at boundaries, and records diagnostics. It separates three signed endpoint contributions:

1. The actual represented UM capacity gap versus rho*a, propagated through the exact nodal endpoint map.
2. Fresh binary64 pressure versus the exact modeled pressure at the actual post-event capacity and EPI.
3. The integrator endpoint versus the exact held stored-pressure update.

Their exact sum is the actual endpoint minus the ideal cycle endpoint. All three contributions are nonzero in this run; none is rounded away to claim exact realization. Existing event and forced-step balances retain metric changes and pressure/execution effects on the mean. The exact-model repetition envelope is reported separately; no uniform runtime error bound is inferred from these finite defects or from finite domain membership.

A decisive production-kernel control prevents that promotion. Let nu0=1+2^-52, nu1=1. The default 0<gamma<1/4 computes nu0 + gamma*(nu1-nu0) as the same binary64 nu0. The positive capacity gap is a represented fixed point, whereas the exact model would reduce it by gamma*2^-52. Default UM is actually admitted and executed in this control, and the capacity-supported relative profile remains nonzero. A positive exact contraction coefficient therefore cannot be transferred to the represented capacity gap. The control's actual EPI is still uniform: a nonzero conditional frozen-profile mismatch is not proof of a positive runtime EPI error floor. Small Euler increments can also round away. This obstruction concerns the capacity map; it does not assert a complete fixed EPI/phase/history runtime orbit.

The runtime controls cover matched preparations, the direct-word refusal, every UM/IL admission, refreshed separator identity, executor inputs, finite box membership, all three error sources, closure and the represented gap fixed point. Reproduce the finite record with:

.venv313\Scripts\python.exe -X utf8 benchmarks/capacity_feedback.py

Its local ignored artifact is artifacts/research/capacity_feedback.json. It carries its own source digest and runtime manifest; the earlier U/P/F artifacts retain their historical source identities. The SHA metric time_to_collapse=+infinity has an explicit numeric_kind: positive_infinity JSON tag; nonfinite structural state or defects still fail strict encoding. B2.d.11 closes this conditional joint recurrence, exact invariant box and finite comparison. The following restricted arithmetic result resolves the unit-capacity boundary without another long word. Full-state admission and action selection remain open. No laboratory correspondence has been tested.

15. The represented capacity class on the unit interval

The same capacity-feedback owner now supplies derive_p2_binary64_coupling_lattice and its detached endpoint observer. The production arithmetic has one shared owner, coupling_capacity_blend in the Coupling kernel. Extraction preserves the existing neighbor mean, subtraction, multiplication and sum in their original order for nonuniform inputs whose intermediate sum is finite. If that sum overflows while the nonnegative mean is representable, the kernel instead rounds the exact represented-input mean before the unchanged blend. The equal-input fixed point now returns its input directly, avoiding a rounded redundant mean; this leaves the present singleton-neighbor unit-binade result unchanged. The shared platform probe also retains REMESH's former private alias; basic format probes are not an exhaustive hardware proof.

Assume IEEE binary64 nearest-even arithmetic with separate operations, fixed neighbor capacity c=1, target capacity 1<=v<=2, and a fixed represented factor g. Write

\[ \delta=2^{-52},\quad u=2^{-53},\quad v=1+j\delta, \quad 0\le j\le2^{52},\qquad T(v)=\operatorname{RN}\!\left(v+ \operatorname{RN}\!\left(g\operatorname{RN}(1-v)\right)\right). \]

The executable reference verifies the strict inequalities

\[ 6g(1+u)<\tfrac12,\qquad 7g(1-u)>\tfrac12,\qquad g<\tfrac18,\qquad g(1-u)>\tfrac5{64}. \]

The actual default factor g=5734161139222659/72057594037927936 satisfies them. They define a proved coefficient class containing that default; they do not cover every allowed UM factor. 5/64 is a rational enclosure used in the proof, not a new coupling setting. The fixed unit interval is a numerical domain restriction, not a physical frequency scale or a scaling theorem for subnormal capacities.

Subtraction is exact on this interval, and every positive product is normal. With m=RN(g*j*delta)/delta, the relative product bound gives g(1-u)j<=m<=g(1+u)j. Final rounding changes the real index j-m by at most one half. Hence indices 0,...,6 are fixed, and every index at least seven strictly decreases. To rule out skipping the terminal index, g<1/8 and representability of j*delta/8 give m<=j/8 by monotonic rounding. For j>=8 the output index is at least seven; j=7 maps to six. These arguments include the upper endpoint v=2. Halfway cases round the final capacity to even; rounding a decrement independently is not equivalent.

The map preserves [1,v]. Its terminal index is therefore

\[ j_\infty=\min(j_0,6). \]

There is also a finite analytic bound. For r=59/64,

\[ j_n\le\frac{32}{5}+r^n\left(j_0-\frac{32}{5}\right),\qquad 59^{512}(5\cdot2^{52}-32)<3\cdot64^{512}. \]

The integer inequality puts the upper envelope below seven after 512 numeric updates for every initial index in the class. Together with the no-skipping result, this proves arrival at six from above. It does not execute 512 graph cycles or admit a repeated UM word; the grammar refusal from section 14 remains unchanged. The result concerns only this fixed numeric capacity map. The 42 lattice controls cover independent endpoints, both directions of final halfway rounding, factor/platform restrictions, cache rebuilding and fabricated endpoint refusal.

In the section 12 capacity-forcing family with independent uniform comparison metric H0=I, the terminal exact frozen-profile mismatch is

\[ R_*^{\rm terminal}=\frac{k^2j_\infty^2\delta^2}{4} \le9k^2\delta^2. \]

Equality holds for initial indices above six. This is a conditional profile calculation after holding the terminal capacities, not a positive lower bound on the EPI that the binary64 integrator will actually store.

16. Uniform stored EPI despite a nonzero exact profile

The finite boundary benchmark prepares five independent unit P2 graphs with zero phase, EPI (0.5,0.5) and capacity (1+j*2^-52,1), for the declared indices 0,1,6,7,2^52. Each graph executes one admitted default UM, one canonically refreshed Euler interval of duration 0.25 with four held-input internal substeps, and SHA after measurement. The uniform comparison reference is prepared independently; it is not an invented ancestor or evidence of a prior perturbation.

Input index	Observed post-UM index	Exact held-profile R_*	Observed post-flow R0
0	0	0	0
1	1	1.248880010e-33	0
6	6	4.495968036e-32	0
7	6	4.495968036e-32	0
2^52	4145214556169080	0.02145926008	0.0003837216997

For indices one and six, UM leaves a positive gap fixed. From seven it reaches six in the observed event. All three then retain EPI exactly (0.5,0.5) through the actual interval, although their refreshed nodal pressures and exact frozen profiles are nonzero. At index six the exact model predicts a total EPI change of about (-1.94e-17,+1.94e-17). The represented pressure contribution plus the integrator endpoint defect cancel that model change exactly. These errors are kept as signed rational values, not discarded with a tolerance. Four smaller stored additions also remain below the local EPI rounding thresholds.

There is no conflict between the two observations: the exact pressure model and finite-precision state storage are different maps. An unchanged EPI array is insufficient to infer zero structural pressure. Nor does unchanged primary EPI/capacity certify a full fixed runtime state: time, derivatives, histories and the later SHA closure have their own effects. The upper-boundary control does change EPI, ruling out a blanket classification of every case as numerical stasis. All cases retain actual gates, factor values, family checks, executor endpoint bindings, diagnostics and separate mean/defect budgets.

The 15 runtime controls also verify strict JSON retention of the tiny nonzero profile mismatch. Generate the local artifact with:

.venv313\Scripts\python.exe -X utf8 benchmarks/binary64_capacity_feedback.py

artifacts/research/binary64_capacity_feedback.json has its own source manifest. B2.d.11 and earlier artifacts remain historical; their source digests are not rewritten after the shared-kernel extraction. B2.d.12 closes this bounded numeric class and its finite EPI distinction, without generalizing to other capacity scales, kernels or complete-runtime invariant domains.

17. Compatible capacity support and differentiated prepared regions

B2.d.13 addresses that structural question using the existing U3 gate and pressure law. Let C be the simple, reciprocal graph of neighbors actually selected by UM at the start of one simultaneous all-target stage. Every row must be nonempty. With its unweighted degree matrix D_C, unit-conductance Laplacian B_C and fixed factor 0<gamma<1, the exact capacity proposal is

nu' = nu - gamma*D_C^-1*B_C*nu.

Thus nu'=nu if and only if capacity is constant on each connected component of C. Distinct components can retain distinct positive constants. Each component preserves its own D_C-weighted capacity mean. These are properties of the configured UM map; the factorization of the nodal equation alone does not select that map or its coefficient. The result concerns simultaneous snapshot proposals, not a sequence of single-target updates.

There is a strict exact energy balance away from that componentwise fixed set. For E_C(nu)=nu^T*B_C*nu/2, p=D_C^-1*B_C*nu and S=(B_C*nu)^T*D_C^-1*(B_C*nu),

E_C(nu') - E_C(nu) = -gamma*S + gamma^2*E_C(p)
                   <= -gamma*(1-gamma)*S.

The inequality follows edgewise from (p_i-p_j)^2<=2*(p_i^2+p_j^2), which gives E_C(p)<=p^T*D_C*p=S. Positivity is preserved by convex averaging. On a fixed finite C, nonconstant normalized-Laplacian modes have multipliers 1-gamma*lambda, of magnitude below one. Conditional repeated exact capacity updates therefore approach each component's conserved mean. Changing phases, new links, represented arithmetic and live grammar are outside that argument.

The common readout is observe_coupling_support; its detached derive_compatible_capacity_balance reuses the existing transport energy and Laplacian kernels. The live reader uses the production phase selector and resolved factors, retains actual neighbor order, and checks reciprocity of the materialized rows. It includes zero-weight support edges in U3 selection even though they carry no EPI conductance. Empty selected rows are recorded as blocked targets with no all-target balance, never as admitted identity updates. The required UM_SYNC_VF flag uses the runtime's Boolean admission: a supported false string disables synchronization and rejects this capacity-update model; an unknown string is invalid. A disabled runtime channel cannot be certified as an active capacity exchange. Full operator admission remains a separate obligation.

A connected pressure graph with two compatible components

The finite control prepares unit triangles (0,1,2) and (3,4,5) joined by edge (2,3). EPI is initially 0.5 everywhere and capacities are (1,1,1,2,2,2). In the split case, the first triangle has phase zero and the second the binary64 value math.pi. These phases and capacities are declared preparation; their formation has not been derived.

The actual U3 selection excludes the bridge and all candidate cross-links. Both selected components are complete triangles, so no compatible missing links remain. One actual default all-target UM preserves phase, capacity, EPI and support. Its raw stored-pressure attenuation is recorded separately; canonical refresh restores the same pressure at unchanged primary inputs. The capacity energy on C is zero while that on the connected full pressure graph is 1/2. Using one graph for both operations would miss this distinction.

The symbolic antipodal preparation does not guarantee an exactly zero represented phasor channel. The current producer retains that channel and reports whether the zero-phase premise actually holds; neither regional synchronization nor a math.pi label establishes it. The capacity gradient is (0,0,1/3,-1/3,0,0) and the topology weight is zero. For the separate capacity-only held model, with captured k=w_vf/e, the existing theorem gives

H = diag(2,2,3,3/2,1,1),     mean_H(nu) = 4/3,
z = (k/3,k/3,k/3,-2*k/3,-2*k/3,-2*k/3),
x_infinity = 1/2 + z,       mean drift = 0.

For fixed coefficients, y=x+k*nu obeys EPI diffusion. The initial range [1/2+k,1/2+2*k] is convex-invariant under refreshed exact Euler steps h<=1/(e*max(nu)). It yields x_A in [1/2,1/2+k] and x_B in [1/2-k,1/2], inside the configured scalar interval at default k. These are conditional exact-model bounds, not a binary64 runtime invariant class or a guarantee of future word admission.

The report's capacity_only_profile_prediction labels this model explicitly. Its applies_to_full_forcing flag requires the extra forcing to vanish exactly. The actual full-forcing reference remains independently retained, including the exact phase contribution, profile difference and weighted mean drift. Even a small nonzero constant drift precludes promoting the capacity-only absolute limiting state to a limit of that full held model. The shared forced balance, rather than a relaxed zero test, supplies the full reference.

Finite runtime discriminator

compatible_capacity_regions.py executes one all-target UM, one physical interval h=0.25 with the shared default integrator's four held-pressure substeps, and an all-target SHA closure after measurement. Both full-word validators, actual target admissions, two-phase stage results, histories, raw/ refreshed pressure, event reference reset, signed flow defects and executor endpoints are retained. An independently reproduced UM proposal is explicitly labeled as a prediction and checked against the actual writes; the executor does not retain those UM proposals itself.

Prepared case	New links	Post-flow EPI, rounded for display	Original-profile error, before -> after
Split 0/math.pi	0	(0.5,0.5,0.504858,0.490283,0.5,0.5)	0.118208 -> 0.113675
Aligned zero	8	(0.508745,0.508745,0.508501,0.483658,0.482510,0.482510)	0.118208 -> 0.099933

The split case's capacity-only conditional exact limit has EPI approximately 0.606103 in the first region and 0.287793 in the second. It is not the absolute limit of full represented forcing with nonzero mean drift. Only the displayed finite interval is executed; no numerical approach to a limit is claimed.

The aligned control starts with the identical graph, EPI and capacity but phases zero everywhere. Its first capacity proposals use the original neighbors: only bridge endpoints change, with exact neighbor means 4/3 and 5/3 before binary64 rounding. Default functional links then add all eight missing edges, producing K6. This is a phase-controlled complete operator comparison with an observed topology change, not a fixed-topology ablation or evidence that one capacity mechanism alone causes the error difference. Both runs keep the original comparison profile; a changed current reference is separately accounted for.

The mathematical and finite runtime controls live in test_coupling_support.py and test_compatible_capacity_regions_runtime.py. Generate the separately manifested local artifact with:

.venv313\Scripts\python.exe -X utf8 benchmarks/compatible_capacity_regions.py

This closes a scoped maintenance discriminator: prepared phase compatibility can protect a capacity contrast while the connected nodal pressure law drives a differentiated EPI profile. Autonomous preparation, repeated complete-runtime maintenance and physical correspondence remain open. In particular, the P2 UM/IL separator result cannot be transferred here: IL's phase-lock neighborhood is not UM's U3-selected neighborhood and includes the bridge. The next dependency is to derive which phase/support conditions survive a complete admitted policy, then connect its preparation to existing birth/attachment or winding mechanisms. No longer horizon, grammar change or fitted restoring pressure supplies that proof.

18. Antipodal phase response under simultaneous UM and IL

B2.d.14 tests that next dependency. The ideal exact antipodal configuration underlying section 17 has a locally expanding mode under the candidate IL separator. This is a limitation of this proposed maintenance mechanism, not a violation of IL's pressure-contraction contract or a universal impossibility result for differentiated TNFR structures.

Exact local composition

Use lifted coordinates (a,a,b,pi-b,pi-a,pi-a) on the same two triangles and bridge. The base is the mathematical antipodal state a=b=0. In a sufficiently small neighborhood, U3 selects only the two triangles, their internal links already exist, and every missing cross-link remains incompatible. All relevant phasor resultants are nonzero. Reflection together with phase conjugation and the within-region interior swaps preserve this two-coordinate chart locally; this symmetry does not prove an invariant neighborhood under repetition.

Let t=UM_theta_push and alpha be IL's phase-lock coefficient. Every bidirectional UM target reads the entire compatible triangle, including itself. Its exact phasor mean is C=Arg(2*exp(i*a)+exp(i*b)). Each receiving node gets three identical proposals, whose shortest-arc displacements are averaged. The derivative is consequently

       [ 1-t/3     t/3   ]
U(t) = [                 ].
       [ 2*t/3   1-2*t/3 ]

IL includes the bridge in its full neighborhood. The interior phasor mean has linear part (a+b)/2. The first bridge vertex sees 2*exp(i*a)-exp(-i*b), whose argument has linear part 2*a+b. Therefore

           [ 1-alpha/2   alpha/2 ]
M(alpha) = [                    ],       P = M(alpha)*U(t).
           [ 2*alpha        1    ]

det(I-P) = -alpha^2*(1-t) - 2*alpha*t/3.

For every 0<alpha<=1 and 0<=t<=1, this determinant is negative. The real monic characteristic polynomial evaluated at one is negative and tends to positive infinity, so P has a real eigenvalue strictly above one. Thus even the preceding UM averaging cannot make this exact local phase fixed point attracting. This is an analytic characteristic-polynomial certificate, not an eigenvalue estimated from the finite probes. At alpha=0, the constant two-coordinate direction is neutral. The algebraic control t=0 is excluded by canonical UM admission; no runtime coefficient is changed to obtain it.

For the predeclared direction a=b=epsilon, UM is exact-real identity and the first-order IL response is (epsilon,(1+2*alpha)*epsilon). The quadratic readout Q=| (a,a,b,-b,-a,-a) |^2/2 = 2*a^2+b^2 therefore has linearized gain 1+4*alpha*(1+alpha)/3, about 1.52 at the default coefficient. Q measures phase deviation from this base; it is neither EPI energy nor a claimed global Lyapunov function. Individual other directions can contract despite the existence of an expanding mode.

The exact reference and tangent observer extend the existing coupling_support.py owner. They rebuild public reference caches, preserve exact or represented coefficient values and infer no graph or execution provenance. The unchanged IL default 0.3 now has one owner in _coherence_stage_kernel.py, shared by direct validation, direct proposals and simultaneous stages. This removes repeated defaults without claiming a new derivation of that coefficient.

Finite controls and the pressure distinction

antipodal_region_phase_response.py predeclares epsilon=0,+/-2^-12,+/-2^-16, regional phases epsilon and pi-epsilon, and otherwise the same preparation as section 17. Each case executes one all-target UM, refresh, one all-target IL, refresh, one h=0.25 nodal interval and an all-target SHA closure after measurement. Both word validators, live admissions, an independent strict IL readiness check, actual two-phase stages, histories, metrics and proposal/endpoint comparisons are retained. Independent proposals are labeled predictions, not sealed executor-retained objects.

Prepared epsilon	Observed post-IL bridge tangent, approximately	Observed phase Q gain
+2^-12	+0.000390624965076	1.519999847413
-2^-12	-0.000390624965076	1.519999847413
+2^-16	+0.000024414062492	1.519999999411
-2^-16	-0.000024414062492	1.519999999411

The exact local scalar response on this direction is epsilon+alpha*(atan(3*tan(epsilon))-epsilon). The runtime controls compare against that independently evaluated expression as well as the linear model. Their observed-minus-linear residual includes nonlinear, binary64 and chart effects; two magnitudes are not a numerical proof of an asymptotic derivative.

In the zero-input control, IL writes approximately 3.67e-17 to node 2's phase. This is a retained represented-arithmetic residue, not exact antipodal invariance; the initial zero phase energy makes its gain undefined (None). Its represented phase-pressure channel can already have a small nonzero residue before IL, and refresh retains the post-event residue as well. Both are compared with the declared finite arithmetic budget, not coerced to zero. The symbolic base uses mathematical pi, not the stored value math.pi.

In all four nonzero controls, IL increases canonical C(t) immediately by contracting stored pressure. After canonical refresh, C(t) instead falls below its pre-IL value because the changed phases produce new pressure. For epsilon=+2^-12, the three values are approximately 0.99352517 -> 0.99508085 -> 0.99351353. This is consistent with the local operator contract: refresh is a separate pressure realization. Neither raw C(t) improvement nor preserved capacity proves preserved phase structure. Capacities, EPI and support remain unchanged through both stages; the following nodal interval changes EPI while holding the captured phase/capacity/support inputs fixed. Source decomposition and signed event/flow budgets retain that connection to dEPI/dt=nu_f*DeltaNFR.

The exact controls independently derive the six-node source/receiver tangent before comparing the reduction. The runtime controls check both signs, scalar response, null residue, actual admission, shared defaults, coherence scopes, flow endpoints and strict artifact serialization. Generate the separately manifested local artifact with:

.venv313\Scripts\python.exe -X utf8 benchmarks/antipodal_region_phase_response.py

Consequence for the next structural test

This heading records the historical dependency developed in the following comparisons; it does not assign a current maintenance campaign.

The mean-of-phasors derivative has entries R_ij=cos(theta_j-m_i)/|sum_neighbors exp(i*theta)| for neighbors, where m_i is their mean argument. Its rows sum to one but need not be nonnegative: the antipodal bridge has coefficients (1,1,-1). Treating this response as ordinary positive diffusion would erase the expanding mode. The existing phase-pressure, winding and coupling-support work should share this local response description before another maintenance policy is proposed.

The uniform-winding C6 comparator under the default all-target bidirectional UM merge is now delivered in the cycle/winding owner. The shared exact Gram/receiver response preserves the antipodal obstruction and gives C6 strict centered tangent contraction plus a separate exact phase-map invariant neighborhood. Its five finite default controls retain winding and nodal response. The same owner now supplies nonlinear phase oscillation contraction and a conditional exact joint phase/capacity/EPI reserve. In its strict Euler regime, EPI tends to its initial mean while phase approaches a rotated winding. Three finite two-cycle controls audit admission; the null retains a numerical phase residue that prevents a defect-free transfer of the homogeneous contraction bound. The additive extension now accounts for those retained errors with finite signed reserve and mean budgets. Its separate uniform range bound requires independently justified future error and signed mean-prefix hypotheses; it does not turn the six observed intervals into a runtime invariant class. The held Euler arithmetic now has an independent local rounding bound; exact addition cells explain the retained null EPI stasis despite nonzero pressure and changing phase. A zero-sum held-input fixture still produces mean drift. Those results resolve local integration arithmetic while leaving phase/pressure accuracy and uniform signed mean control open. The new opposite-pair continuation derives a finite nonlinear phase/EPI partial quotient and a distinct mean-preserving binary64 pure-channel class. The retained full winding execution fails its paired-pressure and common-binade premises, localizing the subsequent historical analysis to joint pressure-pair leakage and compatible rounding cells. A certified two-neighbor midpoint now centralizes the eligible IL and CPU phase-pressure realizations, correcting a retained phasor rounding error. The same finite comparison improves pressure-mean cancellation while retaining nonzero k1 EPI mean drift. Local accuracy does not supply a future signed-mean bound. Exact inverse pressure boxes now show that balanced pressures reproduce the same biased k1 trace, and a two-binade law explains that bias at EPI 0.5. This rules out a pressure-only repair inside those boxes. A separate explicit remainder-carrying nodal reference now bounds executor mean error by the terminal rounding remainder, independently of the supplied finite step count. It preserves reconstructed mean under balanced nodal area; pressure-realization error still requires its own bound. Graph adoption requires explicit pressure-readout, operator/reset, history and certificate contracts (section 20). The restricted flow/event implementation supplies those contracts for EPI-preserving UM/IL/SHA with live visible-pressure refresh, persistent carry and whole-graph rollback (section 21). Its matched finite branch comparison separates actual source changes from executor rounding. Broader jumps and generated-pressure summability remain open. Section 22 now excludes an exact binary64 zero-pressure state throughout the EPI band for one retained fixed phase slice. Its carried-cell horizon separates a visible-state change from a band exit. Joint UM/IL trapping and signed source bounds remain open; fixed-phase nonconvergence does not decide them. The detailed phase continuation in section 23 reaches an exactly closed default UM/IL phase map from the inherited null, with a nonzero periodic source mean. The signed compensation required from refreshed EPI arithmetic is now explicit; phase closure alone does not provide it. The full local analysis in section 24 gives exact generated compensation on one finite cell prefix and its loss at the first refresh after a cell exit. Independent-coordinate traps and permanent residence in the 13-state control class are excluded under the declared per-step conditions; correlated trapping and full-state reachability remain open. The derivation is kept there rather than duplicated in this feedback note. Section 25 adds exact whole-itinerary carry compatibility: a feasible closed visible word is a carried cycle only when its total nodal area vanishes coordinatewise. Two derived cell boundaries show canonical pressure-mean sign reversals, while their carried path remains open. The four observed states retain strict negative node-1 pressure, excluding a carried cycle confined to them despite the changing mean source. Section 26 reaches positive node-1 pressure through a neighboring EPI transition with the inherited carry intact. It centralizes bounded cell-exit replay, derives uniform strict-sector residence bounds and separates index change into nodal area and carry transfer. The first positive-pressure hit still has negative accumulated node-1 area; section 27 follows its two positive steps to the next cell boundary, where pressure becomes negative again before compensating that area. The exact affine budget separates this observed prefix from a hypothetical four-step crossing. A two-level integer constraint rules out short exact return at the fixed step and capacity, without excluding a broader bounded class. Additional canonically reachable pressure levels and correlated trapping must pass the shared six-coordinate budget and carry checks. Section 28 reaches a third node-1 pressure in eleven steps. Node 1 overshoots its earlier loss while the complete vector remains unbalanced. The three-level scalar length constraint is unchanged. Independently, node 4 retains a positive pressure across the three observed visible states, giving a uniform 43-step residence bound with the shared finite drift observer. A larger candidate class must change that drift or fail its confinement test, rather than rely on the recovered node-1 balance. Section 29 makes the frozen-stencil bound executable. A neighbor changes at step 15, ahead of the retained-carry center deadline 21, and lowers node 4's pressure while leaving it positive. The declared eight-boundary sign test ends censored at eighteen steps. The remaining exact gradient cut and the full nodal-area identity constrain any proposed coupled reachability argument. The shared gradient observer rejects uniform area errors that would be invisible to a Laplacian-only comparison. This supplies a return-class discriminator, not a longer live campaign or a stability conclusion; existential carry and reachability stay distinct.

Section 30 completes B2.d.32-B2.d.36 through a canonical centered reference, a carried-step identity, exact spatial contraction, a uniform numerical-defect tube and a separate finite band budget. It reuses the forced-support Poisson solver for w_epi*L*z=A-mean(A)*1 and verifies y_next=(I-h*w_epi*L)*y+h*P*(w_epi*L*r+eta) on all eighteen retained steps, where P removes the arithmetic mean, y=P*X-z, X=x+r, and eta contains the actual EPI-product and channel-assembly errors. At the unchanged default weight and h=1/16, the exact centered norm factor is 573161353023261791/2^59; its square is the homogeneous energy factor. The uniform rounded-defect calculation gives the energy envelope a floor near 6.6560e-31 on [3/8,5/8]. Combining the spatial envelope and a distinct absolute mean bound proves a sufficient conditional numerical-band horizon of 196713720348826219 steps without executing them. This is a fixed-source numerical-map result, not a live phase/grammar or future-runtime certificate.

The retained eighteen-step mean change is exactly -131/(3*2^114), split into phase source -3/2^113, product/assembly rounding -113/(3*2^114) and zero carry feedback. That first shape envelope left the indefinite signed mean budget and node 4's nonpositive-pressure cut unresolved.

B2.d.37-B2.d.40 now connect those questions through the same canonical pressure, carried-state and forced-profile owners. The self-consistent envelope uses its own spatial restriction to close the rounding bound, giving an energy floor approximately 2.958228394578814e-31 and a uniform per-node product/assembly error bound 6.731922543446726e-32. The static seven-point certificate admits seven states at exactly the retained endpoint's reconstructed mean and verifies a positive rational combination of their full pressure vectors equal to zero. This refutes a strictly signed fixed linear drift on that enclosed class; its algebraic coefficients do not define an executable switching rule or prove any of those points reachable.

The finite passage theorem shows that strictly positive node-4 pressure cannot persist through all first 30255 transitions: the derived centered growth would leave the spatial envelope while the independent finite band guarantee still holds. Thus a nonpositive readout must occur at an index no greater than 30254, starting from the B31 endpoint. The decisive continuation with its inherited carry reaches the first hit in 118 steps and 59 visible cell boundaries. Readouts 0..117 are positive; the endpoint has gradient index -22 and pressure -187043320717485/2^105.

The same finite run repays the B27 mean deficit by overshoot at step 59: its net mean area changes from -1/(3*2^114) to 1/2^113. The final local mean area is 41/2^112, split into phase source -59/(3*2^113), product/assembly rounding 305/(3*2^113) and zero carry feedback. Relative to B27 the final mean area is 217/(3*2^114), while every net coordinate area remains nonzero. This closes the declared finite sign and deficit-crossing questions, without an exact vector return or an infinite compensation theorem.

B2.d.41 adds the complete-cell obstruction: any nonempty finite family in B37's gradient class fails forward invariance when each visible tuple admits its entire legal carry fiber. The theorem constructs an outward hypothetical carry; it does not continue or establish escape of the retained state. B2.d.42's stronger claim for B38's particular seven displayed rows comes from their exact temporal graph. Only self-loops and the zero-based edge 5 -> 4 are feasible. Every legal incoming carry leaves that family or fails band admission within 77 steps. Whole-band exit, entry from B40 and a general recurrence theorem are not implied. The static convex pressure balance therefore remains an algebraic certificate whose seven cells cannot themselves form a closed temporal class.

B2.d.43's finite live bridge now executes the original null C6 preparation through 89 UM/IL pairs and terminal SHA, retaining 179 events and 356 pressure-refreshed carried flows. It reaches the same fixed phase source as B40 with its own actual EPI and carry: tail-entry offsets (-3,0,2,-1,6,-5) at time 22, and offsets (-3,0,2,0,8,-5) immediately before SHA at 22.25, all in units 2^-54 from .5. Every flow passes independent replay; all six nodal-balance residuals vanish and the accumulated mean area is 3293/(3*2^114). No detached remainder is imported or reset. SHA then changes capacity from one to 0.9204225284540524, so the observed unit-capacity premise cannot be extended to a later live invocation. The finite bridge closes for this declared carried solver; it does not authenticate B27-B40's different historical detached state or prove indefinite trapping.

The historical proposed extension was forward inclusion of carry subcells restricted jointly with visible shape and signed accumulated vector area, or a recurrent-budget obstruction. A continuation of the B43 branch would have to use its actual tail carry and closure under the fixed-map conditions; a B40 analysis must remain identified as a separate detached branch. Enlarging a complete-cell family inside the same gradient class cannot evade B41. Repeating the node-4 sign search or seeking a fixed strict linear separator over the full certified class cannot remove those obstructions. This is parked research, not the current G3 gate. Detailed proofs and shared owners remain centralized in the winding note; none of these blocks advances the saved B40 endpoint. B2.d.44 now excludes the local energy tube with independent mean bounds around B43's actual pre-SHA mean. Opposite canonical mean increments and exact carry-cell translations provide hypothetical outward points at both boundaries. B2.d.45 preserves the coordinate arithmetic restriction as well: a derived affine carry lift keeps the requested mean and satisfies an analytic energy bound in a slightly smaller local window. Neither result advances the saved B43 state or proves escape of its actual trajectory. An admissible proposed trapping region must couple mean boundaries to shape and individual carries; the independent local mean interval is insufficient even after this arithmetic restriction. The earlier target-only winding theorem remains distinct. Represented joint invariance, autonomous preparation, physical correspondence, eventual antipodal U3 crossing and complete-runtime long-time behavior remain open; none of these open boundaries authorizes extending the failed nominal antipodal run.