Forced regional response and environmental readouts

Regional balances, represented phase reduction and environmental-input geometry.

Part of Forced support balance and model boundaries. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

7. A region and its environment on the same nodal support

The full-network balance does not by itself identify which mechanism supports a particular region. For a fixed proper nonempty node set R, retain the full-graph conductance, strengths d_i, positive capacities and e>0 from section 1. Connectivity is unnecessary for the identities below, but every full strength must be positive. Removing the environment and renormalizing an induced graph changes the model and is not this observation.

Let h_i=d_i/nu_i, Z_R=sum_R h_i,

\[ M_R=\sum_{i\in R}h_i x_i,\qquad m_R=M_R/Z_R,\qquad z_i=x_i-m_R,\qquad V_R=\frac12\sum_{i\in R}h_i z_i^2. \]

M_R is a weighted EPI total, with no identification as physical mass. V_R measures internal EPI contrast about the region's own weighted mean; neither quantity alone defines coherence, identity or biological activity. For an edge oriented from the region to its complement, define the outward EPI current J_ij=W_ij*(x_i-x_j). Direct substitution of the nodal equation and cancellation of opposite internal currents give

\[ \dot M_R=-e\sum_{i\in R,j\notin R}J_{ij} +\sum_{i\in R}d_iF_i. \]

The complementary region receives the opposite cut current. Since sum_R h_i*z_i=0, differentiating the regional mean contributes zero to the fixed-metric variance derivative. Pairing internal edges yields

\[ \dot V_R= -e\sum_{\{i,j\}\subset R}W_{ij}(x_i-x_j)^2 -e\sum_{i\in R,j\notin R}W_{ij}z_i(x_i-x_j) +\sum_{i\in R}d_i z_iF_i. \]

The first sum counts each internal undirected edge once. It is nonpositive. The boundary and non-EPI source terms are signed: either can supply or remove regional contrast. Loops contribute to d_i but carry zero EPI current. Zero-weight support edges contribute no EPI current; their influence on the other canonical channels remains in the independently captured F.

For stored pressure p, retain epsilon_i=p_i-(-e*(Bx)_i/d_i+F_i). The stored-rate balances add sum_R d_i*epsilon_i and sum_R d_i*z_i*epsilon_i, respectively. When a forcing observation is available, split this discrepancy into its fresh-kernel arithmetic defect and stored-minus-fresh pressure residual. Do not redefine F from the observed derivative to remove either term. Linearity also separates the phase, capacity and topology contributions.

These identities use one fixed region and metric. If support or capacity varies continuously, differentiation includes metric-variation terms; birth, membership changes and discrete events require their own endpoint budgets. An instantaneous equality is not a finite-time restoration or invariance theorem. Even exactly balanced source and loss at one instant do not prove that the conditions supplying that balance persist.

The single executable owner is observe_regional_support_balance. It rebuilds snapshot caches, preserves the full source node order and accepts an explicit ordered region plus an independently supplied source vector. The observer checks four exact identities: model and stored-pressure rates for both weighted total and variance. It does not evolve the graph or choose its region. Public detached records carry no causal execution seal.

The independent regional tests include a three-node path with x=(0,1,5), unit edges, capacities (1,2,1), e=1, F=0 and R={0,1}. Here h_R=(1,1), m_R=1/2, internal dissipation is 1, boundary work is 2 and dot V_R=1: internal diffusion can coexist with growing regional contrast. Complementary total fluxes cancel. This control demonstrates why whole-network smoothing cannot replace a regional budget.

Conditional child response is not an autonomous child region

For a selected child set B with no internal edges or self-loops, fixed external EPI and the same held canonical source give, for each child,

\[ \dot x_i=-e\nu_i(x_i-x_i^*),\qquad x_i^*=\frac{\sum_{j\notin B}W_{ij}x_j}{d_i}+\frac{F_i}{e}. \]

The conditional relaxation rate is e*nu_i. This is the diagonal block of the existing nodal generator, without an added restoring force. If the environment changes, this conditional profile changes too and is not a fixed future target. Dependence on the environment is consistent with a relational NFR; the open question is how the coupled system generates and maintains the relevant region and its supporting boundary conditions.

9. Finite regional observation with a held nodal rate

On the same full support and positive metric as section 7, hold the stored nodal rate r_i=nu_i*p_i for one declared duration dt>=0. For an observed endpoint x_plus, let y=x+dt*r and delta=x_plus-y. Both are detached algebraic coordinates; constructing them does not execute a nodal step. Write mean_R(v)=sum_R h_i*v_i/Z_R and v_c=v-mean_R(v) on the region. The exact finite identities are

\[ M_R(x_+)-M_R(x)=dt\,\dot M_R(x)+\sum_{i\in R}h_i\delta_i, \]

\[ V_R(x_+)-V_R(x)=dt\,\dot V_R(x) +\frac{dt^2}{2}\sum_{i\in R}h_i(r_c)_i^2 +\sum_{i\in R}h_i(y_c)_i(\delta_c)_i +\frac12\sum_{i\in R}h_i(\delta_c)_i^2. \]

The dotted terms use the stored-pressure balances of section 7, including the independently identified canonical source and stored-minus-model pressure discrepancy. That discrepancy may contain an intentional IL pressure contraction; it is not automatically floating-point error. The quadratic rate term is nonnegative. Endpoint discrepancy can contain rounding, clipping or a different supplied endpoint and remains signed in its linear cross term. Uniform endpoint shifts change the weighted total but not centered variance. Changed capacity, support or region invalidates this fixed-metric identity and requires additional budgets.

This identity supplies finite accounting, not solver authentication or a future stability theorem. An offline runtime application must separately bind the starting pressure, actual duration, integration boundary and later events. In particular, a phase update after integration can change the future canonical source while leaving the just-completed EPI increment unchanged; it must not be retroactively substituted into the consumed rate. The interruption checkpoint and frozen regional observation contract are in the single execution plan.

The shared executable owner is observe_regional_support_euler. It reuses the instantaneous regional observer, rebuilds both snapshots and rejects changed node order, full conductance, support or capacity. Its 24 focused tests cover direct heterogeneous examples, a mean-only uniform shift, clipping, zero duration, invalid metric/support changes and detached cache tampering. Together with the 79 existing support/forcing tests, 103 tests pass.

Regional contrast is not a universal coherence score

V_R measures dispersion about a region's current weighted mean. It is not the canonical C=1/(1+|DeltaNFR|+|dEPI|), nor error relative to a predeclared nonuniform pattern. Boundary forcing can create a structured difference between children while diffusion dissipates internal differences. Increasing V_R therefore need not be instability, just as decreasing it need not be restoration of a particular form. A finite signed budget identifies those mechanisms without defining an NFR by one favorable scalar trend.

For example, when two independent children have the same fixed capacity, their conditional equation from section 7 gives dot(x_i-x_j)=-e*nu*((x_i-x_j)-(x_i_star-x_j_star)) under held external inputs. A nonzero target contrast can cause the squared contrast first to decrease and then increase when the signed difference crosses zero. This is an algebraic property of the stated stable scalar response, not a proof that the actual moving environment follows the held-input model. Actual IL pressure writes and finite-step defects must still be retained.

11. Phase-source relevance at a fixed regional state

To distinguish a phase-readout difference from a difference in the nodal drive, hold x, nu, W, e, the channel weights and region fixed. Let two declared phase arrays produce captured canonical sources F_a and F_b. The exact source components from the shared forcing owner use represented unit phase gradients; their trigonometric evaluation is not replaced by a linearized phase model. With Delta F=F_b-F_a, the model pressure and nodal rate differences are

\[ \Delta p_{\mathrm{model}}=\Delta F,\qquad \Delta\dot x_{\mathrm{model}}=\operatorname{diag}(\nu)\Delta F. \]

Since the state, region and metric have not changed, neither have their mean, internal dissipation or boundary work. Section 7 immediately gives

\[ \Delta\dot M_R=\sum_{i\in R}d_i\Delta F_i,\qquad \Delta\dot V_R=\sum_{i\in R}d_i z_i\Delta F_i. \]

When only phase changes, Delta F=Delta F_phase; capacity and topology source differences vanish. These identities reuse the full-graph regional owner and the independently captured source. They do not reconstruct a force from an endpoint derivative or introduce a new evolution law.

The fresh represented kernel also has a discrepancy epsilon_kernel relative to this model, so Delta p_fresh=Delta F+Delta epsilon_kernel. If the recorded stored pressure is retained, its actual nodal rate remains unchanged in both detached readings. Its stored-minus-fresh residual changes by -Delta p_fresh. Thus a changed candidate fresh pressure is not a changed historical increment or an observed alternative future. A regional sum can also vanish despite a nonzero pointwise source difference; retain both.

The execution plan fixes two already archived phase alternatives, the original endpoint and nine existing regions. The baseline fresh capture must first reproduce the archived capture exactly. The test concerns the relevance of an established enumeration ambiguity, not another phase-coordination experiment or a maintenance theorem. An exact matching baseline validates that retained input on the current numeric path; it does not make a two-input comparison a uniform robustness bound.

Retained two-phase comparison

thol_phase_source_relevance.py implements that fixed protocol. It authenticates the native, phase-audit and regional-identity inputs, binds the complete original coordination boundary and checks the saved label/enumeration projections. Saved output phases are already aligned to original node identities. Both fresh captures use the original endpoint node and neighbor order, with the retained cached channel weights and admitted default NumPy pressure branch. The full baseline capture reproduces the archive exactly before the alternative is evaluated.

The alternative phase array changes the phase-source component at seven of sixteen nodes. Its largest absolute difference is approximately 0.04914323880773078; the largest difference in the fresh-kernel arithmetic discrepancy is approximately 2.610120553818296e-17. These are separate observed quantities, not a universal floating-point error bound. Capacity and topology contributions, regional metrics, boundary work and internal dissipation remain fixed. All nine source-difference identities close exactly.

Fixed region	Baseline model variance rate	Alternative model variance rate	Difference
Pair 0	-0.399116639750073	-0.399116639750073	exactly 0
Pair 1	+0.007133454707669	+0.000771577106453	-0.006361877601216
Pair 2	-0.400949399259614	-0.397283880240531	+0.003665519019083
Pair 3	+0.000771577106453	-0.005590300494763	-0.006361877601216
Pair 4	-0.399116639750073	-0.399116639750073	+7.997873694454377e-18
Pair 5	+0.000771577106453	+0.000771577106453	exactly 0
Pair 6	-0.399116639750073	-0.399116639750073	exactly 0
Pair 7	+0.000771577106453	+0.000771577106453	-2.14527927667545e-17
All actual children	+0.005813679258035	+0.007568908697630	+0.001755229439595

The pair-3 model rate changes sign. The child cohort's model weighted-total rate also changes, from approximately 1.0523570656281966 to 1.1735385705832913. Tiny nonzero differences are retained rather than classified by a fitted tolerance. The complete captures preserve the other weighted-total rates and every exact coefficient.

Consequently, source invariance under these two previously archived phase outputs is rejected in scope. The known enumeration ambiguity can change the modeled regional drive, not only its tetrad display. This comparison does not rerun the coordination algorithm: it connects the earlier finite ordering witness to fresh pressure at one fixed endpoint. All historical stored nodal rates remain exactly unchanged, including the pair-3 stored variance rate; the sign change above concerns the candidate fresh model. It is not a new trajectory, an observed future transition or a verdict on regional coherence. Source reproducibility remains an unmet prerequisite for the proposed maintenance interpretation.

Validation passes 151 tests: 43 new source-relevance tests plus the reused forcing, regional, identity and phase tests. An independent standard-library recount checks the saved source decomposition and all nine regional rate pairs directly from primitive retained state and conductance; 175 exact checks pass without another kernel call. Both new Python files pass flake8. The study itself uses exactly two forcing-capture calls, zero coordination calls and zero native calls. It changes no engine evolution.

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

The three pinned input identities are retained in the output and the execution plan. Output: artifacts/research/thol_phase_source_relevance_2026_09_18.json, SHA-256 62c4fb07068980acebc75b264f63d715e3a0d28bef6fcc5628441937c01ffb86. Working-source digest: sha256:f3f8609af8ff7033a98c9f0ac449ca4a1343ffece1aa73620340ab98e31e918c. Local validation: artifacts/research/phase_source_relevance_validation_2026_09_18.json. The ignored inputs remain required for the retained run; portable tests use declared synthetic records and do not regenerate the historical studies.

12. Exact reduction of represented phase components

The source sensitivity in section 11 motivates a shared arithmetic contract before changing production phase coordination. Fix a finite nonempty list of materialized binary64 component pairs (c_j,s_j). These are supplied numeric values, not claims that either component is an exact transcendental cosine or sine. Define

\[ a=\sum_j c_j,\qquad b=\sum_j s_j, \]

with exact dyadic arithmetic. The existing exact_weighted_sum_ratio computes each sum with unit weights: binary64 denominators are powers of two, so alignment to their largest denominator and integer addition preserve every bit. Addition of integers is associative and commutative. Hence a, b and their exact joint-zero predicate are independent of input permutation. This is an algebraic argument for every admitted finite list; finite permutation tests validate its implementation rather than prove it by sampling.

If a=b=0, the resultant has no direction. The readout must return an explicit unavailable angle rather than call atan2(0,0). An empty input is a separate invalid request, not evidence of cancellation. For a nonzero pair, set m=max(|a|,|b|)>0 and retain the exact scaled pair (a/m,b/m). This common positive scaling preserves the exact direction. Both components lie in [-1,1] and at least one equals +1 or -1. Materializing each once therefore avoids overflow and joint-zero underflow even when the unscaled sum is outside the finite floating-point range.

Let (u,v) be the resulting binary64 pair and retain the exact defects u-a/m and v-b/m. The minor component can underflow to signed zero; that loss must remain visible. A numerical atan2(v,u) is now defined and uses the same inputs under every permutation of the fixed component list. That determinism is conditional on the same numeric runtime. It is not a certificate of correctly rounded transcendental angle, a cross-library identity, global gauge covariance or continuity near a vanishing resultant. Canonicalizing an exactly zero component to positive zero also specifies the numeric display at the negative-real-axis branch; it does not choose a direction for an exactly zero pair.

There are two distinct zero questions. The sum of supplied rounded components can vanish even when the exact-real sum sum exp(i*represented_theta_j) does not, or conversely differ from the ideal phase preparation. The existing rational resultant enclosures address the represented-angle transcendental question separately. The exact two-neighbor midpoint contract also keeps its own strict chart hypotheses; it is not replaced by this finite component reduction.

The reducer is a mathematical readout. It introduces no phase gain, alignment epsilon, external direction, pressure law or EPI update. Its production use is a separate caller contract, documented below. The single execution plan owns the implementation checkpoint and the subsequent integration boundary.

The implementation is reduce_phasor_components, returning a frozen RepresentedPhasorResultant. Its retained components follow input order, so the complete record is not invariant under a permutation; the derived sums, scale, materialized pair, defects and angle are. Nonempty finite outer iterables are required, and generators are consumed once. The shared finite-real materialization rejects booleans, nonfinite values, overflow and nonzero input underflow; input signed zeros are canonicalized to positive zero. The binary64 rounding precondition is checked before consuming the input. No unit-norm assumption or trigonometric evaluation is introduced. At cancellation, joint_zero=True and all direction/materialization fields are None. This is distinct from the exception for an empty or malformed input.

The 52 independent reducer tests compare with a separate Fraction.from_float summation oracle. They cover all permutations of a cancellation-sensitive four-pair control, exact cancellation without an atan2 call, input generators, signed zeros, axes, subnormals, sums above the finite float range and a minor scaled component that underflows while the exact pair remains nonzero. They also check the explicit input and IEEE-rounding preconditions. With the reused phase-midpoint, unified numerical, circular-semantics and stable-pressure regressions, 225 tests pass. Both new Python files pass flake8. This validated the standalone reducer contract before production integration; no retained research trajectory was rerun. Local validation is retained at artifacts/research/phasor_resultant_validation_2026_09_18.json.

Caller policies remain separate from arithmetic

The existing callers do not have one common degenerate-result contract. Replacing their sums without reviewing these policies would not complete the phase repair.

Existing caller	Current reduction	Empty or degenerate behavior
Legacy global coordination	NumPy means or scalar fsum	Nonempty zero rounded resultant reaches atan2; empty graph returns after earlier history/gain bookkeeping
Local phase list mean	NumPy means or compensated fallback	No usable neighbors return the caller's fallback; nonempty cancellation reaches atan2
Bulk local mean	bincount and division by neighbor count	Isolates retain node phase; nonempty cancellation reaches atan2
Unified circular mean	NumPy means or scalar fsum	Empty input rejects; mean resultant norm at or below the configured numeric tolerance rejects
Default nonfused pressure	Neighbor component averages	The existing 1e-12 small-resultant policy uses the node's phase, giving zero phase pressure
Fused pressure	Indexed component addition	Isolates have zero gradient; nonempty cancellation reaches atan2, except rows handled by the separate certified midpoint path

Those thresholds and fallbacks describe existing code, not derivations from the new reducer. Pairwise U3 phase admissibility uses wrapped separation and needs no resultant mean. The first repair target is global phase coordination, where the retained ordering witness was observed; the public circular mean, local means and pressure branches need their own caller contracts before integration. The reducer alone changes none of them.

Versioned global-coordination integration

coordinate_global_local_phase now accepts global_reduction="exact_components_v1". The default "legacy" retains its previous reduction and returns None. Both paths share the adaptive-gain and local-proposal implementation. The selected version changes the global reduction and its domain policy; it is not a physical coupling coefficient or a new pressure law. The native runtime still uses the default.

For the opt-in path, the global target comes from the exact sum of the materialized cached components. Local means, their neighbor order, phase displacements and phase writes retain their existing owners. The cases are:

Condition	Selected behavior
Nonempty graph, effective kG != 0, nonzero exact resultant	Use the reducer's materialized direction in the existing update
Nonempty graph, effective kG != 0, exact joint-zero resultant	Raise UndefinedGlobalPhaseError and restore graph-owned state
Effective kG == 0	Apply zero global displacement without choosing a global target; local evolution remains active
Empty graph	Return a distinct empty_graph record after the existing history/gain bookkeeping; no resultant is requested

An exact resultant can be tiny and nonzero. No alignment tolerance turns it into cancellation, and this arithmetic decision does not certify its angle as an approximation to the true transcendental sum of the represented phases. At inactive coupling the resultant is still retained as a diagnostic; its available angle, if any, is not selected as the update's global target.

The exact invocation captures the existing GraphTransactionSnapshot before caller gain/job conversion, history changes or cache writes. Failures during admission, proposal computation, phase writes or evidence construction trigger the shared restoration path, retaining the primary exception. The opt-in reader rejects nonfinite gains, components and proposals; failed job-count conversion also propagates rather than taking the legacy fallback. Restoration covers graph-owned state and aliases admitted by that transaction owner, not external I/O or all independently retained Python references. The coordinator does not introduce a second rollback implementation.

The returned frozen GlobalPhaseCoordinationEvidence records the numerical version, empty/applied status, node and neighbor order, primitive phases, materialized-component reducer record, requested/effective gains and their mode, local targets, raw proposals, realized phases and execution branch. It is a public completed-call readout; it is not a sealed causal certificate, and arbitrary hashable node identifiers are not deep-copied into immutable identities. Its numeric evidence concerns that call, not future execution.

Permutation invariance holds for the derived global resultant with the same materialized component multiset. Whole-coordinator invariance would additionally require appropriate invariance of local reductions, adaptive gains and input materialization. Their policies, and the pressure/U3 paths, remain separate. The execution plan owns validation and the next frozen comparison; no historical source output is overwritten.

15. Child-cohort distortion and regional mean-to-shape transfer

thol_child_distortion_audit.py reuses the two authenticated saved reset and regional-response reports for exactly t=2.25 -> 2.5, the actual eight children, and the same two archived branches. It neither advances a graph nor evaluates a reset, pressure or phase kernel. The fixed metric is the restriction of the original full H=diag(d/nu); parent coordinates and their boundary remain in the model.

Staged nodal accounting

Write delta_0, delta_g, delta_i, delta_f for the paired EPI difference before Reception, at integrator entry, after integration and at the endpoint. The stored pressure was generated at delta_0. With admitted common reset map/offset (S,c) and generation coefficients (A,b),

delta_g = S delta_0 + r_reset,

delta_f = (S-hA) delta_0 + r_reset + r_pressure + r_integrator + r_post.

The common source and affine offset cancel only after exact equality checks. The map is S-hA; replacing it by (I-hA)S would change pressure timing. At the integrator entry the stored-minus-current-model rate splits as

A(delta_g-delta_0) + diag(nu) (epsilon_kernel + epsilon_stored).

Here the first term is the intentional held-input lag, while the other terms are differences of the originally captured generation defects. No forcing is inferred from the endpoint derivative. Source components and parent-cut work remain available through the existing regional observer.

For child-centered restriction C_B and increment q, the same observer at zero elapsed time gives the exact reset budget

E_B(delta+q)-E_B(delta) = <C_B delta,C_B q>_H + ||C_B q||_H^2/2.

All sixteen paired local writes are retained, followed by the finite Euler budget at h=1/4 and the postintegration jump. Signed cross terms are essential: a per-node positive quadratic term alone cannot explain the energy change. The later phase update is not assigned to an earlier EPI difference.

A connection to regional identity

Let m_B,m_P be the child and parent H-weighted means of delta_0. The one retained map T=S-hA acts on the exact four-part decomposition

delta_0 = m_P 1 + (m_B-m_P) 1_B + z_B + z_P,

where the two residuals are separately centered and supported on their own cohort. The audit retains each C_B T image, all centered runtime residuals, and the complete Gram energy. These are contributions to one recorded map, not four interventions; their energies are not additive without cross terms. It also retains C_B T 1, so constant preservation is checked rather than assumed for represented coefficients.

This gives a precise mechanism test: a nonzero C_B T 1_B converts a child-parent mean contrast into within-child spatial contrast, even if both initial cohorts are internally uniform and the global constant image vanishes. On a support with no internal child conductance and common child capacity nu_B, (A 1_B)_B = e nu_B 1_B, so C_B A 1_B=0. Any nonzero mean-to-child-shape transfer in this held-pressure step then lies in the admitted Reception map. This is a statement about the combined recorded support and sequential map; it does not isolate update order as a cause.

The transfer complements the earlier family-mean closure obstruction: it measures cohort means feeding unresolved shape on a two-cohort partition. The earlier obstruction measured hidden coordinates feeding eight family means. Neither direction makes an arbitrary subset an autonomous NFR.

Retained result and maintenance boundary

The independent represented-rational recount agrees with every stage:

Paired child state	Centered error energy	H-weighted mean offset
Before Reception	1.162021485807489e-8	0.06635321519
After Reception	4.174981357577767e-7	0.05390852580
After nodal integration	4.894373864559578e-7	0.05132075833
Final endpoint	4.894373864559578e-7	0.05132075833

Reception contributes 4.058779208997018e-7, about 84.9442% of the observed increase. Integration contributes 7.193925069818111e-8, the remaining 15.0558%; the postintegration EPI contribution is exactly zero. These percentages describe signed stage increments in this interval, not universal operator gains. Parent writes do not immediately change the child-only score, but change the inputs later consumed by child writes. Individual child-write increments have both signs and largely cancel; their complete ordered sum, rather than the largest intermediate score, is the Reception contribution.

The integration's first-order parent-boundary work is +9.735871208336425e-8. Its held-generation lag contributes -2.830200479482354e-8, and the generation-kernel discrepancy contributes about +4.447225349707089e-21. Internal child dissipation, paired forcing work and stored-minus-fresh generation work are zero here. The Euler quadratic term is +2.882543409599146e-9; the endpoint-defect linear and quadratic terms are about 3.68103e-20 and 6.16782e-33. Thus the held-input lag partly attenuates the increase; neither that lag nor numerical residuals explains the dominant growth. These are finite Euler accounting terms, not time-integrated flux measurements.

The mean-to-shape mechanism is present. The initial child-parent mean contrast is about 0.05994453363. The eight entries of C_B S 1_B are nonzero, ranging from about -0.00194634 to 0.00426771, while C_B A 1_B is exactly zero. The isolated self-energy of the mapped mean contrast is 1.272946951102293e-7; the mapped parent-centered and child-centered self-energies are about 7.19564e-8 and 5.97976e-9. Their three cross terms are positive and together contribute about 2.84206e-7. All other terms remain in the exact Gram ledger, including the tiny nonzero represented global-constant image. The self-energies alone must not be presented as additive percentages of the endpoint error.

This accounts for the distortion through the existing nodal transport, recorded Reception map and actual boundary. It establishes neither a new physical force nor autonomous regional restoration. Mean relaxation can feed spatial distortion in a region whose environment is coupled differently to its members. That is the useful connection to the NFR identity problem: a maintenance test must retain regional form, mean contrast and environmental response together instead of relying on one global attenuation score.

The historical follow-up criterion, evaluated in section 16, required a restoring response after actual regional form damage under a predeclared canonical perturbation and continuation. Its control must retain measurable nonuniform form, with control drift reported; reduced error caused only by control flattening is not sufficient. Keep the mean-to-shape and boundary contributions visible, and distinguish an existing state-dependent restoring mechanism from supplied operator timing or an imposed target. A failed damage, control or mechanism gate is a reported negative/inconclusive result, not a reason to tune the pressure or select a favorable region. This remains a criterion for that study, not a current assignment to repeat it; the execution plan owns resumption.

Local output: artifacts/research/thol_child_distortion_audit_2026_09_18.json, SHA-256 f3ba30169c969e2b39a0958f3437ce0c6822a3bd7c52006942f184398c1b66b3. The producer ran once on the two saved reports, with zero kernel, pressure capture or native calls. Validation includes 27 new portable tests, 40 existing regional/reset tests, and a separate standard-library/Fraction recount of 277 exact checks. The existing reset suite also invoked its optional historical read-only producer test once; that verification is separate from this producer's budget and did not extend or rewrite evidence.

16. Localized regional form damage and finite configured restoration

thol_regional_restoration.py uses one complete authenticated control replay to t=1.5, then two detached copies of that live source. The shared copy owner rebuilds runtime caches; the study explicitly preserves local neighbor insertion order and the known last-operator instance marker, checks scientific source equality, and checks that execution leaves the opposite branch and retained source unchanged. This avoids interpreting the archival readout's opaque resource descriptions as a complete runtime checkpoint. It does not claim independence of arbitrary external resources or callbacks.

The predeclared intervention is one public default Emission on the first actual child in the retained birth receipt. This is a supplied preparation mark, not an endogenous target-selection law. Both strict read-only admission and the native stage apply. The region stays all eight children, with the original full metric H=diag(d/nu) restricted to them. No factor, target, pressure law or region is fitted to the response.

For a localized EPI jump J at child j, subtracting the regional H-weighted mean gives the exact initial form-damage energy

E0 = (H_j/2) (1-H_j/H_B) J^2, where H_B=sum_B H_i.

Thus a nonzero local jump on a proper multi-node region changes its centered form as well as its mean. The control at that instant fixes the reference independently of the later outcome; the continuation compares the same two branches at seven fixed times from 1.5 to 3.

Shared accounting and decision scope

thol_regional_restoration_accounting.py centralizes the pure reader for this test. It reuses the regional Euler, paired-distance, source-decomposition and record-admission owners. It keeps pre-generation, pre-integration, integration and endpoint EPI budgets, per-glyph EPI and pressure writes, and any intervening residual. Captured pressure generation precedes named glyphs and integration; clocks and full support/capacity are checked at the observed boundaries.

At integrator entry let delta_g be the current paired field, delta_0 its pressure-generation value and delta F_g,delta F_0 the independently captured source differences. The stored-minus-current-model pressure splits exactly into generation kernel discrepancy, generation stored-minus-fresh discrepancy, observed intervening pressure writes, and

e [gradient(delta_0)-gradient(delta_g)] + delta F_0-delta F_g.

The last two terms are held-EPI and held-source lags. The source difference need not vanish in a localized experiment. Each term is evaluated in the same integration-entry regional energy balance; its work is a signed finite Euler contribution, not a measured integrated flux.

For z_i=delta_i-m_B, parent-boundary work is further resolved as

e sum_cut w_ij z_i(delta_j-delta_i) = -e sum_cut w_ij z_i^2 + e sum_cut w_ij z_i(delta_j-m_B).

The first term is nonpositive ordinary relaxation through the boundary; the second retains the incoming parent field. This algebraic split adds no new feedback law. A negative self term alone is not evidence that a regional entity autonomously maintains itself.

The declared finite criterion requires actual positive initial form damage, positive control variance at all seven boundaries, decreased endpoint error, decreased error/control-variance ratio, and nondecreased endpoint control variance. The last condition is a conservative sufficient policy against flattening, not a general definition of pattern identity. Control centered vector drift and mean offsets remain visible. Passing the criterion supports finite configured tracking of an evolving control; it does not establish exact original-form return, erasure of operator history, endogenous scheduling or autonomous long-term maintenance.

Measured result

The retained invocation completes the twelve continuation calls and 26 post-preparation forcing-capture API calls. The single target is actual child 0_sub_0; its native AL jump is 44798133900177/562949953421312. All six predefined gates pass. Both branches execute IL/IL/IL/EN/IL/AL, on unchanged support and capacity.

Time	Centered paired error	Control variance	Error / control variance
1.50, after localized AL	0.006734007748	0.002657876774	2.533604196
1.75	0.006296948745	0.003925631432	1.604060099
2.00	0.005890587815	0.005124599118	1.149472901
2.25	0.005512655227	0.006256037468	0.881173627
2.50	0.002939941119	0.015159875045	0.193929113
2.75	0.002766568872	0.015512596781	0.178343375
3.00	0.002554142631	0.016061734388	0.159020350

The error energy decreases at every sampled boundary and by 62.07099% over the whole window. Its ratio to control variance decreases by 93.72355%, while control variance grows by a factor 6.04307. The normalization therefore benefits from growing control contrast as well as falling raw error; both are reported separately. The paired mean offset falls from about 0.01437493 to 0.00937148, but does not vanish. The control's centered-vector drift energy at the endpoint is about 0.00625356: the control itself evolves substantially.

The exact aggregate ledger is:

Contribution	Child-error energy change / Euler work
Reception EPI writes	-0.002218738764485463
Actual integration, all six steps	-0.001961126353158593
Pre-generation and postintegration EPI changes	0 exactly
Boundary self-relaxation, within integration	-0.002429152538002955
Incoming parent field, within integration	+0.000079302599110432
IL pressure-write correction, within integration	+0.000451285357921915
Held-EPI lag, within integration	-0.000099945741002708
Euler quadratic term, within integration	+0.000037383968814722

The first three rows telescope the measured endpoint energy; the later rows decompose integration and must not be added a second time. Paired non-EPI source-channel vectors, internal child dissipation, held-source lag and stored-minus-fresh generation discrepancies are exactly zero. Generation kernel work is about 5.42265e-19, and combined integration endpoint-defect work about -1.41588e-18; they remain explicit. Every observed intervening pressure write is allocated to its glyph receipt, with zero unassigned remainder. IL and common AL make no immediate paired EPI-energy change.

The positive IL correction is relative to the captured unattenuated-pressure Euler reference; it is not an executed no-IL experiment or a claim that IL violates its nodal coherence contract. Decreasing stored pressure and reducing paired form error are different observables. The negative boundary self term is ordinary coupled transport, while the prescribed EN event supplies the large direct EPI-error reduction.

This gives a useful comparison with section 15: Reception increased the child-cohort error for the earlier all-child perturbation but decreases it for this localized perturbation. An operator label alone therefore does not determine the sign of a regional form-error budget. Sections 17-18 characterize that direction and boundary dependence from the existing maps, without another trajectory sweep.

The scientific outcome is supported_in_scope for finite configured recovery toward the evolving control. It does not establish autonomous NFR maintenance: the source preparation, marked intervention and operator timing remain supplied, and control-form drift is nonzero. No physical emergence claim follows.

Retained output: artifacts/research/thol_regional_restoration_2026_09_18.json, SHA-256 7ebf3b9cd372f70a5b5d170b16e50fae22ec2a56595c73adb3ce82866c596259. The .completed.json checkpoint has identical bytes. Validation includes 58 new portable tests (25 producer/CLI and 33 accounting), 51 reused observer tests, and 289 independent exact checks using only the standard library. Two earlier invocations lost their results to output-format failures; normalization and complete manifest validation now precede execution, and a durable completed-result checkpoint precedes final provenance/output checks. Those technical repetitions are recorded separately in the execution plan and do not count as independent scientific replications.

17. Conditional regional response and environmental input

A condition derived from the admitted nodal map

Fix a positive full-support metric H=diag(d_i/nu_i) and a nonempty proper region B. Let C restrict to B and subtract its H-weighted mean, H_B=diag(H_i:i in B), and M=C^T H_B C. The regional form-error energy is E_B(delta)=delta^T M delta/2. It does not measure the regional mean, phase, capacity, operator history or the whole identity of an NFR.

For two trajectories sharing admitted reset coefficients and generation coefficients, with equal affine offsets and non-EPI sources, the held-pressure interval of section 15 has

delta_after = T delta_before + r, with T=S-hA.

The matrices come from the retained operator/support coefficients, not a fit to the endpoint. The residual r retains reset realization, pressure realization, integration and postintegration changes separately. Common-source cancellation is a checked premise of each pair; the source need not have the same value in two different experiments. Replacing T by (I-hA)S would refresh pressure at a different stage and would change the declared dynamics.

The exact quadratic response form is

Q=T^T M T-M,

E_B(delta_after)-E_B(delta_before) = delta_before^T Q delta_before/2 + <CT delta_before,Cr>_H + ||Cr||_H^2/2.

For the declared exact map (r=0), the sign is determined by the initial difference and admitted coefficients. With a measured residual it is a retained-runtime verification. A prospective binary64 guarantee additionally needs a residual bound supplied independently of the endpoint being predicted.

Existing shape versus incoming mean and parent differences

Write the initial difference in the same four-part decomposition as section 15:

delta=z_B + m_P 1 + (m_B-m_P)1_B + z_P = z_B+w.

Here z_B and z_P are centered on their respective cohorts, extended by zero outside them; Cw=0. The component m_P 1 remains explicit even when its represented image is very small: represented constant preservation must be checked, not assumed. Define

v=CTz_B, u=CTw+Cr,

Z=||C delta||_H^2, A_shape=||v||_H^2, U=||u||_H^2.

The regional error does not increase if and only if

2<v,u>_H + U <= Z-A_shape.

The right side is the attenuation available from the initial regional shape; the left side is the signed effect of mean contrast, parent shape and the realization residual. Either side may have an unfavorable sign. Parent contributions and their cross terms cannot be replaced by an operator label or treated as nonnegative additive percentages of the final error.

A direction-independent sufficient condition for every arbitrary centered input with norm at most sqrt(U) is

Z >= A_shape+U and (Z-A_shape-U)^2 >= 4 A_shape U.

This is exactly sqrt(A_shape)+sqrt(U)<=sqrt(Z), expressed using rational arithmetic. The triangle inequality proves sufficiency; an input aligned with v attains the upper bound. It is not necessary for a particular input direction, because a negative cross term can produce cancellation. Neither the norm nor the signed test derives an autonomous bound on the parent input.

Why a region-only gain can fail

A finite constant q with E_B(T delta)<=q E_B(delta) for every real delta can exist only if T(ker C) is contained in ker C. Conversely, preserving that kernel induces a linear map on the finite-dimensional regional quotient, which has a finite norm. This converse supplies neither q<=1 nor strict contraction. A basis of ker C is 1_B together with the individual outside coordinate vectors; their centered images provide a finite exact test.

In particular, CT1_B!=0 maps zero initial regional shape error to positive shape error. Section 15 already supplies this obstruction for its declared map: CA1_B=0 but CS1_B!=0. Thus a regional variance alone cannot give an unconditional response guarantee for that map. The obstruction is algebraic; it does not assert that every basis vector is a grammar-admitted runtime intervention. Environmental relations or additional state can restrict the admissible inputs, but those restrictions must be derived and checked.

The reusable implementation is regional_response.py. It reuses the exact-coordinate and matrix-product owners, retains both the quadratic form and the signed input ledger, and performs no evolution or operator selection.

Matched retained witnesses

The admission adapter authenticates the earlier reset report and the localized-restoration report. At t=9/4 -> 5/2 all four records share exactly the same S, A, zero affine offset, original full H metric, support/capacities and local EN configuration/order. The sources cancel within each pair; their values also agree across the experiments. These are common declared coefficients: matching configuration does not prove identity of historical implementation sources or a general binary64 kernel theorem. The later record's local reset defect is retained in full without inventing a rounding/clipping split.

The criterion study applies the same form to Reception alone and to the complete held-pressure interval. It first reconstructs every full paired endpoint from the admitted map and its separate residual components. For the complete interval:

Read-out	Earlier all-child perturbation	Localized perturbation
Initial regional error energy	1.162021486e-8	0.005512655227
Final regional error energy	4.894373865e-7	0.002939941119
Isolated child-shape image ratio A_shape/Z	0.5145995631	0.5275976258
Available squared-norm drop Z-A_shape	1.128091474e-8	0.005208382835
Signed input work 2<v,u>+U	9.669152579e-7	0.000062954619
Actual energy change	+4.778171716e-7	-0.002572714108
Combined realization correction to ideal energy change	-3.177557986e-20	-9.007103052e-18
Sufficient input-ball condition	Fails	Passes

The initial perturbation magnitudes differ. The isolated ratios describe images of the two actual child-shape directions, not an all-direction gain bound or separately executed interventions. Both images attenuate, but the mean/parent input overwhelms the first witness's available margin. The localized witness has ample margin and also passes the conservative direction-independent test at its observed input radius. The unknown future environment has not been bounded by that observation. Reception alone has the same opposing signs, and the measured realization corrections do not reverse any of these four signs.

For both S and T, the exact image of 1_B has centered energy about 3.542510399e-5, while CA1_B=0. The same-map obstruction therefore excludes a finite unrestricted regional-shape gain. This identifies an environmental closure obligation, not a contradiction of conditional recovery or of the operator's nodal coherence contract. The child cohort still has no internal conductance and is defined by recorded lineage; this calculation does not promote it to an independently emerged coherent region.

This completes the conditional-criterion delivery with no new graph, trajectory, scalar kernel or forcing capture. The retained result is artifacts/research/thol_regional_response_criterion_2026_09_18.json. There are 82 new portable tests, 84 reused passing observer tests, and 239 independent exact checks using only standard-library arithmetic. The latter reconstruct local reset composition, the nodal generator, source/pressure defects, response matrices and both sign criteria from the saved evidence.

18. Environmental-input geometry and protected read-outs

Image and annihilator in the original metric

Keep the same region B, centering C, positive metric H_B and declared transition T. Let the columns of N be 1_B followed by every outside coordinate vector. They form a basis of ker C. The exact input map is

G=CTN : regional-mean/outside differences -> centered regional shape.

Its columns belong to the (size(B)-1)-dimensional H-centered shape space. A centered vector y represents the scalar read-out y^T H_B z. That read-out is insensitive to every declared environmental input precisely when

G^T H_B y=0 and 1^T H_B y=0.

These are arbitrary independent algebraic inputs. Their individual or joint reachability under the native policy, phase, grammar and source constraints is not inferred from this map. Likewise, input insensitivity alone would not make a read-out constant under the region's self-dynamics or immune to the separately measured binary64 realization residual.

For independent columns V of G, their Gram matrix K=V^T H_B V is positive definite. The image projector is

P_image=V K^-1 V^T H_B.

With local centering P_B=I-1 h_B^T/sum(h_B), the complementary projector is P_protected=P_B-P_image. Both are H-self-adjoint and idempotent. Its image gives every protected centered read-out, with dimension size(B)-1-rank(G). Exact column reconstruction G=V K^-1 V^T H_B G provides a certificate of the image, rather than relying on a numerical singular-value threshold.

The existing regional response owner now shares one validated domain, centering and input basis between the response criterion and this geometry. It reuses the exact Krylov rank and matrix inverse implementations. The weighted image projector factors through two shared exact matrix products, retaining a full-sized zero matrix at rank zero. Coordinate order, exact reconstruction, idempotence and H-self-adjointness checks remain in the observer. No new elimination algorithm, physical parameter or evolution path is added.

Exact classification of the retained child cohort

The retained geometry study authenticates section 17's result, computes each of S and S-hA once, and matches the centering, full metric and input images of both retained perturbations. It does not replay their admission or trajectories.

Declared step	Full input rank	Parent-only input rank	Child-shape dimension	Protected dimension
Reception S	7	7	7	0
Reception plus held-pressure Euler S-hA	7	7	7	0

A separate standard-library proof reconstructs CTN directly and proves the rank using a nonzero 7 x 7 minor from child rows 0--6 and parent columns 0--6. The determinant is approximately 5.34828347915e-9 for Reception and 2.46679963704e-8 for the complete interval; exact rational determinants are retained in the artifact. H-centering supplies the independent rank upper bound seven. The weighted annihilator has only the zero vector, and P_image=P_B, P_protected=0 exactly for both maps.

The parent-only witnesses establish this result without relying on the tiny represented global-constant image. The change of input geometry between the stages is exactly G_T-G_S=-h C A N; it changes coefficients but does not reduce the image dimension. The regional-indicator column is unchanged because CA1_B=0 in this support/capacity configuration.

Thus every nonzero centered linear child-shape read-out can be affected by some declared parent input. This excludes absolute input protection for this particular formed support and lineage-defined region. It neither excludes the conditional recovery of section 17 nor rules out protection under a genuinely derived restriction on environmental inputs. It says nothing by itself about a different geometry, nonlinear native reachability, repeated stability, spontaneous formation or physical particles. Full rank also does not measure the magnitude of sensitivity: weak response and finite conditional robustness are separate from exact input immunity.

Retained output: artifacts/research/thol_regional_input_geometry_2026_09_18.json. Independent reconstruction, exact minors and annihilator certificate: artifacts/research/regional_input_geometry_crosscheck_2026_09_18.json. Validation: 76 new portable tests, 82 reused passing tests, and 100 independent exact checks, including the final report's projectors, Gram inverse, image reconstruction and source/output hashes. All trajectory, kernel and forcing capture counts remain zero.

19. Support symmetry versus the admitted nodal and reset maps

Separate mathematical objects

Let Gamma be the complete automorphism group of the original weighted support, including any zero-conductance support edges. Retain its subgroup that preserves the child set, positive capacity and full metric. For a permutation with P[permuted(i),i]=1, a declared matrix M respects the permutation exactly when PM=MP, equivalently M[permuted(i),permuted(j)]-M[i,j]=0 at every entry. A vector field is invariant when v[permuted(i)]-v[i]=0 at every node.

These are different checks. For an affine map Mx+a, both matrix commutation and Pa=a are needed. The paired difference map can cancel a common a even when each individual affine evolution lacks that symmetry. Here b is the captured contribution of the canonical phase, capacity and topology pressure channels multiplied by capacity; c is the Reception offset. The individual held-pressure interval has offset c+h b.

The exact equivariance owner now centralizes the exact matrix/field checks alongside the existing numerical diffusion diagnostics. It delegates complete enumeration and orbit extraction to symmetry sectors. The explicit cap of 2,000 is an operational limit: exceeding it raises instead of returning a partial group. The owner creates a detached support graph for enumeration, not an evolved TNFR state, and uses no numerical commutation tolerance.

For the common matrix subgroup Gamma_0, the environmental space justified by an assumed invariant input is Fix(Gamma_0) intersect ker(C). Since the group preserves the region, a basis consists of 1_B and the indicator of each outside orbit. This characterizes a conditional input space; group commutation does not itself establish that a particular initial state, source, boundary history or native input belongs to it.

Complete retained classification

The retained symmetry study authenticates the prior response report and preserves its original support, region, metric and coefficients. Exact checks rederive H=d/nu, A, the captured source, and the ordered local-row composition of S. The weighted support has 16 nodes and 24 edges, with no zero-conductance edges in this witness. Its complete group has eight elements. Every one preserves the child set, capacities and full metric.

Object checked within that group	Preserving permutations
Weighted support, child set, capacity and H	8
Nodal generator A	8
Ordered Reception map S	1, identity only
Held-pressure map T=S-hA	1, identity only
Zero Reception offset c	8
Captured canonical source b	1, identity only
Captured generation EPI and phase in each paired record	1, identity only

The eight support actions have four orbits: even parents, odd parents, even children and odd children. The common subgroup for the three declared matrices is the identity, whose sixteen orbits are singletons. Its fixed environmental space therefore still has nine independent inputs: the regional mean and each of the eight parent coordinates. Their image retains rank seven. No nontrivial restriction on those inputs follows from a common symmetry of these matrices.

A compact obstruction uses the cyclic shift by two positions on both parent and child index sets. It preserves the weighted support and commutes with A, but

S[0,0]-S[2,2] = -630642904502747332437308389707 / 162259276829213363391578010288128 ~= -0.003886636972791.

The same entry difference occurs in T because A commutes. Thus this is not solely the tiny represented constant-mode leakage of the previous study. In every retained branch the local EN row family itself satisfies row_permuted(i)[permuted(j)]=row_i[j] for all eight symmetries. Its ordered composition does not. This identifies fixed sequential composition as the obstruction at the declared affine-map level. No alternative order or simultaneous runtime event has been executed by this calculation, and it does not establish a counterfactual binary64 trajectory.

Source and state asymmetry remain distinct. The four recorded source vectors are equal across the experiments but invariant only under the identity; paired cancellation is still valid. Phase invariance here concerns the saved represented coordinates, not equivalence under a global phase gauge. Likewise, matrix commutation does not certify complete history or runtime equivariance. Simultaneous stage semantics cannot be substituted for the captured sequential map without a separate derivation and admission.

The completed result obstructs a symmetry-based input restriction for this particular map and cohort. It does not rule out conditional recovery, correlated inputs derived by another mechanism, or other configurations. In particular it does not decide whether a Platonic graph forms or persists under TNFR dynamics. It establishes the check required before transferring geometric symmetry to a dynamical claim.

Retained report: artifacts/research/thol_regional_map_symmetry_2026_09_18.json. Independent exact enumeration, composition and symmetry witnesses: artifacts/research/map_symmetry_crosscheck_2026_09_18.json. Validation: 113 new portable tests, 140 reused passing tests and 487 independent exact checks, including final report/source identities, signed defects, fixed-input images and scope flags. All native, kernel, forcing capture and historical-admission counts are zero.

20. Same-snapshot Reception and the limit of geometric protection

Comparison using the existing coefficient kernel

For the same retained neighborhood and represented mix, let J contain the local Reception coefficient row of each target. The sequential map S is their chronological single-target composition; J reads every row from one common input. Define U=J-hA with the same pre-generated nodal pressure timing as T=S-hA. The existing comparison reader now exposes this comparison through --snapshot-comparison, reusing the exact symmetry and regional-input owners rather than adding another engine.

All 64 saved rows (16 nodes, two paired experiments) match the shared represented Reception row. Its coefficients depend on mix and input support, not earlier EPI writes. Reception uses an unweighted neighbor mean; A retains the actual weighted transport. The comparison checks configured mix, neighbor membership and the common hard interval. J is a declared unclipped real-affine map: binary64 mean/blend evaluation, clipping, semantic kinds and auxiliary writes remain distinct. Sequential defects and outcomes are not transferred to J.

Exact result and the unmet input condition

Quantity	Sequential S / T	Same-snapshot J / U
Common map subgroup order, including A	1	8
Fixed environmental input dimension	9	3
Centered image rank for those fixed inputs	7	1
Orthogonal centered read-out dimension for those fixed inputs	0	6
Centered image rank for unrestricted inputs	7	7
Image of regional-mean contrast CM 1_B	Nonzero	Zero

The three conditional inputs are the regional constant and the even-parent and odd-parent orbit indicators. Their centered output spans one direction, constant on each child orbit with opposite signs and the ratio required by the original H metric. Thus six independent centered linear read-outs annihilate this restricted input image. This is an exact one-step conditional input-protection result, not a persistence or repeated self-dynamics theorem.

The condition is not met by either recorded experiment. For each actual paired difference delta, decompose delta=z_B+w as in section 17: w retains the parent differences and the H-weighted regional mean. Both retained w vectors preserve only the identity, as do the saved source, EPI and phase. No averaging is applied. Consequently the six-direction result cannot be promoted to either retained environment; unrestricted input rank stays seven. The common-source cancellation still holds, separately from these symmetries.

Runtime boundary and disposition

The criterion report is a projection of historical evidence, not a complete graph-owned stage checkpoint. It lacks replayable grammar histories/debt, EPI kinds and complete graph/source/runtime configuration. Older trace boundaries additionally contain explicitly unavailable resources and named callables, not restorable live owners. The existing two-phase dispatcher requires those preflight and commit conditions. No synthetic history or defaults are inserted to present this map calculation as authenticated runtime execution.

This closes the ordering/symmetry comparison. Same-snapshot semantics remove the exhibited ordering obstruction at the declared-map level, while the actual input restriction remains unjustified. Neither symmetry nor regular polyhedral shape is established as necessary for an NFR. The broader open question concerns a closed endogenous feedback mechanism for differentiated coherence, using the existing nodal channels and explicit state/history; additional symmetry sweeps and historical-state restoration are parked.

Retained report: artifacts/research/thol_snapshot_reception_comparison_2026_09_18.json. Independent arithmetic: artifacts/research/snapshot_map_independent_2026_09_18.json. This delivery performs 64 coefficient-builder calls, zero scalar evolution kernel calls, zero native steps and zero live stage calls. No historical trajectory is rerun or extended. Validation: 14 new portable tests, 149 distinct reused passing tests and 274 independent exact checks, including the final report and working-source identities. The previous symmetry/input reports retain their original bytes.