Causal THOL birth and an admitted transport connection

Status: Exact finite acceleration, edge-reset and declared-action selection obstructions, with bounded canonical runtime controls. The selection/dispatch audit is complete in its stated scope; autonomous generation and persistent self-restoring localization remain open. Research links: B2.d.6/O3.a, S3, S8, S9, S10 and S16.

1. Deriving a birth trigger from actual nodal evolution

Consider a fixed simple unit cycle with uniform held capacity nu>0, scalar EPI x, and a regular phase twist strictly inside the effective U3 gate and the neighbor semicircle boundary. Its phase, capacity and topology pressure gradients vanish. With the existing fully normalized EPI weight e=w_epi>0, define c=e*nu and L=I-W/2. The canonical nodal equation is

\[ \dot x=-cLx. \]

Two successive exact refreshed Euler segments of positive durations h1,h2 give

\[ x_1=(I-ch_1L)x_0,\qquad x_2=(I-ch_2L)x_1. \]

Their secants satisfy s1=-c*L*x0 and s2=-c*L*x1. The same three-point acceleration used by compute_d2epi_dt2 therefore has the exact reference identity

\[ \widehat A=\frac{2(s_2-s_1)}{h_1+h_2} =\frac{2h_1c^2}{h_1+h_2}L^2x_0. \]

The first duration occurs in the numerator because the second refreshed rate is evaluated at the endpoint of the first Euler segment. This is a finite solver identity, not an asserted exact second derivative of the continuous exponential trajectory. The sufficient convex conditions c*h_j<=1 keep all EPI endpoints in the convex hull of the initial field; clipping and runtime arithmetic still require observed residuals.

On an even cycle let v_i=(-1)^i, so Lv=2v, and prepare x0=b*1+a*v. Then

\[ \widehat A_i= \frac{8h_1c^2a}{h_1+h_2}v_i. \]

The default THOL magnitude threshold is crossed when

\[ \frac{8h_1(e\nu)^2|a|}{h_1+h_2}>\tau. \]

This is a derived sufficient preparation condition. It leaves the existing threshold and operator coefficients unchanged and supplies no independent acceleration law. Runtime admission additionally requires the actual operator grammar and parent domain checks.

2. A bounded preparation with the unchanged default threshold

The finite C8 preparation uses alternating EPI levels 2 and 1/2, unit capacity and winding one. Its scalar chart bounds are explicitly [-4,4]; these are declared experiment bounds rather than the default [-1,1] bounds. Here b=5/4, a=3/4. Two refreshed segments of duration h1=h2=1/4 give

\[ \widehat A_i=3e^2v_i,\qquad x_2=\frac54\mathbf1+ \frac34(1-e/2)^2v. \]

Using the materialized default normalized coefficient e=0.18315345241335917, the positive trigger is approximately 0.1006355614, above the unchanged default tau=0.1. The Euler coefficients are below 1/2, so this exact reference remains within the initial interval [1/2,2] and inside the declared chart bounds. The threshold margin is finite; the runtime comparison records its actual three samples and arithmetic residual rather than assuming equality with the symbolic calculation.

Actual Coherence/Dissonance preparation supplies the required operator context. Canonical pressure is refreshed before the two physical segments. The three samples are consequently generated by those executed segments, rather than inserted as a desired threshold-crossing history. Choosing one high-EPI parent for THOL remains an explicit target policy; it is not a claim that the location emerged by spontaneous symmetry breaking.

3. Birth creates a node; Coupling must supply an actual edge

The distinction established in THOL pressure feedback remains essential: public THOL creates an initially isolated child with a triad and hierarchy metadata. The node does not enter its parent's neighbor-pressure walk through the metadata alone. Its initial pressure is zero, capacity is the existing 0.95 multiple of the parent capacity, and phase is inherited. Its EPI is the bounded output of the existing metabolic amplitude map.

This isolated birth also has an exact transport-energy boundary. Old EPI and conductance remain fixed, so for the Dirichlet matrix B=D-W,

\[ B^+=\operatorname{diag}(B,0),\qquad x^+=(x,x_h),\qquad E_D(x^+)=\tfrac12(x^+)^T B^+x^+=E_D(x). \]

The edge-based energy cannot see the isolated child's form. This is not a zero physical cost of creation. Although its capacity is positive, its reversible weight d_h/nu_h is zero until attachment; the connected, positive-metric forced-profile theorem cannot be applied at this intermediate state. The isolated-birth energy test checks the zero-block identity and rejects use of the same-support reset observer across the dimensional jump. After birth, that existing observer can account for UM on two snapshots of the same enlarged node set.

An actual graph-backed Coupling operation can subsequently consider the child as a functional-link candidate. The candidate inventory must include the new node; a retained prebirth sample of graph nodes need not do so. The candidate must pass both the U3 phase condition and the existing compatibility threshold. There is no automatic parent-child edge implied by THOL's hierarchy.

The parent target must already have a compatible neighbor when the hard U3 gate is checked, before functional candidates are considered. Targeting the isolated child, or including it as a target in an all-node stage, fails that gate despite its inherited phase alignment. The successful route targets the parent with its existing cycle neighbors.

The Coupling proposal kernel computes link weight from final merged phases and the snapshot EPI/sense inputs. In its scalar regime this has the form

\[ \omega=\tfrac12\left(1-\frac{|\operatorname{wrap} (\phi_i-\phi_j)|}{\pi}\right) +\tfrac14\left(1-\frac{|x_i-x_j|} {|x_i|+|x_j|+\epsilon}\right) +\tfrac14(1-|S_i-S_j|), \]

with the existing epsilon and sense-index conventions. The implementation evaluates the EPI ratio with scaling to preserve its numeric domain. The default compatibility threshold is pi/(pi+1). The resulting positive conductance is an operator-derived weight, not an independently fitted parent-child coupling strength. Inherited equal phase alone is insufficient: the EPI and sense-index terms can still place the candidate below the compatibility threshold.

Parent-target UM computes its phase, capacity and direct pressure proposals from the old compatible neighborhood before adding the functional edge. The newly considered child is therefore not silently included in that same call's original neighbor capacity mean. Default bidirectional UM can also change the parent's old neighbors' phases. The full before/after triads and the subsequent pressure refresh must be observed; a pure edge-only model does not describe every UM channel.

4. What the new edge changes in the canonical pressure

Keep scalar EPI fixed across a new undirected edge i-j with positive weight omega. For endpoint i, let d_i be its old weighted strength and mx_i its old weighted EPI neighbor mean. If d_i>0, the change of the normalized EPI gradient is

\[ g^+_{x,i}-g^-_{x,i} =\frac{\omega}{d_i+\omega}(x_j-mx_i). \]

For a former isolate, the new single-neighbor gradient is instead g_x^+=x_j-x_i, independently of the strictly positive incident weight. This is normalization of the actual adjacency, not infinite coupling.

The capacity channel uses the unweighted unique neighbor support. With old neighbor count k_i>0 and mean capacity bar_nu_i, its edge-only change is

\[ g^+_{\nu,i}-g^-_{\nu,i} =\frac{\nu_j-\bar\nu_i}{k_i+1}. \]

A former isolate gets nu_j-nu_i. Zero-weight support edges can affect capacity, phase and topology while contributing no EPI conductance; the weighted EPI walk cannot replace those other channel conventions. If UM also changes capacity, evaluate these support-only identities on one declared held triad and account for the capacity write separately.

For the actual unit-cycle parent joined to one isolated child, denote the parent's EPI by H, its old two neighbors' common EPI by J, and the child's EPI by z. For the ideal closed-form reference with parent capacity one and child capacity 19/20, the endpoint gradients are

\[ g_{x,p}^+=\frac{2J+\omega z}{2+\omega}-H,\qquad g_{x,c}^+=H-z,\qquad g_{\nu,p}^+=-\frac1{60},\qquad g_{\nu,c}^+=\frac1{20}. \]

The symmetric parent-target phase update leaves the child and parent phases equal in the exact preparation. Their phase pressures vanish, so the refreshed child pressure is

\[ p_c=e(H-z)+\frac{w_{\nu}}{20}. \]

The production damping coefficient is binary64 0.95, not the exact rational 19/20. For its exact represented capacity difference delta=1-Fraction(float(0.95)), replace -1/60 and 1/20 by -delta/3 and delta, respectively, including the pressure formula.

The default topology weight is zero. For a configured nonzero topology channel, the new degree-neighbor gradients are -4/3 at the parent, 2 at the child, 1/2 at each old neighbor of the parent, and zero elsewhere. These formulas use a simple unit undirected cycle; arbitrary weighted or parallel-edge graphs require the implementation's actual degree and unique-support conventions.

The old neighbors can acquire nonzero phase pressure after bidirectional UM. The newly connected whole graph therefore need not follow pure EPI diffusion, even when the parent and child formulas above simplify. Likewise the previous cycle coordinate x+(w_vf/e)*nu does not generally cancel pressure once weighted EPI transport and unweighted capacity transport differ.

5. Connection has an exact energy reset, not a conservation claim

On fixed ordered support, let W be symmetric conductance, D its strength diagonal and B=D-W. The existing EPI Dirichlet diagnostic is

\[ F_W(x)=\tfrac12x^\top Bx =\tfrac12\sum_{\{i,j\}}w_{ij}(x_i-x_j)^2. \]

Compare the graph after birth but before UM with the graph after the edge commit, including the child in both node orders. If UM leaves EPI unchanged, its exact edge reset is

\[ F_{W^+}(x)-F_{W^-}(x) =\tfrac12\sum_{\{i,j\}}(w^+_{ij}-w^-_{ij})(x_i-x_j)^2. \]

For the single new parent-child edge this is omega*(H-z)^2/2>=0. The change reflects a new conductance in the diagnostic while EPI is fixed; it is not proof of conserved energy or new EPI creation. It is also not a birth-energy theorem across different node supports.

After the edge commit, a held-support nodal rate r=diag(nu)*p has

\[ \dot F=(Bx)^\top r. \]

The EPI-channel contribution alone is -e*(Bx)^T*diag(nu_i/d_i)*(Bx)<=0, with zero mobility at isolates. The capacity, phase and topology terms keep their own signs; a full multichannel decrease does not follow from the EPI term. A subsequent exact held-input Euler interval has the finite identity

\[ F(x+h r)-F(x)=h(Bx)^\top r+\tfrac12h^2r^\top Br. \]

These identities reuse the existing conductance and diffusion geometry. They measure how a created child becomes dynamically connected and how the changed structure enters the nodal channels.

6. A precise boundary for the connected pattern's next evolution

Freeze the connected post-UM support, capacities and phases before any later Silence or structural event. Assume positive capacities and fixed symmetric conductance on a connected graph. With D=diag(d_i), define the held non-EPI pressure contribution

\[ F=w_{\nu}g_{\nu}+w_{\phi}g_{\phi}+w_{\mathrm{topo}}g_{\mathrm{topo}}. \]

The refreshed canonical model, while its scalar chart remains valid, is

\[ \dot x=\operatorname{diag}(\nu)(-eD^{-1}Bx+F). \]

A zero-pressure equilibrium requires e*B*x=D*F. Since the symmetric connected Laplacian has kernel span{1}, the necessary and sufficient compatibility condition is

\[ \mathbf1^\top DF=\sum_i d_iF_i=0. \]

This is an exact balance condition for the actual channel conventions. It is not implied by each channel separately resembling neighbor averaging: the EPI channel's stationary weights are the conductance strengths, while the capacity channel uses unweighted support.

In the ideal reflected C8 preparation, the parent-target bidirectional UM preserves reflection symmetry of the graph and capacities. Its phase gradient is antisymmetric under that reflection, with zero at the fixed nodes, so sum(d_i*g_phi_i)=0 inside the strict chart. With the default zero topology weight and ideal child capacity 19/20, the capacity contribution gives

\[ \sum_i d_iF_i =w_{\nu}\left[-\frac{2+\omega}{60}+\frac{\omega}{20}\right] =\frac{w_{\nu}(\omega-1)}{30}<0. \]

Here the observed link has 0<omega<1. Consequently this frozen ideal structure has no zero-pressure stationary EPI. Reflection cancellation is an exact-model statement; the actual binary64 weighted phase contribution must be measured as a residual. For the exact represented capacity difference delta above, the weighted capacity contribution is instead 2*w_vf*(omega-1)*delta/3, with the same strictly negative sign.

The captured connected state confirms that sign: the weighted capacity pressure is approximately -0.0003191519727040398, while the weighted stored pressure is -0.0003191519727039513. Their weighted phase and arithmetic remainder is approximately 8.86e-17. The corresponding instantaneous represented nodal mean rate is -1.8015381161580725e-5. These quantities are read before the subsequent flow and final SHA; they are not observations of a future limiting drift.

There is a different conditional possibility. Define

\[ \bar v=\frac{\sum_i d_iF_i}{\sum_i d_i/\nu_i}. \]

The weighted EPI mean has derivative bar_v. The equation

\[ eBz=DF-\bar v\,(d_i/\nu_i)_i \]

is compatible and determines a relative profile z up to a constant. Then x(t)=bar_v*t*1+z solves the held model. Subtracting that solution leaves homogeneous fixed-capacity EPI diffusion, so relative deviations decay on a connected graph. This conclusion concerns an unrestricted scalar chart with fixed coefficients. A nonzero mean drift eventually encounters finite EPI bounds; it gives no asymptotic claim for a clipped runtime or for a later UM/SHA policy.

The next bounded question is therefore to distinguish a held-support relative profile with mean drift from restoration under further admitted structural operations. The frozen-support follow-up now implements the exact relative-profile reference, nonlinear forcing read-out and finite comparison. A repeated-policy theorem remains open. Neither birth, one successful connection nor this compatibility calculation establishes a persistent self-restoring localized entity.

7. Shared observations and falsification criteria

support_transport.py centralizes the finite reference budgets on the engine's materialized conductance:

• observe_support_transport reads weighted EPI transport, unweighted capacity and topology gradients, Dirichlet energy, and the exact product of represented capacity with stored pressure. It does not refresh pressure or identify its remainder as a canonical phase contribution.
• observe_support_transport_reset verifies the same-EPI conductance reset on one aligned node set. The child must already be present in both observations; birth itself is outside that identity.
• observe_support_transport_euler separates the held-input drift, quadratic Euler term and observed endpoint defect on fixed conductance and capacity. It recomputes detached state fields before checking the identity; publicly constructible observations carry no causal seal.

These APIs reuse the existing conductance reader and scalar EPI chart. Effective binary64 weights and state values are treated as exact real coefficients after materialization. A zero budget residual proves the displayed algebra for those inputs, not that the caller executed a particular solver, refreshed pressure or preserved a future trajectory.

The bounded campaign in thol_birth_transport.py keeps the complete parent-target word IL, OZ, THOL, UM, SHA, checks both whole-word validators and the live gate before each public application, and records the actual graph mutations. Its birth samples come from one executor-owned refreshed physical partition. Its later fixed-support Euler intervals have explicit canonical refreshes at the captured live endpoints; the final parent SHA follows the measured transport endpoint. Atomicity belongs to each executor invocation, rather than to this entire changing-node orchestration.

The recorded attached case has the following finite results; displayed decimals summarize the captured binary64 states and exact rational budgets.

Quantity	Observed value
Physical acceleration before THOL	0.10063556139279584
Default birth threshold	0.1
Committed parent-child conductance	0.8357695998168526
Refreshed connected child pressure	0.2743391213451684
Child EPI at times 0.5, 0.75, 1.0	0.3869753, 0.4521308, 0.5118328
Same-EPI support energy reset	+0.9177477068340985
Two subsequent flow energy changes	-1.2273590449, -1.0125654108

The support reset and both Euler energy decompositions have exactly zero rational identity residual. The largest captured attached Euler state defect is 15/144115188075855872, approximately 1.04e-16; it is retained in the energy accounting rather than set to zero. These are finite observations, not uniform numerical-error bounds. The observed acceleration differs from the exact represented reference 3*e^2 by approximately -2.04e-15. Full states, fractions and source provenance are retained locally in artifacts/research/thol_birth_transport.json. Regenerate this ignored artifact with python benchmarks/thol_birth_transport.py.

The paired controls disable functional linking or retain the prebirth candidate sample. They preserve the same birth preparation while testing whether an actual transport edge is needed. A successful child creation must not be reported as successful coupling in either control. Both controls keep the child pressure zero and its EPI exactly at the birth value through the measured intervals. The 22 independent exact observer tests are in test_support_transport.py; the 17 bounded runtime checks are in the runtime test suite.

The proposed mechanism would be contradicted by a claimed causal birth whose live history fails the unchanged acceleration threshold, a claimed connection absent from the actual UM commit, or child transport without a pressure-producing adjacency change under the stated isolated control. A disagreement with the exact edge/Euler identities requires checking the captured state, support and numerical defect rather than inferring new dynamics. Persistent restoration requires additional evidence after the finite birth/attachment sequence and must distinguish the mean-drift obstruction in Section 6 from a zero-pressure localized equilibrium.

8. Selection, actual dispatch and the unique-parent obstruction

The bounded selection audit reuses prepare_birth_selection_source from the existing birth benchmark. Every branch executes a fresh IL/OZ and two-segment physical preparation; no history is invented, removed or copied from a postbirth graph. Its target zero is an explicit mark. Observed states are declared projections, not seals of the complete runtime state.

The ordinary runtime refreshes pressure and computes Si before asking its actual built-in selector. Both DefaultGlyphSelector and ParametricGlyphSelector propose IL at all eight nodes in this preparation; the recorded commits are also IL, with eight nodes remaining. Si is 0.9417004454081352, above the unchanged selector si_hi=0.5, even though the parent's three-sample acceleration is above the public birth threshold. The selector's cached acceleration (0.10063556139279795) and the public operator's three-point result (0.10063556139279584) are recorded separately. Neither value is overwritten to obtain a desired decision.

Thus the birth trigger is not sufficient for birth under either tested built-in policy. This is a counterexample to sufficiency for this preparation, not a proof that no other state can select THOL. The heuristic selector thresholds, weights, hysteresis and grammar remain configured constitutive choices; this audit does not derive them uniquely from the nodal equation.

Separate explicitly requested THOL controls show the dispatch distinction:

Parent intervention	Nodes after dispatch	Pressure used by held Euler	Pressure used after canonical refresh
None	8	-0.22671636673672074	-0.22671636673672074
Primitive apply_glyph(..., "THOL")	8	-0.21870804321347506	-0.22671636673672074
Public SelfOrganization()	9, with one isolated child	-0.21870804321347506	-0.22671636673672074

These controls start at the same recorded source. Both THOL routes update parent pressure; only the public route creates a child. With duration 1/4, the held-pressure parent endpoint is 1.8142476224736617; a refresh restores the control endpoint 1.8122455415928502. The public child has zero pressure and unchanged EPI in both isolated cases. Exact held-input predictions and binary64 residuals are retained. The three existing transport controls then verify that a public UM connection and a sample containing the child are needed for its observed positive transport response. That positive word remains an explicit intervention, not an output of the tested selectors.

An independent exact obstruction clarifies what selecting one parent requires. For a deterministic equivariant selection s(gX)=g s(X), let H_X be the subgroup preserving the declared state and relations. For every g in H_X,

\[s(X)=s(gX)=g s(X).\]

Therefore the selected candidate must be fixed by every element of H_X. If the invariant candidate set has no such point, a deterministic equivariant single choice is impossible on that state orbit. An orbit or candidate set can still be selected; this argument does not forbid distributed activation, equivariant random distributions, or selection after an explicitly declared asymmetry. Their execution laws would require their own derivations.

derive_selector_symmetry checks a supplied finite permutation group, exact nodal labels and optional exact directed relation matrix, then computes the joint stabilizer, candidate orbits and fixed candidates. Candidate membership must itself be invariant; an asymmetric restriction must first be declared as state information. Incomplete supplied groups, nonexact labels and exhausted validation budgets fail without a partial result. A smaller supplied group can prove an obstruction, but absence of an obstruction cannot exclude missing symmetries.

For the detached uniform C8 example with empty histories, uniform triad and all dihedral actions, the eight candidates form one orbit and none is fixed. Adding a parent mark at zero leaves fixed vertices zero and four. This removes that obstruction without deriving a unique selector. These examples are not the wound, IL/OZ-marked runtime source. Topology symmetry alone would omit phase, history, target, sample order and policy information.

The extended relabeling test compares two independent marked preparations under one explicit label map, extending it to newborns by parent and birth ordinal. It checks raw/refreshed birth and coupling, all node attributes except declared wall-clock telemetry, edge attributes, neighbor order, hierarchy, causal histories and the value projection of authenticated preparation receipts. The later SHA timestamp is checked for format rather than equality. This finite test retains insertion order and the unlimited-candidate policy; it does not establish all-state covariance. Its closure follows UM directly, without the two later transport intervals of Section 7. The old four-channel comparisons alone could not test this newborn correspondence.

Reproduce the audit with python benchmarks/selection_birth_closure.py. Its source-bound local report is artifacts/research/selection_birth_closure_2026_09_18.json; the versioned integration tests and exact action tests preserve the contracts. The report records two actual selector branches, six explicit dispatch/refresh controls, three transport controls and two detached exact examples. It is a completed mechanism audit with scoped obstructions and positive marked controls. Autonomous generation, active maintenance and physical identification remain open. The sole task queue remains the execution plan.

9. Set-valued eligibility and explicit finite dispatch

observe_self_organization_eligibility implements the all-current-node relation implied by the preceding audit. For the complete declared graph state X, its eligible set is

E(X) = {v : A_v and G_v and P_v and B_v},

where A_v is an available three-sample acceleration strictly above the resolved threshold in magnitude, G_v is incremental THOL grammar admission, P_v is the public application's configured precondition check, and B_v is a valid complete public proposal containing an actual child. The last condition includes depth, hierarchy, amplitude, metric-input and structural checks; an executable pressure-only THOL is not a birth candidate. The result preserves the independent checks and errors instead of reducing them to one score. Grammar admission alone does not check a complete future word.

Physical history precedence, endpoint matching and unavailable acceleration come from the shared observation owner. Public application preconditions and SelfOrganization._prepare_execution retain their existing implementations. In particular, the optional gate is active only when both its graph switch and the request switch are enabled. The execution window controls history retention; it does not replace the runtime grammar's own recent-context window. Neither invalid nor short histories are replaced with desired samples.

The relation returns all candidates in snapshot rank, with set-valued membership. It does not choose a single representative. On the eligible subset, observation and execution reuse one private merge/validation owner in network_stage.py: child-name collisions are allocated in snapshot rank and the merged hierarchy/support is checked on a detached graph. Empty eligibility is vacuously jointly viable but causes no stage execution. Joint planning does not invoke monitors and cannot guarantee later execution checks will succeed.

The separate execute_eligible_self_organization_stage function implements the explicit policy all eligible parents once. It captures graph-owned state before materializing its restricted options mapping, freezes option values once, restores input side effects, and recomputes the relation. It does not accept a caller-supplied observation as authorization. One built-in two-phase public THOL stage performs the actual commit; fallback schedules are rejected. Successful dispatch checks exactly one new isolated child per eligible parent and unchanged original node/edge support, including directed and parallel-edge identities. This is not a check of every old attribute. Failures restore graph-owned state, including late monitor failures. External I/O and resources outside that ownership are not covered. Empty dispatch skips even the stage's schedule metadata writes.

from tnfr.operators import (
    observe_self_organization_eligibility,
    execute_eligible_self_organization_stage,
)

observation = observe_self_organization_eligibility(graph)
# Explicitly chosen policy; the function rechecks the current graph itself.
dispatch = execute_eligible_self_organization_stage(graph)

The bounded C8 study reuses the unchanged causal IL/OZ preparation and its actual physical samples from Section 8. It makes no post-dispatch pressure refresh, UM connection or flow.

Evidence	Configured public policy	Explicit counterfactual enabling the optional gate
Nodes with observed magnitude above tau=0.1	All eight	All eight
Complete proposals containing a child	All eight	All eight, independently of gate admission
Incremental THOL grammar admits	Node 0 only	Node 0 only
Application preconditions admit	All eight; optional gate disabled	Odd nodes only; node 0 has negative pressure
Eligible set	{0}	Empty
Actual dispatch	One child 0_sub_0, degree zero; 9 nodes	No stage, no child; 8 nodes

The marked parent's stored pressure is -0.22671636673672074, so forcing positive-pressure preconditions would change the tested public policy. The counterfactual declares that change explicitly. It does not lower a threshold, invent history, choose a replacement parent or dispatch IL instead. Both observations preserve the captured structural state and all nodal data. The live positive dispatch obtains its own fresh relation inside its transaction.

These results establish a finite eligibility/dispatch contract, not an endogenous generation law. Node zero already carries the preparation's IL/OZ history. Executing the eligible set remains a supplied policy, and isolated creation does not establish active transport or persistence. The adapter tests also check finite relabelings with transported histories/marks, simultaneous multiple parents, name collisions, stale observations, late failures and input side effects. Their supplied-state fixtures are distinct from the causal C8 experiment; neither group proves all-state numerical covariance.

Reproduce the source-bound report with python benchmarks/thol_eligibility_dispatch.py; its default local output is artifacts/research/thol_eligibility_dispatch.json. The study tests retain the eight-proposals/one-admission distinction and the empty-policy control. The unresolved origin of preparation asymmetry and dispatch choice is now explicit; the next bounded discriminator remains in the sole execution plan.

The retained run is artifacts/research/thol_eligibility_dispatch_2026_09_18.json, with SHA-256 f16d65ec2ed3e84cc540ecf2acbee2184896ac97ee6e9b92783a0c2dd726ffa5. Its manifest identifies the dirty source scope and records both finite branches. The delivery passed 200 targeted operator/API/integration tests; validation metadata is retained in artifacts/research/thol_eligibility_validation_2026_09_18.json. These ignored local artifacts are reproducible outputs, not dependencies of the versioned implementation or proofs of future execution.

10. Preparation dependence and distributed explicit birth

The bounded preparation study compares three actual prefixes on the same initial C8: none, IL/OZ on one parent, and IL on all eight nodes followed by OZ on all eight nodes. Each all-node operator uses the existing simultaneous public stage, with live incremental admission. No history is assigned. The shared prepare_birth_selection_source retains the initial checkerboard, capacity, phase, support, factors, birth threshold and refreshed physical partition (0.25, 0.25). Each continuation executes its preparation independently; source projections must match before the policy comparison proceeds.

Real preparation	Above-threshold complete birth proposals	Grammar-admitted / eligible parents	Explicit all-eligible births	Default / Parametric runtime births
No IL/OZ prefix	8	None	0	0 / 0
Single-parent IL/OZ	8	Prepared parent only	1	0 / 0
Common all-node IL/OZ	8	All eight	8	0 / 0

The public optional precondition gate remains disabled, as configured in the retained preparation. The observed reference-parent acceleration remains 0.10063556139279584, above the unchanged tau=0.1. Every case has three executor-recorded physical samples. The preparations share the same exact checkerboard-mode reference and capacity. Their represented EPI, pressure and acceleration need not agree exactly: IL changes a phase representation from 0 to represented 2*pi, and finite pressure arithmetic can retain that difference. The report keeps each refreshed Euler boundary and the signed endpoint defect relative to the common exact reference; no bitwise neutrality is assumed. Node data and grammar histories also differ. A real prefix is therefore a joint physical/history intervention, not a general guarantee of phase neutrality or a synthetic change to grammar marks alone.

The all-node case commits eight public births in one two-phase stage. Each parent receives one isolated child; the graph grows from 8 to 16 nodes with its original edges intact. This removes the need to nominate a unique parent for this preparation and dispatch policy. It does not derive the common prefix, the initial nonuniform state, or the decision to dispatch. Both actual built-in selectors still propose and commit IL on every original node, in all three preparations. Eligibility is consequently distinct from execution under either built-in policy; no Si value, threshold or factor was tuned to obtain a birth.

Each case is repeated under i -> (i+3) mod 8, transporting the initialized triad, mark and insertion/neighbor order before execution histories. Initial pressure construction has already populated rebuildable caches; rotation uses detached cache ownership and recomputes that initial pressure. Both graph-copy operations can rebuild undirected adjacency order, so the helper explicitly preserves the original neighbor order. The finite checks find exact binary64 equality of the declared source/eligibility projections, explicit dispatch states, runtime proposals, pre-integrator states and whole-step endpoints after inverse transport. Newborn names are compared by parent lineage. This is one transported-label control, not phase-gauge invariance, arbitrary insertion-order invariance or a complete-state theorem.

The source and immediate dispatch boundary are at t=0.5. Ordinary runtime also advances one Euler step and updates phase/capacity, ending at t=0.75; its endpoint is not compared as though it had the same horizon as immediate public birth. The policy record now includes those phase/capacity settings and the actual prefix/target set, reusing the existing selector instrumentation. The accompanying backend check found and repaired a separate phase update discrepancy: the NumPy path used unwrapped target differences where the scalar path used circular differences. Both now use the existing circular-difference owners. This restores the same phase convention without changing the selected forces or introducing a new dynamical term.

The delivery is complete with preparation dependence and explicit distributed birth supported in this finite scope. The claim that threshold crossing or eligibility suffices for these built-in policies to generate children is rejected in this scope. Autonomous initiation, distributed newborn transport/feedback, active maintenance, perturbation recovery and physical identification remain separate open obligations. The sole queue places the next bounded test at the newborn-support/pressure feedback boundary.

Reproduce with python benchmarks/thol_preparation_policy.py. The default source-bound local artifact is artifacts/research/thol_preparation_policy_2026_09_18.json; the preparation/policy tests preserve source replay, real prefixes, all-node dispatch, selector controls and transported finite projections. Earlier artifacts retain their original source identities and are not recertified by the phase repair.

The retained report has SHA-256 57fc31c4dc368d2245af00f060db2828acbadd01e8e6182b40616a03bbd145b5. Its scoped dirty-source identity is sha256:a17a8669065aeb0869e627dcc023bd3ce29e2334d1383e28bdd8781ed3bcc398. The delivery passes 135 distinct preparation, transport, selector and circular phase tests. Validation metadata, including the exact commands and checked documentation links, is retained in artifacts/research/thol_preparation_policy_validation_2026_09_18.json.

11. Distributed generated support and pressure feedback

The distributed continuation starts from Section 10's actual common IL/OZ preparation and all-eligible public birth stage. Its eight parents remain the UM targets; the eight new children enter the refreshed candidate inventory. An isolated child cannot itself be an initial UM target because the existing neighbor phase gate precedes functional-link proposals. The pure production proposal and the actual simultaneous public stage are recorded separately; no edge or preferred child is assigned by the experiment.

The unchanged default policy produces 16 new edges. Every child is linked to its own parent. Each child of an even parent also links to the two adjacent odd parents; these children have degree three, while children of odd parents have degree one. The original parents consequently have degrees three and five, respectively. The resulting 16-node graph has 24 edges. Distributed attachment is therefore not eight independent copies of the single-parent control: the same compatibility kernel admits cross-parent links.

Disabling functional links or retaining the pre-birth candidate sample gives no new edges. Both controls still execute UM's phase, capacity and direct pressure path on the original parents. The isolated children's freshly computed default pressure is zero in every channel. An explicit new fixed-support physical partition follows structural execution, with boundaries at 0.5, 0.75 and 1.0. The pre-birth eight-node execution certificate is not reused to claim evolution on the new sixteen-node support.

Energy accounting and the capacity source

The isolated birth extends the old Dirichlet form by eight zero blocks, while preserving the original EPI coordinates and conductance. Its energy is unchanged. The separate same-node-support reset from after birth to after UM uses Section 5's exact sum over all new conductances. It is not a conservation claim or a valuation of the new nodes. Each subsequent Euler budget keeps the held initial pressure, refreshed endpoint pressure and binary64 state defect distinct.

The observed support also exposes a useful exact channel identity. Let every original parent have capacity one, every child have the same represented capacity c=Fraction(float(0.95)), and delta=1-c. Each parent has two old unit-weight neighbors, m_i new child neighbors and total new conductance s_i. The children have only parent neighbors. UM's first capacity blend uses the old parent neighborhood, so these capacities are unchanged in this stage. The shared unweighted support convention gives

g_vf(parent_i) = -delta*m_i/(2+m_i) and g_vf(child) = delta.

For strengths d_i and the represented capacity-channel coefficient w_vf, the weighted source is exactly

\[ \sum_i d_i F_{\nu,i} =2w_{\nu}\delta\sum_{i\in\mathrm{parents}} \frac{s_i-m_i}{2+m_i}. \]

Every admitted new conductance is below one in this finite witness; w_vf>0, c<1 and at least one new edge make this capacity-channel sum strictly negative. It follows from weighted EPI transport and unweighted capacity neighborhoods; no extra force is introduced. It is not the full pressure sum: the canonical phase channel and pressure assembly error must retain their separate measured contributions. The topology coefficient is zero in the retained configuration.

On the connected positive-capacity support, H_i=d_i/nu_i. In a declared fixed-coefficient reference, the weighted mean rate is sum(d_i*F_i)/sum(H_i) because the symmetric EPI diffusion contribution cancels. This identity identifies possible collective drift; finite transport does not imply a zero-pressure equilibrium or recovery of a fixed pattern. The existing non-EPI forcing capture derives its channels from phase, capacity and support, independently of stored pressure or a measured EPI derivative. The existing forced-support balance and fixed-target observers remain the owners for subsequent persistence and recovery questions.

Retained finite outcome and next boundary

The distributed study gives the following results over the two post-UM Euler segments. Parent parity refers to the original C8 labels; all eight children are retained in every control.

Observation	Refreshed candidates, links enabled	Links disabled	Pre-birth sample retained
New UM edges	16	0	0
Child degree, even / odd parent	3 / 1	0 / 0	0 / 0
Child EPI change, even / odd parent	+0.0574718321 / +0.0209938443	0 / 0	0 / 0
Parent EPI change, even / odd parent	-0.1166867145 / +0.0350430770	-0.1081676977 / +0.1081676977	Same as disabled links
UM conductance-energy reset	+3.9581016870	0	0
Subsequent two-step Dirichlet-energy change	-2.2444930823	-1.9551208711	Same as disabled links

Every reset and segment accounting residual is exactly zero as a rational identity on the represented data. Actual Euler state defects are retained; the largest coordinate defect is about 1.04e-16. Executor-owned binary64 held-input replays are identified. Exact-affine identification, pure EPI diffusion and a global disagreement contraction are not certified by these multichannel observations. The enclosing preparation/birth/UM/flow chain is not one transaction; the new physical invocation has its own graph-owned transaction and finite causal evidence.

The connected case's exact capacity-weighted source is approximately -0.0032498335468833964. Its contribution to the instantaneous H-weighted mean rate is -7.382632026173776e-5; the phase, topology and EPI weighted source sums are exactly zero in this captured witness. Kernel assembly adds about 5.676e-18 to that rate, and the refreshed stored-pressure residual is zero. The captured non-EPI coefficients are unchanged at the end of the two-step partition. These results exhibit collective drift alongside local child growth; they do not establish an equilibrium. The full disconnected controls explicitly have no positive-H reference, since their isolated children have zero degree. They are not reduced to their old component to force admission to the connected theorem.

This closes the finite distributed birth -> support -> refreshed-pressure response test. The explicit preparation, birth policy and UM target policy remain premises. Persistent nonuniform identity, fixed-target recovery and autonomous operation remain open. Section 12 freezes a reference before an additional admitted event, keeping shape recovery distinct from mean drift; the sole task queue remains the execution plan.

Reproduce with python benchmarks/thol_distributed_transport.py. The retained source-bound output is artifacts/research/thol_distributed_transport_2026_09_18.json, SHA-256 51a8ce4db20c6aa1291fae36beefb83bb4819e8b15b94ac2871532774baa9880, with source identity sha256:e3247f76dd1b94347ac0a6e84dd272915ca0944cf5806418cc8142580233e3b7. The integration tests independently check actual topology, full retained support, causal boundaries, Euler/energy accounting and channel-source identities. These local artifacts are reproducible outputs, not a physical measurement or a future guarantee.

The delivery passes 125 distinct focused tests: 30 distributed integration and accounting cases, plus 95 shared support, forcing, coupling-stage and physical-partition cases. Validation metadata is retained in artifacts/research/thol_distributed_transport_validation_2026_09_18.json.

12. Fixed original model profile under a later UM stage

The fixed-target study continues the same connected sixteen-node case. It reuses the relative-profile derivation and original-target compatibility calculus. It adds no pressure law, recovery score or replacement profile solver.

At the first refreshed post-UM boundary, t=0.5, the independently captured phase/capacity/topology channels determine one exact held-model reference. Its centered relative profile z0, metric H0 and predicted collective drift are fixed and content-addressed before the baseline flow. The profile is the relative equilibrium of that declared model. It is not the EPI pattern actually created by THOL; the initial difference is explicitly measured.

Each branch independently replays the preparation, birth, first UM and common physical partition to t=1.0. The frozen references, complete retained prefix records and declared baseline projections must agree before branching. One branch continues without another operator. The other executes one actual simultaneous UM stage on all sixteen current nodes, with refreshed candidates, live admission and unchanged factors. Both then execute just one further (0.25, 0.25) physical partition, ending at t=1.5.

The original score is always

R0(x) = 1/2 * ||P0*x-z0||^2_H0,

where P0 removes the original H-weighted mean. New references describe the post-event held model and its event/flow accounting, but do not replace z0 or H0. In particular, UM's zero EPI jump leaves the immediate original score and original mean unchanged even when capacity, support and the current model's coordinates change. A change in a current-model error at that event is not measured recovery of the original shape.

The existing target observer evaluates the current model's residual at z0:

r = P0 * (diag(nu_current)*F_current - A_current*z0).

Nonzero r excludes compatibility with the old relative profile in that unchanged exact model. It can coexist with a decreasing finite R0, because the evolving state can initially approach an incompatible old target. Zero r does not require zero collective drift and does not certify future runtime persistence. The signed old-metric rate split, channel cross terms, pressure realization and Euler defects retain their separate meanings.

Finite comparison and immediate rate decomposition

The same initial error is R0(0.5)=7.9301241975. The baseline flow reduces it to R0(1.0)=6.2231811440 in both independently executed branches. The second UM adds eight actual edges, bringing the support to 32 edges. It is admitted without changing the configured factors or reconstructing history.

Observation	No second event	All-node second UM
Immediate original-target error at t=1.0	6.2231811440	6.2231811440
Original-target error at t=1.5	4.8924378794	5.0044383605
Current held model compatible with z0	Yes	No
Exact compatibility energy, displayed approximately	0	5.6844308511e-7
Current relative profile's error in the original metric	0	1.5067976770e-5
Current model's uniform drift coefficient	-7.3826320262e-5	-4.7603761320e-5
Total original-metric mean change, 0.5 to 1.5	-7.3826320262e-5	-0.0064284882

The current-profile error is an algebraic mismatch between two derived relative equilibria. It is the limiting error only under the corresponding exact held-model convergence assumptions; it is not a forecast certified for the future binary64 runtime. The original-metric mean uses H0, whereas the changed model's drift coefficient uses its own H1. The event observer retains the mean reweighting, and each physical step separates its current-model drift, pressure realization and Euler defects. These two mean quantities must not be equated after the event.

The unchanged state at the event also allows an exact signed comparison of the old-target energy rates. The stored nodal rate changes from approximately -2.9140469721 to -2.6594895413, a change of +0.2545574308: it remains negative but is less negative immediately after UM. The shared target observer splits this change as follows:

Signed contribution to the rate change	Value
Homogeneous generator term in the original metric	+0.2540293456
Original-target source term	+0.0005280852
Stored-pressure realization defect term	+1.5201530622e-16

Their sum equals the total exactly as rational arithmetic on the represented inputs. The change in the homogeneous term dominates this instantaneous comparison. It does not assign the finite endpoint difference to a separately executed channel intervention. The compatibility-energy allocation is also signed: the EPI and capacity contributions are about -2.0009575391e-6 and +2.5694006242e-6, respectively, with exact zero phase/topology contributions. Cross terms are retained; these are algebraic contributions, not ablations.

This completes the finite test. The later all-node UM fails original-profile compatibility despite reducing the short-horizon error. The no-event branch's compatibility and error decrease do not establish autonomous maintenance. The useful next discriminator is the scale of the actual UM dispatch, using the original-parent and generated-child cohorts supplied by retained THOL ancestry. Its execution and acceptance conditions belong only to the plan.

Reproduction and validation

Run python benchmarks/thol_distributed_target.py. The source-bound output is artifacts/research/thol_distributed_target_2026_09_18.json, SHA-256 09757606385536245ae35b2224d0d1932e8f33b465f37ab22209ef7dd6689574, with source identity sha256:848d7de8343b5d35703e9bc0a64c429199848c7e42a0294074bb15cedef733f9. The original reference has SHA-256 0adabc2cc60bf03d898b84af1d2b88774d1b7b77de02dc67685fc220c24af27b in both independent branches. The earlier Section 11 artifact keeps its historical source identity; the preparation/flow helper is now shared by both studies.

The new integration tests independently check the Poisson/gauge references, fixed original metric, matched causal sources, actual event, exact reset and eight step budgets, signed channel/rate identities and complete JSON serialization. Serialization reuses the existing literal projection, preserving deque contents and retention limits. Fourteen new cases, 30 prior distributed cases after the shared-helper refactor, and 83 shared forced-support/event/target cases pass: 127 distinct focused tests. Validation metadata is retained in artifacts/research/thol_distributed_target_validation_2026_09_18.json.

No long campaign, threshold search, independent physical observation or new pressure law is part of this result. The declared preparation and dispatch policies remain supplied inputs. Live admission/read-out failure aborts the bounded study without a tuned retry.

13. Lineage-scoped coordination on the generated support

The lineage study asks whether the second UM response depends on coordinating the original nodes, their generated children, or both together. The two cohorts come from the actual THOL birth pairs and retained parentage; they are disjoint and exhaust the sixteen-node support. Cohort choice is a declared intervention, not an autonomously selected law.

Each new branch reuses the causal preparation, reference freeze at t=0.5 and baseline flow to t=1.0 from Section 12. The original z0/H0 remain fixed. The retained no-event and all-node controls are historical evidence with their original producer identity; they are usable only after exact agreement of the new branch's complete retained baseline and observation inputs. The two additional interventions target either every original parent or every actual child once, with unchanged factors, fresh candidate sampling, live admission and one subsequent (0.25, 0.25) refreshed physical partition.

Target membership and write scope are different. UM capacity updates belong to selected targets, but bidirectional phase proposals can write neighboring nodes outside that set. Simultaneous execution merges overlapping proposals from one immutable snapshot. An all-node stage is therefore not assumed to equal a sequential or additive composition of the two cohort stages. Raw phase writes and wrapped circular changes must also remain distinct at the represented 0/2*pi boundary.

This comparison contributes to the primary generation/persistence question: it tests the effect of coordination scope after genuine node birth and support formation. Creating a child ring, retaining a derived relative profile, restoring a damaged pattern and maintaining it autonomously are separate claims. Interaction laws and independently tested physical identification remain later obligations in the single execution plan.

Matched outcome and the child ring

Both new branches pass the actual admission gates and reach t=1.5. Their complete retained baselines and original references match both historical controls. Every selected target's local proposal also matches its corresponding proposal in the all-node control, before the simultaneous merge. The unchanged pre-event original error is 6.2231811440 in all four cases.

Observation	No event, retained	Parents only	Children only	All nodes, retained
New edges	0	0	8	8
Original-target error at t=1.5	4.8924378794	4.8939252624	5.0030976710	5.0044383605
Current held model compatible with original profile	Yes	No	No	No
Compatibility energy	0	2.0900858401e-7	1.7504227884e-7	5.6844308511e-7
Current-profile mismatch in original metric	0	2.4075802416e-6	5.7409527832e-6	1.5067976770e-5
Current model's uniform drift coefficient	-7.3826320262e-5	-7.1234944722e-5	-4.9530683966e-5	-4.7603761320e-5

The child-only stage creates the same eight child-child edges as the all-node stage, each with represented weight 0.871053763203346. Those edges form a connected degree-two ring on the eight children. Parent-only targets cannot propose an edge with two child endpoints because a candidate edge must have a selected source. They create no new edge in this execution. This identifies a structural consequence of the declared target set, rather than selecting the target set by its eventual score.

Parent-only UM decreases the capacities of even/odd original parents by about 0.0013262912 / 0.0023873241, respectively, while child capacities stay fixed. Child-only UM increases every child capacity by about 0.0039788736, while parent capacities stay fixed. Phase proposals reach all eight untargeted nodes in each branch; there are twelve multiply proposed phase recipients in the parent branch and four in the child branch. In both executions, the only raw phase differences are the parent-zero/child-zero writes from 0 to represented 2*pi. Their wrapped differences are about -2.4493e-16, not a large circular phase displacement. The exact raw writes remain retained rather than erased.

EPI and its original-target score do not jump at either UM event. The original mean likewise stays fixed immediately, while a current-metric mean can change by reweighting. The total original-metric mean changes from t=0.5 to 1.5 are about -6.6998959883e-5 for parents and -0.0064446596 for children; neither is silently substituted for the current-model drift coefficient.

At the same pre/post-event EPI, the change in the original-target nodal energy rate separates exactly into the following signed terms:

Rate change	Parents only	Children only
Homogeneous generator term	+0.0033640238	+0.2506653218
Original-target source term	+0.0001999095	+0.0006907930
Stored-pressure realization defect	+3.8818e-17	+1.1512e-16
Total	+0.0035639333	+0.2513561149

All sums are exact rational identities on represented values; the table rounds their display. Both rates remain negative but become less negative at that event. This is instantaneous accounting, not a channel-isolated causal explanation of the finite endpoint differences. Profile mismatches are conditional held-model limiting discrepancies, not pointwise error floors or guarantees about future clipped execution.

The declared cohort comparison is complete. Neither additional cohort preserves compatibility with the original profile; neither supplies recovery or autonomous maintenance. No-event has the smallest finite endpoint error in this comparison. That does not turn inaction into an autonomous generation law, nor does the rejection cover all possible schedules or target sets. Further subset search would not by itself resolve the macro question. Section 14 instead tests the sufficient observed state of actual parent-child families using the existing quotient and derived-memory identities.

Evidence and reproduction

The output artifacts/research/thol_lineage_coordination_2026_09_18.json has SHA-256 8b7d6e3e217d57566788350955aa8afbba747add07a49b341167c0626b0b848a and source identity sha256:30b40b98e87b1ad9026e5932338965e640562b7a55b886141246c0ad23a8adaa. It retains the Section 12 control file's exact SHA-256, producer manifest and source scope. Only the two new cohort trajectories execute; historical control outcomes are not assigned the new producer's identity.

Run python benchmarks/thol_lineage_coordination.py when the pinned Section 12 artifact is available. In a fresh checkout without local artifacts, generate complete new controls and explicitly declare their new byte identity:

python benchmarks/thol_distributed_target.py --output artifacts/research/target_controls_reproduced.json
$controlDigest = (Get-FileHash -Algorithm SHA256 -LiteralPath artifacts/research/target_controls_reproduced.json).Hash.ToLowerInvariant()
python benchmarks/thol_lineage_coordination.py --controls artifacts/research/target_controls_reproduced.json --expected-control-sha256 $controlDigest

Those regenerated controls have their own producer identity; they do not replace the historical file. The CLI refuses to overwrite its control input. Digest admission alone is insufficient: full retained baseline and observer comparison still applies. Per-node grammar, hard U3 and enabled optional-gate refusals are recorded without another event or continuation. Joint-stage refusals and unexpected implementation/accounting failures abort the study; they are not converted into ordinary rejected branches.

The integration tests generate portable real control fixtures rather than requiring ignored historical artifacts. They check ancestry and refusal boundaries, complete source matching, actual stage effects and independent exact profile/event/step accounting. The historical artifact has a separate read-only rational recount and retains its original identity. Validation covers 35 new lineage cases, 14 prior fixed-target cases after the shared refactor, and 81 shared coupling/event/target cases: 130 distinct focused tests pass. Validation metadata is recorded in artifacts/research/thol_lineage_coordination_validation_2026_09_18.json.

13.1 Full-triad identity and boundary exchange on the retained child ring

The read-only cohort audit reuses Section 13's child-only UM branch and its authenticated no-event control. Their complete pre-event state at t=1 agrees. It runs no producer, operator, pressure-generation kernel, phase evolution or new trajectory. The historical manifests retain their original source identities.

Exact record admission is centralized in recorded_support.py. It rebuilds support, forcing and held references through the existing physics owners and checks every archived field. The earlier family/regional readers use compatibility wrappers, so historical and new audits share those checks. Raw represented triad values are distinct from exact rational coefficient records: booleans, nonfinite values and textual raw phase coercion are refused. A separate detached graph view supplies the current winding and tetrad read-outs. The historical record did not contain a complete tetrad: these are explicitly offline observations, including correlation-fit/fallback provenance and neighbor order reconstructed from saved edges.

Observation	Immediately after child UM, t=1	Retained endpoint, t=1.5
Numerical child-cycle winding	+1	+1
Child capacity	0.9539788735772974	Same represented value
Child H-weighted mean EPI in the post-event metric	Approximately 0.43906174	Approximately 0.46065218
Child centered variance in the fixed post-event full-graph metric	Approximately 0.00065625348	Approximately 0.00410622688
Projection on the actual post-event centered form, in the common pre-event metric	1	Approximately 2.50141449

The phase pattern is inherited and its cycle is newly connected. Before UM, child-cycle winding is undefined, not zero. UM immediately leaves EPI unchanged, raises child capacity from 0.95, and adds eight internal child edges. The subsequent form is nearly proportional to the post-event child shape, but not exactly: the squared orthogonal residual divided by the squared form norm in that common metric is approximately 4.19086e-30. Exact represented arithmetic retains that positive residual. This projection is a descriptive amplitude measurement, not a selected dynamical gain, acceptance threshold or proof of an NFR identity. Actual form drift from the first attached preparation remains separate from error relative to the original solved profile z0.

The existing full-graph regional owner explains the increase. At the two Euler segment starts, its variance-rate terms are approximately:

Start	Internal dissipation, subtracted	Parent-boundary contribution	Explicit non-EPI source	Stored nodal variance rate
1	0.00024351903	+0.00436140411	+0.00006485451	+0.00418273959
1.25	0.00078611839	+0.00731692546	+0.00011652462	+0.00664733169

Represented-pressure discrepancies remain separate, of order 1e-19 in these rates. Both exact finite regional mass/variance balances close with zero residual. The two-step child variance increase is approximately 0.00344997340, while full-graph Dirichlet energy decreases approximately 1.60652170. Thus global relaxation and increasing regional contrast coexist: the parent boundary supplies variation while internal child edges dissipate it. Finite Euler terms are held-step accounting, not exact continuous-time integrated fluxes. The fixed-metric assumption applies after the event; the UM metric/support reset is retained separately.

The no-event comparison also nearly preserves its normalized centered shape direction, while its amplitude changes. Its final original-model target error is about 4.89243788, compared with 5.00309767 for child UM. Neither number measures recovery of the born pattern. The audit compares actual form using one common frozen regional metric, rather than comparing the unequal post-event metrics as if they were the same score. Against the actual initial attached form, the ring's centered-error measure is smaller instead: which branch is closer depends on the reference and quantity being compared. Neither comparison establishes restoration. The distinctive topological evidence is created cyclic support carrying inherited winding. A matched damage/recovery trial is absent for this ring-connected branch; later regional-restoration records have different support/history and cannot fill that gap.

Evidence: artifacts/research/thol_child_ring_identity_2026_09_19.json. Integration validation: 341 passing tests in 11 modules, including 98 new controls; two optional pinned-history tests were explicitly excluded. The shared-reader, exact shape, record-mismatch and regional-continuity controls are portable and do not require ignored research artifacts. Checkpoint: artifacts/research/cohort_identity_integration_validation_2026_09_19.json. The single G3 gate owns any prospective finite-identity protocol. This result supplies no new force or autonomous phase/event law, and establishes neither future maintenance nor physical-particle correspondence.

14. State sufficiency of actual parent-child families

The completed analytical comparison uses the four retained held references: the original no-event model and the parent-only, child-only and all-node post-UM models. No runtime trajectory, partition search or parameter fit is added. One family consists of each actual parent and its actual THOL child, as recorded by birth receipts and retained parentage. These eight disjoint pairs are a declared observation partition, not an assumed set of autonomous objects.

The shared exact affine closure criterion retains the canonical forcing and asks whether the H-weighted family means determine their own derivatives for every microscopic EPI state. A negative answer means the chosen aggregate omits dynamically relevant information; it does not reject the complete nodal dynamics or establish a physical memory law by analogy. Current-model coordinates used for this calculation remain separate from the frozen original profile used to score retention in Sections 12-13.

Exact obstruction and memory

All four models have nonzero R*A*Q and Q*A*P, so none closes on the eight family means for all real scalar EPI states. Each exact observation retains a deterministic witness delta=Q*e_0: the states x and x+delta have identical R*x, but different projected derivatives. These detached rational states need not satisfy a live graph's chart, clipping or grammar constraints. The result rejects all-state affine closure of this observation; it is not a pair of newly executed trajectories.

The source of the obstruction is already visible in the original support. Parent 1 has neighbors in other families, whereas its child connects only to that parent. Applying A to the indicator of family 0 gives approximately -0.0726326622 at parent 1 and exactly zero at its child. Thus a lifted family field does not remain block-constant. Reversibility converts this failure into the nonzero projected hidden-state term.

For every model the exact Gram identity Hbar*K(0)=(Q*A*P)^T*H*(Q*A*P) also holds, with K(0) != 0. The table displays rounded values from the exact rational matrices. Here RAQ_inf is the maximum absolute row sum, and K0_max is the largest absolute entry, not a matrix norm. The separate binary64 cross-check uses the existing spectral 2-norm residual diagnostics and does not decide exact closure.

Held model	Eight-mean affine closure	RAQ_inf	K0_max	Maximum absolute same-EPI change in family observation
Original	Rejected	0.0508396000	0.00312297573	0
Parent-only UM	Rejected	0.0506375593	0.00310284723	0.000433054959
Child-only UM	Rejected	0.0206040003	0.000256228121	0.171042708165
All-node UM	Rejected	0.0205261381	0.000251303493	0.170629235097

These numbers are not an improvement ranking: the model, metric and observer change between rows. The exact nonzero decision is the closure result. The zero-lag identity identifies a derived-memory coupling; no history kernel is fitted, truncated or evaluated with a matrix exponential in this study.

Source, centering and observer resets

Every model retains its captured b=diag(nu)*F. Known constant forcing is compatible with a closed affine projection when R*A*Q=0; Q*b=0 is the additional condition for invariance of the lifted affine subspace, not an extra requirement for projected autonomy. The implementation checks this distinction with a positive test control.

The homogeneous error coordinates use each model's own relative profile and weighted mean: u=x-m_H(x)*1-z, giving u'=-A*u. The original frozen z0/H0 score from Sections 12-13 remains a separate read-out. At the retained UM boundary the fine EPI is identical across models, while their family means change by exactly (R1-R0)*x. The corresponding current-H global mean resets are +9.5688660174e-5, -0.108022140967 and -0.107841721248 for parent, child and all-node UM. These are changes of weighting at fixed EPI, not elapsed drift or a nodal EPI jump. The centered reset also subtracts the change of global mean and the projected relative-profile shift; all residuals are zero.

Section 15 determines the minimal sufficient linear observation that retains these means, using existing invariant row-space and P5 minimality tools. Extra coordinates may be internal or nonlocal contrasts. They do not alone establish autonomous entities, full macro-triad dynamics, recovery or physical correspondence. The sole execution plan owns the next gate.

Implementation, evidence and reproduction

The shared exact observer observe_forced_support_closure rebuilds the forced-support reference and checks projection, reversibility, memory Gram and current-rate identities. The detached study authenticates both historical input files, producer identities, actual ancestry, complete common state and captured forcing. It does not construct a new graph or run a solver. Its manifest therefore has no new seed or timestep.

With both pinned historical artifacts available:

python benchmarks/thol_family_closure.py --output artifacts/research/thol_family_closure_2026_09_18.json

For a fresh checkout, regenerate the Section 12 controls and Section 13 lineage comparison first, using explicit new output paths and the controls' new digest. Then pass both new paths with --controls and --lineage, and declare their byte identities with --expected-control-sha256 and --expected-lineage-sha256. The study retains those producers rather than claiming they are the historical ones. It rejects an output path equal to either input. No ignored artifact is required by the portable integration-test fixtures.

The completed report artifacts/research/thol_family_closure_2026_09_18.json has SHA-256 13642d936260e849ac5e588f32a6ea6716b74d733ab8232e4376f5f48d621ae1 and source identity sha256:c3b5cea441d735fa28b06cd943e0b4677fbbe769926aa85a1c789085169e2263. It preserves both input hashes and producer manifests from Sections 12-13. Independent rational reconstruction verifies the generator, projection, Gram identity, profiles, four witnesses and four observer resets without importing the producing calculations. Review corrected a numerical-output label: the shared diagnostic measures the matrix spectral 2-norm, not the infinity norm. This changed labels and added a regression, not the exact decision.

The exact-observer tests and study tests cover independent matrix reconstruction, positive closure and affine-source controls, hidden witnesses, stale/malformed input, ancestry, model-state admission, observer reset and CLI behavior. Existing memory/morphism regressions remain included. Validation covers 147 distinct focused tests: 33 exact-observer cases, 28 study cases and 86 existing memory/morphism cases. The initial combined run passed 146; after the norm-label correction, all 28 study cases passed. Validation metadata is retained at artifacts/research/thol_family_closure_validation_2026_09_18.json.

15. Minimal sufficient linear state for the retained family observations

The minimal-state construction resolves how much extra information Section 14's eight family means need. It keeps the same actual ancestry partition, four held models and captured forcing. Each model supplies an exact A, b and R; no new graph, trajectory, partition, target or fitted coefficient is introduced.

Rank and sufficient state

All four models give the same sequence of exact row-space ranks:

Included observations	Original	Parent-only UM	Child-only UM	All-node UM
R	8	8	8	8
R, R*A	15	15	15	15
R, R*A, R*A^2	16	16	16	16
Through R*A^3	16	16	16	16

The complete third-power level adds no rank. Deterministic row selection retains all eight rows of R, rows 0 through 6 of R*A, and row 0 of R*A^2. These sixteen rows form C; every microscopic column is a pivot. The realized state s=C*x satisfies s'=-G*s+C*b and recovers the family observation through R*x=D*s. Exact identities C*T=I, C*A=G*C and R=D*C are checked on full matrices. Here T=C^(-1) because the rank equals the microscopic dimension, and D selects the first eight coordinates.

Thus eight additional linear coordinates are necessary and sufficient for every model in the stated all-state domain. This is an exact change of coordinates, not a reduction of the sixteen-dimensional EPI state. The extra rows are algebraic observations of EPI derived from its generator; they are not independent forces or eight newly discovered physical objects. For example, R*A*x is related to the projected nodal derivative by R*A*x=R*b-y'; this identity does not claim that such derivatives have been measured independently.

The minimality proof covers every linear observation retaining these family means and closing for every scalar initial state of the held affine model. It does not rule out an approximation with a quantified error, a reduction on a justified restricted state class, or a description retaining history. It also gives no robustness guarantee for reconstruction from noisy data.

What this resolves for the generative route

Birth and ancestry do not alone justify replacing each family by a single autonomous scalar. The exact state requirement is now known for all four retained models, so repeating an exact compression search on those same observations would add nothing. The derived-memory representation remains available as an alternative way of retaining the missing information. The formation, persistence/recovery and interaction questions still concern the actual TNFR dynamics. Physical correspondence requires its separate independent observation and prediction protocol.

Implementation, evidence and reproduction

The shared realization observer observe_forced_support_realization reuses the prior closure geometry, rectangular exact products, rational rank and inverse. Every model needs 49 rank calls and nine matrix products in the realization stage. Maximum materialized coefficient bit sizes are respectively 582, 635, 828 and 884; these counters exclude reference/closure reconstruction and internal elimination costs. The operational rank-call limit is 4096 per model, and exhaustion returns no partial certificate or approximate substitute.

The detached study authenticates the Section 14 report and both of its input files, replays its entire scientific payload, and compares every retained model, witness and observer reset. Input file locations may move; byte identities and historical producer metadata remain fixed. The new manifest marks the realization as derived, with no new seed or timestep.

With the three pinned historical input files available:

python benchmarks/thol_family_realization.py --output artifacts/research/thol_family_realization_2026_09_18.json

A fresh checkout can regenerate Sections 12-14 first. Supply their explicit paths with --controls, --lineage, and --family, together with the three corresponding --expected-*-sha256 values. Regenerated inputs retain their own producer identities. Output cannot overwrite any input, and admission or resource failure does not produce a successful partial report.

The completed report artifacts/research/thol_family_realization_2026_09_18.json has SHA-256 cf83a848f753c55c307bded156fda635441d735924ea178a777042af6e2e5daa and source identity sha256:b65a48f9bc51c3b977ee91b7b9a8b593501af93195159f3c4fa012be0b25c407. Independent Fraction arithmetic reconstructs every generator and observation from retained conductances/capacities, repeats row selection and rank, and checks all four realizations and projected rates without importing production calculations.

The owner tests include an exactly closed quotient, the existing minimal P5 reduction, full-rank controls, tiny rational symmetry breaking, complete-level stopping, affine forcing and work-limit failure. The study tests use portable fixtures and check complete replay, rejected tampering, relocated inputs, CLI behavior and unchanged historical metadata. The final combined run passes 210 tests, including 63 new cases and the prior closure, family, memory and morphism regressions. A separate baseline run also passes the 55 existing P5 reduction cases. Validation metadata is retained at artifacts/research/thol_family_realization_validation_2026_09_18.json.

16. Full-EPI response to one child Emission

The exact additional coordinates from Section 15 are already available. Retaining them permits a mechanism test without first quantifying the error of omitting them. The optional eight-mean approximation study is parked. Here "full state" means all sixteen EPI coordinates of the held affine model; phase, capacity, support, operator policy and lineage remain separately recorded parts of the engine state.

Declared intervention and reserved prediction

The protocol selects two independently prepared branches of the existing distributed THOL/first-UM source. Both use the same refreshed (.25,.25) baseline to the common physical boundary t=1. One branch has no event; the other requests one default public simultaneous Emission on the eight actual born children. No response score selects the operator, cohort, factor or horizon. The source's original z0/H0 reference stays fixed.

Emission is used because its canonical direct structural write is EPI; it preserves capacity, phase and stored pressure. Live grammar and preconditions still decide admission, and its actual clipped jump and lineage metadata must be retained. The shared strict precondition reads EPI_LATENT_MAX and VF_BASAL_THRESHOLD from configuration; their current defaults are both 0.5. The actual children at the retained boundary have EPI about 0.43-0.44 and capacity 0.95. No gate is weakened to obtain a perturbation. A refusal or absence of fixed-target damage remains a valid result.

The native pointwise receipt's stronger full-network affine certification is not required for this child-only intervention. That flag also requires every graph node to be targeted and zero exact affine realization defect. The study retains its actual value and failed conditions, verifies the receipt seal and proposal/gate replay, binds every before/after EPI by node, and independently checks the represented affine jump plus its signed rounding residual. It does not promote a partial finite stage to a global jump-gain certificate.

After the event, a read-only kernel capture verifies that phase, capacity, support, EPI coefficient and independently captured forcing agree with the original model. The full-EPI forecast then fixes the next two .25 steps from the actual post-event initial state, before that branch's runtime continuation advances to t=1.5. Every predicted EPI coordinate follows the same derived C/T/G/Cb realization; the reference, source and coefficients are not fitted to the endpoint.

This forecasts the declared Euler discretization. Actual pressure realization, rounding and clipping remain separate measured defects, propagated through the complete Euler matrices. The graph itself is evolved only through the existing pressure-refreshed executor. Its first boundary refresh replaces the stored pre-event pressure; the forecast does not mistake that stale value for the new model pressure. Each physical partition receives its fresh entry snapshot; a frozen reference's earlier EPI is not its entry state.

Response observables and claim boundary

Let d=x_AL-x_control on the common held support and let m_d=sum(h_i*d_i)/sum(h_i), with H=diag(h_i) from the original model. The full separation, mean offset, and centered spatial energy E_d=1/2*sum(h_i*(d_i-m_d)^2) answer different questions. In ideal arithmetic the common forcing cancels, d_next=(I-h*A)*d, and m_d is conserved. The spatial difference can decrease while a uniform offset persists. Runtime defects and their propagated effects are reported separately.

The same energy can be expressed through the complete chart using the inherited metric T^T*H*T. An unweighted Euclidean norm of C*x would mix coordinate scales and is not substituted for the nodal energy. Each branch also keeps its original-target error, positive capacity and actual nodal activity. If AL immediately reduces target error, the event has not damaged that target and its continuation cannot be called recovery from such damage.

Finite attenuation of an actual perturbation is a test of the held nodal response. It does not erase the Emission history, prove absolute-state return, derive autonomous maintenance under changing support/phase/capacity, or establish physical particles. These distinctions keep the experiment connected to the macro objective without promoting a limited result beyond its evidence.

Measured finite response

Both independent preparations agree in every retained common-source field. Strict and native admission accept AL on all eight children. Their represented EPI increments are approximately 0.0795774715459476; no clipping acts and every unselected EPI stays unchanged. The native full-network affine flag is false for its two stated reasons: partial target coverage and a nonzero exact affine rounding residual. The finite event evidence above remains valid.

Observable	Immediately after event, t=1	Endpoint, t=1.5
Paired centered H energy	0.030645699389185228	0.02348125740481349
Paired full H energy	0.045497084867582895	0.03833264288321107
Paired H-weighted mean offset	0.02597607536317475	0.025976075363174672
Original-target error, control	6.22318114397756	4.892437879433556
Original-target error, AL branch	5.747522706029956	4.503852112409268

The centered difference energy decreases by 23.3783%, while full EPI separation persists. The measured mean-offset change is about -7.70e-17; the ideal held-model recurrence preserves this mean exactly. All sixteen nodes retain positive capacity and nonzero refreshed nodal rate at each reported activity boundary. This is active finite response under held coefficients, not freezing.

The maximum absolute EPI discrepancy over both forecast frames is 1.414e-16 for the control and 6.227e-17 for the AL branch. Independently propagated pressure-realization and integrator defects account for every coordinate exactly. These observations are not a uniform floating-point error bound or a continuous-time solver accuracy estimate.

AL immediately changes the fixed-target error by -0.47565843794760393. It improves this particular score rather than damaging it, so neither its further reduction nor its lower final value establishes recovery from a damaged pattern. The target and outcome rule were fixed before this result. The finite response/forecast delivery is complete and supported in its stated scope; autonomous maintenance and physical correspondence remain untested.

Evidence and reproduction

The paired study reuses the actual distributed birth/transport preparation, public stage executor, exact forced-support/event observers and full-state prediction owner. It does not implement a second graph integrator. AL's complex lifecycle metadata now uses the existing exact metadata projection, extended with separately retained binary64 real and imaginary components. Configured AL thresholds replace stale hardcoded examples in the operator docstrings; operator behavior is unchanged.

python benchmarks/thol_full_state_response.py --output artifacts/research/thol_full_state_response_2026_09_18.json

The completed report has SHA-256 5d95176d28ac633351337d90f3bac0c0fc7edd3e24cf68cc04a5df7de125987e and source identity sha256:bfd61a88c903ef3b2f4f72a087bb6bf21ddacf100cbdde4501c4e507c09c7052. The prediction tests and causal study tests verify the rational recurrence, prediction-before-continuation order, complete paired preparation, partial-stage evidence, actual defects and refusal path. The final validation passes 53 new tests and 210 existing regressions (263 distinct cases). A separate standard-library Fraction audit reconstructs the model from raw conductance, capacity and forcing, checks all forecast frames and actual defect recurrences, and recomputes paired/target metrics: 281 exact checks pass. The detached audit does not authenticate serialized runtime seals or independently prove execution chronology; the instrumented runtime tests supply the chronology check. Validation metadata and the independent audit are retained in artifacts/research/thol_full_state_response_validation_2026_09_18.json and artifacts/research/thol_full_state_response_independent_check_2026_09_18.json.

17. One native composite runtime response

Predeclared protocol and interpretation

Section 16 establishes a finite response with phase, capacity and support held. The next bounded test retains both actual causal preparations and runs exactly one existing native runtime step from each endpoint at t=1.5, with dt=.25, use_Si=True and apply_glyphs=True. Both prior scientific branch payloads must replay before either continuation. Defaults, selector, histories and stable counters remain as produced; there is no threshold, operator, cohort or horizon search. The original target z0/H0 is never replaced to improve the score.

Emission records a fresh UTC activation label in addition to the graph's structural clock. Historical replay therefore preserves and compares old/new UTC labels at explicitly identified child lifecycle paths, checks their internal alias/origin consistency, and requires exact equality elsewhere. Physical _t, timestamped EPI history, activation counts and all nodal state remain part of exact replay admission. No wall-clock label is rewritten on the graph, and no generic timestamp field is silently dropped. Agreement modulo those declared labels is not byte equality of the complete report.

Before each step, the existing full-EPI predictor freezes one held-model Euler counterfactual. Read-only trace wrappers then observe the actual native schedule: callbacks, sampling/history boundaries, pressure and Si preparation, selector decisions and actual glyph application, integration, clamps, phase coordination, capacity adaptation, optional math engine, REMESH history and decision, validators and final callbacks. Instrumentation delegates each operation once to its existing owner and restores the wrapped call sites. It is not a new integrator, selector or transaction around the entire step.

The prepared Si is evaluated before glyphs. Capacity adaptation later reads that retained Si and the then-stored pressure; it does not perform a fresh endpoint coherence test. Its configured thresholds and counters are part of the declared policy, not new constants derived by this experiment.

The exact counterfactual ledger separates pre-integrator EPI writes, changed current-model rates, stored versus fresh pressure, pressure realization, integration residual and later EPI writes. Report the old-target score only while its original node space remains available. A changed phase, capacity or support is a model boundary even if the same EPI coordinates still permit a score; changed node identity/order makes the old vector comparison unavailable. A valid refusal, inactive mechanism, worsening or changed domain is retained, not tuned away.

This tests one execution of the existing composite state-dependent policy. It can locate active feedback and its immediate consequences. It cannot isolate one mechanism causally, establish autonomous generation or indefinite maintenance, or establish recovery from damage that the AL event did not cause. Physical correspondence remains a separate open obligation.

Measured response and actual active mechanisms

Both native steps complete at t=1.75. Each freshly computes Si, makes sixteen outer selector decisions and commits Coherence (IL) once at every node. The resolver's nested internal calls are retained separately; they are not extra decisions or operator executions. Actual IL changes stored pressure only in this witness. There is no EPI jump before integration or afterward, and the clamp interval between integration and phase coordination has zero EPI change.

Observable	Control: before -> after	Child Emission: before -> after
Original z0/H0 error	4.892437879433556 -> 4.470889149742858	4.503852112409268 -> 4.111168480693482
Stored-state total coherence C(t)	0.8360164407909924 -> 0.8736409592473426	0.8439378325321703 -> 0.8800051393409857
Capacity and support	Preserved	Preserved
Phase and captured non-EPI forcing	Changed	Changed

Paired centered H energy decreases from 0.02348125740481349 to 0.021254184497818077, a further 9.48447% over this one step. The mean offset remains approximately 0.02597607536; no full EPI return occurs. Coherence above reads the actual stored pressure and derivative. It is not a claim that fresh kernel pressure after phase motion equals the stored one.

At integrator entry, the exact current model still agrees with the original model. IL's intentional stored-pressure contraction supplies the dominant departure from the frozen Euler counterfactual: maximum absolute nodal impulses about 0.011256 and 0.011011. Fresh-kernel realization terms are below 7.0e-18, and the generic integration remainders are below 8.4e-17. The exact full-vector ledger closes for each branch. Calling the roughly 0.011 displacement a floating-point error would therefore misidentify the mechanism. Subsequent global phase coordination changes phase by up to 0.1672005576644261 radians and changes the forcing for future evolution.

Si was absent at entry and is populated by the native preparation, with ranges approximately [0.9312,0.9836] and [0.9346,0.9811]. Capacity adaptation nevertheless remains inactive: every absolute stored pressure exceeds its actual 0.001 stability threshold (minimum 0.03598 and 0.02640), so every stable counter is zero afterward. Its configured count requirement is five consecutive qualifying evaluations. No counter or gate was adjusted. The math-engine and REMESH stages introduce no structural change here, and the callback registry is empty. Their trace records preserve those negative observations rather than claiming they were never considered.

This locates actual state-dependent negative feedback and phase evolution in the generated structure, while identifying dormant capacity feedback. It is still one finite invocation of a supplied policy. The prior AL event improved the target score, so this continuation does not establish damage recovery.

The endpoint phase arrays and independently captured forcing are exactly equal across the two branches. Their common source cancels in a paired rate comparison. This does not preserve the original individual relative profile: the exact endpoint residual P_H0*(diag(nu)*F_end-A0*z0) has maximum absolute component 0.03145148206669386, with the same nonzero vector in both branches. Its uncentered uniform rate is approximately 0.00027302155397510133. Thus decreasing finite old-target error and compatibility of that target are different facts here as well. A persistent evolving pattern need not be a fixed EPI profile; any alternative notion of identity still needs an explicit derived observation and prospective test, not a retrospectively moved target.

Shared implementation, evidence and reproduction

The native study uses replay_response_branch from the preceding study to obtain the real live endpoint. The old mapping-only API remains unchanged. The same shared predictor, forcing readout, fixed-target pattern observer and exact metadata projection are reused. Structural tetrad telemetry is evaluated on detached copies and checked not to alter the live trace.

python benchmarks/thol_native_runtime_response.py --output artifacts/research/thol_native_runtime_response_2026_09_18.json

The default input is the pinned Section 16 artifact. To use a relocated or regenerated prior report, pass --prior-input and --expected-prior-sha256; historical producer metadata is retained. Output cannot overwrite that input. The final native report has SHA-256 71252d116d8933d15a797707ed9f44ed865406422da6db492b48f6989d739c95 and source identity sha256:724ec096bb2a361ee49fa12b5a6f06665d469a34a21826e8e6320a69e959fc96.

Native study tests cover the exact discrepancy ledger, actual chronology, source and capture binding, narrow UTC exception, node-space boundaries, refused partial calls, same-time paired admission and serialization. 35 new tests and 22 prior response tests pass; a separate 56-test baseline passes for native pressure refresh, stability-gated adaptation and phase coordination. The 113 distinct cases do not constitute an exhaustive engine audit. An independent standard-library audit passes 25,410 exact/structural checks, reconstructing raw-state ledgers, replay admission, paired energies and endpoint target compatibility without importing production calculations. It also verifies the conditional common-IL paired map with maximum absolute paired residual 3.531e-17. This is finite evidence for the recorded common-policy witness, not a repeated-runtime theorem. The independent report and overall validation are retained at artifacts/research/thol_native_runtime_response_independent_check_2026_09_18.json and artifacts/research/thol_native_runtime_response_validation_2026_09_18.json.

Global phase direction near the symmetric input

A read-only follow-up identifies a conditioning boundary in this retained trajectory. In the ideal duplicated regular twist phi_k=k*pi/4, opposite unit phasors cancel exactly, so the global resultant is zero and has no direction. The stored binary64 angles are distinct exact real inputs; their transcendental resultant is not assumed zero. Canonical phase normalization also occurs between the entry record and actual coordination input.

For each exact represented angle x, rational Taylor polynomials through degrees 80 for cosine and 81 for sine enclose their real values with errors at most abs(x)^82/82! and abs(x)^83/83!, respectively. The next odd/even coefficient vanishes, and every real sine/cosine derivative has magnitude at most one. Summing these intervals gives a rectangle excluding zero at both recorded boundaries. At actual coordination input its midpoint has mean approximately (-4.15476e-17, -9.38127e-17), norm 1.02601e-16, and direction -1.98771 radians. Re-evaluating the NumPy trigonometric/reduction path on those same angles gives direction -2.03444; the scalar compensated readout gives -2.01397. Norms and angles here are approximate midpoint/readout displays; the retained rational rectangles are the exact enclosures.

For nonzero resultant z, the differential of arg(z) has norm 1/abs(z). Consequently the observed small resultant is strongly sensitive to input, trigonometric and reduction changes. This does not invalidate the recorded native calls or their common-source paired identity. It does prevent treating the common phase trajectory as demonstrated robust symmetry breaking. The engine was not changed and no threshold, alternate direction or new trajectory was introduced by this audit. Its read-only helper and evidence are retained at artifacts/research/analyze_thol_phase_resultant.py and artifacts/research/thol_phase_resultant_audit_2026_09_18.json. Numerical phase-direction robustness must be addressed before promoting the individual phase evolution to a structurally persistent mechanism.

18. Five additional native policy steps

Fixed protocol and shared implementation

The predeclared window extends both authenticated Section 17 endpoints from t=1.75 by exactly five additional .25 native calls, with unchanged configuration, selector, capacities and retained history. Its length was chosen from the observed VF_ADAPT_TAU=5 before the results, without assuming that the stability gate would qualify. A refused call would stop that branch with its actual partial state; no retry, retuning or horizon extension is permitted. Both prior scientific replays are admitted before either new window begins, retaining only the enumerated actual-child UTC-origin exception and preserving the original input bytes.

The window runner reuses the existing native trace owner. An opt-in observation captures the generated pressure immediately after native preparation, separately from pressure consumed by the integrator; the previous runner's default trace schema and behavior are unchanged. The live continuation entry checks its complete projected endpoint record, including configuration and counters. Opaque resources retain explicit unavailable markers, rather than a claim of complete internal serialization. The shared forcing, target-compatibility, pattern and tetrad observers remain the calculation owners. All original-target comparisons retain z0/H0.

Repeated JSON trees are stored once in a SHA-256-addressed record pool. Expansion validates content digests and returns detached values; each produced step is checked against its complete uncompressed record. This preserves inactive stages, refused calls and exact scalar literals. The report contains 9,511 pooled records and occupies 13,209,097 bytes, including the replayed native steps and all ten new calls.

Actual response and policy gates

Both branches complete all five calls through t=3.0. The same glyph is applied to all sixteen nodes at each step, in the existing sequential runtime order. The observed sequence is IL, IL, EN, IL, AL.

Endpoint time	Actual glyph	Paired centered H0 energy	Original-target error, control	Original-target error, prior child AL
1.75 (entry)	Previous IL	0.02125418450	4.470889150	4.111168481
2.00	IL	0.01924351210	4.077236952	3.743658257
2.25	IL	0.01742768716	3.712979695	3.403352021
2.50	EN	0.00643978687	1.563335194	1.421563646
2.75	IL	0.00585202028	1.424752976	1.293568728
3.00	AL	0.00515489141	1.265146793	1.147332908

Paired centered energy falls 75.74646% over this window. Its weighted mean offset is approximately 0.02597607536 before EN and 0.02701964019 after EN; it remains nonzero. Neither full EPI equality nor equality modulo a uniform offset is attained. Both original-target errors decrease at every recorded endpoint, while the original profile remains incompatible throughout. Its common endpoint compatibility residual has maximum absolute value 0.07990084718 at t=3.0. Improvement in the fixed score is therefore not preservation of that relative equilibrium.

The changes of glyph have a concrete policy source. In the native selector, lag counters increment before decisions: since_EN>EN_MAX_LAG=3 forces EN at t=2.5, and since_AL>AL_MAX_LAG=5 forces AL at t=3.0. Those steps have no outer selector requests; the three IL steps have sixteen each. These are configured history-driven actions, not evidence of a newly derived phase transition or autonomous initiation. No new node or edge is created in the window.

Capacity adaptation never qualifies: every actual stored-pressure magnitude at its gate exceeds .001, although Si is high. The closest observed magnitude is approximately 0.00116717 in the prior-AL branch at t=2.75. All stability counters remain zero, no capacity update becomes eligible, and support/capacity stay fixed. Si and pressure still come from different stages as documented in Section 17. Stored-state coherence is not monotone: it falls at the EN and final AL calls, with endpoint values approximately 0.899716 and 0.904565. This diagnostic uses the retained pressure/rate, not a pressure refresh after the later phase motion.

The conditional IL identity is admitted at t=2.0, 2.25, 2.75, with generated source, actual retention multiplication, consumed pressure and EPI-write boundaries checked separately. Its maximum combined paired residual is below 1.78e-16 in those three records, with exact zero accounting residual. The checker explicitly abstains at EN and AL: both write EPI while leaving the generated pressure unchanged. Their sequential reset and held-pressure integration need their own existing operator contracts; substituting a common-IL map would be false. Generation and endpoint phase/forcing arrays agree exactly across branches at all five steps, as do the captured support, capacities and EPI coefficient. AL adds the same represented 0.07957747154594763 to each EPI in both branches; EN has different sequential per-node jumps. No post-reset pressure refresh occurs in either step. The accounting in Section 19 therefore preserves the pre-reset drive, rather than inserting a freshly evaluated drive at the reset EPI.

The result establishes a finite response under the actual configured policy. It does not close indefinite maintenance, generative initiation, a phase transition or empirical emergence. The near-singular initial phase-direction conditioning in Section 17 also remains relevant to any robustness claim.

Reproduction and retained evidence

python benchmarks/thol_native_policy_window.py --output artifacts/research/thol_native_policy_window_2026_09_18.json

Both input artifacts are pinned by digest; relocation/regeneration requires the explicit CLI input and expected-digest arguments. Output cannot overwrite either historical input. The window artifact has SHA-256 37907325606a10551c65302fcbc18ef0fc23e0f46868cc222b3bc524b0faeedb, with source identity sha256:084a85f74d66f5f6a89a3646c1930e304725fb60b95c1f82b17d42544cba310e.

The independent standard-library audit authenticates all 9,511 pooled records and passes 1,628,987 exact/structural checks, mostly complete replay and codec field comparisons. It independently reconstructs target compatibility, paired energies, admitted IL ledgers, forced lag decisions and stability counters. This is finite retained-evidence validation, not an exhaustive runtime audit. Its report is artifacts/research/thol_native_policy_window_independent_check_2026_09_18.json.

Portable window tests cover the exact horizon, adjoining live records, mixed-policy admission, old-target domains, narrow UTC exception, ancestry, refusal, frozen-history protection and lossless serialization. The validation summary is retained at artifacts/research/thol_native_policy_window_validation_2026_09_18.json. Validation covers 43 new tests, 35 previous native-response tests and a 56-test runtime baseline, totaling 134 distinct cases. Two initial failures were synthetic test-fixture mistakes; their four affected guard/refusal controls pass after correction, without changing or rerunning the measured trajectory. These cases do not constitute an exhaustive engine audit.

19. Retained reset accounting and phase conditioning

This closes the supporting interpretation block using the authenticated Sections 17-18 records. No native step, longer trajectory, alternate forcing, threshold change or engine modification is introduced. The existing canonical operators remain the transformations under study; their selection and timing are separate from their action contracts.

Sequential EN/AL and the pressure actually consumed

The offline reset audit binds the historical artifacts and original target, checks complete retained boundary continuity, and reads the generated pressure before the glyphs. It evaluates each recorded scalar EN/AL kernel on its actual detached input, including neighbor order, configured factors and clipping. A later EN sees the preceding node's EPI write. These are sequential calls, not a simultaneous snapshot stage.

Let the generated drive be -A*x+b, and let S*x+c denote the composed affine reset built from the shared represented coefficients. Every local kernel-evaluation and clipping defect is retained and propagated through the later reset rows. Binary64 row coefficients are rationalized as represented; the audit does not assume that the resulting S is exactly stochastic. The subsequent nodal integration still consumes the pre-reset pressure:

x_after = (S - h*A)*x_before + c + h*b
          + propagated_reset_defect + pressure_term
          + integration_remainder + postintegration_change.

The integration remainder is observed, not assumed to consist only of rounding. Source capture comes from the independent channel readout; pressure is never reconstructed from the observed endpoint to fit a model. All four branch vector identities and the two paired identities close with exact zero rational accounting residual. For the two branches, common S,c,A,b give the paired map S-h*A; common AL has S=I. This implements the already derived pre-generated-pressure identity on these records, without claiming a general runtime theorem.

Retained step	Paired centered H0 energy before reset	After reset	Native endpoint
EN, endpoint t=2.5	0.0174276871647	0.00783124789728	0.00643978686669
AL, endpoint t=3.0	0.00585202027919	0.00585202027919	0.00515489141424

EN itself increases the paired mean offset by approximately 0.00104356482712: the branches have unequal mean reset jumps, -0.00927390658874 and -0.00823034176162. Their final mean offset is approximately 0.02701964019. The all-node AL jump is identical and uniform in both branches, so its immediate paired mean and centered-energy changes are exactly zero. The later energy reduction belongs to the consumed drive and its measured defects, not to a differential AL reset. Maximum propagated EN reset defects are below 1.75e-16; AL defects are approximately 4.16334e-17 in each branch. All 64 local clipping defects are exactly zero. Retaining the kernel-evaluation defects is what permits exact accounting; small defects alone would not prove the identities.

Thus EN explains the large local attenuation and the mean-offset change; common AL does not erase the remaining branch difference. The configured lag counters explain why these actions occurred. They do not derive that schedule from coherence dynamics or establish maintenance of the original incompatible profile.

One held-gain phase calculation and its ordering control

The phase audit makes the earlier rational resultant enclosure reusable and selects the first retained native coordination boundary. It reconstructs only the fields needed by detached coordination and diagnostics. Effective gains are held at the captured values kG=0.05929429797231117, kL=0.1496560821254262. The baseline NumPy calculation must reproduce the archived phase endpoint exactly. This binds the calculation, not a replay of adaptive history or a complete native step.

The predetermined rotation is k -> k+1 on the eight parents and their actual children. All comparisons map phases and per-node diagnostics back to the original labels. Two distinct ordering controls are retained:

• Transporting both labels and node/neighbor enumeration gives exact node-aligned phase equality. A label change alone does not fail this test.
• Rebuilding the same relabeled mathematical input in the original canonical node enumeration, with transported neighbor order, changes the global reduction order. Its anchor phase shift is approximately 0.02749166 radians, but the maximum residual between anchor-relative patterns is 0.372557061819 radians. This is not merely a common phase rotation.

The global direction changes from approximately -2.03444 to -pi/2. At original parent 2 and its child, the wrapped global displacement changes branch; the relative residual is approximately -2*pi*kG. Shared readouts also change: maximum aligned gradient and curvature differences are 0.149022824728 and 0.203531276820, respectively. Structural potential is unchanged; the coherence-length difference is about 4.44e-16.

The scalar arithmetic comparison instead has a common shift of approximately 0.00121371 radians and a maximum relative residual of 4.44e-16. Gradient/curvature differences are at rounding scale. This comparison switches the coordination/trigonometric arithmetic only; it is not a NumPy-free whole-engine trajectory or an exact gauge-invariance theorem.

Exact Taylor bounds certify a nonzero resultant for the represented input, with midpoint norm approximately 1.026e-16. The ideal regular twist has exactly zero resultant. These are different mathematical inputs. The tested ordering sensitivity rejects robustness of this particular numerical phase pattern; it does not establish that every phase preparation or canonical operator is unstable. No arbitrary zero-resultant epsilon or replacement phase force is introduced.

Evidence and return to the generative question

python benchmarks/thol_retained_reset_audit.py
python benchmarks/thol_retained_phase_audit.py

The outputs are local research artifacts under artifacts/research/:

Report	SHA-256
thol_retained_reset_audit_2026_09_18.json	6257b5ce03a8620afe7c982f68cf2913c4b6759610a99ef047f2c878818310ec
thol_retained_phase_audit_2026_09_18.json	b9e957bbfbc21ac68c3dfa31c818a9d46db3dbae08da3879b6d33c7991f7371a

The reset and phase source identities are respectively sha256:3df78190f7088ca9b36ae40655b9f2081068004d15711b8dd6037f08cdd60c8f and sha256:0954e6149032793e9137db562b6df74e97572568dfefae55dcf0d1e2e89c1355. Historical input bytes remain unchanged. Portable reset controls and phase controls retain admission, arithmetic and label/order boundaries without running a native trajectory. The local validation summary is artifacts/research/thol_retained_mechanism_validation_2026_09_18.json. Validation passes 33 new tests and 41 Reception/Emission/phase regression tests, totaling 74 distinct cases. An initial reset-test fixture encoded exact component values incorrectly; the corrected final 16-test reset suite passes, including its independent retained-record arithmetic check. No measured trajectory was rerun to fix those synthetic fixtures. These artifacts and tests do not constitute a whole-repository or empirical validation.

This supporting block is complete. The principal unresolved chain is how an existing nodal configuration requires a transformation that generates and sustains a new coherence region. Canonical action contracts are retained; activation, targets and ordering still need their structural justification. The single execution plan owns that bounded derivation or underdetermination test. No longer native window follows automatically from the present result.