Self-organization pressure, refresh and structural feedback

Status: Scoped operator and nodal-integration identities; finite runtime observations distinguish pressure retention from refresh and hierarchy birth. Research links: B2.d.5/O3.a, S3, S8, S9, S10 and S16.

1. What the canonical THOL operation actually changes

The public SelfOrganization implementation prepares the complete action before committing it atomically. On the selected parent, let x be scalar EPI, nu capacity, phi phase and p stored DeltaNFR. For an actually admitted THOL action, its primary channel proposal is

\[ p_T=p+a\widehat A,\qquad x_T=x,\qquad \nu_T=\nu,\qquad\phi_T=\phi, \]

where a=THOL_accel>0 is the existing operational coefficient and A_hat is the signed structural acceleration reconstructed from active EPI history. The cached acceleration attribute is telemetry; public THOL does not treat an independently assigned cached value as its history. The displayed algebra treats supplied coefficients and states as exact real values; executed binary64 arithmetic is identified with its measured residuals rather than assumed equal to that reference. The pressure write occurs whether or not the child-creation threshold is crossed. A threshold crossing abs(A_hat)>tau permits a child proposal subject to the configured hierarchy-depth and other domain checks.

The instantaneous local pressure magnitude can increase or decrease:

\[ \tfrac12(p_T^2-p^2) =a p\widehat A+\tfrac12(a\widehat A)^2,\qquad |p_T|\leq|p|\iff a\widehat A(2p+a\widehat A)\leq0. \]

Thus the U2 stabilizer role does not establish a universal local pressure contraction, a Lyapunov decrease or an attractive EPI profile. Those are separate dynamical claims. The glyph itself leaves the parent's EPI fixed; an EPI response to its pressure requires an actual subsequent integration interval or another named operator action.

2. Structural acceleration is a measured history diagnostic

The shared compute_d2epi_dt2 owns the acceleration calculation. With an active timestamped history, the last three samples (t0,x0),(t1,x1),(t2,x2) give

\[ h_1=t_1-t_0>0,\quad h_2=t_2-t_1>0,\qquad s_1=\frac{x_1-x_0}{h_1},\quad s_2=\frac{x_2-x_1}{h_2},\qquad \widehat A=\frac{2(s_2-s_1)}{h_1+h_2}. \]

The active samples must be finite, their times strictly increase, and the final EPI must match the current parent state. A timestamped source takes precedence over legacy histories. Fewer than three active samples return zero as an unavailable-acceleration result; that return alone does not prove that a measured trajectory has zero curvature. The legacy epi_history and _epi_history paths retain unit operator-step spacing, so they use x2-2*x1+x0. They are not silently assigned physical times.

The nodal equation explains what this diagnostic measures. On an unclipped interval with no additional EPI jump,

\[ s_j=\frac1{h_j}\int_{t_{j-1}}^{t_j}\nu(t)p(t)\,dt. \]

For two exact held-input intervals this reduces to

\[ \widehat A= \frac{2(\nu_2p_2-\nu_1p_1)}{h_1+h_2}. \]

The diagnostic therefore compares previously accumulated nodal rates. On a smooth interval the ordinary product rule also gives x''=nu'*p+nu*p'; it does not introduce a new acceleration law or inertial degree of freedom. Finite samples can straddle changes of held pressure or capacity, and no derivative-limit theorem follows merely from their three-point estimate. The standard physical boundary recorder restarts history at a detected same-time EPI jump, excluding that jump from the following physical secants. The reader alone does not certify this causal provenance for caller-prepared histories. Clipping, unrecorded jumps and binary64 arithmetic require their own measured residuals.

3. Hierarchy birth does not itself connect a child to the parent walk

When birth is admitted, THOL creates a graph node with a parent reference, hierarchy path and level, and synchronized sub_nodes, sub_epis and graph-level hierarchy records. It initializes child pressure to zero, child capacity to 0.95*nu_parent, and child phase to the parent's wrapped phase. Its bounded EPI amplitude uses the existing 0.3 parent scaling and, when enabled, measured neighboring EPI and circular-phase inputs via compose_subepi_amplitude. These are declared implementation coefficients, not newly derived universal physical constants.

The commit adds a node and hierarchy metadata, without adding a graph edge. The child's identity is an operational nested coordinate, not an additive conserved EPI mass. THOL does not subtract its amplitude from the parent or propagate it into neighboring EPI. The legacy propagation flag is rejected by the canonical birth path; form propagation requires its own explicit admissible operator composition.

This birth path belongs to public SelfOrganization and its shared staged implementation. The primitive THOL node-protocol dispatch implements the pressure action without invoking that hierarchy commit. An ordinary runtime selection of the primitive glyph is therefore not, by itself, evidence that new child nodes were generated.

The default pressure law in dnfr.py reads actual graph neighbors, EPI, capacity, phase and local degree. It does not use parent_node, sub_nodes, sub_epis or hierarchy as implicit transport edges or a child-pressure source. In exact arithmetic, adding these isolated children therefore leaves every original component's canonical structural pressure unchanged, provided its triad, edges and pressure configuration are fixed. The isolated children themselves have zero default neighbor pressure.

On an unchanged unit undirected cycle with j isolated children, write L_Cn=I-W/2 and e=w_epi for the effective channel coefficient. The EPI-channel nodal flow has block form

\[ \dot x=-\left[ e\operatorname{diag}(\nu_{\rm parent})L_{C_n}\ \oplus\ 0_j \right]x. \]

Each child contributes a disconnected zero mode. Its nonzero EPI can persist at positive capacity because its default pressure is zero, while the parent-cycle diffusion modes are unchanged. The resulting nonuniform field across disconnected components is not a localized profile maintained inside a coupled component. This is the existing graph-diffusion nullspace identity applied to the actual child support, not a new confinement law.

This is a pressure-kernel statement, not universal invisibility. Node counts, hierarchy queries, aggregate diagnostics and later selectors may observe the new nodes; explicitly declared later operations can change their graph connections or channels. Added nodes may also change a runtime vectorization path, so exact component independence does not by itself establish identical binary64 reductions. Any claimed parent feedback must identify such an actual channel or policy dependency.

4. Refresh order determines whether the pressure increment reaches EPI

Let R(G) denote the default structural pressure recomputed from the graph state, and let p0 be the stored pressure just before THOL. The stored value can include earlier operator writes and need not equal R(G). Immediately after THOL the parent pressure is p0+a*A_hat, while its triad and original neighbor support are unchanged.

If the default refresh runs before physical integration, it overwrites the stored value with R(G_T). By the component independence just proved, this equals R(G) on the original component in the exact fixed-scope model. The direct additive THOL term is absent from that following refreshed nodal rate. An uncoupled hierarchy birth does not retain it through the default neighbor-pressure kernel.

If the pressure is instead held over a positive interval h, the classical unforced, unclipped branch of the shared nodal integrator gives the exact reference increment

\[ x_H-x=h\nu(p_0+a\widehat A). \]

A matched control that holds p0 over the same interval satisfies

\[ x_H-x_{\rm control}=h\nu a\widehat A. \]

This difference derives directly from the existing pressure write and nodal law. It adds neither a force nor an autonomous feedback rule. Relative to an immediately refreshed reference, the same held increment contains both earlier pressure mismatch and THOL's contribution:

\[ h\nu\bigl[p_0-R(G)+a\widehat A\bigr]. \]

Those contributions cannot be attributed to THOL as a whole when the preparation includes other pressure-changing operators.

Once a held interval has changed EPI, a later pressure refresh does not reverse that accumulated change. On the fixed unit cycle with common capacity and unchanged phase, two states differing only by delta_x have refreshed pressure difference -e*L*delta_x. Thus the transient operator pressure can leave an ordinary structural EPI perturbation if it is actually integrated. This surviving perturbation is not evidence that an isolated child feeds pressure back to its parent or maintains a localized restoring field.

For several compatible held intervals the finite accounting is

\[ x_N-x_0=\sum_j h_j\nu_jp_j+\sum_j r_j, \]

where each p_j is the pressure actually held by the declared integrator call and r_j records its EPI arithmetic/clipping defect. Named EPI jumps, if any, must be included separately. Only intervals whose retained pressure contains the THOL increment include its direct contribution. Refresh boundaries are consequently part of the causal specification, not a presentation detail.

5. Falsification criteria for a stronger feedback claim

A claim that THOL's hierarchy sustains parent-component structure through the default pressure law must expose a reproducible dependency beyond the uncoupled birth just described. Useful discriminating checks are:

• Record the executed glyph and grammar context. A request for THOL that falls back to a different admissible operator is not a THOL experiment.
• Keep the parent triad and its actual neighbor support fixed while varying only isolated child values. A changed exact default parent pressure would contradict the present component-independence claim and identify a previously unaccounted pressure input.
• Capture pressure immediately before THOL, after its commit, after every refresh and at each integration boundary. An EPI response attributed to an erased pressure increment would contradict the held-input accounting.
• Separate histories generated by earlier executor-owned flow from a caller-prepared threshold-crossing fixture. The latter tests the admitted birth path and its effects, not spontaneous generation of its trigger.
• Compare child creation with suppression by the existing depth boundary while retaining the same parent pressure update. This can distinguish hierarchy effects from the signed pressure increment itself.
• Identify and exercise any later actual transport edge, parent/child aggregation or adaptive-policy dependence before claiming persistent feedback. Merely observing a nonzero child count, a global diagnostic change or a parent EPI transient does not establish that mechanism.

These checks narrow the next structural question: whether an explicit canonical interaction between a created hierarchy and the parent network can maintain and restore a localized pattern while reorganization remains active. They do not assume that such an interaction already belongs to the default THOL pressure path.

6. Shared pressure preparation and the runtime evidence boundary

The shared _thol_pressure.py prepares the signed pressure proposal and checks it before committing pressure or acceleration telemetry. Public SelfOrganization and primitive dispatch use this same arithmetic. Graph-backed primitive THOL reconstructs acceleration from the shared history reader; graph dispatch prepares it before creating a cached node adapter and reuses that same-call proposal. Invalid history or a nonfinite proposal cannot be replaced by stale cached curvature. A graphless node protocol intentionally retains its explicitly supplied acceleration input; it supplies no graph-history provenance.

Pressure-history regressions exercise graph routes, public/primitive pressure agreement, signed acceleration, unavailable physical history, stale telemetry, invalid history, finite-domain failure and graphless compatibility. The existing structural-acceleration tests cover unequal physical spacing, legacy precedence, endpoint matching and read-only diagnostic evaluation. This centralization aligns pressure semantics without promoting primitive dispatch to public hierarchy birth or replacing the declared runtime refresh policy.

The finite runtime tests and benchmark supply the following discriminating observations with default THOL factors:

• On C8, actual Coherence/Dissonance preparation followed by two refreshed physical segments of lengths 0.125 and 0.375 produces three recorded EPI samples. The shared physical acceleration is 0.012579445174099646, while the naturally produced cached value is 0.008386296782733171. No poisoned cache is needed to expose the difference. Using the shared physical history removes a real mismatch between graph-backed primitive and public THOL pressure inputs.
• The stored parent pressure changes from -0.07935727615962375 to -0.07835623571921803. Holding it for h=0.25 yields EPI 0.9358019009519275, versus 0.9355516408418261 in the matched held baseline. The exact observed extra EPI is 2254142677197/9007199254740992 (approximately 0.0002502601101014301), accounted for by the represented-acceleration prediction, pressure arithmetic and Euler arithmetic residuals separately. Public and staged refresh controls instead recover the baseline pressure and endpoint EPI exactly in this finite observation.
• Ordinary runtime execution with the declared target selector consumes primitive THOL's pressure before integration and matches the held public EPI endpoint. Later ordinary phase evolution is recorded separately; this observation does not assert that the complete ordinary runtime map is identical to the public operator-plus-integrator control.
• A distinct P2 fixture with explicitly prepared physical history has acceleration 0.4 and creates one child at the default birth threshold. The child is isolated and no edge is added. Canonical refresh restores parent pressures (-0.03180163034256735, 0.03180163034256735), gives the child zero pressure, and its EPI remains unchanged under the next shared Euler interval. This tests the birth contract and lack of immediate parent feedback; the prepared trigger is not a causal emergence result.

These are reproducible engine observations, not laboratory validation or an autonomous localization theorem. The next constructive test must first generate a birth-triggering history through actual canonical dynamics, then identify an admitted parent/child interaction and measure whether the resulting connected structure feeds back through the existing nodal channels. The isolated hierarchy result does not supply that interaction implicitly.