THOL birth on a certified maintained-parent history

One supplied canonical THOL occurrence now creates an isolated child from the retained local-capacity history. Independent prediction and all 21 frozen checks pass. The existing component keeps every continuous coordinate and edge; the declared frozen-isolate successor admits the new node. This is a finite mathematical/software result under a supplied preparation, event time and constitutive completion. It does not establish autonomous occurrence or a coupled enlarged organization.

The research program owns status. The hierarchy contract owns the general THOL reset, metabolic construction and continuation obstruction; the hierarchy-to-contact owner owns the explicit frozen-isolate extension. Its earlier two-node coefficients and synthetic history are not substituted for this twenty-node retained parent.

1. Preparation, independent prediction and one reserved event

Restore the original endpoint at structural time 7+2^-19, parent 2, capacity 128, twenty-node order and 45 existing edges. Re-admit the primitive source, live Gamma registry, complete law, original three samples, clock, stored pressure and actual VAL history with U2 debt one. The admission owner independently recomputes the source and history certificate. Cached verdicts and previously reported bounds do not replace those checks. Immediately before execution it also rejects intervening changes to typed persistent state and rebuilds the consumed mathematical evidence.

The observed acceleration remains 0.2513427734375, its certified represented lower bound remains above 0.12484884, and the unchanged default threshold is represented 0.1. No VAL, numerical integration or old source producer is replayed. The original source and response bytes remain preserved.

The protocol and prediction were frozen before any THOL proposal on this state. The archive and manifest retain the implementation used for the one attempt. Hashes associate bytes; they do not authenticate acquisition or chronology.

The independent predictor uses primitive neighbor forms and real phase lifts, rational interval sine, cosine and square root. It invokes neither THOL's proposal nor its metabolic implementation. The runtime's represented coefficients are treated as exact inputs to this independent ideal algebra. The fixed absolute acceptance allowance is 2^-44 for amplitude, signal and raw-pressure comparisons; this is a prospective numerical budget, not a universal libm theorem.

For bare neighbors N(2)=(0,1,4,5), define

\[ \bar x_N=\frac14\sum_{j\in N(2)}x_j,\qquad V_N=1-\left|\frac14\sum_{j\in N(2)}e^{i\theta_j}\right|. \]

The default construction predicts the interior child amplitude

\[ z=sx_2+g(\bar x_N-x_2)+kV_N, \quad s=\operatorname{fl}(0.3),\quad g=\operatorname{fl}(1/(2\pi)),\quad k=\operatorname{fl}(1/(4\pi)). \]

These symbols denote the actual stored default coefficients. Runtime intermediate arithmetic has its separately checked defect. The predicted amplitude interval, including that allowance, is approximately [0.14881439697539223, 0.14881439697550590], strictly inside the child rails. Acceleration selects eligibility; it does not multiply this amplitude again.

The event adapter executes only step 1 of the validated VAL,THOL,SHA context through the shared atomic stage, on the single target 2. Strict policies and grammar rejection by raising remain active. No hierarchy or link is inserted by the study. The deterministic wall-metadata timestamp is distinct from both the structural clock and the archived execution receipt.

2. Actual child, hierarchy and pressure handoff

The retained response and verdict record:

Quantity	Actual represented result
New node	2_sub_0, the sole appended child
Child EPI	0.14881439697544907
Child capacity	121.6, the represented default damping product
Child phase	1.2554098388612607, equal to the in-range parent phase
Child pressure and degree	0, 0
Hierarchy	parent 2; level 1; path [2]
Parent raw pressure after THOL	0.0201219358794979
Parent fresh pressure after handoff	0.00012071347799567027
Event structural time	unchanged 7+2^-19

The recorded metabolic mean is approximately 0.3129882811925492, its form gradient 0.00048825172194233435, and circular phase variance 0.6909830055042361. All pass the independent frozen comparisons. The hierarchy record retains the actual acceleration, threshold, operator step 2 and child association. Existing forms are not depleted: this is a configured new coordinate and hierarchy relation, not an additive matter split.

Before refreshing pressure, the callback captures the complete raw graph. The parent jump is the checked implementation of p_2^+=p_2+k*A_hat; other old pressures are unchanged. The supplied constitutive refresh then restores every old pressure to its pre-event value. The raw pressure reset and fresh law are different retained states; consuming the raw value for a later interval would require another complete law.

THOL changes the parent's glyph history to VAL,THOL and discharges its U2 debt from one to zero. SHA has not occurred, so the prospective word is not complete. The parent's timestamped history remains the same pre-birth record. The child receives one legacy EPI sample and empty glyph history; its temporal acceleration observation is unavailable. Neither a model's predicted zero rate nor the parent's history supplies a measured child rate.

3. Complete successor and exact work accounting

The connected-only reader rejects the actual 21-node successor, as predicted. No denominator at zero degree is evaluated. The explicitly selected completion retains the existing sine rows separately on every nontrivial component and supplies p=xdot=thetadot=0 on an isolate, with its form, phase and capacity held. The shared detached frozen-isolate reader admits this law; the event callback provides its pressure handoff without installing a native dispatcher or advancing a trajectory.

With the existing edge-only storage

\[ H=\frac12\sum_{\{i,j\}}W_{ij}(x_i-x_j)^2 +\sum_{\{i,j\}}[1-\cos(\theta_j-\theta_i)], \]

the complete event has exactly zero storage jump. Every old edge argument is exactly unchanged and the child has no incident edge. This establishes equality directly; subtracting two independent interval enclosures would not establish it. An intrinsic child-storage term would be an additional premise with a different budget.

The old component's two weighted means, with weights d_i/nu_i, are also exactly unchanged. Its full continuous rows coincide before and after restriction, so its admitted identity and maintenance law retain their original scope. The child is outside that twenty-node identity. A zero isolate mean weight merely ignores the child; it does not define a complete positive mean metric for the enlarged graph.

The new frozen coordinate has no coupling or independently acquired response. The hierarchy is structurally admitted, but numerical U5 needs separate change-rate observations and an explicit coefficient. The existing U6 comparison needs the same complete node support before and after; birth does not supply a missing child baseline. No numerical U5 or cross-birth U6 certificate is reported.

4. Reuse, verification and remaining boundary

The shared retained-state owner now admits exact integer and nonempty-string node identifiers consistently across nodes, edges and neighbor order. This preserves the actual THOL child identifier and its hierarchy without relabeling the old component. Booleans, floating aliases, empty strings and unsupported objects reject. Earlier integer captures and all archived evidence remain unchanged. A maintained old-evidence test permits this explicit compatible reader migration while still verifying every archived producer byte.

Verification combines admission controls, independent prediction controls, synthetic wiring and retained-evidence checks. The latter reconstruct primitives and independently compare the numerical records without executing the producer or any operator. Existing atomic THOL, grammar and frozen-isolate controls retain their own responsibilities.

This result closes actual birth admission for this certified parent. A later connection changes degrees, both continuous rows, phase pressure, support storage and observation weights. The canonical default parent-target UM policy also considers other compatible nonneighbors, so its support cannot be assumed to consist of a single new parent-child edge. Its actual selection, complete reset, work and successor identity need their own analysis. Neither the old small-source composition nor the present zero-work isolated birth settles that question.