The connected child's own temporal capability and action boundaries

The actual child 2_sub_0 admits a short genuine three-sample acceleration window with positive constitutive pressure and preserved common principal geometry. Its form stays below the unchanged strict THOL minimum throughout that window, and its empty executed history independently blocks U4b. Temporal capability therefore does not establish complete action admission. This is an analytical result from retained primitives, with no new trajectory, operator event, pressure refresh or installed numerical history.

A separate retrospective whole-cell observation admits the joint form, positive-pressure and acceleration conditions later in the already retained continuation. The child's missing measured history and executed U4b context remain independent obligations.

The research program owns the next task. The THOL compatibility owner defines acceleration, three-sample evidence and the separate event contracts; the ontology distinguishes sufficient continuous state from grammatical execution state.

1. Actual source and complete law

Use the committed SHA endpoint of the actual parent word VAL,THOL,UM,SHA. Re-admit its primitive event evidence and all nineteen endpoint checks through the actual UM source. Retain all 21 nodes, 53 positive-conductance edges, signed forms, real phase lifts, positive capacities, node/neighbor order and structural clock. The shared complete law is

\[ S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i),\quad p_i=\frac{S_i}{\pi d_i},\quad M_i=\frac{\nu_i}{\pi d_i},\quad \dot x=MS,\qquad \dot\theta=ML_Wx. \]

Support, conductances and capacities are held, with zero forcing and loss. This remains the detached weighted-form, bare-neighbor phase comparison; neither native relational execution nor another pressure mixture is substituted. The preparation is the actual represented point, not a newly chosen family.

The child has index 20 in the retained order and sole neighbor parent 2. Its source form is 0.14881439697544907, capacity 121.6, and fresh pressure approximately 5.213909427865455e-5. Its hierarchy level is one, parent path [2], and structural time

\[ t_* = 7+2^{-19}=\frac{3670017}{524288}. \]

It retains one legacy EPI sample, no acquired timestamped history and an empty glyph history. These are the child's own records. The parent's samples and completed word do not transfer to it. Current acceleration observation is explicitly unavailable because there are insufficient samples.

2. Whole-window bounds from primitive state

Use elapsed structural time s=t-t_*; zero below denotes the source endpoint. The shared temporal-window owner rebuilds the complete motion bounds from primitive coordinates and coefficients. It accepts no cached motion verdict as a premise. For each original oriented edge e=(i,j), retain

\[ \delta_e(0)=\theta_j(0)-\theta_i(0)-2\pi k_e,\qquad v_e(0)=\dot\theta_j(0)-\dot\theta_i(0). \]

The complete-law derivation uses A=M*L_W*M and |S_k|<=d_k to give the global bound |delta_e''|<=B_e=sum_k|A_jk-A_ik|d_k, without a future phase-tube assumption. Twice integrating that bound gives valid simultaneous enclosures on [0,T]:

\[ D_e=\delta_e(0)+v_e(0)[0,T]+[-B_e,B_e]T^2/2,\qquad V_e=v_e(0)+[-B_eT,B_eT]. \]

Orient these intervals toward each target's neighbors before summing. Then the actual nodal pressure, form rate and second form derivative satisfy

\[ p_i([0,T])\subseteq\frac1{\pi d_i}\sum_{j\sim i}\sin D_{ij},\qquad \ddot x_i([0,T])\subseteq M_i\sum_{j\sim i}\cos D_{ij}\,V_{ij}. \]

If this last interval is A_i, a form enclosure is x_i(0)+xdot_i(0)[0,T]+A_i[0,T^2/2]. This is an integral remainder bound, not an extrapolation from the initial acceleration alone. Original principal branches are retained when pi.lo-D_e.abs_max is positive, where abs_max is an upper bound on the magnitude throughout the interval. This margin may be strengthened by the independently rebuilt global energy margin for the same horizon. Neither endpoint rewrapping nor stored labels supply this proof.

For the child, put delta=theta_c-theta_2; its branch shift is zero. Then

\[ p_c=-\frac{\sin\delta}{\pi},\qquad \ddot x_c=-M_c\cos\delta\,\dot\delta. \]

At the source, delta is approximately -0.00016379979628311503, delta_dot approximately -5.346318925921182, and M_c approximately 38.70648215994894. Fresh global edge acceleration is bounded by 3417. The source child acceleration is approximately 206.93719535146448; a whole-window conclusion still requires the integrated bounds above.

3. Fixed true-history window and prospective observation budget

Choose exact structural spacing h=2^-16 and T=2h=2^-15, with prospective sample times t_*+(0,h,2h). All these timestamps are exactly representable in binary64; no laboratory-time conversion is supplied. Outward rational interval evaluation of the actual source proves, throughout this window,

\[ -0.000329<\delta<-0.000162,\qquad -5.451<\dot\delta<-5.242,\qquad \ddot x_c>202,\qquad p_c>\frac1{20000},\qquad x_c<\frac{149}{1000}. \]

Every one of the 53 original principal branches has a positive independently rebuilt short-window margin. This proof needs no saved trajectory steps. The earlier half-unit certificate remains separate compatible evidence; none of its jets, remainders or step boxes are consumed here.

The positive-kernel identity expresses the genuine three-sample acceleration as a nonnegative unit-mass average of the second derivative. Consequently

\[ \widehat A_{\rm true}= \frac{x_c(t_*+2h)-2x_c(t_*+h)+x_c(t_*)}{h^2}>202. \]

This exceeds the unchanged represented THOL threshold tau=0.1. It proves the existence of true samples with this property; it neither generates numerical sample values nor installs them in the runtime history.

For prospective certified value errors epsilon_0,epsilon_1,epsilon_2, the shared sample-budget owner gives the exact-arithmetic observation allowance

\[ |\widehat A_{\rm samples}-\widehat A_{\rm true}| \leq\frac{\epsilon_0+2\epsilon_1+\epsilon_2}{h^2}. \]

The proposed uniform allowance epsilon_i<=2^-30 gives error at most 16, so the conditional sampled acceleration still exceeds 186. This is a loose sufficient budget chosen from the analytical margin, not an achieved solver accuracy or a newly measured response. Clock association, solver error and reader arithmetic must be certified in any later acquisition.

The adapter also derives the genuine global third-derivative bound M_c*sum_e(Vmax_e^2+B_e) for its incident edges and passes that actual bound to the shared budget owner. It uses only the sample-error component here: the owner's separate initial-derivative truncation allowance is not subtracted again from a theorem that already bounds acceleration over the whole window.

4. Independent form, history and grammatical obstructions

The strict THOL form minimum remains the represented default 0.2, read through the same policy resolver used by the runtime gate. The proved inequality

\[ x_c(t)<149/1000<1/5\leq\mathrm{THOL\_MIN\_EPI} \qquad(0\leq t-t_*\leq2^{-15}) \]

excludes strict THOL execution on this entire short window even if accurate samples were acquired. This is a configured execution-domain restriction, not a physical inability to have large acceleration or a different definition of signed EPI. The result does not exclude later form growth elsewhere in the already retained half-unit interval.

Live grammar independently rejects THOL through U4b: the child has no recent executed destabilizer. The event-free complete law changes x and theta, not glyph history. Therefore merely waiting or measuring the trajectory cannot remove this grammatical obstruction, including over the retained [0,1/2]. The child's own nonzero form meets U1's continuation premise; this supplies no past U4b event. A future word cannot manufacture a previously executed prefix. Recency windows are configured operator-position policies, not a laboratory relaxation time derived from this flow.

Capacity and connectivity currently meet their strict minima, and the child is below the configured hierarchy-depth limit. Those facts do not replace history, form or grammar. A future birth would also require its own hierarchy, identifier, metabolic-output and atomic-stage checks, followed by a complete law for the initially isolated new coordinate. No second birth is selected.

Thus the result is positive temporal capability with explicit action obstructions. It separates dynamical response, acquired observation and canonical event admission without lowering a threshold, copying a parent record, supplying a capacity boost or inserting stabilization.

5. Implementation and verification boundary

The actual-child adapter re-admits the source, reconstructs primitive branches, applies the shared window bounds and independently reads current temporal and grammatical state. All calculations are detached. Cached reports, changed live primitives or fabricated history associations cannot replace the retained source.

Kernel controls use independent small weighted systems, differentiated fields and invalid domains; actual-source controls verify the strict rational margins, sample budget and distinct obstructions without replaying events or flow. The shared policy controls retain default/override behavior, exact threshold boundaries and invalid-value rejection. These checks establish implemented arithmetic and contract behavior; the stated bounds depend on the complete-law derivation above.

6. Joint conditions on retained whole-time enclosures

The source-local result above excludes form admission only on its stated short window. A separate retrospective observation uses the already evaluated post-SHA certificate to ask whether form, fresh pressure and acceleration become jointly admissible later. It changes neither that certificate's target nor its original verdict.

The retained reader re-admits the actual SHA source, complete field, original branches and fixed 256-cell chronology. It checks the frozen artifact associations, source primitives, strict Picard inclusion, recorded Taylor arithmetic, endpoint carry and retained summary. Recorded derivative jets and remainders keep their frozen producer provenance; this read-only calculation does not independently regenerate them. Byte associations do not authenticate acquisition.

For an admitted whole-time box Z_k on the original cell J_k, the shared bound_node_history_box rebuilds both constitutive rows on that same box. With delta=theta_c-theta_2, its conditional ranges are

\[ X_k=(Z_k)_{x_c},\quad P_k=-\sin(\Delta_k)/\pi,\quad A_k=-M_c\cos(\Delta_k)(\Omega_c-\Omega_2), \qquad \Omega=ML_WX. \]

The source's held positive capacities and conductances, full 42 coordinates, unwrapped phase lifts and original clock are retained. Reusing a pointwise derivative or a box from another law would not justify these ranges. Box endpoints need not occur jointly; universal bounds throughout one common box are nevertheless sufficient. Overlap of favorable marginal ranges is not.

The fixed criterion admits a whole cell only when

\[ \inf X_k\geq x_{\min},\qquad \inf P_k>0,\qquad \inf A_k>\tau\ \text{or}\ \sup A_k<-\tau, \]

and every original principal margin is positive. Here x_min and tau come from the unchanged runtime policy resolvers. Original tube margins may be strengthened by the freshly rebuilt source-energy margins for the same complete law and half-unit horizon. A nonpositive sufficient branch margin is an abstention, not evidence of crossing. Both acceleration signs are admitted by the canonical absolute threshold.

The positive three-sample kernel then bounds every genuine three-point acceleration wholly inside the chosen cell in A_k. This supplies a joint continuous domain for future observation; it does not supply samples, live execution state or U4b context. Form equality passes the configured gate, whereas pressure and acceleration inequalities are strict.

The observer checks cells in chronological order and stops at the first complete certificate. Earlier inconclusive boxes do not exclude a smaller valid window or determine the first actual threshold crossing. The fixed observation definition forbids cell refinement, longer horizons, policy changes, new trajectories and events. Its result is a new observation of existing evidence, not a reserved prospective prediction.

7. Retained result: a joint continuous window

The definition, archived observation source and byte associations precede one observation attempt. The original trajectory was already evaluated; this ordering does not turn the new observation into a prospective prediction or independent acquisition.

The result selects cell index 12, after inspecting the first 13 of 256 original cells. Its elapsed structural interval is

\[ J_{12}=\left[\frac3{128},\frac{13}{512}\right] =[0.0234375,0.025390625]. \]

Throughout that entire interval, the independently rebuilt ranges give the following conservative rational bounds:

Quantity	Certified whole-cell bound	Unchanged admission
Child form	0.2008 < x_c < 0.212	x_c >= represented 0.2
Fresh pressure	0.034 < p_c < 0.037	p_c > 0
Form acceleration	111 < x_c'' < 129	Absolute true three-sample acceleration exceeds represented 0.1
All original principal margins	Each exceeds 1.58	Each strictly positive

The tighter recorded lower endpoints are approximately 0.2008503787, 0.0343352332, 111.0743952 and 1.5814125089, respectively. Decimal approximations summarize the exact outward rational records; they are not replacement inputs to admission.

Cells 0 through 10 exclude the form requirement on their whole enclosures. Cell 11 is inconclusive about form; all three dynamical conditions are jointly certified on cell 12. Thus the result locates the first certified whole cell under the fixed criterion, not the first actual admissible instant or a minimal waiting time. The pressure bound on cell 0 is also inconclusive; this does not contradict the smaller analytical window in section 3. Later cells need no new child-condition observation for this gate.

The generated child therefore reaches a jointly admitted continuous domain under the held post-SHA law without a capacity boost, extra stabilization, policy change or new trajectory. This finite consequence of one supplied complete law is neither a universal emergence theorem nor autonomous event selection. The event-free continuation leaves its own glyph history empty: U4b still requires an actually executed destabilizer. Mathematical tubes do not supply a runtime state, timestamped samples or numerical reader accuracy.

The retained-result controls reconstruct the inspected prefix using the independent degree-one pressure and acceleration formulas on the re-admitted complete tubes, and verify the fixed criterion and source associations. No event, trajectory, jet producer or installed history is executed. The separate OZ and timing owner admits a conditional canonical event/refresh and reader budget on this cell. The execution plan owns the remaining execution and observation obligations.