Acquired local-capacity history with independently certified error

One frozen represented preparation now supplies an actual local VAL event and a numerical three-sample record whose errors are independently bounded against the complete continuous law. The recorded acceleration exceeds the unchanged THOL threshold, with positive fresh pressure and the required live grammar context. No THOL proposal or child event was executed. This is a finite mathematical/software result under supplied preparation, law and event timing; it establishes no autonomous occurrence or physical identification.

The local-admission theorem owns the open source domain and whole-window proof. The research program owns status. The frozen protocol, source, freeze, response and verdict retain this single evaluated case. The source archive and manifest associate its implementation bytes; they do not authenticate ancestry.

1. Admitted represented preparation and actual VAL

The source is a new rounded declaration of the unsplit aligned contact at common form origin 5/16, with the fixed twenty-node order, 45 edges and actual represented conductances. It does not replay the earlier contact producer. In particular, the retained IL/OZ source had a different split and phase offset and cannot be substituted into this narrow source domain.

The source's exact represented coordinates define the mathematical initial condition. Its relation to the ideal nominal is checked separately through the preceding theorem's direct-output corollary. The detached reader rebuilds primitive form, phase, capacities, support and weights, then checks coordinate and weight deviations, its own radius and its own storage. On the prospective capacity endpoint the certified upper bounds are approximately

\[ Z^2<7.741\,10^{-6}<1/256,\qquad H-H_*<9.445\,10^{-6}<1/8192. \]

These are upper enclosures, not measured errors or substituted nominal values. Mathematical pi and retained real phase lifts define the reference. Rounded angles are not identified with exact multiples of pi.

The actual atomic VAL stage targets only node 2, with factor 128 and explicit EDGE_AWARE_ENABLED=False. Strict preconditions and unoverridden VAL/THOL policies remain in force. Its incoming capacity is one, below the default VAL input limit. It changes only that capacity among the primary continuous coordinates; form, phase, support and conductances remain unchanged. The storage jump is consequently zero by the event map. The target's initially empty glyph history becomes ['VAL'], with U2 debt one and the actual recent destabilizer needed by U4b. A prospective word supplies sequence context; only its VAL prefix is executed.

2. Complete numerical evolution and recorded observations

Hold nu_2=128, nu_0=nu_1=1/2, all other capacities one, and all actual weights and support. Retain the structural clock, e=0,w=beta=1, and no live Gamma, forcing or loss. The complete rows are

\[ \dot x_i=\frac{\nu_i}{\pi d_i} \sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad \dot\theta_i=\frac{\nu_i}{\pi d_i} \sum_{j\sim i}W_{ij}(x_i-x_j). \]

The study adapter evaluates both rows with independent scalar binary64 arithmetic and math.fsum, using the frozen edge order. The shared rk4_coupled_step reevaluates the complete immutable state at four stages. It reuses the existing Euler and RK4 quadrature kernels. The ordinary graph integrator continues to hold pressure during its own calls; its existing rk4 method is not reinterpreted as this complete-field scheme.

There are exactly two steps of h=2^-20, with samples at

\[ (t_0,t_1,t_2)=(7,7+2^{-20},7+2^{-19}). \]

Times and stage clocks are exactly representable. These are structural times, not laboratory seconds. There is no continuous clipping or phase wrapping. For example, node 10's final lifted phase is approximately -1.976e-10; adding represented 2*pi would be a different numerical operation with its own error obligation.

The numerical parent samples are

Structural time	Materialized parent EPI
7	0.3125
7+2^-20	0.3125000147351891
7+2^-19	0.31250002947060684

The response retains all forty coordinates at each sample, all eight RK4 stage states and rates, exact represented quadrature residuals, rate and pressure defects, storage changes and the two weighted-mean changes. These residuals are retained rather than assigned zero by conservation labels. Pressure is refreshed from phase and bare support, never inferred from the observed form response. Its separate 2^-46 evaluation budget uses the shared dyadic128 trigonometric kernel: the general snapshot's cheaper pressure enclosures can be too wide for this precision check even when the materialized value is accurate.

The three actual values are installed in the parent's timestamped history. The final EPI matches live state, and the adapter independently checks that the final history timestamp equals the graph's structural clock. The ordinary acceleration reader does not authenticate that clock association by itself.

3. Independent continuous-law certificate

The certificate adapter re-admits source and records before constructing an exact singleton initial box. The existing weighted support field with the identity permutation retains all forty coordinates; it introduces no twin reduction or uniform clock change. Two independent direct-from-source calls to validated_box_taylor_step enclose the exact solution at durations h and 2*h.

Order four, dyadic128 arithmetic, Machin pi, trigonometric jets and the strict Picard tube policy were fixed before response evaluation. The complete sine field is smooth on these boxes. No capacity-at-most-one metric bound, point derivative extrapolation or numerical RK4 stage is used as a Taylor remainder certificate. Each remainder is evaluated on its admitted whole-time tube.

Every retained numerical coordinate is compared to its corresponding exact endpoint enclosure. The parent-form error bounds are

Sample	Certified absolute error upper bound
t_0	0 relative to the declared represented initial condition
t_1	less than 7.067e-18
t_2	less than 1.233e-17

All are below the frozen 2^-52 allowance. A small enclosure radius alone would not suffice: the numerical point's distance to the entire interval is included. Source representation, the continuous-law enclosure, numerical materialization and runtime observation remain distinct evidence.

The preceding theorem gives true whole-window acceleration greater than 1/8. The shared sample-budget owner turns the actual endpoint bounds into an acceleration allowance below 2.909e-5. The independent conservative reader arithmetic allowance is 2^-13. Therefore the certified represented lower bound is approximately

\[ \widehat A_{\mathrm{binary64}}>0.12484884279679245>0.1. \]

This lower bound is distinct from the observed value 0.2513427734375. The finite-window theorem already controls acceleration variation; no second Taylor truncation penalty is subtracted for that same comparison. The retained certificate includes the complete endpoint boxes, series, remainder intervals and strict Picard margins for independent review.

4. Endpoint admission and the remaining boundary

The exact endpoint pressure enclosures remain positive. Their approximate values increase from 0.00012070973312992051 at the source to 0.00012071347799567383 at the final sample. The materialized pressure, form, capacity, degree and live history pass the unchanged strict THOL reader; the independent grammar reader admits THOL after the actual VAL prefix. Threshold crossing is checked separately because the strict precondition reader only warns when a history is below the child threshold.

All 13 frozen checks pass. They include endpoint accuracy, the independent crossing bound, the actual observed crossing, clock association, fresh positive pressure, grammar, preserved debt and absence of child/support changes. Endpoint checks leave persistent state unchanged.

The result closes the gap between a possible exact-history domain and one associated, certified numerical record. It does not execute a child proposal or settle hierarchy/metabolic construction, support-event work, the newborn's continuing law, subsequent contact or enlarged-system maintenance. The old identity barrier continues to apply only while its admitted support and laws remain in force. An admissible future event is not an autonomous occurrence.

Verification uses the retained-evidence controls, source and certificate controls, detached wiring controls and coupled numerical-kernel controls. They consume retained evidence or independent fixtures; the frozen numerical producer is not rerun for routine tests. The archive retains its evaluation-time test sources, including a preliminary pressure-test alias typo subsequently corrected in the maintained test; no evaluated implementation, input, response or verdict was rewritten for that test-only correction.