A supplied form probe of the acquired two-port C9 pair¶
Question, source handoff and scope¶
The capture result places the entire prepared thirty-six-coordinate family in the compatible two-port basin at slow time \(\sigma_*=1025\). This result asks whether a common, explicitly supplied form input produces a resolvable receiver response in that acquired pair and an unjoined control, while preserving the winding identities and accounting for the input's storage jump.
The discriminating mechanism is transport through supplied contacts. A nonzero receiver response does not distinguish the winding classes, prove that the acquired organization is necessary for transport, or identify a physical interaction. Acquisition matters here as the justified source of the complete pre-probe state and its retained identity. The response theorem itself also applies to other states meeting its explicit endpoint bounds. The simpler form-diffusion model \(x_\tau=-Ax\) already has the positive contact-transfer term proved below, without any phase coupling. The proposed sign contrast therefore cannot select the sine law against that alternative. Its additional conclusions concern a controlled input applied to the previously acquired complete state, bounded work, finite readout error and preservation of its winding identity.
The runtime certificate is conditional on that endpoint domain. Its fixed research protocol additionally binds the domain to the previous capture theorem and its retained execution. Neither a cached verdict nor a small nominal-reference error alone supplies this handoff. No actual state is reset to the compatible target.
Complete rows, endpoint domain and common input¶
Retain the capture owner's two unit C9 cycles, donor nodes 0,...,8,
receiver nodes 9,...,17, and unit contacts (0,9) and (1,10). Write
\(L\) for this joined Laplacian, \(M\) for its degree matrix,
\(A=M^{-1}L\), and \(f(\theta)=M^{-1}S(\theta)\), where
\(S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i)\). Each component has
degree mass twenty and the total mass is forty. The ports have degree three;
all other degrees are two. Keep held capacities one, beta one,
\(e=1023/1024\), \(w=1/1024\), and
Between events the complete law remains
There is no continuous input during the response window. Both rows are retained; the heat semigroup used below is only a variation-of-constants tool for the form row of (1).
Let \(P_M\) remove the full degree-weighted mean and let \(\theta_*^m\) be the compatible target with the actual state's conserved continuous phase mean. Its correlated root, winding periods \((2,1,0)\) and criticality are those of the compatibility owner. Use the same continuous lifts; no independent component rotation is made. At the pre-probe handoff require
The previous capture theorem proves strict bounds inside (2) for every member of its original independent nodewise form and radian phase error family, with both source radii \(1/65536\). These deliberately larger endpoint bounds retain that actual family without treating its sharp saved bounds as new source coordinates.
At the fixed handoff supply the hybrid event
This is a uniform signed form increment on all donor nodes. It is an external, instantaneous preparation action, not a derived event selector or a finite-duration pressure flow. The same nodal action is supplied to the unjoined control. Its internal coefficient rows are given by (1) with that control's own Laplacian and degree matrix.
The control consists of the two isolated unit C9 cycles from its initial preparation onward. It starts with the same nominal eighteen nodal phases, zero nominal form and the same independent source error budgets as the capture experiment, and evolves to the same original handoff time. Removing contacts from a previously joined state is not this control preparation. The argument permits either paired source errors or independently selected members of the declared family. No control equilibrium reset is needed.
Measure elapsed fast time \(s=\tau-\tau_*\) after (3), with fixed \(h=1/4\). Thus the added original time is \(h/e=256/1023\), and the added slow time is \(1/(4\cdot1023^2\pi^2)\). The handoff itself remains
These are structural clocks; no laboratory-unit identification is supplied.
Work and preservation of the acquired identity¶
The normalized storage and loss are unchanged:
Define \(b=e_0+e_1-e_9-e_{10}\). The actual joined support gives
Consequently the exact event work, in these storage units, is
This accounts for the jump separately from continuous loss. It does not infer a reservoir, event passivity or a physical energy scale. For (2)--(3) the lower work bound is positive, so the declared action supplies storage. The upper work bound obeys
The full form mean increases by \(a/2\). Since \(\|P_Mr_D\|_M=\sqrt{10}<19/6\), the post-event relative norm is at most
The phase mean and target representative are unchanged. Criticality and the normalized Laplacian upper bound two give the pre-event estimate \(H^- -H_*\le X^2+Y^2\). The capture owner's local ball has radius \(R=1/12\) in the combined relative-form and phase norm and boundary storage at least \(B=1/648000\), provided the target's minimum acute margin exceeds \(1/8\) radian. The two strict admissions after (3) are
For the chosen constants the second margin is exactly
Thus the same loss and boundary argument traps the post-event state for every later uninterrupted evolution. It retains periods \((2,1,0)\) and converges to \(\theta_*^m\) and uniform form equal to the new mean. This preserves the established mathematical identity through the supplied probe; it does not establish autonomous maintenance against arbitrary or repeated inputs.
For the unjoined control, (3) is constant on each connected component and therefore lies in its Laplacian kernel. Its work is exactly zero and its relative motion is unchanged by the input. The original control preparation also retains each isolated winding identity. To see this directly, each ring has degree mass eighteen, diameter four, and normalized gap at least \(1/18\). The uniform class-two twist has acute margin \(\pi/18>1/6\), and class one's margin is larger. In the relative radius \(1/12\), the phase Hessian is bounded below by \(L/25\). The ring boundary storage is consequently at least \(1/129600\). After subtracting each ring's own means, its initial combined squared norm and excess storage are at most \(36/65536^2=9/1073741824\). They are strictly inside that radius and barrier. The same loss and acute uniqueness argument traps each component; the donor's common form shift at (3) does not change these conclusions.
Receiver observation and the exact control result¶
For the joined graph define
The event leaves receiver coordinates unchanged, so the same baseline can be read immediately before (3). Use the corresponding actual-degree normalization, mass eighteen, for the unjoined receiver, and call its increment \(J_0\). The observation rule is common: the receiver's degree-weighted mean form increment. Its normalization is part of the declared measurement map rather than an invented shared degree matrix. The fixed coefficient vectors differ between the two supports; this is the same support-dependent rule, not an identical linear functional.
Summing (1) over an isolated receiver with its actual degrees cancels both Laplacian and sine terms. Therefore
for every control initial state and every positive observation duration. This equality does not use small errors, relaxation, equilibrium or a linearization. The control identity argument above is a separate claim.
There are four scalar readings: before and after in each model. Allow independent signed readout errors of magnitude at most \(\delta=2^{-26}\) per reading. If \(\widehat C\) is the observed joined-minus-unjoined increment, then
No cancellation, independence in a probabilistic sense, or averaging advantage is assumed. Common form origins and their source errors cancel from each increment.
Full-law finite-window bounds¶
Let \(T(s)=e^{-sA}\). It is a contraction in the degree metric and preserves constants and weighted means. Globally on continuous lifts, \(\|A\|_M\le2\), \(f(\theta_*^m)=0\), and \(\|f(\theta)-f(\theta_*^m)\|_M\le 2\|\theta-\theta_*^m\|_M\). The latter follows from the sine Jacobian \(-M^{-1}B\operatorname{diag}(\cos)B^T\); its metric operator norm is at most that of \(A\), even outside the acute chart.
Write \(Q\) and \(P\) for the suprema over \([0,h]\) of the post-event centered form norm and target phase distance. Variation of constants for the form row and integration of the phase row give
For \(c=2\bar\gamma h<1\), use the explicit enclosing bounds
These estimates retain the nonlinear phase response and original form. They do not identify phase with the auxiliary gradient trajectory or treat the compatible state as the actual preparation.
The exact form variation of constants splits (12) into three terms:
The first is a supplied-kick heat term, the second the pre-existing form background, and the third the actual nonlinear sine feedback. None is discarded from the result.
For the first term, \(P_{\rm walk}=I-A\) is the actual nonnegative random-walk matrix. Since \(\ell_Rr_D=0\) and \(\ell_RP_{\rm walk}r_D=1/10\), the nonnegative exponential series proves, for \(0<h\le1\),
For an upper bound put \(v=r_D-\tfrac12\mathbf1\). Reversibility gives
Its derivative is nonnegative and its second derivative nonpositive, by the nonnegative spectrum of the self-adjoint \(A\). Its initial value is zero and its initial derivative is \(1/10\), hence
Since \(-\ell_RA=b^T/20\), metric contraction and (6) bound the second term by
The mean-free restriction of \(\ell_R\) has exact dual norm \(1/\sqrt{40}<1/6\). Every \(f(\theta)\) has weighted mean zero, so contraction and (15) bound the last term in (16) by
In particular this estimate does not require cancellation of the actual contact sine currents. Define
Equations (13)--(20) give the complete-state response enclosure
For the fixed constants,
and the lower side of (21) is exactly
With the four readout errors retained, the declared observation threshold \(C_{\min}=1/262144\) has strictly positive margin
Thus every state in (2), including the whole acquired source family, produces \(\widehat C>C_{\min}\) while meeting the work and identity requirements, conditional on the supplied input and observation contracts. Equation (22) is an analytic sufficient prediction, not a measured response or a trajectory simulation.
Prospective protocol and admission boundary¶
Before the first new report assessment, freeze the following:
- The joined support, complete rows, capacities, coefficients, source and three structural clocks are the unchanged capture model. Its entire original error family reaches the handoff time (4), and (2) is justified by that theorem's retained first execution. The unjoined control is prepared separately from the same declared source on its own support.
- The single supplied action is (3), with amplitude \(1/2048\), applied to both models at the fixed handoff. There are no resets, contact events, further inputs, searches over amplitude, or response-selected times.
- The duration is \(h=1/4\) in fast time. The observation is precisely (12)--(14), using actual-degree receiver means, four independent error bounds \(2^{-26}\), and strict contrast threshold \(1/262144\).
- The work allowance is \(1/2000000\). Rebuild (7)--(11), including both work endpoints, the strict post-event radius and storage margins, and the same target's fresh minimum acute margin above \(1/8\) radian.
- Re-admit the primary compatible
(2,1)root using thirty-two outer and sixty-four inner strict-sign refinements. Its enclosure supplies target admission; it does not replace the actual endpoint by a root midpoint. - Rebuild all rational bounds (15)--(22) from admitted primitive constants. Preserve each failed strict obligation as unavailable. Equality with a strict threshold is unavailable; a cached passing flag cannot supply it. There is no trajectory integration, adaptive search or response-fitting budget in this analytical certificate.
- Archive the prospective proof, protocol and complete producing source before assessment, then retain the report and manifest separately. Link the unchanged capture artifacts and declare their retained execution as a premise of source acquisition. A read-only consistency audit can re-admit source, law, coordinates, work policy and saved evidence without rerunning that producer. Such an audit does not independently reproduce every validated Taylor remainder or authenticate execution chronology.
The generic endpoint-domain API may be used without a particular acquisition history, but then certifies only the conditional theorem for (2). The fixed research record additionally requires the explicit capture-source handoff. Do not promote one scope to the other through report labels or filenames.
At this prospective declaration no new probe report has been evaluated. Any retained result must be appended separately. A successful assessment would certify supplied-contact transmission with a finite readout margin, bounded supplied work and retained winding identity. It would not establish a winding-specific response, an autonomous input or contact rule, physical binding, or an experimental measurement bridge.
Retained certificate after export recovery¶
The original frozen execution completed its certificate calculations but failed while exporting the auxiliary control-identity dictionary: the shared projector rejected a dictionary passed as one value. No primary report was saved. The first-attempt record preserves that software failure; it is neither a failed mathematical inequality nor an available first response. The original attempt is not represented as a successful retained assessment.
A separately frozen export-recovery wrapper projects the dictionary's individual values and records the evaluation history. Because the first process retained no report, the wrapper repeats the deterministic analytical assessment with the identical scientific inputs and runtime. It changes neither the prospective inequalities nor the source, law, input, observation, time, target-refinement budget or thresholds. The original frozen evaluator, archive and prospective proof remain unchanged. The earlier capture producer is not rerun.
This retained recovery assessment returned certified_probe, with no
unavailable reasons. All ten fixed stopping conditions passed, including
the acquired-source handoff, fresh compatible target, four-error observation
margin, positive supplied work within allowance, joined identity and
recovery, and the original unjoined control's separate identity bound.
The saved report contains exact rational endpoints. The following values
are approximate displays of those bounds, not measured responses or
replacement thresholds:
| Retained quantity | Approximate bound |
|---|---|
| True joined receiver mean increment | [7.34855715e-6, 1.40137475e-5] |
| Joined-minus-unjoined contrast with all four readout errors | [7.28895251e-6, 1.40733522e-5] |
Lower contrast margin above 1/262144 |
3.47425524e-6 |
| Supplied joined storage jump | [1.68879827e-7, 3.07957331e-7] |
| Post-input storage margin below the capture barrier | 2.66677068e-7 |
The unjoined receiver's true increment and the unjoined event work are exactly zero. The observed contrast includes its two readout errors as well as the joined model's two errors. Its observation uses the same actual-degree mean rule with its own coefficient vector; the two models are not assigned an identical linear functional.
The source handoff audit
re-admits the original source and complete law, reconstructs the target
from strict root brackets, checks the retained reference chain and
recomputes the complete-state handoff bounds. The admitted original-form
and phase norms are approximately 5.39860959e-5 and 3.51060503e-4,
inside the larger budgets (2). The audit retains the explicit premise
that the earlier Picard inclusions, local remainders and endpoint centers
are valid evidence from the archived execution. It does not independently
reproduce every Taylor remainder or authenticate that execution.
Consequently the conditional probe conclusion covers every member of the unchanged original thirty-six-coordinate source family after its justified handoff at \(\sigma=1025\). The joined state retains periods \((2,1,0)\) through the input and subsequent uninterrupted flow. It converges to the same phase geometry with its original phase mean and form mean increased by \(a/2\). Each unjoined component retains its winding; the control donor's form mean instead increases by \(a\). These are analytic full-family conclusions, without a probe trajectory, an endpoint reset or a measured physical response.
The retained
protocol,
original producing source archive,
recovery report
and manifest
bind the unchanged prior capture artifacts and the separate failure and
recovery records above. The original source archive SHA-256 is
d54a607c76bdd992bedf26f92618360ed506739e2c2e70ed0b260fc704bda578,
based on revision f1ce7f196a50708e5e5910cdf68d0143265059a1 with the
three declared runtime overlays. The prospective text remains a
byte-identical prefix of this owner. Content hashes support association
and source recovery, not independent chronology authentication.
The positive contrast establishes the declared supplied-contact transmission, bounded work and retained mathematical identity. The pure form-diffusion alternative already transmits the same donor input, as (17)--(18) demonstrate. This baseline probe therefore does not select the sine law, show a winding-specific signature or establish that acquired organization is necessary for transport. No autonomous input or contact selection, physical binding or laboratory observation bridge follows.