An interior phase probe of the acquired two-port deformation¶
Question and independent preparation¶
The uniform form probe certifies supplied-contact transmission and retained identity, but pure form diffusion also transmits that input. This result instead supplies the same interior phase dipole to the acquired two-port pair and an independently evolved unjoined pair. It asks whether the different donor bulk angles produce distinguishable finite local form increments. A separate complete phase-blind alternative tests whether ordinary form diffusion alone can supply that new contrast under the same preparation and input.
This is a new experiment from the unchanged original capture preparation. The earlier uniform form intervention is not inserted into its trajectory. There is no equilibrium reset, reduced source uncertainty or response-selected warmup. The capture handoff and its explicit retained-execution premise supply the joined state at \(\sigma_*=1025\). The source still contains all independent nodewise form and radian phase errors of magnitude at most \(1/65536\).
Supply the same original nominal nodal phases and error allowances to the unjoined control, whose support consists of the two isolated C9 cycles from preparation onward. Independent errors are allowed in the two experiments. Common phase origins are removed only in proof coordinates; the components are not physically rotated or reset. The control is not obtained by deleting contacts from an evolved joined state.
Complete laws, means and existing trapping domains¶
The joined graph has donor nodes 0,...,8, receiver nodes 9,...,17,
unit cycle edges, and unit contacts (0,9) and (1,10). Its ports have
degree three and other nodes degree two. The unjoined graph omits the
contacts and has degree two at every node. In either graph let \(L\)
be its own Laplacian, \(M\) its own degree matrix, \(A=M^{-1}L\),
and \(f(\theta)=M^{-1}S(\theta)\), with
\(S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i)\). Keep unit held
capacities, beta one, \(e=1023/1024\), \(w=1/1024\), and
The complete sine rows between events remain
All thirty-six form and phase coordinates of each experiment remain.
Let \(P_M\) subtract each connected component's conserved weighted mean. The joined graph has one form and one lifted phase mean; the control has separate form and phase means in each C9. Write \(y=\theta-\theta_*^m\) for the difference from the corresponding exact target with those means. The joined target is the unique compatible \((2,1,0)\) geometry. The control target consists of the two uniform twists with periods two and one. On these respective relative spaces use the degree norm and the constants
The joined normalized spectral bounds are \(\lambda I\preceq A\preceq\Lambda I\). Each isolated ring has the stronger lower gap \(1/18\), by its degree mass eighteen and diameter four; the weaker common \(\lambda\) remains valid on the control's componentwise relative space.
The joined target's minimum acute margin exceeds \(1/8\) radian. The uniform control donor's margin is \(\pi/18>1/6\), and the receiver's is larger. Thus, in either combined relative ball
every phase segment from the target has edge cosine at least \(\sin(1/24)>c_*\). The phase Hessian in the degree metric lies between \(c_*A\) and \(A\). With the unchanged storage
criticality gives
within that ball. Its boundary storage is at least \(B\).
The previous joined capture theorem places the entire joined family inside this strict trapping domain at \(\sigma_*\). For the full unjoined control, the initial combined norm squared and storage are bounded by
This uses both components together after their own mean projections. The same first-exit storage argument traps the full control from its initial time onward. Consequently both experiments are already forward trapped at the common warmup start
The common coarse bounds \(X_0=Y_0=R\) below apply to these trapped states. They do not assert that every point of the independent product \(\|P_Mx\|_M\le R,\ \|y\|_M\le R\) is in the trapping ball. A generic warmup certificate retains forward trapping as an independent hypothesis. Source acquisition and this hypothesis are not supplied by a cached passing flag or by the two radii alone.
A finite warmup in the original coordinates¶
The strict modified-energy mechanism extends to this degree metric without any commutation assumption. The following isometry makes the reuse explicit. On the appropriate relative space put
The nullspace of \(\widehat A\) is removed componentwise. Thus these inverse square roots are well defined, and Euclidean norms of hatted coordinates equal the original degree norms. Define
Differentiating both complete rows gives exactly
No companion row or original form coordinate is discarded.
Use the rational bounds
The trapped-chart Hessian satisfies \(\mu I\preceq\nabla^2W\preceq M_WI\). Set \(\epsilon=\lambda/4\) and
Young's inequality and cancellation of the mixed damping terms yield [ \begin{aligned} a_-&=\mu/2+\lambda^2/16,\ a_+&=M_W/2+\epsilon\Lambda/2+\epsilon^2,\ \tfrac14|v|^2+a_-|\xi|^2 &\le V\le\tfrac34|v|^2+a_+|\xi|^2,\ V'&\le-(\lambda-\epsilon)|v|^2 -\epsilon\mu|\xi|^2\le-\kappa V,\ \kappa&=\min{4(\lambda-\epsilon)/3,\epsilon\mu/a_+} =\frac1{2233865700000}. \tag{7} \end{aligned} All derivatives here are in fast time \(\tau\). This is a bound for the actual nonlinear potential throughout the retained trapping domain, not merely for the tangent matrix at equilibrium. The shared \(\_sine\_lyapunov\) arithmetic kernels consume precisely these isometric Euclidean coefficients and norms.
For initial norm bounds \(X_0,Y_0\), the modified-energy upper bound is
With \(X_0=Y_0=R\), this is \(4512863/2592000000\). Fix the finite common warmup
Since \(\exp(1)>2\), the exact bound \(\exp(-\kappa T)=\exp(-128)<2^{-128}\) needs no numerical exponential evaluation. Retain that rational upper bound directly; do not round it through an interval grid and then silently restore discarded precision. The original returned norms satisfy [ \begin{aligned} |P_Mx(T)|M^2 &\le\frac{4V_0 2^{-128}}{\underline\eta\lambda},\ |y(T)|_M^2 &\le\frac{\Lambda V_0 2^{-128}}{a-}. \tag{9} \end{aligned} Exact rational comparison makes both quantities strictly less than \(2^{-80}\). Their approximate upper bounds are respectively \(2.03\times10^{-32}\) and \(1.33\times10^{-36}\); these displays are not replacement thresholds. Hence use the common pre-probe allowances
The warmup retains the entire original source family. These are proved finite-time output bounds, not narrowed preparation errors.
The probe time is \(\tau_p=\tau_*+T\), or \(t_p=t_*+1024T/1023\). Equivalently its slow time is \(1025+T/(1023^2\pi^2)\). The large conservative structural dwell makes no claim of practical speed or laboratory duration.
Identical interior event and local observation¶
Fix
At the common \(\tau_p\), supply one event in each sine experiment:
This is an externally supplied instantaneous phase action. It is not an
autonomous event selector, finite-duration pressure pulse or source reset.
Nodes four and five have degree two, the same neighbors, and the same
three affected donor edges (3,4), (4,5), (5,6) in both supports.
Consequently
The event preserves all relevant weighted form and phase means, including the control's separate component means.
At elapsed fast time \(h\), observe the same linear functional in both experiments:
where \(J\) denotes joined and \(0\) unjoined. Thus this probe uses the identical local coefficient vector, not the different receiver-mean weights of the previous experiment. The arbitrary common form origins cancel exactly. Four scalar readings of \(q^Tx\), two per experiment, have independent absolute errors at most
Here the error contract is per scalar form-difference reading, not per individual node sensor. For the recorded contrast,
No favorable correlation of those errors is assumed. The added original readout duration is \(1024h/1023=1/1023\).
Exact local coefficient and a finite full-law error¶
Let \(A_*\) be the joined donor short-arc coordinate in turns. The compatible donor's uniform bulk angle is
Both targets are critical. At either target, the phase jump changes the three affected oriented bulk gaps to \(b+a,b-2a,b+a\). Directly summing the two actual degree-two sine rows gives
Thus the exact-equilibrium leading contrast is
This local coefficient depends on the acquired bulk deformation. Equal local adjacency fixes this immediate calculation; it does not make the two full-graph propagators equal at finite time.
The previously certified strict root bracket has upper endpoint \(6573559493/38654705664<7/40\) turns. Consequently
For the fixed \(a\), every angle between \(b_0-a/2\) and \(b_J-a/2\) lies below \(\pi/2\) and within \(1/5\) of it: \(\pi/18+a/2<1/5\). Therefore its sine exceeds \(49/50\), and
This is an analytical design bound from the existing root evidence. The new fixed certificate also requires fresh target admission and an outward positive bound \(\Delta_c>1/32\); it does not treat an old target box or rounded midpoint as a new equilibrium.
To retain the actual residual states, put \(g=1/3069>\gamma\), \(c_h=2gh\) and
The denominator is positive. On each graph the semigroup \(T_G(s)=e^{-sA_G}\) contracts its own degree norm, \(\|A_G\|\le2\), and \(f_G\) is globally two-Lipschitz in that norm. Variation of constants for (1) gives whole-window relative form and target-phase bounds \(Q_{\max},P_{\max}\), as in the preceding probe. For \(\psi_0=\theta_*^m+a q\), it also gives
Since \(\|T_G(s)-I\|\le2s\), \(\|f_G(\psi_0)\|_M\le4a\), and \(\|\theta(s)-\psi_0\|_M\le Y+2ghQ_{\max}\), the three nonleading terms have combined absolute bound
This includes the pre-existing form background, phase error, nonlinear phase motion and both actual supports' finite propagation. No linearized law, identical heat kernel, hidden equilibrium reset or favorable source symmetry is substituted for the complete flow.
Combining (14)--(20) gives
For a prospective sufficient rational check, use \(\gamma>1/3216\) and \(\sin(3a/2)>(5/4)a\), the latter following from \(\sin z\ge z-z^3/6\) with \(0<z=3a/2<1\). Equations (16)--(17) imply
Exact rational substitution of the fixed constants proves
For scale, this deliberately loose observed lower bound is greater than \(5.17\times10^{-12}\), while the threshold is approximately \(3.64\times10^{-12}\). These are derived design margins, not an evaluated response or a claim of sensor resolution in physical units.
Event work, winding identity and later recovery¶
The event changes only phase storage. At the target its exact work is
Criticality cancels the first variation. Along the target jump segment, \(q^TLq=6\) and the common cosine bound give \(3c_*a^2\le W_*\le3a^2\). For any actual pre-probe phase error of norm at most \(Y\), the global Hessian norm bound two gives
Therefore each sine experiment has the signed event-work enclosure
At the fixed constants its lower endpoint is positive, and its upper endpoint is strictly below the common allowance \(W_{\rm allowed}=2^{-21}\).
The post-event combined radius and storage satisfy the strict admissions
These inequalities apply to both the joined graph and the full unjoined control on their respective mean leaves. Continuous loss then traps each post-event family for all later uninterrupted times. The periods remain \((2,1,0)\) in the joined graph and \((2,1)\) in the control. LaSalle's argument on the compact relative sublevel, followed by uniqueness of the acute target in each component or sector, gives recovery to the same phase geometry and conserved form means. This event shifts none of those means.
The work in (24) is supplied separately from the loss in (3). A positive work allowance is not an inferred reservoir or passive event. One admitted phase probe does not establish autonomous maintenance against arbitrary inputs, contact selection or physical binding.
A complete phase-blind alternative and its bounded background¶
On each of the same supports consider the explicitly different complete law
It has the same capacities, clocks and original preparation family, and receives the same phase event (11) at \(\tau_p\). Its form row is independent of phase. Thus kicked and un-kicked trajectories with the same pre-event state have exactly identical form for all later times. This is an exact causal null; an arbitrary before/after raw increment need not be zero because an initial form background can still relax.
That background is independently bounded for the unchanged source. Initially \(\|P_Mx\|_M\le7/65536\) on either graph, with componentwise projection for the control. Since \(\kappa\le\lambda\), (8) implies \(\lambda T\ge128\). Without using a sine-law warmup to certify (26), its own heat contraction therefore gives
Each raw increment (13) has magnitude at most \(2hX\). Including the four same readout-error allowances, its recorded joined-minus-unjoined contrast obeys
Thus the finite contrast separates the stated sine-law experiment from this phase-blind alternative under the common preparation and observation contracts. No winding or storage-trapping conclusion for (26) is needed for its form response bound, and the sine potential is not assigned as that alternative's conserved or dissipated storage.
This separation is not unique identification of the sine law. Other phase-sensitive laws or preparations can share such a response. The result relates one coefficient to the acquired internal deformation and bounds the complete finite-time confounders; it does not define a topology-free material constant, identify a physical constituent or supply an independent laboratory measurement bridge.
Prospective protocol and stopping boundary¶
Freeze the following before the first new response-certificate assessment:
- Retain the original thirty-six-coordinate preparation uncertainty, support-specific complete sine rows, held capacities and all clocks. Re-admit the original joined capture handoff without rerunning its producer; retain the declared Picard, remainder and endpoint-center execution premise. Separately admit the full unjoined source and its initial trapping bound (4).
- Both source families continue unprobed to (5). Use the already-trapped norm envelopes \(X_0=Y_0=1/12\), common constants (2), rational \(\eta\) bounds above, \(N=128\), \(T=N/\kappa\), and the exact upper decay \(2^{-128}\). Require the original-coordinate squared returns in (9) to be strictly below \(2^{-80}\). No root, source, dwell, precision or error-budget search is admitted.
- At the fixed \(\tau_p\) supply exactly (11), with \(q=e_4-e_5\) and \(a=2^{-12}\), in both sine models and the declared phase-blind alternatives. Form and phase means retain their own support-dependent conservation laws; no relative state is reset.
- Use \(h=2^{-10}\), the identical local form-increment map (13), per-scalar-readout error \(2^{-50}\), contrast threshold \(2^{-38}\) and per-event work allowance \(2^{-21}\). Keep all four errors.
- Rebuild the compatible
(2,1)target with thirty-two outer and sixty-four inner strict-sign refinements. Require the fresh target's analytic criticality and acute margin above \(1/8\), and the outward local cosine contrast \(\Delta_c>1/32\). The implicit root remains correlated; its midpoint or interval product is not an actual target. - Rebuild the full-law error (18)--(21), strict response margin (22), positive work and allowance (24), both post-event trapping admissions (25), and the alternative's independent warmup and upper response (27)--(28). Equality at a strict threshold is unavailable. A candidate bound without its target, source or trapping premises cannot supply a passing result.
- Use shared exact rational and outward dyadic128 trigonometric arithmetic for the fresh target and coefficient enclosure. The warmup uses the proved elementary exponential bound, not numerical integration. There is no new trajectory, adaptive response search or repeated capture run.
- Archive this prospective proof, protocol and complete producing source before assessment; retain the response, verdict and any failure separately. An unavailable sufficient bound is not observed dynamical failure and cannot be replaced by a longer dwell, smaller source or different probe without a separate prospective experiment.
The endpoint-domain and warmup mathematics is conditional on its stated trapping premises. The fixed research record additionally binds those premises to the original full source families. At this declaration no new dipole response report has been assessed; any actual retained outcome must be appended separately. The analytical design margins above do not replace that execution record or alter the preceding gates' frozen evidence.
Retained first dipole result¶
The first assessment of the unchanged frozen protocol was saved successfully
as certified_dipole, with no unavailable reasons. All thirteen stopping
conditions passed. They include the actual joined source handoff, the full
unjoined source's trapping bound, finite warmup, fresh target and local cosine
contrast, both nonlinear error budgets, four readout errors, separation from
the phase-blind alternative, positive supplied work within both allowances,
and preservation and recovery of both winding identities.
The retained report contains exact rational and outward interval bounds. The following approximate displays describe those bounds; they are not measured responses, sampled trajectories or replacement thresholds:
| Retained quantity | Approximate bound |
|---|---|
| Joined-minus-unjoined recorded contrast | [8.39509270e-12, 9.62315088e-12] |
Lower response margin above 2^-38 |
4.75711390e-12 |
| Absolute recorded phase-blind contrast upper bound | 7.10542736e-15 |
| Lower separation from the phase-blind bound | 8.38798728e-12 |
| Local cosine difference | [0.0404805704426, 0.0404805704629] |
| Complete-law remainder per sine model | 3.05238185e-13 |
| Joined supplied phase-jump work | [2.37655527e-8, 2.38874643e-8] |
| Unjoined supplied phase-jump work | [3.10650433e-8, 3.10650451e-8] |
| Joined post-event capture-storage margin | 1.51932241e-6 |
| Unjoined post-event capture-storage margin | 1.51214483e-6 |
The full warmup returned squared-norm bounds of approximately
2.02615606e-32 for original relative form and 1.32621120e-36 for
phase. Both are strictly below 2^-80, so the declared full-family
pre-probe allowances X=Y=2^-40 are admitted. These bounds follow from
the unchanged finite duration 285934809600000 in fast time and the
strict modified-energy argument. They do not shrink the original source
errors, substitute an asymptotic target for the actual state, or imply a
short laboratory recovery time.
The positive contrast concerns the identical interior phase action and the identical local form-difference observation on both sine supports. Its leading coefficient depends on the acquired donor bulk-angle deformation; the retained remainders account for both complete nonlinear flows and their different full-support propagators. The recorded lower bound remains above the threshold after every permitted initial residual and all four observation errors. The phase-blind alternative's own source-to-probe contraction bounds its remaining form background below the separate comparison ceiling.
Consequently every member of the original prepared sine family, with
its own conserved means, satisfies this finite discrimination after the
proved acquisition and warmup. The joined periods remain (2,1,0) and
the unjoined periods remain (2,1) through the phase input and subsequent
uninterrupted flow. Both recover to their respective phase geometries and
unchanged form means. Supplied event work is positive and bounded in each
model, and remains distinct from later continuous dissipation.
The source handoff retains the earlier capture execution as a premise. Its read-only audit re-admits primary preparation and law data, strict target brackets, the saved metric chain and the complete-state tail; validity of the archived Picard inclusions, local remainders and endpoint centers remains explicit. This gate ran no new capture trajectory or probe trajectory, and did not rerun the original capture producer.
The retained
protocol,
producing source archive,
exact report
and manifest
preserve this first assessment and its thirteen stopping conditions.
The source archive SHA-256 is
68ea24ad6301a53fa492fa6b8f53409cb910e700f7d0b9ec7e12e4b59285fb50,
based on revision 4183b788ee63a61e1c44afc08b7b3950a0bdd311 with the
two declared runtime overlays. The prospective proof remains a
byte-identical prefix of this owner. Source hashes establish recoverable
content associations, not independent chronology or execution authentication.
The result establishes conditional finite sensitivity to the acquired internal geometry and separates the declared phase-blind alternative. Other phase-sensitive laws can share this response. It does not select a unique fundamental law, remove the dependence on supplied topology, derive an autonomous probe or contact rule, or identify physical binding or a laboratory measurement map.