Finite contact between actually formed sine classes¶
Prospective interaction claim and frozen protocol¶
The central-interface admission result proves an exact null for the original nominal C9 preparations at zero relative origins. This experiment supplies one common nonzero relative phase origin from the initial preparation. It asks whether the actual formed donor classes then produce distinct receiver-form responses while the complete joined state retains both cycle identities. The limiting geometries below are comparison states, never resets of reached states.
Freeze the following before evaluating the response certificate:
- Prepare two initially disjoint simple unit C9 cycles, each with local node order \(0,\ldots,8\), with unit capacities, beta one, \(e=1023/1024\) and \(w=1/1024\). The complete continuous rows are \(\dot x=-eKLx+(w/\pi)KS(\theta)\) and \(\dot\theta=(w/\pi)KLx\), where \(K=\operatorname{diag}(1/d_i)\), \(L\) and \(d_i\) use the live support, and \(S_i(\theta)=\sum_{j\sim i}\sin(\theta_j-\theta_i)\). Declare \(\tau=et\) and \(\gamma=1/(1023\pi)\).
- Compare donor winding classes \(k=1,2\) against the same receiver class one. Retain the exact rational slope \(m=(2046/9)(355/113)^2\). Initial donor forms are \(km(j-4)\); initial receiver forms are \(m(j-4)\). Donor nominal phases are zero and receiver nominal phases are the common origin \(\varphi=1/1000\), supplied from initial preparation in both donor comparisons. Both nominal common form origins are zero.
- Retain per-coordinate initial form and continuous phase-lift error bounds \(10^{-10}\), with each error row summing to zero separately on each component. The two component errors and the two donor comparisons may vary independently within these constraints. Each component retains its original sixteen-dimensional relative source family and common initial storage ceiling \(10^9\); unequal nominal preparation costs are unchanged. The receiver phase shift changes neither its internal source storage nor its formation proof.
- Retain acquisition at \(\tau=100\), the original full relative radius \(r=1/12\), and the complete two-class formation admission. Continue the actual unprobed trajectories for the fixed additional dwell \(D=10^{13}\). Contact occurs at \(\tau_c=100+D\), or original time \(1024(100+D)/1023\). No intervening phase probe, reset, feedback, forcing or coefficient change occurs.
- Before contact, require each actual component to have Euclidean form norm and phase-error norm, relative to its own target and original common phase origin, at most \(\epsilon=10^{-32}\). Use the nonlinear strict Lyapunov estimate below with freshly reconstructed coefficients. Admit the exact rational decay upper bound \(2^{-512}\) only when its derived exponent satisfies \(512\le\kappa D\le4096\).
- Supply one unit bridge between the two local nodes \(4\), without changing their states or any other nodal value. Use degrees three at the two ports and two elsewhere afterward. Declare event-work allowance \(10^{-6}\), and account for the exact support-event storage jump separately from continuous dissipation. Hold the joined support, capacities and coefficients with no later event or forcing.
- At the fixed elapsed scaled time \(h=1/100\) after contact, read actual receiver-port form. Each readout has independent additive error of magnitude at most \(d=10^{-30}\). Predict the recorded contrast, donor class two minus donor class one, to exceed \(10^{-25}\). The same phase-origin premise, contact rule, clock, receiver preparation and readout apply in both cases.
- Require both complete joined source families to enter and remain in the radius-\(r\) acute quotient neighborhood of their aligned joined target. Retain donor winding \(k\) and receiver winding one for all future uninterrupted joined flow, with recovery on the actual postevent conserved-mean leaf. The disconnected comparison must have no donor-dependent receiver response under a matched actual receiver preparation; its independent preparation and readout errors retain their explicit control bound.
- Use exact rational primitives and the shared outward dyadic128 trigonometric arithmetic for reported intervals. Keep \(2^{-512}\) and the warmup squared-norm estimates as exact rational proof bounds; do not round that decay upward to the dyadic128 grid before checking the endpoint tolerance. Use the fixed fourth-order correlated response expansion, remainder and perturbation bounds proved below. No trajectory propagation, horizon or offset sweep, fitted coefficient, source selection or precision search is admitted.
- Pass only if fresh formation, warmup, event-work, full joined identity and strict recorded-contrast obligations all pass. Otherwise retain the insufficient bound or separately proved obstruction under this unchanged protocol. Preserve the producing source and protocol before the first reserved evaluation, and record its response separately from this declaration.
The long structural dwell and small readout error are supplied mathematical premises. No laboratory clock, preparation precision or sensor bridge is established. A supplied contact with a finite response does not derive support birth, a binding force, autonomous composite selection or a uniquely selected fundamental law.
The actual formation images reach the contact tolerance¶
The receiver's supplied common phase shift is an exact isolated-component symmetry. Subtracting it leaves precisely the original class-one formation family, including all initial errors and their zero-sum constraints. On either isolated component put
where the receiver target includes its common origin \(\varphi\). The formation admission at \(\tau=100\) gives \(\|x\|_2^2+\|y\|_2^2<r^2\), and traps this complete state in its acute chart. Both component means relative to this target are zero. This starting claim uses the actual formation image, not an independently selected state near the target.
Apply the strict nonlinear Lyapunov estimate to this uninterrupted unprobed flow. Its use here does not include any of that owner's supplied maintenance jumps. With
take fresh rational bounds \(\underline\eta,\overline\eta\) for \(\eta\), and a positive rational lower bound \(\underline c\le c\). Define
For \(\xi=A_0^{-1/2}y\), \(v=\gamma A_0^{1/2}x\), that estimate retains the complete nonlinear potential and proves
The initial combined-radius constraint therefore gives the uniform bound
The reusable calculation retains the freshly derived component bounds \(b_{x,k},b_{\theta,k}\) and can use the sharper \(V_{0,k}=3\overline\eta\Lambda b_{x,k}^2/4+ a_+b_{\theta,k}^2/\lambda\le V_0\). Its formation admission has already checked \(b_{x,k}^2+b_{\theta,k}^2<r^2\). For any admitted decay upper bound \(\rho_D\ge\exp(-\kappa D)\),
No small tangent residual replaces the nonlinear trapping premise.
The following rational envelopes explain the fixed dwell:
The trigonometric bounds are independently available from the fixed constants. For example,
The upper inequality uses \(\pi<22/7\) and \(\sqrt2>7/5\). The lower bounds \(\pi>25/8\), \(\sqrt2<10/7\), and \(\sin t\ge t-t^3/6\) give \(c>1/20\). Fresh outward intervals must still satisfy the stated rational envelopes; a cached numerical value is not their premise.
Equation (1) then implies
Since \(\exp(1)>2\), the guarded choice \(\rho_D=2^{-512}\) exceeds the exact decay whenever \(\kappa D\ge512\). Equation (2) becomes
These are exact rational inequalities; for example, \(2^{510}>10^{153}\) follows from \(2^{10}>10^3\). They establish the declared Euclidean endpoint bounds \(\epsilon\) on both actual components. Keeping the chosen decay exact avoids losing this conclusion through an unrelated absolute interval grid.
One joined state, event work and both retained identities¶
Write \(a,b\) for the donor and receiver central ports. Put \(A=K^+L^+\) after joining, with \(K^+_{aa}=K^+_{bb}=1/3\) and \(K^+_{ii}=1/2\) elsewhere. All later rows are exactly
The actual contact states satisfy
where \(u\) is the phase error from each component's limiting twist and its original common origin. Each of these four error rows has zero unweighted mean before contact.
Choose the aligned joined critical phase geometry \(\Theta_k\): donor twist \(k\alpha(j-4)\), receiver twist \(\alpha(j-4)\), with \(\alpha=2\pi/9\) and both port phases zero. Internal sine sums cancel and the bridge sine is zero, so \(S(\Theta_k)=0\) on the full joined support. It is a reference for the retention proof, not the actual postevent state. With \(P=I-\mathbf1\mathbf1^{\mathsf T}/18\), the actual squared quotient distance to this reference satisfies
Indeed, centering the two nominal origins gives offsets \(-\varphi/2\) and \(+\varphi/2\) on nine nodes each. Their cross terms with the phase errors vanish by the retained component zero-sum constraints. No independent favorable error alignment has been imposed.
Let
The exact supplied support-event work is
and obeys
Strict positivity follows from \(0<\varphi-2\epsilon\le|\varphi+u_b-u_a| \le\varphi+2\epsilon<\pi\). There is no assumption of a passive contact or an undeclared reservoir paying this work.
On each original cycle, \(\lambda_{\max}(L_{\mathrm{C9}})\le4\). Its target is phase-critical, so Taylor's integral bound gives internal phase excess at most \(2\|u\|^2\); internal form storage is at most \(2\|x\|^2\). Adding the bridge exactly yields
This is a complete joined-state bound, not subtraction of unrelated rounded preevent and postevent storage intervals.
The joined graph has eighteen nodes and diameter nine. If \(z\) has zero mean and range \(R_z\), then
The second inequality follows along a shortest path between a maximum and a minimum. Hence \(\lambda_2(L^+)\ge2/81\). Throughout the relative radius-\(r\) chart, every edge has positive cosine at least \(\cos(4\pi/9+\sqrt2r)>1/20\). For closing cycle edges, this statement uses their fixed turn offsets in the retained local lift chart. It does not replace a real closing gap by an unrelated representative during evolution.
Interpolation to \(\Theta_k\) stays in the convex chart. The form quadratic and the integrated phase Hessian therefore give the strict boundary barrier
The coefficient is strictly greater than \(1/1620\), so the last inequality is strict on the boundary \(Z=r>0\). The frozen bounds satisfy
Along the entire joined continuous flow, (4) gives exactly
The mixed terms cancel using the symmetric diagonal mobility. A first exit through \(Z=r\) would contradict (7)--(9). Thus every actual joined member remains in the acute chart, retaining donor winding \(k\) and receiver winding one.
The postevent conserved means use the new degrees, whose total is \(38\). They are
The internal twists have zero degree-weighted mean by reflection. Equation (10) implies \(|\bar x^+|\le\epsilon/19\) and \(|\bar\theta^+-\varphi/2|\le\epsilon/19\). The original isolated means cannot replace these event-reweighted quantities. In particular, every perturbed member need not recover to absolute form zero.
On each fixed postevent mean leaf, the trapped set is compact. Zero dissipation forces \(L^+x=0\); remaining in that set also requires \(S(\theta)=0\). Strict phase convexity in the chart leaves only \(\Theta_k\) modulo its common phase. LaSalle's argument therefore gives asymptotic recovery to uniform form \(\bar x^+\) and the aligned twist with common phase \(\bar\theta^+\). This conclusion concerns the supplied joined support; it provides no dynamical law for its formation or removal.
The first donor-dependent receiver derivative¶
For the response comparison only, start (4) at zero form and at \(\theta_k^0\), obtained by adding the receiver origin \(\varphi\) to the receiver block of \(\Theta_k\). This defines an ideal comparison flow; the actual state remains the one bounded in (3). Put
The common vector \(f\) is independent of donor class. Let \(B_k=K^+L_{\cos(\theta_k^0)}\) be minus the derivative of the normalized sine field at the comparison state, and define
Here \(L_{\mathrm D}\) contains only the donor's nine internal edges, embedded in the full eighteen-node matrix. Thus \(B_2-B_1=-d_cD\), including the correct degree-three donor port.
All derivatives in this section are in the scaled clock. Direct differentiation of the complete rows at time zero gives
The terms containing a first phase derivative vanish here; nonlinear phase derivatives have not otherwise been discarded.
For \(Y_k=e_b^{\mathsf T}x_k\), the first three derivatives are class-independent. More explicitly,
The receiver row of \(D\) vanishes. The relevant fine-row values are
Only the bridge entry \(A_{ba}=-1/3\) connects the receiver row to the donor block. Equation (11) consequently proves
The class changes the donor's phase feedback before it changes the receiver. Equal initial port currents therefore do not imply equal later responses. For \(\sigma=0\), the ideal state is instead stationary and the central contact null applies.
A correlated finite-horizon remainder¶
Rescale the comparison form as \(x=\gamma v\), set \(\eta=\gamma^2\), and write \(y=\theta-\theta_k^0\). Then
The matrix \(A=I-P_{\mathrm{rw}}\) has a row-stochastic neighbor matrix \(P_{\mathrm{rw}}\). Hence \(H(t)=e^{-At}\) is nonnegative and row-stochastic, with \(\|H(t)\|_\infty=1\). Globally,
and scalar sine Taylor remainder gives
No acute-phase premise is needed for these response bounds.
Define the same first comparison for both classes:
Variation of constants gives
For \(0<h\le1\), the fixed \(\eta\) satisfies \(\eta h^2\le3/4\). A Volterra bootstrap using (13) yields, for \(0\le t\le h\),
Indeed, with \(V_h=\sup_{0<t\le h}\|v_k(t)\|_\infty/t\), the integral equations imply \(V_h\le s/3+(2\eta h^2/3)V_h\). The same equations give
Substituting these into (14), using (13), proves
This controls the full nonlinear flow, including receiver backreaction.
Subtract (14) before bounding it. Its first-\(\eta\) contrast is \(\eta d_c\int_0^hH(h-t)Dz(t)\,dt\). The absolutely convergent matrix series for that integral is
Every \(p=0\) term vanishes at the receiver. The \(p=1,q=0\) term is exactly (12) divided by \(4!\) after restoring \(\gamma\eta d_c\). For \(n=p+q\ge2\), there are \(n\) remaining terms. Their receiver tail is at most
The successive positive-term ratio is \(2h(n+1)/(n(n+4))\le h/2\). The common support and common forcing have been retained in this subtraction; two unrelated marginal remainder estimates would lose the cancellation that identifies the class response.
Combining (15)--(17) gives the complete ideal-flow bound
The extra power of \(h\) in the tail distinguishes an actual fourth-order class response from an unbounded formal derivative.
Actual prepared states, observation errors and the disconnected control¶
Use the maximum norm on the full original \((x,\theta)\) state, without rescaling the form error by \(\gamma\). The field in (4) is globally Lipschitz with constant at most \(2+2\gamma<3\). At contact, each ideal comparison differs from its actual joined state by at most \(\epsilon\) in this norm, by (3). For \(0<h<1/3\), the integral comparison therefore gives
For example, taking the supremum in the integral inequality gives \(E_h\le\epsilon+3hE_h\). This bound retains every initial form and phase error in both components and all later feedback. There is no ideal-target reset and no new state-error injection.
The two donor comparisons may have independent admitted preparation errors. Their contrast error from (19) is at most \(2\epsilon/(1-3h)\), and their readout errors contribute at most \(2d\). In the frozen case \(\sigma=s>0\), a sufficient recorded contrast lower bound is consequently
No cancellation of preparation or measurement errors is assumed.
Removing the bridge leaves the receiver's complete field independent of every donor coordinate. Thus a fixed actual receiver preparation has exactly the same disconnected trajectory under either donor label. For independently varying receiver preparations within the admitted endpoint families, the disconnected recorded contrast instead lies in the explicit control band
An exact null for matched states is not an assertion that all independently noisy records coincide.
Analytic design margins and the scope of the prediction¶
The fixed constants satisfy
For the cosine difference, \(2\pi/9<7/10\) gives \(\cos\alpha>151/200\), while \(\cos2\alpha=\sin(\pi/18)<11/63\); their difference exceeds one half. The sine bound follows from \(t-t^3/6\). At \(h=1/100\), rational substitution gives
Therefore the ideal part of (20) exceeds
The last two terms of (20) sum to less than \(10^{-29}\). These sufficient analytic bounds predict \({\cal L}>10^{-25}\), while the disconnected control (21) stays below \(10^{-29}\). They explain the frozen design without evaluating or fitting a response. The reserved implementation must still check the actual outward bounds, all formation and warmup premises, event work and both joined-family retention obligations.
The observed signature concerns internal donor class under a matched interface premise; the unequal original preparation costs are retained, so this is not an equal-cost causal comparison. The receiver's initial port current is the same in both cases, and the discrimination arises only through later complete-law feedback. Both cycle windings persist, but the transient form contrast is not an indefinitely stored form record. The bridge and its positive work remain supplied, and no independent claim about laboratory distinguishability follows from the small mathematical error budgets.
The primitive-only assess_sine_formed_class_contact returns SineFormedClassContact. It freshly reconstructs formation, nonlinear relaxation and the complete contact bounds from explicit inputs; no incoming report or cached verdict is admitted as a premise. Its reusable response flag requires strict positivity, while this frozen protocol additionally requires the threshold \(10^{-25}\). The contact contract and independent controls retain its implementation and observation boundaries.
Retained evaluation of the unchanged protocol¶
The complete saved report returns certified_formed_class_contact with no unavailable reasons. Both actual-state warmup handoffs pass. The response, whole joined identity and event-work allowance flags all pass. The four retained warmup squared-norm upper bounds are strictly below \(10^{-64}\), establishing the declared \(\epsilon=10^{-32}\) endpoint budgets without changing the original \(10^{-10}\) preparation errors.
The recorded contrast has exact lower endpoint
approximately \(8.5036397851\,10^{-25}\). The disconnected control band has absolute upper endpoint below \(10^{-29}\). For compactness, every endpoint in the following table is the displayed integer divided by \(2^{128}\). The common joined margins apply to both donor cases.
| Retained quantity | Lower numerator | Upper numerator |
|---|---|---|
| Recorded receiver contrast, donor two minus donor one | 289363867352485 | 397865303326900 |
| Disconnected recorded contrast | -687580866 | 687580866 |
| Joined radius-squared margin | 2361540721855372884298960699209391844 | 2361540721855372884298960699209391845 |
| Joined excess-storage margin | 1288545231667010464184699045007455 | 1288545231667010464184699045007456 |
| Supplied bridge work | 170141169282037749306974721162487 | 170141183460469231731687303722690 |
| Event-work allowance margin | 170141183460469231731687303709078 | 170141183460469231731687303709079 |
The bridge work is strictly positive and below its supplied allowance. It is not inferred from prior dissipation. The contact time remains \(\tau_c=10000000000100\), or original time \(930909090918400/93\), followed by the unchanged readout offset \(h=1/100\). The saved decay exponent is approximately \(2089.4361\), inside the declared guard, and the chosen exact decay bound remains \(2^{-512}\).
The frozen protocol, producing source archive and evidence manifest retain the original source and evaluation provenance. The archive includes this owner's protocol and derivation as they stood before the reserved evaluation. Those contents remain unchanged; the current owner adds this result documentation in place of the former unevaluated-status paragraph. The table was transcribed from the saved report through the shared JSON reader, without reassessment or producer regeneration.
This is a conditional analytic interaction certificate for actual formed families under the supplied complete law. It retains the structural relaxation duration \(10^{13}\), readout error \(10^{-30}\), positive contact work and both source families. No trajectory measurement, laboratory-time interpretation, realizable sensor precision or autonomous contact mechanism has been established.