Acquired mediator class and a transmitted form response¶
Status: conditional finite-response theorem and prospective protocol under the supplied complete sine law. The finite estimate below supplies a single preparation, probe and error budget for separate reserved assessment. Any retained assessment is recorded separately below; previous frozen reports remain unchanged. Task status belongs to the execution plan.
The question is whether an acquired mediator's internal organization changes a donor-induced response in a different receiver while all three patterns retain their identities. This is a generative interaction claim: the class-dependent response must follow from the full nodal dynamics and its prepared organizations. A physical force, particle property or material identification is not inserted into the observation. The ontology retains those later physical obligations.
F1: full state, source pairing and observation¶
Use three simple unit C9 components in order donor, mediator, receiver, indexed \(i=0,1,2\), with local node order \(j=0,\ldots,8\). Their class tuple is \((1,k,1)\), with \(k\in\{1,2\}\). The state consists of all 27 signed real form coordinates and all 27 circular phases, represented by declared continuous lifts during the proof. Capacity is held at one at every node. The supplied contacts join \((0,4)\) to \((1,4)\), and \((1,4)\) to \((2,4)\). Thus the joined central degrees are \((3,4,3)\), and every other degree is two. Both class comparisons use precisely this same support.
Let \(\alpha=2\pi/9\) and \(\Theta_{i,j}=k_i\alpha(j-4)\), with \(k_i=(1,k,1)_i\). The common source origins are aligned at zero. This is a declared choice of the initial preparation, permitted by the isolated components' common-phase symmetry; it is not a rotation or reset of an already reached state. The earlier composition protocol used origins \((0,1/1000,0)\) and remains unchanged.
Retain the actual formation preparations
The original per-coordinate error bounds are \(10^{-10}\), with zero component sum separately in each channel. All sixteen relative source coordinates per component remain admitted; no symmetry constraint is added to these errors. Each complete family starts with zero winding and must reach its own class target by the existing two-class formation theorem, with its primitive premises freshly admitted.
For each admitted source and each class, compare two continuations of the same reached state: one receives the declared donor form jump and one receives no jump. The observation is the receiver central form increment between these paired continuations,
The class contrast is \(R_1(h)-R_2(h)\), with its orientation fixed before evaluation. Pairing is exact within each probe/baseline pair; a uniform claim across classes must cover the independently admitted source residuals or explicitly retain any stronger cross-class pairing premise. An unmeasured baseline cannot be replaced by zero merely because the aligned nominal target is stationary.
All coordinates, event amplitudes and readouts use the same normalized structural units as their owners. Angles are radians, and the response horizon is in the structural clock declared below. A recorded version would need an independently supplied observation law and error allowance; the mathematical form readout supplies no laboratory-unit calibration.
F2: complete law and supplied events¶
Retain beta one, \(e=1023/1024\), \(w=1/1024\), \(\tau=et\) and \(\gamma=1/(1023\pi)\). On each current support, write \(L\) for the unit Laplacian, \(K\) for inverse actual degree, and \(S_a(\theta)=\sum_{b\sim a}\sin(\theta_b-\theta_a)\). The complete rows are
Primes mean derivatives with respect to \(\tau\); every consumed coordinate obeys this same clock. No refreshed pressure inferred from the observed response, hidden input or altered mediator coefficient is added.
Continue each isolated formation trajectory through its unprobed dwell. The handoff retains acquisition at \(\tau=100\), relative radius \(1/12\), and the existing additional dwell \(D=10^{13}\). The actual-state handoff must supply per-component form and target-phase Euclidean errors at most \(\epsilon=10^{-32}\) at \(\tau_c=100+D\), including the guarded decay calculation. These are bounds on the reached states, not new source tolerances or target initial conditions.
At \(\tau_c\), add both contacts simultaneously without changing any state coordinate. The probe then supplies a positive form increment at the donor central node,
Its amplitude \(a\), horizon \(h\) and response/error allowances are fixed in the prospective protocol below. The unprobed member receives only the same contact event. The joined support, capacities and both rows remain fixed afterward; no further event or controller is implicit.
F3: source cost, work, identity and cancellations¶
The source information is supplied, and its class-specific cost must be retained. Equation (1) has nominal component storage \(36k_i^2m^2\); the whole-family ceiling includes both residual channels as in the formation owner. Changing only mediator class therefore does not mean equal preparation cost. The prospective comparison changes acquired internal organization under a common declared preparation budget.
For each support, use its actual storage
Continuous joined flow obeys \(H'=-(Lx)^{\mathsf T}K(Lx)\le0\). The contact work is the sum of the two newly installed edge storages; aligned nominal origins make its nominal value zero, without making the actual full-family work vanish. On the joined support the exact form-jump work is
Its linear term retains the actual pre-jump state. Contact and probe work need separate prospective allowances; previous dissipation is not an undeclared reserve that pays either event.
The degree-weighted form and lifted-phase charges are conserved between events. The probe changes the joined form charge by \(3a\), or its weighted mean by \(3a/58\), while leaving the phase charge unchanged. Any recovery statement must use this new conserved-mean leaf rather than silently imposing the unprobed common form. Contact changes the degree weights without resetting the states.
The joined acute-storage argument applies to both classes and both continuations as follows. Write \(u_i=\theta_i-\Theta_i\) at the handoff. The inherited component sums are zero, and \(\|x_i\|_2,\|u_i\|_2\le\epsilon\). The aligned contact has work in \([0,8\epsilon^2]\), and its joined storage excess is at most \(22\epsilon^2\). These bounds follow from the two added edge storages and the per-cycle Hessian bound four.
After the probe, the squared Euclidean distance on the quotient by common form and phase, the storage excess, and the probe work obey
For the first bound, arithmetic centering replaces the kick by \(a(e_{(0,4)}-\mathbf1/27)\), of squared norm \(26a^2/27\); the original component zero sums remove the other common-mode cross terms. For the last two, \(|(Lx)_{0,4}|\le6\epsilon\).
The joined fine diameter is at most ten, so \(\lambda_2(L)\ge2/135\). In the radius \(r=1/12\) quotient ball, every edge cosine exceeds \(1/20\); hence the acute-storage coercivity coefficient exceeds \(1/2700\). If
the first-exit and zero-loss arguments retain all three windings for all subsequent unforced joined time and give recovery to the unique local target geometry on the actual conserved-mean leaf. The unprobed member satisfies the smaller bounds obtained by setting \(a=0\). The preprobe degree-weighted form and target-phase means lie within \(4\epsilon/58\) of zero, since only the additional central degree weights remain after the component sums cancel. The probe adds the already stated \(3a/58\) to the form mean; no mean is reset.
Retaining these identities is a separate calculation from separating the readout. A small transient response or a correct tangent coefficient alone proves neither.
The aligned nominal unprobed target is critical. Leading diffusion can be class blind even when later form/phase feedback is not. A reflection fixing the central interface can also hide winding orientation. The finite proof must preserve these cancellations and may conclude that its selected observation or error budget is insufficient. If the degree-aware reduced port method is used, its internal linearization, actual port degrees and full nonlinear remainder all remain explicit; its earlier frozen error certificate is not a certificate for the new probe.
Finite response from internal mediator feedback¶
Set \(A=KL\), \(\eta=\gamma^2\), and let \(C_k\) be the degree-normalized phase Hessian at the aligned target. Its cycle edges have coefficients \(\cos\alpha\), \(\cos(k\alpha)\), \(\cos\alpha\), while the two contact coefficients are one. Let \(D_M\) be the mediator's internal cycle Laplacian normalized by the actual joined inverse degrees, extended by zero outside that component. Then
These coefficients are derived from the fixed nonlinear law and target geometry, not inserted as fitted interaction parameters.
For the ideal comparison only, start at the aligned target and apply the same donor form jump. This reference is used to bound the reached family; it does not replace the original full-state coordinates of that family. Put \(y=\theta-\Theta\). The tangent evolution in proof coordinates \((x,y/\gamma)\) has matrix
Here \(D=(0,4)\) and \(R=(2,4)\) in subscripts of coordinate vectors; the scalar denominator introduced below is written \(D_h\). The first four tangent form derivatives are
Their receiver differences vanish through order two. Since \(e_R^{\mathsf T}D_M=0\), the order-three difference is \(a\eta d_c e_R^{\mathsf T}AD_MAe_D\). The mediator central entry of \(Ae_D\) is \(-1/4\), its internal-neighbor entries are zero, and the central diagonal of \(D_M\) is \(2/4\). The receiver contacts that support only through its central neighbor, with coefficient \(-1/3\). Therefore
The order-zero through order-two cancellations isolate later mediator feedback from the common initial diffusion response. The same third derivative holds for the ideal nonlinear receiver. Its additional term is \(\gamma D^2(KS)_R(\Theta)[\theta'(0),\theta'(0)]\). Every receiver internal edge has zero initial phase velocity at both ends, while its contact edge has zero base gap and hence zero sine second derivative. That additional receiver term vanishes exactly.
For \(0<h<1/3\), \(\|M_k\|_\infty\le3\) and \(\|M_1-M_2\|_\infty\le2\eta d_c\). Telescoping the matrix powers gives \(\|M_1^n-M_2^n\|_\infty\le2\eta d_c n3^{n-1}\). The absolute tangent contrast tail starting at order four is bounded by
Indeed its first bound is the numerator, and each successive term has ratio at most \(3h/4\). Independent marginal tangent remainders are not substituted for this correlated class-difference estimate.
The full nonlinear remainder retains every fine coordinate. The diffusion semigroup is a maximum-norm contraction; normalized sine is globally Lipschitz with constant two. For the ideal nonlinear probe trajectory,
The first inequality follows by substituting the phase integral into the form Duhamel formula. Every normalized sine Taylor defect is at most \(2\|y\|_\infty^2\), so its form forcing is at most \(8\gamma^3a^2t^2/D_h^2\). Comparing the nonlinear and tangent equations in (10), their whole-window maximum error is bounded by its forcing integral plus \(3h\) times that error, using the proof coordinates \((x,y/\gamma)\). This also bounds the needed form readout. Each class contributes at most \(8\gamma^3a^2h^3/[3D_h^2(1-3h)]\). Both ideal baselines are exactly stationary, giving the pair bound
No invariant nonlinear reflection-even subspace is assumed.
Return to the actual full source family in the original coordinates \((x,\theta)\). Its complete field has maximum-norm Lipschitz constant less than three. Each actual probe or baseline differs from its ideal reference by at most \(\epsilon/(1-3h)\). This comparison does not divide phase uncertainty by \(\gamma\). Four histories and four scalar recording errors of magnitude at most \(\delta\) therefore give
Here the two readings defining each \(\widehat R_k\) retain the same actual preprobe source. Cross-class source errors may be independent.
The conditional sufficient separation criterion is that the right-hand error in (16) be strictly smaller than \(P(h)-c_{\min}\), for a prospectively declared positive threshold \(c_{\min}\). Failure of that sufficient inequality alone does not prove that the responses agree.
A postcontact phase-blind control¶
Declare the comparison law after the same actual source handoff and contact, with the same jump and paired baseline,
Both rows are specified. This removes phase feedback into form while retaining the identical degree-normalized support and clock. Its paired form response is \(a\exp(-Ah)e_D\) for any reached source, so its receiver response is independent of mediator class. The noiseless paired class contrast is exactly zero, including independent source residuals between classes. For the reserved decision use the conservative outer null allowance
which includes the same source/readout allowances as (16). Require the sine-law contrast lower bound to exceed \(B_0\) as well as its fixed positive threshold. This is a declared postcontact phase-feedback ablation. It supplies no alternative account of the preceding formation, no phase-blind identity/recovery certificate, and no exclusion of all other complete laws.
F4: one prospective finite protocol¶
Use the source family (1), all aligned origins, both mediator classes, formation time 100, additional dwell \(10^{13}\), guarded decay exponent 512, and the actual component handoff allowance \(\epsilon=10^{-32}\). Require the original whole-family source storage ceiling \(10^9\) separately on each component, the original per-coordinate source errors \(10^{-10}\), and the radius \(r=1/12\). Fix
Contact work has allowance \(10^{-12}\); probe work has the separate allowance \(2\,10^{-6}\). Require the actual full-family contact work, both probe-work intervals and both probe/baseline identity conditions in (8) to pass. Require the recorded class-one-minus-class-two contrast lower bound in (16) to exceed \(c_{\min}\) strictly. Require it also to exceed the phase-blind bound (18).
These values have an analytic margin before evaluating a response. The elementary bounds \(1/3216<\gamma<1/3069\) and \(d_c>1/2\) hold: for the latter, \(\cos\alpha>3001/3969\) and \(\cos(2\alpha)<59/243\) follow from \(3<\pi<22/7\) and the alternating cosine bounds. At (19), \(B_T/P<13/100\), \(B_N/P<1/1000\), and \(P>3\,10^{-25}\). The remaining source and four-reading allowance is less than \(5\,10^{-30}\), leaving a strict margin above the selected threshold. Equations (7) and (8) hold, with positive probe work below its allowance; the contact bound is below its separate allowance. These analytic checks specify a single design, not a horizon or precision search or a retained assessment outcome.
Before assessment, freeze the producing source and protocol, including primitive scalar admission, outward arithmetic precision and stopping rules. Rebuild every consumed formation/handoff, work, identity and response bound from admitted primitive evidence. A cached verdict is not a premise. Unknown errors give unavailable evidence rather than a favorable contrast. No numerical trajectory is required by (16); any implemented assessment must retain its analytic provenance.
Retain the first evaluated outcome, including an insufficient bound or failure. The old formation, contact and composition evidence stays frozen; only matching primitive premises and proved reusable mechanisms may enter the new calculation. Reserved assessment requires the prior source and protocol freeze; this theorem supplies neither a parameter search nor a physical identification.
Retained first assessment¶
The first reserved analytic assessment completed under the frozen protocol
and passed all nine declared stopping conditions. The
saved report
returns certified_class_mediation, with no unavailable reasons or
evaluation error. This is a complete-family conditional bound calculation;
no trajectory was simulated or measured and no response was fitted.
The exact retained recorded contrast is
whose lower endpoint exceeds the frozen threshold \(10^{-25}\). The following decimal intervals round outward from the saved exact fields:
| Retained quantity | Outward lower bound | Outward upper bound |
|---|---|---|
| Recorded class-one-minus-class-two contrast | \(3.465025154359\,10^{-25}\) | \(4.500763923538\,10^{-25}\) |
| Phase-blind postcontact recorded contrast | \(-4.040012003677\,10^{-30}\) | \(4.040012003677\,10^{-30}\) |
| Separation from the phase-blind upper bound | \(3.464984754239\,10^{-25}\) | \(4.500723523418\,10^{-25}\) |
| Supplied donor probe work | \(1.499999999999\,10^{-6}\) | \(1.500000000001\,10^{-6}\) |
The assessment freshly admitted both formation classes and the actual unprobed handoff. It retained the original \(10^{-10}\) coordinate uncertainty, the structural formation and dwell times, and the paired probe/baseline source association. Contact and probe work remain below \(10^{-12}\) and \(2\,10^{-6}\), respectively. Both baseline and post-probe full-family identity certificates pass, including recovery on each actual conserved-mean leaf. None of these conclusions is inferred merely from the positive contrast.
Changing the acquired mediator class changes the finite donor-induced receiver response under the same interface, complete law and clock. The common early diffusion and the declared phase-blind postcontact law do not account for this class distinction. The result retains different original preparation costs, supplied contact and probe events, a long structural dwell and an extremely small mathematical readout allowance. It establishes neither an autonomous interaction selector nor laboratory resolution, a unique physical law or a mass, charge or spin identification.
The protocol,
source archive
and manifest
retain the first outcome and its reconstruction boundary. The base revision
is ca4350535c9ef4014e96695136e0edecbe38a573, supplemented by the two
listed runtime overlays. The archived prospective proof has 18,460 bytes
and SHA-256
76c4f28f2d8353759744353e074caba905a7fc2fee574f1a5b56cca5fd932d8b;
its content is preserved above. The archive SHA-256 is
98eeb475b07e986b9985888b78b3822904c2b458ee126737635868a2dac5ed1b.
These hashes associate retained bytes; they are not provenance authentication.