Finite collective coupling across two registered interfaces¶
Prospective interface and decision declaration¶
This admission tests a purely multiplicative real source/probe factorization
of a finite response. It uses the actual acquired and observed sources of the
joint-resource theorem, whose resource
declaration was frozen at 0ca8630d. The second interface below changes
the subsequent complete support. Its response is not supplied by the
same-slot certificate, an onset coefficient or a winding label.
| Item | Fixed choice and scope |
|---|---|
| Sources | The paired acquired outer-zero and outer-one host families after outer formation and the entire sequential observation word, and one independent recovered C9 donor. Retain all original source errors and actual reached states |
| Resource box | The admitted box from 0ca8630d: twelve primitive component allowances in [0,1e-6], quarter formation 200, quarter recovery 8e10, handoff Euclidean allowance q=1e-8, decay power 80, and sequential raw errors at most 1e-14 |
| Relative preparation | Same actually prepared donor member in all four counterfactual contacts, zero common donor form and donor phase origin beta=pi/2; both actual host means remain those of the carried history |
| Node order | Host node 9*c+j, c=0,...,4, j=0,...,8; donor node 45+j. All original 99 internal and host-outer edges remain unchanged |
| Same-slot interface | For every c and j, add the unit edge (9*c+j,45+j) |
| Registered interface | For every c and j, add the unit edge (9*c+j,45+((j+k_c) mod 9)), with signed representative k_c=2*(c-2), hence offsets (-4,-2,0,2,4) or residues (5,7,0,2,4) |
| Independent normalization | Each interface has 45 unit contacts, total contact conductance 45, host degree 5, donor degree 7, the same donor/readout normalization and the same clock. Neither interface depends on unknown source errors or readings |
| Complete law | Each actual graph uses x_tau=(-L*x+gamma*S(theta))/4, theta_tau=gamma*L*x/4, gamma=1/(1023*pi), held unit capacities, smooth full-circle sine and no continuous input or later event |
| Clock and observation | tau=(1023/1024)*t in both rows; donor mean signed-form increment from 0 to h=1e-3. Rows of the response matrix are outer zero, outer one; columns are same-slot, registered |
| Reading model | Two raw readings per matrix entry, hence eight raw errors altogether, each arbitrary and bounded by eta=1e-9; retain correlations and the initial-reading error even though the true initial donor mean is zero |
| Source quantifiers | Choose one actual paired host preparation and one actual donor member, then compare its four counterfactual contacts. A row uses the same precontact host member in both columns. No physical copying or repeatable exact preparation is asserted |
| Hypothesis | R(s,p)=a(s)*b(p) for one real source factor and one real probe-only factor, with the donor, horizon, readout and clock fixed. Source factors may depend on the full actual host state. The hypothesis includes no additive baseline or arbitrary nonlinear dependence on a scalar |
| Decisions | Obtain outward true-entry intervals T, observed-entry intervals T+[-2*eta,2*eta], and uniform error-correction envelopes T+[-4*eta,4*eta]. Require the true and error-correction determinant intervals to exclude zero; threshold exactly zero |
| Work and identity | Re-admit new contact work below 100, joined present storage below 200 for each source/interface, and all six internal and nine old outer windings throughout the contact. Preserve all earlier preparation and observation payments separately |
| Arithmetic and budget | One assessment at the maximum resource-box corner, exact rational primitives and existing outward dyadic128 arithmetic. No registration or origin scan, trajectory, frozen producer, precision change or replacement candidate |
| Incomplete result | An unresolved determinant or missing source/work/identity premise leaves this test unavailable. It does not prove factorization or authorize more interface candidates |
Commit this declaration and its consuming source before the single assessment. The registration is fixed by the marked child index. Adding an integer turn to a slot representative changes no circular field, while the actual stored lift and its conserved mean are not reset. Earlier protocols and evaluated responses remain unchanged.
1. Complete support, normalization and state¶
Let \(p=0,1\) index the same-slot and registered interfaces, and define
For each c this is a permutation of all nine donor slots. Thus every host node gains exactly one contact and every donor node gains exactly five. The complete graph is undirected, connected and has 54 nodes and 144 unit edges, with degrees five and seven. No contact is repeated. The internal C9 supports and the original outer C5 slot cycles are identical between the two contacts; the mixed contact cycles can differ.
All signed forms and primitive phases remain in the state. With \(L_p\) the actual graph Laplacian and \(S_p\) its complete sine current, the rows and storage are
Both global form and lifted phase sums are conserved. At contact they are zero and \(9\beta\), respectively. The uniform quarter factor is unchanged in both rows; it is not inverse degree. The two contact graphs define different subsequent full vector fields. Equal degrees or an abstract graph relabeling do not establish equality at the same retained source and marked readout. The isolated origin symmetry does not erase relative registrations after attachment.
For an edge from host \((c,j)\) to donor \(\sigma_{p,c}(j)\), write
The host receives \((f,g)\) and the contacted donor node receives \((-f,-g)\). These are both boundary channels, including backreaction. Their edge storage is \(u^2/2+1-\cos\delta\); its derivative plus both regional derivatives gives (2). No auxiliary scalar forcing or one-sided donor waveform replaces these reciprocal rows.
2. Permutation matching preserves the exact mean equation¶
Set \(R_D=(1/9)\sum_jx_{D,j}\), and
Every internal current cancels in the donor mean. Bijection of each \(\sigma_{p,c}\) gives \(\sum_{c,j}x_{D,\sigma_{p,c}(j)}=45R_D\). The host form total is \(-9R_D\), from the conserved zero whole form. Summing (3) therefore gives, for arbitrary admitted fine states,
Here \(\Psi_D\) is the donor's actual lifted phase mean. The matching permutations preserve the exact form cancellation; an arbitrary contact weight profile would not follow from this argument. The nonlinear moment \(\mathcal A_p\) still consumes the full boundary phase state.
Let \(X=1/100\), \(T_o=1/5000\), \(q=10^{-8}\), \(\gamma_+\ge\gamma\), and
The carried host has \(\|x_H(0)\|_\infty\le B_o\), and the donor has \(\|x_D(0)\|_\infty\le q\). The graph heat semigroup contracts the maximum norm and \(\|S_p\|_\infty\le7\), so \(\|x(t)\|_\infty\le B\) up to h for either interface. The phase row bounds host and donor rates by \(5\gamma_+B/2\) and \(7\gamma_+B/2\). Hence
Since \(|\mathcal A_p|\le5\), the nonnegative damping kernel in (5) gives \(|R_D(t)|\le5\gamma_+t/4\). Integrating the complete mean equation, including damping and phase motion, yields
Thus the previous form of the finite remainder is justified on each new graph by its own permutation, conservation and degree premises. It is not transferred merely because the contact counts agree.
3. Actual source correlations and finite matrix entries¶
The joint-resource proof rebuilds all six original sources, their uninterrupted recovery, actual outer acquisition and the complete sequential word. Let \(Z\) be its full host phase error about the acquired carrier and let \(P_{w,c}\) be the actual carrier phase enclosure at outer time 200. For the outer-zero control \(P_{0,c}=\{0\}\); for outer one these are the finite reached intervals, not its limiting twist. The word adds coordinate drift at most \(m_o=2\gamma_+B_oT_o\). Set \(e=Z+m_o+q\).
Write \(\psi_j=\alpha(j-4)\), \(\alpha=2\pi/9\). The actual host offsets belong to \(P_{w,c}+[-Z-m_o,Z+m_o]\), while the donor has offsets \(\beta+z_{D,j}\), \(|z_{D,j}|\le q\). For an integer \(n_{p,c,j}\),
The turn term drops out of sine and cosine exactly. No actual lifted coordinate or phase total is changed by using the smaller signed k in the trigonometric argument. At the supplied \(\beta=\pi/2\),
This is a source-family enclosure of the actual slot correlations. It does not assume that different nodes' errors are independent, that interval corners are attainable, or that child means close the fine law. Using the same source member for two contacts introduces correlations between columns; enclosing them independently only enlarges the possible response set.
Let \([\gamma]\) be its shared outward interval. The true finite response matrix \(R\), with rows \(w=0,1\) and columns \(p=0,1\), satisfies
All four entries describe different complete continuations of retained precontact states. The same actual donor member is fixed in all four. The paired host members retain their shared primitive preparation errors; their distinct reached states are never identified with one another.
4. Work, present storage and fifteen finite identities¶
The support event leaves all coordinates unchanged. Each of its 45 new edges has form difference at most \(B_o+q\) and phase cost at most two, so its actual supplied work obeys
The bound applies to each permutation without asserting equal actual works. With \(V_n=n[1-\cos(2\pi/n)]\) and the freshly reconstructed outer-one endpoint excess U, the two present-storage upper bounds are
These retain continuous loss and the observation word's cumulative absolute work bound. Six original preparation costs, earlier outer attachment and all observation payments remain separate historical premises. The four counterfactual accounts are not one autonomously executed four-contact path.
Set \(D_o=2m_o\), \(D_j=7\gamma_+Bh\), \(\zeta=10^{-6}\), and \(A_o=7/5\). Require \(W_+<100\), both bounds (12) below 200, and the five strict margins
The old internal and outer edges are unchanged, and their phase rates have the same degree bounds. Thus (13) preserves all six internal winding-one cycles and all nine old outer windings throughout either contact. No new mixed contact loop has an assigned winding. Neither old product capture nor same-slot control maintenance is asserted on the registered support beyond h.
5. Scalar factorization, noise and admissible correction¶
Fix one actual paired host preparation and one donor member. If real functions \(a(s)\) and \(b(p)\) represent the four true responses by \(R(s,p)=a(s)b(p)\), then
The source factor may depend on every coordinate of the actual host member; the probe factor may consume the fixed donor and protocol, but cannot depend on which host is presented. Excluding zero from an outward determinant interval constructed from (10) contradicts (14). This tests multiplication of two real factors; a pure complex product also has zero determinant and is excluded by the same argument. It does not exclude an affine response, arbitrary nonlinear dependence on one scalar, multiple channels or memory-dependent descriptions. In particular, a real-part observation of a complex product need not have rank one: \(\operatorname{Re}(ab)=\operatorname{Re}(a)\operatorname{Re}(b) -\operatorname{Im}(a)\operatorname{Im}(b)\) is a sum of two real products.
Let the observed increment matrix be \(Y_{w,p}=R_{w,p}+n_{w,p}\). Each increment uses two raw errors, so \(|n_{w,p}|\le2\eta\), with \(\eta=10^{-9}\). All eight raw errors can be correlated. Define
Then \(Y\in O\). For any particular allowed observed record, every candidate corrected true matrix in its componentwise error box satisfies
The extra enlargement in K accounts for possible corrections of a noisy record; it introduces no additional readings or actual physical errors. For a matrix of intervals J define its outward determinant interval by \(\mathcal D(J)=J_{0,0}J_{1,1}-J_{0,1}J_{1,0}\). The prospective strong decision requires
The second condition makes the first redundant mathematically, but both evidence levels are reported. It ensures that every allowed observed record has no rank-one correction under the declared error model. A nonzero determinant of one noisy point matrix, by itself, would not reject a true rank-one response. Likewise, a determinant interval containing zero proves neither factorization nor existence of a jointly realizable correction.
These uniform statements concern the same-source counterfactuals specified above. Four independently prepared laboratory records need a separate preparation/matching admission: different states cannot silently share the same source factor. The theorem supplies neither physical copies nor record authentication. Missing law, source, work or identity premises keep the combined result unavailable even if an algebraic determinant is separated.
6. Assessment and scope boundary¶
Re-admit the maximum corner of the joint resource box before constructing the four fresh source enclosures. Its source radii and cost bounds are nondecreasing in primitive allowances. The fixed q bounds every admitted recovered member; the supplied observation word does not depend on its reading errors. All downstream intervals therefore enclose the smaller allowance families as well, while (15) grows with contact reading error. No assessment of smaller corners or alternative registrations is needed.
The detached proof consumer belongs in
_organized_interface_factorization.py.
It must rebuild primitive source and complete-support evidence, retain the
true/observed/correction distinction, and preserve explicit unavailable
conditions. Shared arithmetic and field kernels do not establish source
identity, an occurrence law or an independent physical measurement bridge.
A separated determinant would identify a concrete restriction on carrying one real multiplicative coupling across these normalized probes. It would not identify electric charge, disprove intrinsic properties generally or establish that every physical interaction must satisfy (14). An unresolved test would remain a finite admission result. Neither outcome selects a unique TNFR law, supplies indefinite outer-one maintenance or initiates a new hierarchy of closure moments.
7. Retained finite factorization obstruction¶
The declaration and complete consumer were committed at 9548b8e1 before
the single assessment. The attempt
retains the committed source inventory, declaration and arithmetic. The
first result
is finite_scalar_factorization_excluded. Both complete supports and
all acquisition, observation, work, storage and finite-identity prerequisites
pass. The true and error-rectification determinant intervals both exclude
zero. No alternative registration, trajectory or second candidate was run.
The following outward decimal summaries enclose the recorded exact-rational true increments; rows share actual precontact source members, not merely winding labels:
| Acquired host | Same-slot contact | Registered contact |
|---|---|---|
| Outer zero | [-3.893e-7, -3.886e-7] |
[4.109e-8, 4.169e-8] |
| Outer one | [5.130e-9, 5.730e-9] |
[-3.834e-7, -3.827e-7] |
The larger response belongs to different organizations in the two columns. This change cannot be explained by multiplying both source responses by one common probe gain. More precisely, the retained outward bounds satisfy
These have squared form units. The second bound includes both the eight actual raw-reading errors and every correction consistent with their error model, as defined in (15)--(16). It does not substitute the nonzero rank of a noisy point matrix for a true-response argument.
The work and present-storage ceilings remain below 90.003 and 169.740,
respectively, for every contact (the outer-zero storage bound is smaller).
All six internal and nine old outer windings survive through h=1e-3.
Actual contact work need not agree between interfaces. Each counterfactual
history retains its earlier preparation and observation payments; summing
them does not define an executed four-contact experiment.
Thus the single purely multiplicative source/probe factor is excluded uniformly over the declared full preparation and noise box. The claim is stronger than different winding labels having different onsets: the source factor was allowed to depend on the complete actual host state, and the decision concerns finite continuations with retained uncertainty and work. It does not exclude an affine response, a nonlinear scalar encoding, real observations of complex products, or a richer retained boundary description. It does not derive physical charge, spin, a quantum state or a unique law. The execution plan alone selects further investigation.