Independent full-law check of distinct-neighbor nonadditivity

The distinct-neighbor theorem predicts a finite response of the mediator that separately matched nonlinear single-neighbor functionals cannot reproduce by addition. This owner admits one independent complete-law calculation and fixes its source transfer, numerical budget and observation decisions before that response is evaluated. The execution plan owns admission and subsequent evaluation status.

The shared producer evolves all 54 original coordinates. Its inputs contain no analytic coefficient, predicted sign, source-radius policy or observation error. The independent forward interval is never intersected with the prediction. This is a conditional structural test; no laboratory clock, sensor calibration, autonomous event selection or physical identification is supplied by its success.

Complete model, preparation and intervention

Use three unit-conductance C9 components on ordered nodes 0 through 26, with contacts (4,13) and (13,22), unit capacities and actual joined degrees. The central degrees are (3,4,3). All forms precede all continuous phase lifts in the state. The class tuple is fixed to (1,2,1). The complete rows and structural clock remain

\[ x'=-Ax+\gamma D^{-1}S(\theta),\qquad \theta'=\gamma Ax,\qquad A=D^{-1}L,\quad \gamma=\frac1{1023\pi},\quad \tau=\frac{1023}{1024}t. \tag{1} \]

Both rows use this clock, with continuously refreshed pressure and no additional continuous forcing. Forms are signed; phase lifts are not wrapped or recentered during execution.

Retain the original acquired-family premise, including its preparation, support formation, storage costs, common origins and original component zero sums. Each component's form and target-phase Euclidean residual has norm at most \(\epsilon=10^{-32}\). These are conditional handoff bounds, not a newly acquired source or a conclusion from the forward response. Every actual initial residual is the same in all four counterfactual histories. Different independently prepared residuals would require a different protocol and error bound.

With \(m=7/10000\) and \(T=1/8\), the four histories \((00,10,01,11)\) apply the time-zero jump

\[ x(0^+)=x(0^-)+i m e_4+j m e_{22},\qquad \theta(0^+)=\theta(0^-),\qquad Y_{ij}=x_{13,ij}(T). \tag{2} \]

The events commute because they translate distinct form coordinates. Every hidden coordinate carries unchanged through these instantaneous jumps. There is no intervening flow or reset. The prospective statistic and alternative are

\[ M=Y_{00}-Y_{10}-Y_{01}+Y_{11},\qquad Y_{11}^{\rm add}=Y_{10}+Y_{01}-Y_{00}. \tag{3} \]

Each single-neighbor term in the alternative is the same-source full nonlinear response under (1). The alternative's true mixed statistic is zero by its definition. Rejecting this additive functional does not reject pairwise microscopic laws: (1) itself is pairwise.

The theorem supplies separate simultaneous work and whole-state identity certificates at radius \(1/12\), contact allowance \(10^{-12}\), and each single jump allowance \(10^{-6}\). Those certificates keep the actual family and its correlations. A successful smooth-domain numerical enclosure does not certify work or identity for arbitrary corners of an outer box.

Nominal reference and common-source transfer

For component \(c\), local node \(j\), and \((k_0,k_1,k_2)=(1,2,1)\), the exact reference is

\[ x=0,\qquad \Theta_{9c+j}=\frac{2 k_c(j-4)}9\pi. \tag{4} \]

The shared source recipe uses the certified 256-bit Machin pi enclosure. For each signed multiplier \(q=2k_c(j-4)/9\), store the ordered pair \([\min(q\pi_-,q\pi_+),\max(q\pi_-,q\pi_+)]\). Materializing it on the shared outward dyadic128 grid supplies the absolute phase input. Negative multipliers reverse endpoints. This is an enclosure of the nominal reference, not the phase-deviation variable and not a new preparation.

Separately retain actual-family coordinate covers: form pairs \([-\epsilon,\epsilon]\) and phase pairs from (4) expanded by that radius. The original component balls and zero-sum correlations imply containment in these boxes; the converse is not claimed. Only the nominal reference enters the forward calculation. Pi uncertainty and interval rounding already belong to its numerical enclosure and are not charged again as source error.

Define the residual \(z_0=(x(0^-),\theta(0^-)-\Theta)\) relative to the original declared common origins. Residual zero denotes the nominal wound target; it does not denote zero absolute phase. The full nonlinear initialization bound applies before any cubic approximation. For each history \(h\), with \(A_h\in(0,m,m,2m)\), the actual-minus-nominal form difference is

\[ x_h(T;z_0)-x_h(T;0)=[\exp(JT)z_0]_x+e_h, \qquad \|e_h\|_\infty\le E_{\rm init}(A_h,T,\epsilon), \tag{5} \]

where \(J\) is the common full linearization at (4). Put \(g=1/3000\), \(\ell=1-2gT\), \(D_T=1-2g^2T^2\). The bound is

\[ E_{\rm init}(A,T,\epsilon)= \frac{4gT\epsilon}{\ell D_T} \left(\frac{gA}{D_T}+\frac{\epsilon}{\ell}\right). \tag{6} \]

The common linear term in (5) cancels exactly with the four weights in (3). No symmetry of an arbitrary actual residual is assumed. Summing all four remaining defects, including the nonzero zero-input defect, gives

\[ |M(z_0)-M(0)|\le S =\frac{16gT\epsilon}{\ell D_T} \left(\frac{gm}{D_T}+\frac{\epsilon}{\ell}\right) <1.556\,10^{-42}. \tag{7} \]

Thus a nominal forward interval \(F\) gives the actual-family enclosure \(I=F+[-S,S]\). This is the only source transport allowance in the forward comparison. Cubic recoupling, static-cone and fifth-order amplitude errors belong to the analytic prediction; a full-law forward interval receives none of them. The proof of (7) does not consume any predicted endpoint.

Shared execution and fixed numerical policy

The four-history observation owner accepts optional selectors first_probe_node, second_probe_node and readout_node, each an ordinary integer in 0 through 26. The new protocol uses (4,22,13); existing calls retain (4,4,22). Admission precedes field construction. Explicit selector metadata is retained and reconstructed from admitted primitives. Old reports with no selector fields can be read only under the legacy defaults; invalid or partial explicit fields never fall through to those defaults. Existing donor/receiver aliases must agree.

Two shared prefixes and four suffixes use the unchanged complete sine field and source-box Picard/Taylor kernel. At zero delay the prefixes apply their events without a flow step. Every suffix evolves a complete inherited state. For general admitted selectors the signed sum of repeated second-event readout jumps is zero, as is the signed sum of the shared prefix values. Therefore the primary mixed observation may retain only the four suffix flow increments. In this selected protocol neither jump acts at node 13, so even individual readout jumps are zero. Raw endpoint intervals and their separate mixed interval are retained, not substituted after seeing widths.

Quantity	Prospective value or rule
Event delay / final time	0 / \(1/8\)
Fixed step	\(1/128\), clipped only at declared boundaries
Taylor order	16, with certified order-17 remainder
Unique attempt cap	64: four suffixes of 16 steps, two zero-duration prefixes
Arithmetic	Shared exact rational and outward dyadic128 intervals
Flow evidence	Full source box, strict Picard tube, series, remainder and full endpoint
Failure policy	First numerical or budget failure stops all later steps and events
Adaptation	No retry, width-driven extension, order change or source narrowing
Numerical postcondition	Transported actual interval width at most \(10^{-30}\)

The width condition is a required postcondition, not a claim that the unexecuted method attains it. The short fixed horizon and high order reduce truncation, while wrapping, pi materialization and finite arithmetic remain in the forward evidence. All attempted steps consume the global budget, including the first failed attempt. Partial evidence remains inspectable; there is no complete mixed endpoint until all four histories finish.

Independent comparison and observation decisions

Rebuild the analytic nominal band from the static direct cubic heat term, complete cubic correction and amplitude tails. Its actual-family band also includes (7). The static theorem puts the actual response inside the coarse open band \((-3.9\,10^{-29},-1.0983\,10^{-29})\). The comparison uses the freshly rebuilt rational closed bounds, not rounded summaries or stored passing flags.

Preserve completion, both nominal and actual prediction overlaps, numerical width, true sign, recorded sign and independent-additive separation as distinct decisions. Closed bands overlap when their intersection is nonempty, including shared endpoints. This comparison differs from testing membership in the displayed coarse open band. The forward band itself remains unchanged.

For \(I=[L,U]\), each of four readings has a supplied independent error allowance \(\delta=10^{-30}\). The full-law recorded statistic lies in \([L-4\delta,U+4\delta]\), while a separately recorded additive alternative lies in \([-4\delta,4\delta]\). Consequently

\[ U<0,\qquad U+4\delta<0,\qquad U+8\delta<0 \tag{8} \]

are respectively the negative true-sign, recorded-sign and independent separation criteria. The third requires a strictly positive margin \(-U-8\delta\); equality does not pass. This allowance is inherited from the earlier nonlinear studies, not from the later \(10^{-8}\) memory comparison. It is not a demonstrated laboratory capability.

The assessment owner reconstructs support, law, source, selectors, ancestry, complete state carry and each retained Taylor arithmetic step before consuming the mixed interval. Derivative and Picard generation remain execution premises: arithmetic reconstruction and hashes do not authenticate their construction. Its status distinguishes missing completion, disjoint prediction bands, excessive width, unresolved separation and all conditions met. Missing observations retain unavailable predicates. A wide band is not a scientific counterexample; disjoint rigorous bands require an audit of the theorem, source and numerical premises before a scientific interpretation.

Immutable protocol and subsequent evaluation boundary

The source base, complete runtime environment, exact primitive inputs, source covers, rebuilt static prediction, observation policy, scripts and model owner are pinned under class-neighbor-forward-v1. Only .protocol.json, .source.zip and .freeze.json belong to this admission. The common source inspector has an explicit schema adapter; the archive is associated with a full Git base without runtime overlays. The original acquired family remains a conditional proof premise, so no earlier observed response is required as a numerical input.

Freezing evaluates no selected time coefficient or trajectory and creates no attempt or outcome. The historical receipt field evaluation_status_at_freeze="not_evaluated" retains that meaning after a later evaluation. Byte association does not authenticate chronology, acquisition or mathematical correctness.

The separately admitted evaluation makes at most one attempt. Its preflight checks frozen inputs, script bytes, runtime source and environment before creating the attempt marker. It retains completion or failure, all available evidence and its independently rebuilt decisions. Exceptions and export failure preserve the attempt and prevent an automatic retry. Neither an unresolved result nor a consistency conflict permits silently revising the policy. Any correction requires separately identified evidence.

Retained prospective freeze

The completed admission is pinned to complete source commit 8af5b1932308b81e5fbb4e519de0162b8e82ccfc, with no runtime overlays. The protocol, source archive and receipt retain the policy above and 30 archived files plus their manifest. The protocol has 152589 bytes and SHA-256 bcdc0160bc91e628bb1259139f5451772d2af61298f88e068d55232fef498179; the archive has 149640 bytes and SHA-256 2317c3a54dd83704d99ad1bcf9ff976d6f25e5cd7cc2754fa5e38726769733ec. The receipt SHA-256 is c2dec8230791e8b5377283cd088b1414c105f010d3ada82a4e5e075d8ddc0cdf.

The supplemental evaluator is build/class-neighbor-forward-freeze/evaluate_neighbor.py. The shared restorer can recover its full pinned runtime and archived supplements. No prior observed response is consumed; the original source remains the stated conditional acquisition premise. At freezing there was no attempt, response or export-error artifact. The read-only freeze audit checks this association and the sealed prospective text without importing the archived worker or regenerating scientific evidence. The synthetic evaluator controls separately check preflight, attempt and export behavior with unrelated records.

First reserved full-law result

The exclusive attempt and retained response record the one execution in a restored workspace at the pinned source base. The complete runtime, environment, evaluator bytes and primitive inputs matched the frozen declaration. No source, observer, horizon, step, order or budget was changed. Both zero-duration prefixes and all four full-state suffixes completed: 64 planned, attempted and validated steps, six completed segments and no failed or unattempted branch. The recorded calculation and assessment took approximately 125.504 seconds of wall time; the structural horizon remains \(1/8\). No retry or export-error artifact was produced.

The read-only response audit admits exact artifact and member bytes before decoding. It reconstructs every 54-coordinate event carry, retained Taylor increment and endpoint before rebuilding the observation. Original derivative/Picard generation remains an execution premise; the audit does not rerun the flow or authenticate acquisition. Independent source and recording arithmetic agree with the saved assessment without trusting its cached verdicts.

The exact primary nominal mixed interval reconstructed from the four suffix increments is

\[ F=\left[-\frac{1456691539}{2^{126}}, -\frac{2913382991}{2^{127}}\right],\qquad \operatorname{width}(F)=\frac{87}{2^{127}}. \tag{9} \]

Apply only the common-source allowance (7). Conservative outward summaries of the exact rational decisions are:

Quantity	Retained bound
Actual-family mixed response \(I=F+[-S,S]\)	\([-1.71233267499,-1.71233262385]\,10^{-29}\)
Nonlinear source allowance \(S\)	\(<1.55568520680\,10^{-42}\)
Actual-family interval width	\(<5.11343153979\,10^{-37}\), below the frozen \(10^{-30}\) ceiling
Four-reading recorded interval	\([-2.11233267499,-1.31233262385]\,10^{-29}\)
Separate additive recorded interval	\([-4,4]\,10^{-30}\)
Strict independent-additive separation margin	\(>9.12332623850\,10^{-30}\)

Both nominal and actual closed prediction overlaps pass. The independently computed forward intervals are retained in full, without intersection with either prediction. All fixed conditions pass: completion, numerical width, prediction consistency and strict separation after the two independent four-reading error budgets. True and recorded negativity remain separately reported predicates, not substitutes for the eight-error comparison.

The response ZIP contains the exact original JSON outcome as its sole member, without reserialization. Its associations are:

Retained item	Bytes	SHA-256
Attempt	762	3ed3763c37a7ddc83f075eb1d2ee5cd369d5b88991091f441e6598e96b7bc537
Response ZIP	2,016,937	c62c4edcdf8765028f4af9c9c41b7ff35bc02a9d89bdddfed8c14b34501b6c68
Outcome member class-neighbor-forward-v1.json	29,737,680	e308c8c9432d94372e515f68673471b1e056258ba28f829b79eb9a9048abea97

This finite result supports the prospective distinct-neighbor prediction under the supplied complete law and original source premises. Separately matched nonlinear single-neighbor responses do not reproduce the joint response by addition. It does not exclude pairwise microscopic dynamics, establish a difference between classes, determine an energetic interaction law or identify a physical force. The laboratory observation bridge remains open. The historical not_evaluated freeze field continues to describe the earlier admission stage; this section owns the subsequent result.