Certify a coherent-contact cost ceiling for the retained 640-input interface (no new κ)

research/sol1337-contact-frontier/.gitattributes

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+vendor/dynamic_neighbors.cpp -text
+vendor/neighbors.hpp -text

research/sol1337-contact-frontier/BUDGET.md

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+# Optimistic bill and changed model before discovery
+
+Canonical discovery base: 624de9dc66858af0945e5f8a0942488d62280d43.
+Predecessors: Cinna Davis #4694/#4698, Cao Yifan #4610, Cinna Davis #4690/#4696/#4700, and Dugongue #4841 at 6debdcdac8fe1c87958f8cd42ccb7db73c0a0eef. Substantial OpenAI Codex assistance.
+
+Question: does retaining the eight formerly owned centers as arbitrary independent inputs permit a strict paid-profile improvement by changing their scalar contact charts to literal general Clifford charts?
+
+The #4841/#4690 graph freezes any scalar contact touching an owned center and keeps all eight rank-eight center calls as a fixed charge. The changed graph uses #4694's invertible 640-input coupled interface: copy becomes addition, clear is removed, each center has its own fixed original entrance chart and fixed empty exit chart. It exposes actual center transitions as paid edges and unfreezes center scalar contacts while including every incident old-field transition. All 640 endpoint charts, original main/adapter boundaries and all 408 helper targets remain fixed. No center output receives independent productive Gaussian credit. This changes the frozen-owned-contact restriction and may change Lagrangian subspaces, unlike #4696 fixed-subspace phase changes and #4700 monotone canonical chart changes.
+
+Baseline paid histogram is {1:2201,2:412,3:510,4:127,5:40,8:8}; independent helper targets are {4:28,7:7,8:47,9:326}. The optimistic gain test is Delta(theta)=sum_r(new_paid_r-old_paid_r)*r^theta<0, retaining every helper and every other paid call. Native compiler free-adapter passes, scalar operations, routing, precision, source-uniform compiler and complete consumer funding remain additional positive costs, not zero. No new outer kappa is inferred from this optimistic component test.
+
+The first bounded experiment checks all 1,022 distinct single-transvection neighbors at newly variable center contacts only, with actual mapped incident edges and frozen boundaries. Reject immediately if no profitable complete paid change occurs. No unchanged 632-role graph search, random seed campaign, scalar-only saving or root refinement is authorized by this gate.
+
+A positive component would require literal phase-correct inverse-image charts, chronological all-640 arbitrary-input primal/raw-dual verification, independent helper recovery, source-bound complete caller admission and the fully charged compiler/outer transfer. An unchanged #4694 coupled ABI alone does not supply #4610's owned-center caller contract.
+
+The immutable single screen covers 188,048 candidates and finds 18 exact delta classes. Every class has S_j=sum_r delta_r min(r,j)>=0 for j=1..9 and S_1>0; summation by parts proves strict loss for all 0<theta<1. It does not exclude coordinated changes: simultaneously applying the same transported transvection to two adjacent charts keeps their shared edge rank fixed, whereas the two separate moves can each pay for that edge. The optimistic interaction refund is the sum of those two single-endpoint shared-edge increases. Every other incident edge remains charged.
+
+The new contact region has 264 internal edges, 256 of rank zero and eight of rank eight. The complete paired-neighbor formula therefore gives 1,022 and 196,607 nontrivial pair representatives respectively (the 2,043 case applies to distance one and is absent here). One immutable adjacent-pair sweep is permitted because it tests barrier removal absent from the single screen, still using fixed real boundaries and no new helper reward. It will stop after this finite deterministic domain, with no random restart expansion.

research/sol1337-contact-frontier/LICENSE

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research/sol1337-contact-frontier/NOTICE.md

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+Apache-2.0. New graph binding, complete changed-center screens, cut certificates
+and written paid-moment argument prepared for Chris R with substantial OpenAI
+Codex assistance. Retained exact-byte source files in vendor/ come from Cinna
+Davis #4694/#4690 and Dugongue #4841; original headers and assistance disclosures
+are preserved. The native payload is Cao Yifan #4610's frozen round-three word.
+Retain all original #4470, native Gaussian/Clifford, helper, scalar, precision,
+compiler and outer contributor attribution. No predecessor scientific admission
+or independent human proof review is asserted by this package.

research/sol1337-contact-frontier/PROOF.md

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+# Proof of the scoped paid ceiling
+
+Let `d_r` be a complete candidate paid-histogram change, including every
+incident old-field edge and all center work, with the helper invoice fixed.
+Zero-rank events have no recursive charge. For `1<=j<=9` define
+
+    S_j(d) = sum_{r=1}^9 d_r min(r,j).
+
+For `0<theta<1`, put `q_j=j^theta-(j-1)^theta`. These numbers are positive
+and strictly decreasing. Finite summation by parts gives
+
+    sum_r d_r r^theta
+      = sum_{j=1}^8 S_j(d)(q_j-q_{j+1}) + S_9(d)q_9.
+
+Thus componentwise nonnegative S_j certify a nonnegative paid change for every
+such theta. S_1>=1 makes the change at least `q_1-q_2=2-2^theta>0`.
+This is an elementary identity for this finite invoice, not an assumption
+about asymptotic supplier composition.
+
+## Complete local neighborhoods
+
+The pinned #4841 single-neighbor theorem gives exactly 1,022 distinct changed
+Lagrangians from each nine-dimensional chart. Its paired theorem gives
+`2^(n+r+1)-2^(2r)-1` representatives for two charts at distance r. In this
+new graph the 264 internal edges have distances zero (256 edges) and eight
+(eight edges). The complete pair count is therefore
+
+    256*1022 + 8*196607 = 1834488.
+
+The source graph's every actual incident rank is recomputed by binary row
+elimination after each move. A pair applies the same transported transvection
+to both endpoints, which preserves the shared edge rank, but every other edge
+is also recomputed. Independent axes are a different, unsearched grammar.
+
+The resulting 18 single and 46 pair classes all satisfy
+
+    S_j(d) >= S_j(d*) for every j,
+    d* = {1:+1,7:+1,8:-1}.
+
+The actual classes include d*. Consequently the exact minimum over each
+nontrivial tested family is `1+7^theta-8^theta`. Strict subadditivity of
+`x^theta` proves this is positive. The identity chart/no change is excluded
+from a nontrivial-neighbor family and of course has zero change.
+
+## Arbitrary coherent blocks
+
+Fix one original chart L, its alternative L'=T_a L at distance one, and any
+subset of the variable contacts with original chart L. Retag precisely that
+subset to L'. All other contacts remain fixed. Every internal edge is the
+identity map with original rank zero. Its new rank is zero when the two labels
+agree and one when they differ. This holds also for noncontiguous subsets and
+for edges between contacts of different retained centers. Every external edge
+contributes a unary cost determined by its full actual mapped rank.
+
+For tariff `h_j(r)=min(r,j)`, the complete changed invoice therefore has form
+
+    S_j = sum_v unary_j(v) x_v + sum_{uv internal} |x_u-x_v|.
+
+The coefficients are integers. This is exactly an s-t cut energy. Positive
+unaries use vertex-to-sink edges; negative unaries use source-to-vertex edges
+and add their value to the constant offset; internal edges have unit capacity
+in each direction. No mixed-original-chart or mixed-alternative submodularity
+is assumed.
+
+The only group of size greater than one has 176 original empty-chart contacts.
+The exact replay checks all 1,022 distinct alternatives and all nine tariffs.
+Every flow has nonnegative capacities, satisfies conservation and has a value
+equal to an independently recounted attaining cut capacity. Including the
+negative-unary offset gives zero in every case. Feasible flow and equal cut
+capacity prove optimality; the empty subset directly attains zero.
+
+For every first-prefix problem, all variable vertices can reach the sink in
+the residual graph. Any minimum cut has no positive residual edge leaving its
+source side. A variable vertex placed on that side would therefore force the
+sink onto the same side, a contradiction. The unique minimum is the empty
+subset, so every nonempty subset has integer S_1>=1. Since its other S_j are
+nonnegative, the summation identity proves loss at least `2-2^theta`.
+
+The other eight original-chart groups are singletons, already covered by the
+complete single-neighbor classes. No extra independent center output is
+rewarded in any equation. None of these graph arguments asserts a complete
+native phase, caller, compiler or outer-multiplication theorem.