Prove a two-region moment obstruction after carried-frame swaps (no new κ)

research/swap-contact-fusion-gate/.gitattributes

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+inputs/* -text

research/swap-contact-fusion-gate/PROOF.md

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+# A two-region retiming gate after carried-frame field swaps
+
+This is a source-scoped abstract transport result, not a physical supplier
+or a universal frame-optimization theorem. It uses the unchanged source at
+PR4971 `0130b196145a43a7d305de41f49e56e8e8556378`, including the actual
+forward and backward scalar words. Substantial OpenAI Codex assistance;
+Apache-2.0. The original swap construction and endpoint proof are Dugongue's.
+
+## The declared replacement class
+
+Keep the scalar contact order, weighted field exchanges, signed native address
+operators, source/target owners, start/end charts and all eight center queries.
+Keep each center's copy/query/clear interface at its original charts. A field
+exchange carries its chart. Between consecutive scalar contacts on each field
+strand, fold all intervening signed native address maps into their actual
+invertible binary map M. The strand contributes the containment constraint
+
+    M F_u ⊆ F_v
+
+and one price `(dim F_v - dim F_u)^t`, where `0<t<1`, whenever that
+difference is positive. Zero differences cost zero. Here t=1-C in the
+recursive moment. Grant every containing binary subspace, without imposing
+additional native phase restrictions; this enlarges the candidate class.
+Fixed scalar/tape/owner work is unchanged and has no negative refund.
+
+The source graph's nodes are scalar ADD contacts and fixed start, final and
+owned-center boundaries. A contact has one chart shared by both participating
+fields. A SCALE is chart-independent, so it needs no new node. Each current
+strand records its most recent contact and intervening binary map. An ADD
+consumes two strands and creates two outgoing strands at one node. A center
+copy creates a fixed node for its source and owned cell, retaining the source
+strand; center queries and clear operations fix the owned-cell chart.
+
+All physical warp events are first checked against their literal signed
+G-columns. Arbitrarily changing an event's frame labels is not an allowed
+operation. The zero-to-full mutation from the #4571/#4979 lineage is rejected.
+This binary consistency check does not re-prove absolute Clifford phases.
+
+A **region** is a connected component of graph edges that have identity
+binary map and zero rank in the SOURCE schedule. Every node in one such
+region has the same source chart. Its internal identity edges must remain
+zero in the replacement class: one common chart is assigned to all nodes
+of the region. A region containing any fixed boundary is fixed. At most two
+other regions may change chart. All remaining region charts stay at their
+source values. Thus splitting a region, opening one of its zero edges or
+changing three regions is outside this theorem.
+
+## Exhausting the chart choices
+
+For one changed region, with the outside charts fixed, let
+
+    L = sum of M F_u over incoming boundary edges,
+    U = intersection of M^-1 F_v over outgoing boundary edges.
+
+Every allowed common chart F satisfies L⊆F⊆U. Internal nonidentity edges
+also require M F=F; these are checked separately. Each source orientation
+has 768 free regions: 642 have dim U-dim L=0 and 126 have difference one.
+There are therefore only one or two subspaces to consider per region.
+The checker enumerates both and checks the complete affected-edge histogram.
+
+For two adjacent free regions, remove BOTH from the outside chart constraints.
+Construct separate outer intervals [L_a,U_a], [L_b,U_b]. Enumerate all subspaces
+in each interval and check every internal and connecting edge literally.
+To enumerate an interval, start with L and repeatedly adjoin every vector of
+U outside the current span, retaining each resulting subspace once. Every
+intermediate subspace of U/L has a basis, so this process reaches every one;
+using only a chosen coordinate basis of U would be insufficient. The replay
+checks a two-dimensional quotient fixture containing its noncoordinate line.
+The source intervals all fit the enforced four-dimensional quotient bound.
+
+For each of 1,474 adjacent free-region pairs per orientation, this gives
+59,231 candidate assignments in total and 2,090 feasible assignments. Count
+each affected strand edge once, including edges between the two regions.
+The inequality certificates below settle every feasible assignment. The
+specific forced-common-chart fusion is impossible for all 1,438 pairs: the
+joint L is not contained in the joint U. Each saved witness supplies an
+explicit vector in L outside U.
+
+If two free regions are not adjacent, their affected-edge sets are disjoint,
+and their feasibility constraints and price differences separate. The
+one-region result therefore settles those pairs without enumerating them.
+
+## Exact prices for every 0<t<1
+
+For each feasible assignment, form sorted old and new lists of affected
+nonnegative integer edge ranks, retaining zeros. First test that the old list
+majorizes the new list, with equal total rank. Concavity of x^t on [0,∞)
+then gives
+
+    sum old_rank^t ≤ sum new_rank^t.
+
+The remaining differences, NEW minus OLD price, have exactly three forms:
+
+    A(t) = -1 + 3*2^t - 3*3^t + 4^t,
+    B(t) = 12 - 8*2^t - 4*3^t + 4*4^t,
+    D(t) = 12 - 9*2^t + 3^t - 3*4^t + 3*5^t.
+
+They occur respectively 24, 6, 1 times in the one-region enumeration and
+136, 58, 10 times in the adjacent-pair enumeration, in each orientation.
+Every other feasible assignment has a majorization certificate. No floating
+moment evaluation or sampled exponent grid is used in the certificate.
+
+A is the third forward difference of x^t at x=1. Its integral representation
+is the integral of `t(t-1)(t-2)*x^(t-3)` over a unit three-cube, so A>0
+for 0<t<1.
+
+Put `b(t)=B(t)/4=3-2*2^t-3^t+4^t`, l=log 2, and p=log_2 3.
+Since `3^5<2^8`, p<8/5. For 0≤t≤1,
+
+    b'''(t)/l^3 = 8*4^t-p^3*3^t-2*2^t
+                 ≥ (8-p^3-2)*4^t > 0.
+
+Thus b' is convex and lies below the chord joining its endpoint values.
+Both are negative: b'(0)=-log 3 and b'(1)=log(16/27). Therefore b decreases
+to b(1)=0, and B(t)≥0 on [0,1], strictly positive for t<1.
+
+For D, put q=log_2 5. The integer inequality `2^23<5^10` gives q>23/10.
+Ignoring a positive term and enlarging the negative terms gives
+
+    D'''(t)/l^3 = 3*q^3*5^t + p^3*3^t - 24*4^t - 9*2^t
+                ≥ (3*q^3-33)*5^t > 0.
+
+The last sign follows from 23^3>11*10^3. Again D' is convex. Its endpoint
+values are negative because
+
+    D'(0) = log(375/32768) < 0,
+    D'(1) = log(5^15*27/2^42) < 0.
+
+Consequently D decreases to D(1)=0 and is nonnegative on [0,1]. All numerical
+premises here are exact integer inequalities checked by the replay.
+
+This proves that every declared one- or two-region replacement has local
+recursive cost at least the source cost for every 0<t<1. Keeping all other
+supplier terms fixed cannot turn it into a higher-saving invoice. No new
+productive target, supplier admission, selected frontier or κ follows.
+
+## What would change the question
+
+The result permits different charts but keeps each source region intact and
+changes at most two of them. It leaves larger simultaneous changes, splitting
+a region with new paid transitions, frame descents, changed scalar dependencies,
+earlier output access, changed center contracts, new native operators and new
+owner resources open. Even a positive abstract graph improvement in a larger
+class would still need exact phase-aware physical lowering, actual all-input
+return, the full finite/tape bill, completion/compiler, BIT and outer transfer.

research/swap-contact-fusion-gate/README.md

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+# No one- or two-region moment gain after PR4971's carried-frame swaps
+
+**Exact scoped obstruction with executable witnesses; no new κ.** On the
+changed scalar representation of [PR4971](https://github.com/CrocSwap/integer-mult-bounds/pull/4971)
+at `0130b196145a43a7d305de41f49e56e8e8556378`, recharting at most two
+source-defined constant-frame regions cannot lower the recursive moment for
+any `0<t<1`, where t=1-C. This is stronger than a failed floating search and
+narrower than a global optimum. The selected conditional κ stays 0.00302015.
+
+The source replaces 534 three-ADD swaps per orientation with weighted field
+exchanges carrying their charts, including 43 swaps with unequal incoming
+charts. That opens a changed representation for rank-shape optimization. We
+keep this source scalar order, signed address maps, center queries and boundary
+contracts, then test the complete local cost of changing chart regions.
+
+| Exact check | Forward | Backward |
+|---|---:|---:|
+| Contact graph nodes / strand edges | 15,959 / 29,966 | 15,959 / 29,974 |
+| Source constant-frame regions | 1,045 | 1,037 |
+| Movable regions | 768 | 768 |
+| Adjacent movable-region pairs | 1,474 | 1,474 |
+| Candidate two-region assignments | 59,231 | 59,231 |
+| Feasible assignments, all non-improving | 2,090 | 2,090 |
+| Adjacent identity-edge pairs admitting constant-chart fusion | 0 / 1,438 | 0 / 1,438 |
+
+All 768 one-region intervals have quotient dimension at most one: 642 are
+rigid and 126 have two chart choices. The two-region enumerator includes
+every subspace in each actual outer interval, including noncoordinate lines,
+then enforces every connecting and internal operator constraint. Nonadjacent
+two-region changes separate into the already checked single-region problems.
+Sorted integer rank majorization and three explicitly proved power inequalities
+certify every feasible price comparison at every exponent in the stated range.
+[PROOF.md](PROOF.md) defines the replacement class and proves the implications.
+
+Both orientations keep the complete recursive profile
+`{1:2201,2:412,3:510,4:127,5:40,8:8}`, rank mass 5,327. The unchanged
+predecessor replay checks all 632 arbitrary input rows and all 408 helpers;
+the new graph code recounts exactly that paid profile and checks signed
+operator-to-chart consistency before pricing it. This is not a new physical
+Clifford or all-column audit.
+
+## Reproduce
+
+Python 3.12, standard library, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/swap-contact-fusion-gate
+python3.12 -I -O -B research/swap-contact-fusion-gate/verify.py
+```
+
+The registered hook runs both normal and optimized variants. Default verification
+is read-only and compares the complete fresh result and every witness against
+`RESULT.json` and `WITNESSES.json.gz`. Compressed-container bytes are not a
+mathematical equality test: the canonical decompressed witness bytes are compared.
+`--write` explicitly regenerates only these two new artifacts. The original
+retained verifier never receives the new command's arguments and cannot enter
+its own authoring mode.
+
+Checks reject a false zero-to-full native warp, missing unequal-chart carry,
+broken graph containment, an omitted paid rank and a profitable fusion being
+misclassified as a negative result. A complete quotient fixture includes the
+noncoordinate line. The original swap sign/frame mutations also replay.
+All checks use explicit exceptions and run under `-O`.
+
+## Sources, scope and limits
+
+Scientific base is canonical main `b91956876e5b4122fd89179f1b57557ff8378f15`,
+priority revision 7. The rank-shape card was proposed with seed
+`b139e45c9b653201499030c31dc4fe8f`. This changes its historical decoder premise
+to the concrete carried-swap representation, with that change recorded in
+[TASK.json](TASK.json). [SOURCES.json](SOURCES.json) pins the selected context,
+card and exact retained inputs. `inputs/ORIGINS.json` maps every unchanged
+predecessor file to the public PR4971 head and SHA-256; these bytes are committed
+locally before the new analysis so replay needs no side-branch Git history or
+network. The main-tree #4948 source word is the same compressed payload, retained
+as provenance. This checkout is shallow; full-history ancestry audit is not claimed.
+
+A region is an identity-map zero-edge component in the SOURCE graph, and its
+internal identity edges must remain zero. The theorem changes charts on at most
+two intact free regions. It keeps the owned-center copy/query/clear boundaries,
+scalar dependencies, field exchanges and start/end charts fixed. It does not
+exclude a larger simultaneous change, splitting regions by new paid transitions,
+nonmonotone transports, changed centers, changed native operators or a new
+scalar word. No target credit, complete supplier, fixed-tape implementation,
+completion/compiler extension, BIT/outer transfer, Lean proof, human review or
+higher κ is supplied. The inherited physical phase premises remain explicit.
+
+Initial exploratory scripts also found the smallest-frame chronology fragments
+the paid calls and loses moment. Those temporary scripts and floating comparisons
+are not certificate dependencies or claims of optimality. The committed result
+uses exact enumeration and written all-exponent inequalities only.
+
+## Open next questions
+
+Try one fully specified interaction among at least three intact free regions,
+or change the scalar dependency graph or split a region with a charged new
+transition. First check exact containment and the full affected-edge histogram;
+then lower a positive candidate physically with absolute phases, arbitrary-input
+returns and every scalar/movement/precision cost. A new graph optimum would
+still need complete supplier and outer accounting. Structured first steps and
+blockers are in TASK.json. The present result gives no reason to rerun broader
+one/two-region searches on the same source.
+
+## Credit
+
+Prepared by Cinna Davis / 1337isnot1337 with substantial OpenAI Codex assistance;
+solo, no subagents, no independent human review. Apache-2.0. Credit Dugongue for
+PR4971's changed word, construction and replay, and PR4935/4948 source binding;
+Cao Yifan / intqwq for the round-three source beyond PR4610; the Cinna Davis
+scalar/helper and PR4440/4470 lineage; zxv, Jacob Alan Sussman and all authors in
+the unchanged retained NOTICE. Credit James Reed / jwr404 for #4571's
+operator/frame gap, #4979 for its exact event guard, and James Reed's independent
+PR4971 review. Existing fixed-eight-array and monotone-frame obstructions are
+distinct precedents, not global barriers applied to this changed graph.

research/swap-contact-fusion-gate/RESULT.json

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+{
+  "scope": "Exact at-most-two-region moment and adjacent identity-edge fusion gate; inherited physical maps, no new kappa.",
+  "source_payload_sha256": "cfcd36b40326d7a5815e1db5313444379b3fb756b27d5201c561574c085cc0dd",
+  "orientations": {
+    "forward": {
+      "nodes": 15959,
+      "edges": 29966,
+      "regions": 1045,
+      "free_regions": 768,
+      "paid": {
+        "1": 2201,
+        "2": 412,
+        "3": 510,
+        "4": 127,
+        "5": 40,
+        "8": 8
+      },
+      "single_region_counts": {
+        "five_four_three_two": 1,
+        "four_three_two": 6,
+        "gap_0": 642,
+        "gap_1": 126,
+        "majorization": 863,
+        "third_difference": 24
+      },
+      "adjacent_region_pairs": 1474,
+      "two_region_counts": {
+        "candidate_assignments": 59231,
+        "feasible_assignments": 2090,
+        "five_four_three_two": 10,
+        "four_three_two": 58,
+        "majorization": 1886,
+        "third_difference": 136
+      },
+      "swaps": 534,
+      "unequal_swaps": 43,
+      "mutation_controls": [
+        "zero_to_full_operator_mismatch",
+        "dropped_unequal_frame_carry",
+        "broken_edge_containment"
+      ]
+    },
+    "backward": {
+      "nodes": 15959,
+      "edges": 29974,
+      "regions": 1037,
+      "free_regions": 768,
+      "paid": {
+        "1": 2201,
+        "2": 412,
+        "3": 510,
+        "4": 127,
+        "5": 40,
+        "8": 8
+      },
+      "single_region_counts": {
+        "five_four_three_two": 1,
+        "four_three_two": 6,
+        "gap_0": 642,
+        "gap_1": 126,
+        "majorization": 863,
+        "third_difference": 24
+      },
+      "adjacent_region_pairs": 1474,
+      "two_region_counts": {
+        "candidate_assignments": 59231,
+        "feasible_assignments": 2090,
+        "five_four_three_two": 10,
+        "four_three_two": 58,
+        "majorization": 1886,
+        "third_difference": 136
+      },
+      "swaps": 534,
+      "unequal_swaps": 43,
+      "mutation_controls": [
+        "zero_to_full_operator_mismatch",
+        "dropped_unequal_frame_carry",
+        "broken_edge_containment"
+      ]
+    }
+  },
+  "additional_controls": [
+    "noncoordinate_quotient_line",
+    "profitable_fusion_rejected",
+    "omitted_paid_rank_rejected"
+  ],
+  "witness_payload_sha256": "cc8ea97062750b2485bbf741493eb5de45a672ff6190a0899985c72db0e354ed"
+}