Audit P104 cross-axis encoding mismatch and exact boundary transition

research/p104-encoded-boundary/PREDECESSORS.json

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+{
+  "credit": "See README and source notices; substantial OpenAI Codex assistance.",
+  "context_only_pending_heads": {
+    "3882": "9d56eeedbed5e3e9bbbc8c189d66f14cf15b6067",
+    "3940": "51ba88c007cfbcc3d9cf22dd321f5224be3a5bb1",
+    "3943": "2cba526a4ebe0ef6c3c0143498adcf67dca97941",
+    "3954": "f1e2839819959cd0534d008be91ed5b79ec55a6a",
+    "3956": "f73eef845623a42363ec8d883058fab98fb1335d"
+  },
+  "new_scope": "Axis/role orientation encoding mismatch and exact transition; no adapter optimality, private COPY retirement or coverage optimization."
+}

research/p104-encoded-boundary/PROOF.md

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+# A row-to-column encoding is a separate physical interface
+
+## Scope and the actual source boundary
+
+The pinned `notes/copied-centers-complex.tex`, subsection **Two-stage phase
+endpoint and its paid correction**, specifies a forward scalar shear followed
+by an inverse second shear. It gives physical data outputs
+`A=-F y`, `B=F T^-2 x+E y`, a paid correction, phase/sign/bank wrappers, and
+**distinct** auxiliary banks. `notes/structured-bulk-complex.tex` states the
+completed all-role C contract. Those sources do not contain a chronological
+P104 replacement with two adjacent completed P104 calls.
+
+This audit therefore identifies the actual *stage boundary* and tests the
+natural proposed P104 row/column interface at it. It does not claim to have
+found the missing adjacent completed-call implementation. PR3954's positive
+control says that a common completed C commutes with a fixed scalar bank
+encoding. The new question is whether the same encoding is present on both
+sides when the active fibers and source/target orientation change. It is not.
+
+The row and column maps below are explicitly proposed contracts. In particular
+`S A_c S` is the contract obtained by swapping the source and target roles of
+the forward encoding; it is **not** a replay or admission of an inverse P104
+word. An actual inverse caller may choose another encoding, whose compatibility
+must then be derived separately.
+
+## Fixed physical slots and exact transition
+
+There are two data banks `X_(i,j),Y_(i,j)`, `0<=i,j<104`. Ordering each port
+index by the eight pinned pivots followed by the 96 nonpivots, put
+
+```
+D = [I B; B^t B^t B-I],    H=diag(0_8,I_96),    J=I-H,
+D^-1 = [ I-B B^t    B ] = [ -11I    B ]
+       [ B^t       -I ]   [  B^t   -I ].
+```
+
+The canonical fixture gives `B B^t=12I`.
+Define `D_r=D tensor I`, `D_c=I tensor D`, and likewise `H_r,H_c,J_r,J_c`.
+This is a genuine change of which physical owners mix; it is not a permutation
+of names inside an already matched interface.
+
+Let `T_r,T_c` be diagonal families of invertible address operators, commuting
+with their own H masks. For either axis `a`, define
+
+```
+L_a = [D_a H_a T_a; 0 I],
+Pi_a = [J_a H_a; H_a J_a],
+A_a = Pi_a L_a.
+```
+
+`Pi_a` swaps the X/Y slots at nonpivot ports of that axis. It records literal
+PR3882 occupancy. Let `S=[0 I;I 0]` exchange source/target orientation. The
+candidate reversed column encoding is `Abar_c=S A_c S`.
+
+A physical handoff preserving the same underlying unencoded fields must be
+
+```
+K = Abar_c A_r^-1
+  = S Pi_c L_c S L_r^-1 Pi_r.
+```
+
+No factors commute past each other in this formula. It is exact on arbitrary
+address functions, regardless of the noncommutation of D with the chart family.
+`K A_r=Abar_c` follows by adjacent inverse cancellation. Every map is extended
+by identity on both disjoint dirty-helper banks. These identities preserve all
+helper values, but do not execute their required completed F transformations
+or supply an owner schedule. Fixed output slots remain the named X and Y slots.
+
+On the common chart sector `T_r=T_c=I`,
+
+```
+A_r = [J_r D_r H_r; H_r D_r I],
+A_r^-1 = [D_r^-1(J_r-H_r) D_r^-1 H_r; H_r J_r],
+Abar_c = [I H_c D_c; H_c J_c D_c].
+```
+
+Consequently the four blocks of K are
+
+```
+K11 = D_r^-1(J_r-H_r) + H_c D_c H_r,
+K12 = D_r^-1 H_r       + H_c D_c J_r,
+K21 = H_c D_r^-1(J_r-H_r) + J_c D_c H_r,
+K22 = H_c D_r^-1 H_r       + J_c D_c J_r.
+```
+
+Row and column operators commute only when they act on different port factors;
+the displayed order within each factor is retained. These are interface
+identities, not native rank-cost estimates.
+
+## A physical constant-input counterexample
+
+Take `X_(0,0)=2` and every other data field zero, with arbitrary dirty helper
+values. Both index zero ports are pivots. Under `Abar_c`, this stays exactly
+`X_(0,0)=2`: the column H mask kills that pivot source. Under `A_r`, the same
+X field survives and **48** additional Y fields appear at nonpivot row indices
+with column index zero, each equal to +1 or -1. This follows from the first
+row of B having 48 signed halves. The verifier reconstructs the canonical E8
+roots and punctures and checks the count and signs; it does not import PR3954's
+matrix code.
+
+This is not merely an artificial choice of globally identity native operators.
+For the positive physical convention `C_u=((1+i)I+(1-i)X_u)/2`, every C_u,
+its inverse, and F fix constant address functions (the zero Fourier character).
+Thus the counterexample lifts to that actual invariant physical sector by
+assigning the displayed constants to entire arrays. It disproves identity
+handoff for these named encodings before endpoint phase wrappers. It does not
+price a constant-sector operation as a native child or assert every orientation
+or changed encoding has the same defect.
+
+A common completed operator C applied on every data role commutes with both
+common-chart encodings separately. Nevertheless `A_r != Abar_c`. Therefore
+`C A_r=A_r C` does not imply `Abar_c^-1 C A_r=C`; the change of representation
+still occurs. The independent checker includes both commutation identities as
+positive controls and rejects identity handoff, omitted Pi and omitted reversal.
+
+## Explicit formal word and complete cost decision
+
+For one common-chart fiber, D factors as two triangular updates:
+
+1. `u_J += B u_N`.
+2. `u_N = B^t u_J - u_N`.
+3. `u_N += w_N`.
+4. Swap the 96 nonpivot source/target slots.
+
+The inverse reverses those operations. There are 384 signed-half terms in each
+B traversal. With temporarily writable donor arrays, each traversal halves
+its donor bank, performs signed unit additions, and restores donors by doubling.
+Thus either direction has 864 unit ADD/SUBs, 104 halves, 104 restoring doubles,
+96 signs, and 96 explicit in-place slot swaps. This factor word uses no scratch
+array. The signs and swaps remain separately charged operations. This is not
+a new optimum or a comparison to PR3940/3943/3956's differently fused adapters.
+
+Applying the inverse to 104 row fibers and the reversed encoding to 104 column
+fibers gives a formal common-chart transition invoice:
+
+| Operation | Count | Status |
+|---|---:|---|
+| Unit array ADD/SUB | 179,712 | Constructive exact scalar count |
+| Array halve | 21,632 | Constructive exact scalar count |
+| Restoring array double | 21,632 | Constructive exact scalar count |
+| Nonpivot array sign | 19,968 | Constructive exact scalar count |
+| Explicit slot swap | 19,968 | Constructive exact logical-slot count |
+| Extra scratch arrays | 0 | For this factor word |
+| Nonuniform chart transport, physical swap/routing and fixed-slot return | unknown | No native realization supplied |
+| Dirty-helper completion, lifetime/holding, width/precision/control and recursive admission | unknown | Retained obligations |
+
+Role reversal is expressed by choosing source and target operands in the column
+word; it does not require two extra whole-bank exchanges. Its legal physical
+availability is still an obligation. The finite invoice is **not** a physical
+upper bound, since native chart changes and ownership have not been realized.
+
+For the unchanged PR3954 scaffold, grant all new interface work zero cost.
+The named hypotheses are 208 local calls, rank mass 2576 each, persistent
+progress at most 2312 each, 104^2 data pairs, 72 helpers per fiber in two
+disjoint banks, ambient rank 81, entrance rank one per data source, and native
+rank tariff at least remaining persistent progress. Then
+
+```
+W = 2*104^2 + 2*104*72 = 36608,
+remaining >= 81W-104^2-208*2312 = 2473536,
+bill >= 208*2576+2473536 = 3009344,
+capacity = 81W = 2965248,
+excess >= 44096.
+```
+
+Adding K as external work cannot repair this failed gate. Unknown nonnegative
+costs are not zero; zero is only the most favorable rejection relaxation.
+For 0<=a<1 and every child rank r<=81, `(r/81)^(1-a)>=r/81`, so failure also
+holds for the corresponding positive-saving moment. This is inherited gate
+arithmetic under explicit hypotheses, not an independent proof of a global
+coverage law. A changed owner schedule, merged operations or profile requires
+a new complete ledger rather than a deduction of 44,096 units from this paper.
+
+Decision: **narrow and stop implementation expansion**. The exact unresolved
+item is an actual completed-call pair with one shared physical encoding and
+legal helper/owner returns, or a priced changed transition that also changes
+the failed complete gate. The current canonical source does not supply it.

research/p104-encoded-boundary/README.md

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+# P104 row/column encoded-boundary audit
+
+The natural row-fiber encoding and reversed column-fiber encoding do not match
+at fixed physical slots, even on the constant-address sector where all positive
+C_U charts act as identity. A two-unit pivot input creates 48 extra signed unit
+Y fields under the row encoding and none under the column encoding.
+
+This adds the missing cross-axis condition to [PR3954](https://github.com/CrocSwap/integer-mult-bounds/pull/3954)'s
+common-completed-C commutation control. The exact replacement is
+`K = S Pi_c L_c S L_r^-1 Pi_r`; all dirty helper values are retained. A direct
+common-chart factor word is counted explicitly, with physical transport,
+ownership, helper completion and precision left as unknown obligations.
+The unchanged favorable whole-scaffold gate still exceeds capacity by 44,096.
+
+**Scope:** a source-bound interface lemma and concrete physical-sector
+counterexample for one proposed row/column encoding. The retained caller
+supplies an actual two-stage boundary; it does not supply an actual adjacent
+completed P104 pair. The reversed column encoding is a proposed swapped-role
+contract, not an independently replayed inverse native word. No supplier,
+new kappa, physical runtime gain or selected-frontier change is claimed.
+
+See [PROOF.md](PROOF.md) for equations, the full obligation ledger and stop rule;
+[RESULT.json](RESULT.json) for exact finite counts; [TASK.json](TASK.json) for
+scope and next questions; [PREDECESSORS.json](PREDECESSORS.json) for credit.
+
+## Reproduce
+
+Standard-library Python 3.12:
+
+```
+python3.12 -I -B research/p104-encoded-boundary/verify.py
+python3.12 -I -O -B research/p104-encoded-boundary/independent.py
+python3.12 -I -O -B research/p104-encoded-boundary/test_verify.py
+python3.12 -B scripts/research_package.py --package research/p104-encoded-boundary
+python3.12 -B scripts/check_research_commit.py --package research/p104-encoded-boundary --check-source-history
+```
+
+The primary checker reconstructs the canonical 104 roots and B, checks 208
+local inverse basis columns (all fibers are disjoint), the physical-sector
+counterexample and complete conditional gate. The separate dense checker
+assembles four small whole-grid operators, including varying invertible chart
+blocks, without importing the primary implementation. It checks all columns
+and both literal occupancy and orientation failure controls. Tests check
+optimized execution, bad result/source bytes and an offline read-only export.
+Finite checks support the displayed algebraic proof; no human or kernel review
+or full selected supplier replay is claimed.
+
+## Open next questions
+
+The actionable question is an actual chronological pair of completed calls
+whose exit and entrance share one physical encoding on every role. Begin with
+one named owner schedule and its source/target orientation, not a new scalar
+adapter. Its helper maps, chart transports, fixed returns and complete cost
+must be specified before native expansion. Alternatively, change the owner or
+coverage contract and derive a new favorable gate first. The sibling copy and
+coverage studies are separate investigations, not premises of this result.
+
+## Credit and source history
+
+Chris R / Clear1nsight, with substantial OpenAI Codex assistance. Apache-2.0.
+PR3882/3940/3943 and PR3954 retain credit for residual encoding, actual caller,
+cleanup, and physical two-axis compatibility/gate. PR3956 independently matches
+the 960/1152 adapters; this package does not repeat that comparison. Rohan Arun
+(PR1972), Chafik Boukhalfa (PR1991), Adam Gall (PR1940), Josie Elliston/E8,
+Sagar Thakrar/PR1012, Douglas Colkitt, Dominik Scholz and the retained native,
+compiler, occupancy and terminal-floor lineage retain their original credit.
+The three live source inputs are canonical-main files. Pending PRs are credited
+context; no unmerged Git object is required for replay.

research/p104-encoded-boundary/RESULT.json

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+{
+  "conditional_gate": {
+    "capacity": 2965248,
+    "excess": 44096,
+    "rank_bill_lower_bound": 3009344,
+    "remaining_rank_lower_bound": 2473536,
+    "width": 36608
+  },
+  "formal_transition_invoice": {
+    "explicit_in_place_slot_swaps": 19968,
+    "full_grid_bank_swaps": 0,
+    "halves": 21632,
+    "nonpivot_signs": 19968,
+    "restoring_doubles": 21632,
+    "scope": "homogeneous common-chart factor word only; not physical upper bound",
+    "scratch_arrays": 0,
+    "unit_add_sub": 179712
+  },
+  "local_inverse_basis_columns": 208,
+  "physical_dc_witness": {
+    "column_extra_slots": 0,
+    "first_leak": [
+      1,
+      8,
+      0,
+      1
+    ],
+    "input_slot": [
+      "X",
+      0,
+      0
+    ],
+    "input_value": 2,
+    "row_extra_Y_slots": 48
+  },
+  "pivot_ports": [
+    0,
+    1,
+    3,
+    6,
+    22,
+    31,
+    93,
+    103
+  ],
+  "ports": 104,
+  "schema": "p104-encoded-boundary/1",
+  "scope": "exact fixed-slot interface mismatch and formal transition; no actual global P104 pair, native supplier or frontier change",
+  "small_global_basis_columns": 50
+}

research/p104-encoded-boundary/SOURCES.json

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