research: prove P104 whole-fiber coverage and helper-sharing gate

research/p104-coverage-gate/PROOF.md

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+# Whole-fiber coverage and the conditional helper-sharing gate
+
+## Named model and source boundary
+
+This is a scoped theorem for selecting whole calls from the proposed 104 by
+104, nine by nine tensor scaffold of PR3954, head
+`f1e2839819959cd0534d008be91ed5b79ec55a6a`. It does not embed that scaffold in
+the selected h=28 producer. The pinned canonical native-channel witness has
+104 X ports, 104 Y ports and 72 arbitrary dirty helpers. A whole local word
+uses every port. Its source and target dimension increments are eight each,
+and its helper increment is nine: P=104*8+104*8+72*9=2312. The witness has
+rank mass2368. We conditionally grant the corrected mass M=2368+208=2576
+from PR3882/3894/3940, including the unproved opposite orientation. No native
+correction is independently recompiled here.
+
+As in PR3954, let p_i be the 104 distinct odd physical root labels in F2^9.
+Each grid pair is the nondegenerate line span(p_i tensor p_j), in dimension
+m=81. A first-axis call fixes j and uses every i; a second-axis call fixes i
+and uses every j. Swapping the names row and column has no effect below.
+This study uses only those exact incidence sets. It neither chooses a new
+physical gauge nor grants a free permutation, source decoder or owner move.
+
+The retained caller in notes/copied-centers-complex.tex fixes two logical
+stages S(x,y)=(x,x+y), R(x,y)=(x-y,y); the required composition is
+R S(x,y)=(-y,x+y), before its paid physical correction and phase wrappers.
+We grant the corrected calls this ideal unencoded logical interface to make
+the pruning test favorable. Actual encoded boundaries remain obligations.
+
+## Whole-call closure theorem
+
+Select a row set A and column set B, of sizes a and b (0<=a,b<=104).
+Their whole-word incidence union is
+
+    U = (A x [104]) union ([104] x B),
+    N = |U| = 104(a+b)-ab.
+
+Exactly ab pairs receive both calls. Exactly
+
+    d = 104(a+b)-2ab = a(104-b)+b(104-a)
+
+pairs receive one call. These are dangling endpoints. On any such pair the
+logical map is S or R rather than R S. The differences are nonzero on arbitrary
+inputs: at (x,y)=(1,0), S=(1,1), R=(1,0), and R S=(0,1). Chart changes cannot
+turn this logical endpoint equality into a proof; any changed physical
+completion must be established and billed separately.
+
+Consequently, if every touched pair must receive the retained two-stage
+endpoint and no replacement completion is supplied, d=0 is necessary and
+sufficient at the incidence level. Since each summand in d is nonnegative,
+nonempty A forces b=104 and nonempty B forces a=104. The only closed masks
+are empty or all208 calls. This is not an impossibility theorem for paid
+native completion or a new partial-port word. A rectangle A x B is not the
+support of these whole calls: removing its other incident ports changes the
+word/interface. Treating them as padding still incurs their acquisition,
+return and physical inventory obligations.
+
+## Most favorable complete-cost gate for every mask
+
+Now permit arbitrary replacement completion of dangling endpoints. Grant one
+initial rank unit at every touched pair, including pairs whose retained
+second stage is absent. This is an optimistic relaxation, not an established
+physical initial/final boundary for the pruned caller. Let H be any number of
+physical helper owners and W=2N+H. Helper owners may be reused at disjoint
+times, provided the stated entrance/exit and progress hypotheses are retained.
+
+The named hypotheses are: every persistent owner must finish under full F;
+touched data sources receive the N initial rank credits; each whole call has
+mass at least M and net persistent progress at most P; remaining work has rank
+tariff at least its net persistent dimension progress; child ranks are <=81;
+and the equal physical-role normalization is1/W. Unknown acquisition,
+intertwiners, completion, copy/add/disposal, routing, holding, control, tape,
+precision and admission costs are nonnegative and additional. This is the
+same conditional progress accounting used by PR3954; changed translated
+frames that invalidate its P cap require a new proof, not this bound.
+
+For c=a+b calls, persistent progress required is81W-N. Remaining paid rank is
+at least max(0,81W-N-cP). Therefore
+
+    total rank >= cM + max(0,81W-N-cP)
+               >= 81W + c(M-P)-N.
+
+The excess over capacity is at least
+
+    Delta = 264(a+b)-[104(a+b)-ab] = 160(a+b)+ab.
+
+Every nonempty mask has Delta>=160>0. Thus pruning whole calls cannot rescue
+the unchanged corrected profile even with optimistic initial credits and free
+new boundary work. For 0<=alpha<1 and 0<=r<=81,
+(r/81)^(1-alpha)>=r/81, so the complete moment also exceeds one.
+The full mask gives44096, reproducing the named source model as a boundary
+case of the new all-mask theorem, not as its only evidence.
+
+H cancels from Delta. Hence reducing the number of helper owners by reuse,
+while retaining these hypotheses, cannot reverse the rejection. This does
+not establish legal helper reuse. In the actual retained caller arbitrary
+dirty helpers are original inputs and must receive full F, including after
+their last local use. Calling their local last use 'retirement' does not erase
+the remaining endpoint. Deleting b such owners while retaining their b
+independent arbitrary input arrays would contradict injectivity; transferring
+their information to other owners changes capacity and requires a paid map.
+Aliasing independent helpers across different calls reduces the declared
+input space and is a different owner contract, which the conditional H-bound
+already grants if all other hypotheses hold. Private COPY disposal is a
+separate question and is not audited or newly lower-bounded here.
+
+## Exact criterion after an actual profile change
+
+Let e_R=M_R-P_R and e_C=M_C-P_C be proved per-call excesses for the two axes.
+Let A>=0 denote additional paid rank with no already-counted progress. The
+same relaxed numerator is
+
+    Delta(a,b) = (e_R-104)a+(e_C-104)b+ab+A.
+
+At A=0 this is bilinear. Over [0,104]^2 its minimum occurs at a corner, by
+minimizing each affine coordinate in succession. Equivalently its value is
+the bilinear convex interpolation of its four corner values. Therefore a
+nonempty mask can have negative relaxed Delta iff e_R<104 or e_C<104.
+If at least one call from each axis is required, the corresponding corners
+are (1,1),(1,104),(104,1),(104,104). Testing those four is exact. If the retained
+two-shear endpoint must be incidence-closed, only the full corner remains:
+
+    Delta_full = 104(e_R+e_C-104)+A.
+
+For a common excess e, a two-axis partial mask can pass the relaxed test iff
+e<207/2=103.5, whereas an incidence-closed caller requires e<52 (and must pay A).
+The first statement follows from ab/(a+b)>=1/2 for positive integers a,b,
+with equality only at a=b=1; the second follows from a=b=104.
+
+A concrete discrimination control is e_R=e_C=100. Selecting one call per axis
+produces relaxed Delta=-7, yet leaves206 dangling pairs. The complete closed
+mask has Delta=9984. This is a warning against treating a passing sparse
+relaxation as a constructed caller, not a lower bound on every possible paid
+completion of those206 pairs. Even forgiving all208 correction units per call
+leaves e=56, for which the closed-mask excess is832. That conditional arithmetic
+is not a new COPY floor; PR3045 and the concurrent private-COPY audit own the
+literal lifetime question. The earlier #3620 threshold concerns a different
+caller and is not used in this proof.
+
+## Decision and remaining obligations
+
+NARROW: stop expansion of unchanged-profile partial-fiber and inventory-only
+reuse variants. A reopening result must supply a changed excess/coverage law,
+or disprove one named progress hypothesis with an actual translated call.
+For sparse candidates it must also implement each dangling endpoint and all
+encoded/physical owner boundaries, and include those costs in A. No scalar
+incidence theorem proves physical supplier execution, signed-floor or finite
+precision correctness, recursive admission, a new kappa or a universal P104
+barrier. No selected files or shared verification controls are changed.

research/p104-coverage-gate/README.md

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+# P104 whole-fiber coverage and helper-sharing gate
+
+**Scoped result:** selecting partial sets of whole P104 row/column calls cannot
+rescue PR3954's unchanged corrected profile. The exact touched-pair count is
+`N=104(a+b)-ab`; the optimistic rank excess is
+`264(a+b)-N=160(a+b)+ab>0` for every nonempty mask. Reducing the helper-owner
+count does not alter this numerator while the named progress/endpoint
+hypotheses remain valid. No new kappa, native supplier or frontier change.
+
+There is also a separate endpoint obligation: the retained two-shear caller
+has `104(a+b)-2ab` dangling pairs. Without a changed paid completion, the only
+whole-call sets closed under its endpoint are empty or all208 calls. A smaller
+rectangle is not the support of an unchanged104-port word.
+
+The proof gives an exact four-corner test for asymmetric changed profiles.
+It distinguishes a passing sparse relaxation from an actual closed caller:
+with hypothetical excess100 per axis, one call on each axis has relaxed
+excess-7 but206 unresolved endpoints, while the full closed caller has
+excess9984. These hypothetical profiles are discrimination controls, not
+constructed improvements. The all-correction-free closed value832 is only
+conditional arithmetic, not a new private-COPY lifetime floor.
+
+See [PROOF.md](PROOF.md) for the general incidence proof, source assumptions,
+helper-potential cancellation, exact optimization criterion and stop decision.
+[RESULT.json](RESULT.json) records the finite evidence. The conditional gate
+explicitly grants full-array initial rank credits and leaves new physical
+boundary costs nonnegative; it does not infer actual preparation for a
+one-sided caller or physical legality of shared helpers.
+
+## Reproduction and evidence
+
+Python3.12, standard library only, from the repository root:
+
+```sh
+python3 -B research/p104-coverage-gate/verify.py
+python3 -B research/p104-coverage-gate/independent.py
+python3 -O -B -m unittest discover -s research/p104-coverage-gate -p test_verify.py -v
+python3 scripts/research_package.py --package research/p104-coverage-gate
+```
+
+The primary checker authenticates three canonical source inputs, recounts the
+literal native witness's frame dimensions/rank histogram, and checks all11025
+count pairs. The separate checker imports no producer or primary verifier:
+it explicitly forms incidence sets for21844 masks on grids of sizes1..7,
+checks actual pair maps and rational asymmetric corner criteria. Seven tests (including six actual optimized CLI faults)
+reject rank undercharging, missing ports and helper retirement, and distinguish
+overcounted initial credits and an unclosed sparse pass. These finite tests
+support the proof; they are not a formal theorem or independent human review.
+Checks use explicit exceptions and unittest, and remain active under `-O`.
+
+Source baseline is canonical main
+`20ecf2891d80b5acf18d980b2164fb44ea7faa90`, priority revision6. Frozen inputs
+and the selected record remain unchanged. Exact source pins are in
+[SOURCES.json](SOURCES.json); no unmerged Git object is required by replay.
+Current task/overlap scope and remaining questions are in [TASK.json](TASK.json).
+The registered workflow is the publication check; local results do not imply
+hosted CI success until its exact published head passes.
+
+## Predecessor credit and what is new
+
+- [Rohan Arun's PR1972](https://github.com/CrocSwap/integer-mult-bounds/pull/1972)
+  owns the retained scalar/frame word. [PR1991](https://github.com/CrocSwap/integer-mult-bounds/pull/1991)
+  supplies its native realization and2312 persistent progress/2368 mass.
+- [PR3882](https://github.com/CrocSwap/integer-mult-bounds/pull/3882),
+  [PR3894](https://github.com/CrocSwap/integer-mult-bounds/pull/3894) and
+  [PR3940](https://github.com/CrocSwap/integer-mult-bounds/pull/3940) supply
+  the correction and caller context; their208 correction units are granted.
+- [PR3954](https://github.com/CrocSwap/integer-mult-bounds/pull/3954), head
+  `f1e2839819959cd0534d008be91ed5b79ec55a6a`, defines the proposed two-axis
+  scaffold, physical endpoint obligations and full-mask44096 rejection.
+  The new contribution is the all-mask closure/coverage theorem, the exact
+  asymmetric changed-profile criterion and the inventory-only reuse consequence.
+- [PR3943](https://github.com/CrocSwap/integer-mult-bounds/pull/3943) and
+  [PR3956](https://github.com/CrocSwap/integer-mult-bounds/pull/3956) own the
+  residual consumer and matched return-adapter comparison; neither is redone.
+  [PR3045](https://github.com/CrocSwap/integer-mult-bounds/pull/3045) and the
+  concurrent private-COPY study concern literal copy lifetimes, a distinct scope.
+- Douglas Colkitt and the credited authors of
+  `notes/copied-centers-complex.tex` supply the retained all-role caller and
+  paid physical endpoint semantics.
+
+Clear1nsight, with substantial OpenAI Codex multi-agent assistance. Apache-2.0.
+
+## Open next questions
+
+1. **A changed partial-port word or dangling completion.** Produce one actual
+   completed pair with all owner/encoded boundaries and invoice its work.
+   This could change the sparse gate; merely removing whole calls cannot.
+2. **A changed translated progress law.** Exhibit one actual legally translated
+   call whose persistent progress exceeds2312, including preparation and
+   eventual helper return. This is blocked by a source-bound physical owner
+   schedule, not by the finite incidence search.
+3. **A favorable changed profile.** Supply proved per-axis excesses and additional
+   boundary bill A satisfying the closed gate, then check the full positive-
+   exponent moment. A passing rank relaxation alone does not admit a supplier.
+
+Decision: NARROW. No further native expansion is justified by pruning or
+inventory reduction under the unchanged conditional profile.

research/p104-coverage-gate/RESULT.json

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+{
+  "schema_version": 1,
+  "scope": "conditional whole-fiber incidence and progress gate",
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "witness": {
+    "ports": 104,
+    "helpers": 72,
+    "progress": 2312,
+    "native_mass": 2368,
+    "granted_correction": 208,
+    "corrected_mass": 2576
+  },
+  "mask_count": 11025,
+  "nonempty_masks": 11024,
+  "closed_masks": [
+    [
+      0,
+      0
+    ],
+    [
+      104,
+      104
+    ]
+  ],
+  "all_masks_sha256": "56552e880f665d0af7f5ec986d06e852eabd53db9dae4243cf98d4c9d7ea334f",
+  "minimum_nonempty_excess": 160,
+  "full_inventory_controls": [
+    {
+      "helpers": 72,
+      "width": 21704,
+      "remaining_rank": 1266312,
+      "lower_rank": 1802120,
+      "capacity": 1758024,
+      "excess": 44096
+    },
+    {
+      "helpers": 144,
+      "width": 21776,
+      "remaining_rank": 1272144,
+      "lower_rank": 1807952,
+      "capacity": 1763856,
+      "excess": 44096
+    },
+    {
+      "helpers": 14976,
+      "width": 36608,
+      "remaining_rank": 2473536,
+      "lower_rank": 3009344,
+      "capacity": 2965248,
+      "excess": 44096
+    }
+  ],
+  "changed_profile_controls": [
+    {
+      "row_excess": 264,
+      "column_excess": 264,
+      "minimum_two_axis_relaxation": 321,
+      "closed_excess": 44096
+    },
+    {
+      "row_excess": 100,
+      "column_excess": 100,
+      "minimum_two_axis_relaxation": -316,
+      "closed_excess": 9984
+    },
+    {
+      "row_excess": 56,
+      "column_excess": 56,
+      "minimum_two_axis_relaxation": -4936,
+      "closed_excess": 832
+    },
+    {
+      "row_excess": 51,
+      "column_excess": 52,
+      "minimum_two_axis_relaxation": -5460,
+      "closed_excess": -104
+    },
+    {
+      "row_excess": 103,
+      "column_excess": 104,
+      "minimum_two_axis_relaxation": 0,
+      "closed_excess": 10712
+    },
+    {
+      "row_excess": 104,
+      "column_excess": 103,
+      "minimum_two_axis_relaxation": 0,
+      "closed_excess": 10712
+    },
+    {
+      "row_excess": 0,
+      "column_excess": 300,
+      "minimum_two_axis_relaxation": -10516,
+      "closed_excess": 20384
+    },
+    {
+      "row_excess": 300,
+      "column_excess": 0,
+      "minimum_two_axis_relaxation": -10516,
+      "closed_excess": 20384
+    }
+  ],
+  "sparse_discrimination_control": {
+    "a": 1,
+    "b": 1,
+    "excess_per_axis": 100,
+    "touched": 207,
+    "dangling": 206,
+    "relaxed_excess": -7,
+    "closed_excess": 9984
+  },
+  "selected_frontier_changed": false
+}

research/p104-coverage-gate/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
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+    },
+    {
+      "path": "notes/copied-centers-complex.tex",
+      "usage": "input",
+      "git_blob": "fddd7bc7f2a2f749ba6eada19d153bf9243ee31c",
+      "sha256": "18eac6d5c5755e53230c0c12a633f38b6ce9666ef0322c1aff79394f2952af72"
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+    {
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+    {
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+}