Prove stage-reuse and intersection cuts for complete callers

research/shortest-controller-stage-capacity/README.md

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+# Query Gram cuts for complete caller words
+
+This package gives two necessary cuts for a changed global caller. They bound
+both the number of independent outputs that can use one stage axis and the
+number that must share two axes. The proof allows changed paid inputs and
+arbitrary constant binary bases throughout the word. No earlier caller input
+is held fixed. **No new kappa is claimed.**
+
+The cuts turn two promising price relaxations into scoped obstructions:
+
+| Caller family | Proven consequence for both pinned P104 bills |
+| --- | --- |
+| Shortest query count L=n+1, n>=4, arbitrary overlapping commuting involutive axis sets, full/thin calls, no idles | Complete target-minus-paid moment <= -8 for every normalized nondecreasing concave tariff; the bound is sharp |
+| L=n+2, all full, separate ports, each disjoint eight-axis stage block used twice, no idles | Full stage capacity is impossible for n>=6; no caller with 2<=n<=50 passes a power-tariff saving >= .003 |
+| Fourteen calls on twelve fields in the second family | Complete moment <= -368 for every such concave tariff; the optimistic raw crossing near .003393385 is not realizable |
+
+The shortest-family statement has no upper size limit. The longer-family
+capacity statement also has no upper size limit; its complete price rejection
+is bounded as stated. These are written proofs with exact finite controls,
+not extrapolations from search failures.
+
+Section 9 also proves the shortest result for arbitrary scalar mixing
+across all 104 ports, with total main dimension N=104n and L=n+1 calls.
+Its block-simplex proof derives the required Gram matrix. The longer
+paired-stage price statement keeps ports separate. The written arguments
+extend to fields of characteristic two; the executable controls use F2.
+
+## What the cuts measure
+
+Normalize the central-query tangent terms as R_i=u_i v_i, with sum R_i=I.
+The query Gram matrix K has entries v_i u_j. If axis x occurs in call set S,
+has h_x charged one-field idle occurrences, and reaches r_x main jobs, then
+
+```text
+r_x <= 2|S| + h_x - rank K[S,S].
+```
+
+For two axes, put delta_x=2|S|+h_x-r_x and define delta_y in the same way.
+Their target sets must have at least
+
+```text
+rank K[S,T] - delta_x - delta_y
+```
+
+common main jobs. At the shortest count the first cut becomes
+`r_x <= |S|+(|S| mod 2)+h_x`. At one additional query, separate axis tests
+can pass while their required intersections are impossible. Sections 2, 7
+and 8 of [PROOF.md](PROOF.md) give the general cuts and this distinction.
+
+There is also an inventory design condition. Write the net local rank mass
+as `2*b*v+d`, where v is the port count and b is the transverse rank. If the
+paid-minus-helper hinge sums remain nonpositive, a shortest no-idle caller
+needs `d<v/2` before it can have a positive concave moment. Both pinned P104
+bills have `d=56`, while `v/2=52`. Section 5 gives the proof. Satisfying this
+necessary threshold would not by itself supply a caller.
+
+## Use the filter
+
+[cuts.py](cuts.py) accepts a JSON query frame and proposed target sets. Vectors
+are binary integers. `u[i]` and `v[i]` describe the rank-one query R_i. Each
+axis lists its call indices, target indices and optional `idle_terms`.
+An axis can represent one whole disjoint stage block when all its axes have
+the same occurrences and target set.
+
+```sh
+python3 research/shortest-controller-stage-capacity/cuts.py research/shortest-controller-stage-capacity/examples/six-query-positive.json
+python3 research/shortest-controller-stage-capacity/cuts.py research/shortest-controller-stage-capacity/examples/fourteen-query-intersection-rejected.json
+```
+
+The first command returns `PASS_NECESSARY_CUTS_ONLY` and exit code zero.
+The second returns `REJECT_NECESSARY_CUT` and exit code one. It passes every
+separate capacity test but fails the intersection test. The proof excludes
+all target assignments with full capacity in that family, not just this file.
+
+A passing filter does not supply compatible native projectors, an operator
+endpoint, paid scalar bases, helpers, ordinary maps or a compiler budget.
+The retained six-query example is an actual binary operator word; it is a
+positive control for the cut, and its complete P104 bill still loses.
+
+## Evidence and reproduction
+
+[RESULT.json](RESULT.json) records the exact checks. Two literal interpreters
+agree on the five-query sharp word and the six-query longer control. The
+replay checks 4,352 two/three-projector combinations, simplex identities
+through 64 fields, the AB/BA kernel identity, intersection inequalities,
+target-incidence maxima, complete helper/paid histograms and exact integer
+root intervals. A deleted query, an invalid query frame, a non-idempotent
+sum and the false extension of the shortest cut are discriminating controls.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/shortest-controller-stage-capacity
+python3.12 -I -O -B research/shortest-controller-stage-capacity/verify.py
+```
+
+Expected marker: `PASS_CALLER_STAGE_CAPACITY_CUTS`. The replay needs only
+Python 3.12 and the standard library. It reads and compares the committed
+receipt. `--write-result` is an explicit regeneration option. The direct
+verifier also works in a source export containing the five declared inputs.
+It authenticates those bytes before it loads the credited interpreter.
+
+The local inventories are original P104 and PR5599 at
+`09dd0965711ca7eeb169057a420abab1c9052137`. They are exact historical inputs,
+not a claim about the strongest pending frontier. Both have complete net
+rank mass 1720. Source pins and hashes are in [SOURCES.json](SOURCES.json).
+Native and outer admission of those sources is inherited, not rerun here.
+The negative necessary moment leaves no compiler or ordinary-work saving
+to establish. [VALIDATION.md](VALIDATION.md) records the checks actually run.
+
+The initial advancing-operator and repeated-stage searches are saved in
+[discovery/README.md](discovery/README.md). Solver `unknown` stays unknown;
+solver `unsat` is not a separately checked proof certificate. Those searches
+are not dependencies of the theorem replay.
+
+## Scope and coordination
+
+The model is binary with independent commuting involutive coordinate axes.
+Helpers are separate independent inputs. The principal obstructions permit
+no idle maps. The generic cuts allow explicitly counted one-field idle
+terms, but their concave cost needs a separate bill. Arbitrary complex or
+even-denominator codes, noncommuting stages, unbounded-order advancing maps,
+joint main/helper codes and different target contracts are outside the proof.
+
+This is a user-directed change from the QQ16-specific card. It does not
+complete that card's installed-protocol question. [TASK.json](TASK.json)
+preserves the original card and states the changed scope. Canonical main was
+refreshed from `5e0052ca0aa3036a3afa8b0438927377c8d97d53` to
+`d7633ce230196a1d5d061347de7a0fd8be512ed6`; the frozen source pins remain.
+Priorities and the selected record did not change at that checkpoint.
+The final refresh to `6cf7a683f447b9f1e2333791907fd92be5009a63` also found
+no change in those authorities. The source pins stay fixed.
+[Discussion5593](https://github.com/CrocSwap/integer-mult-bounds/discussions/5593)
+is the canonical investigation. Local contact, helper-basis and G20 claims
+are separate. No selected-frontier change or scientific acceptance is implied.
+
+## Open next questions
+
+- Extend the intersection bound to a quantitative rank loss that grows with
+  caller size. Start from the rank-two complement and allow unsaturated
+  stage classes and permit coupled ports. This could close more of the
+  longer-caller domain. The separate-port proof only forces one lost
+  stage occurrence; its complete size bound does not cover coupled ports.
+- Search a changed caller with more queries, mixed calls or partially
+  overlapping axis blocks. Apply both cuts before a full polynomial search,
+  then verify the whole endpoint and complete bill. The filter is necessary
+  only; no such favorable word is supplied here.
+- Test the advancing operator route in characteristic zero. The recorded
+  rational four-field screen is negative and the six-field screen timed out.
+  An infinite-order stage is outside the involutive-axis obstruction. A
+  surviving word still needs native maps, helpers and complete pricing.
+- Independently review the normalization, private-axis extension and joint
+  incidence argument. Start with Sections 2, 4, 7 and 8 of the proof. Finite
+  controls do not replace a proof audit or a formal kernel.
+
+## Credit
+
+Cinna Davis / 1337isnot1337 supplies the minimum-query simplex, private-stage
+projector obstruction, five-query sharp word and controller lineage. The
+new contribution is the rank/intersection cut, its application to coordinate
+reuse, the paired-stage packing obstruction and the complete concave price
+test. Elementary rank and convexity tools are standard; no claim of worldwide
+priority is made for those tools.
+
+Credit the P104 productive-helper and source-word authors, the PR5565 cascade
+and PR5599 helper-split authors, Jacob Alan Sussman's productive recursion,
+the native suppliers, Cao Yifan/zxv compiler and moment work, and all retained
+precision and analytic sources. Source-specific notices remain unchanged.
+The equality cuts in Ali Romman's PR5588 motivated this whole-word search.
+Prepared for Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.