[aliromman]

A scoped enabling proof and negative result for complete caller words. No new kappa. An optimistic fourteen-call/twelve-field P104 profile passes the earlier equality cuts and has a floating raw crossing near .003393385. New query Gram and target-intersection cuts exclude that profile in the paired-stage family below. The contribution also closes shortest callers at every size in the stated model and supplies an executable filter for future searches.

Proof, exact checks and complete prices, source pins, replay evidence, and filter examples. Exact contribution head: 6c088037e13b87faf7cfe3233c53d0ae51850dd8. Canonical coordination: Discussion #5593.

Results and scope

For normalized query terms R_i=u_i v_i with sum R_i=I, let K_ij=v_i u_j. If axis x occurs in call set S and reaches r_x independent main jobs, then

r_x <= 2|S| + h_x - rank K[S,S].
r_xy >= rank K[S,T] - delta_x - delta_y,
delta_x = 2|S| + h_x - r_x.

Here h_x counts charged one-field idle occurrences; r_xy counts jobs that contain both axes. These are necessary cuts for whole words with changed paid inputs and feedback. Passing them does not construct compatible projectors or an endpoint.

• Shortest callers: with L=n+1, n>=4, arbitrary overlapping commuting involutive axis sets, mixed full/thin calls and no idles, the complete P104 target-minus-paid moment is at most -8 for every normalized nondecreasing concave tariff. A real five-query word attains -8 at the linear tariff. Section 9 permits arbitrary scalar mixing across all 104 ports at total main dimension N=104n; the block simplex is derived, not assumed from separate ports.
• Longer paired stages: with L=n+2 all-full calls, separate ports, disjoint eight-axis blocks each used exactly twice, and no idles, full stage capacity is impossible for n>=6. No such caller with 2<=n<=50 passes a power-tariff saving >= .003. The fourteen-call/twelve-field and eighteen-call/sixteen-field bounds are -368 and -176 for every concave tariff.
• A local design threshold: if the paid-minus-helper hinge sums stay nonpositive, a shortest caller needs local rank debt d<v/2. Both pinned P104 bills have d=56 while v/2=52. Redistributing local paid ranks while retaining these conditions cannot reopen that route.

The proofs use characteristic-two field algebra, independent commuting axis variables and autonomous fresh helpers. The executable filter uses F2. The longer paired-stage price result does not cover arbitrary coupled ports. Complex or even-denominator codes, noncommuting or unbounded-order advancing stages, main/helper mixing, other main dimensions and changed contracts need separate proofs. No selected-frontier update or full multiplication theorem is asserted.

Verification

Python 3.12 and the standard library:

python3.12 -W error -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. Registered normal and optimized replays pass. A read-only Git-free export passed in about 1.23 seconds with PATH unavailable; source corruption was rejected before loading code. Two literal interpreters agree on the sharp five-query word and an actual six-query positive control. Checks cover 4,352 projector combinations, 47,904 simplex pairings, 21,808 intersection-factor cases, the 104-port block rank formula, incidence maxima and integer-root price intervals. The longer example passes every separate rank cut but fails the joint cut.

The written arguments establish the unbounded claims under their hypotheses. Finite checks are not independent human review or a formal kernel. Native source admission, the huge uniform word and the full multiplier were not rerun. Full-history provenance preflight was not run in the partial checkout; the declared source objects and bytes were checked. Optional solver searches and timeouts are retained as discovery evidence and are not dependencies of the proof replay.

Sources and credit

Work began on canonical main 5e0052ca0aa3036a3afa8b0438927377c8d97d53 and the isolated branch advanced to d7633ce230196a1d5d061347de7a0fd8be512ed6. The final refresh checked 6cf7a683f447b9f1e2333791907fd92be5009a63; authoritative guidance, priority revision 7 and selected result were unchanged. The QQ16-specific original card is retained with an explicit user-directed scope change. Active p2 leakage, P104 contact/helper and G20 claims concern separate contracts.

The complete price inputs are original P104 at the frozen base and PR #5599 at 09dd0965711ca7eeb169057a420abab1c9052137. The earlier cut is PR #5588 at 97db3fb45cd93a350e300781d75440d14e6a1f73. These exact inputs are not claims about the best current pending frontier.

Cinna Davis / 1337isnot1337 supplied the simplex, private-projector obstruction, sharp word and controller lineage. The new work is the rank/intersection cuts, stage-reuse and paired-stage applications, block extension and complete concave price test. Credit the P104/helper authors, the PR #5565 cascade, Jacob Alan Sussman's productive recursion, Cao Yifan/zxv compiler and moment work, and all retained native, precision and analytic sources. Elementary rank and convexity tools are standard; no worldwide-priority claim is made for them. Prepared for Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.