[Matarener]

Requested by Mark

Change and mathematical scope

Scoped negative result; no new κ, no supplier change. Adds research/p104-controller-income-bound: in the literal Laurent caller alphabet of the retained p16 controllers (research/p104-productive-controller/caller.py), no caller word can twist more outputs than it has paid odd shears, minus one (even-part rank argument: free events preserve the rank of the even parts, a paid shear changes one row so the rank drops by at most one, and the first paid shear leaves it unchanged). Consequence: the controller's rank-mass income per port is at most the number of full calls minus one, and the three retained controllers attain it (32 twisted outputs from 33 shears; income 17 = 18 − 1). Hence the first-order mass surplus obeys M ≤ calls × (ports − centre) − ports (PR5907 word: 18 × 48 − 104 = 760). Status: proof (PROOF.md) plus an exact replay of the three literal controllers over ℚ. Remaining obligations: none inside the alphabet; a multi-bank or cross-port paid primitive is outside it.

Task and overlap

Base commit 721c03822c421b7f7fcbb2598993b947c4d88419. Inputs (pinned in SOURCES.json): research/kappa-backward-controller-20261023/CONTROLLER.json, research/joint-helper-lattice/CONTROLLER.json, research/p104-productive-controller/CONTROLLER.json and caller.py. Task card joint-query-controller (scope refined: the card asks for a controller that beats the retained one; this package proves the mass side cannot be beaten with the same number of full calls, see TASK.json).

Nearest overlap: Cinna Davis's research/productive-controller-basis-search (GLOBAL-PROOF.md, Theorem B) proves queries ≥ 2p + 1 for 2p independent monomial outputs over F₂[Q]/(Q² − 1), constant binary boundaries, no idle work, and states that the binary theorem is not asserted for complex words; its PROOF.md uses the even-part rank change as a search pruning rule. This package is the complex-side counterpart: the same count in the rational Laurent alphabet with idle markers, output by output, stated as a bound on the complete invoice's income, checked on the literal controllers; it answers that package's first open question for the rank mass (queries beyond 2p + 1 cannot raise the income). research/p104-thin-query-port-floor (#5996) is a different statement (15 thin requests are necessary with the 18 LOCALs fixed); PR5554 covers the moment side for path controllers.

Validation

python3.12 -B research/p104-controller-income-bound/verify.py
python3 -B scripts/research_package.py --package research/p104-controller-income-bound

Standard library only, 0.3 s. Evidence: even-rank trace 32, 32, 31, …, 0 over the 33 paid shears for θ = Q² ∈ {7/3, 11/5} on all three controllers, all 32 declared diagonal outputs odd monomials, income 17 recomputed from the literal word. Controls: the two-bank Whitehead word with three odd shears reaches diag(Q, Q⁻¹) with trace 2, 2, 1, 0 and its two-shear prefix leaves an even part; dropping the last QUERY_SHEAR of the p104 controller leaves a positive even rank and its declared diagonal is rejected; a free Q^{±1} is rejected by the alphabet. RESULT.json compared in full. Finite checks support, and do not replace, the written lemma. Not run: no selected-result replay, no outer ledger (nothing changes).

Credit

Controllers by Cinna Davis (p16/a7 lattice), Ali Romman (PR5398 wrapper, PR5544 all-forward caller) and the P104 lineage; Cinna Davis's productive-controller-basis-search for the binary 2p + 1 theorem and the even-part pruning idea; zxv (PR5405/PR5554) for the path-family ceilings; fragmentation-ceiling for the moment side. Prepared by Matarener with Claude (Anthropic) via Claude Code; substantial AI assistance. Apache-2.0.

Open next questions

1. Does the bound extend to paid primitives that are not odd central shears of one pair (a paid rank-r Gaussian touching several banks or ports)? Such a primitive is the only way to raise the controller income above full calls minus one. First step: state it as a physical job in caller.py terms and extend the tracker. Blocked by: a new physical primitive.
2. Is the bound attained with more full calls, and does the resulting invoice ever beat p = 16? Income grows by (ports − centre) per call but target ranks and idle chunks grow too (PR5907 word: p = 20 gives 0.003117). Blocked by: a non-path caller word.

Workflow feedback (optional)

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https://claude.ai/code/session_011VhYm9w66QL9GsMRvFmYvy

[JosieElliston]

Independent audit in #6120 (research/p104-controller-income-audit/), at head 3183c655.

The theorem and the 17 = 18 − 1 numbers check out with a separate engine. Two suggestions for PROOF.md:

1. Step 4 uses zero rows of L·M, not of M. A left-kernel argument repairs it, and L need not be invertible.
2. "Every stage that appears in an output was paid once as a lift" follows from the determinant, stage by stage.

The scope sentence calling a multi-bank/port primitive "the only way to escape" seems too strong. A paid one-bank Q^{±1} escapes, and so does a boundary that already carries odd Q content. One-row jobs with any number of sources and an even nonzero self-coefficient do not escape. PROOF.md A4 there gives the per-job bound, (n − rank A_e) + rank A_o, and exact controls.

[Matarener]

Thanks @JosieElliston for the independent audit in #6120. All four points are correct; addressed in 1ad66cc (numbers unchanged, 17 = 18 − 1):

1. Left boundary (A1). The last inequality now uses the left kernel: the twisted outputs are rows of L·M, so the matching rows of L annihilate E_final and are independent because the outputs are; L need not be invertible. The verifier also records the left-boundary rank (32 on all three controllers).
2. Lift cancellation (A2). PROOF.md now derives #{i : s ∈ S_i} = ℓ_s from the determinant, stage by stage, and income() checks it for every stage plus Σ q_i = net Q_POWER (194 row-lifts on each controller). RESULT.json regenerated.
3. Markers (A3). D_0, …, D_32.
4. Scope (A4). The "only escape" sentence is gone from README, PROOF.md and TASK.json. It now states that one-row paid jobs with a nonzero even self-coefficient obey the bound for any number of sources, lists the three escapes (multi-row job, paid odd one-bank translation, odd boundary) and cites your per-job bound (n − rank A_e) + rank A_o.

verify.py and scripts/research_package.py --package research/p104-controller-income-bound pass.

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