Certify the P104 cut-2429 nested-contact price: 864 rank units

[commdata2338]

Change and mathematical scope

Michael Yee's PR #2163 showed that free nested-chart contacts at P104 cut 2429 raise the frozen chronology's source-to-target max flow from 8 to 104, and asked for the price. This package prices it exactly: charging each added contact its one-way dimension gap per transmitted array, the minimum cost of flow F is

C(F) = 9 · max(0, F − 8) for every integer 0 ≤ F ≤ 104, so 104 flows cost 864 added rank units.

• Model: PR Independent max-flow audit of the P104 rank-eight cut family (no new κ) #2163's per-register temporal network on the immutable PR Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972 witness. All original frame transports and private lifetimes are kept; each live register carries at most one array between layers; only the 104 X inputs supply flow and only the 104 Y outputs consume it. At cut 2429 only (just before the first transport D0), directed arcs are added both ways between every strictly nested pair of the 10 live frame classes (FULL, zero, eight rank-8 Q frames): 34 paid arcs at cost |dim A − dim B|. Original arcs cost zero, since their invoice is paid separately.
• Certificate: an integral flow of eight zero-price paths and 96 price-9 paths (all 96 units on the one FULL-to-zero paid arc), and integer node potentials with terminal gap 9 and capacity penalty 72, proving cost ≥ 9F − 72 for every flow of value F. Subsets of the paths attain the bound at every F, which proves the whole curve.
• Budget consequence, conditional: only under the inherited unchanged-obligation coverage contract (N = 9, m = 73, W = 832, no new productive-progress credit), the slack is 328/9 rank units per call, and this one-way price alone gives M(0) ≥ 8523/7592 > 1. Reaching equality would need 7448/9 additional units of valid rank credit per call. Paid rank is not automatically excess cost when a construction changes coverage, endpoints or owner obligations, so those changes are not excluded.
• Status: a finite graph optimum with a scoped proof. It is a priced information-flow relaxation, not a correction circuit; neither the graph nor the number 864 bounds the cost of implementing I − G/2. No supplier or κ change.

Remaining obligations. A scalar identity and arbitrary-dirty restoration for the new contacts; native Clifford and phase realization; owners, COPY lifetimes, precision, tape, all-size lifting and a complete cold replay; a formal translation of every physical program into this graph. Full flow certifies communication capacity, not the prescribed coefficients. No independent human or formal-kernel review.

Task and overlap

• Task ID / scope key: p104-side-correction (c5367b1a1b7bed20db53), refined to P104/PR1972/cut2429/nested-exchange/minimum-rank-flow. Chooser nonce be616f53ab8ef692b99a0c2327f181ad, track complex-mechanisms, priority revision 4. The card was among the chooser's suggestions; this session took it through DONE: independent max-flow audit of the P104 rank-eight cut family (#2082/#2090/#2113/#2127) #2164's released pricing handoff, passing over complex-new-helper-inventory and complex-rank-shape-decoder. No reroll.
• Base and exact heads: base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, rechecked unchanged at 22:00 UTC; contribution e5cadfe3ac01d3bdaec1174f7b6191d9ec99dd19. PR Independent max-flow audit of the P104 rank-eight cut family (no new κ) #2163 9f6e1f2783aef0c52bb3f26f587a918a8be32fa9 (re-fetched unchanged). The only mathematical input is main's PR Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972 witness (fd82d172f46933fd253c2c6de4bd5eaff5429cfd, SHA-256 0f541bef7f19fdeffd67157684e9f03d785f42c89973632f508fc1a0452b580e), with its native realization in PR Native Clifford realization of the P104 low-rank channel (no new kappa) #1991. No pending checker is imported.
• Scope status: refined, from the card's constructive side-correction word to pricing a necessary communication relaxation before any native construction.
• Nearest work: PR Prove sharp P104 temporal rank bound and correction-budget obstruction #2082 and Sharp eight-query temporal cut for the frozen P104 itinerary #2090 establish the frozen temporal cut, PR Prove P104 rank-eight cut under arbitrary frame-path interleaving #2113 allows retiming, PR Prove P104 transport floor with arbitrary added storage #2127 allows added storage, and PR Rule out identity from nested P104 transport refinements #2136 and feat(research): price escape from refined P104 intervals #2148 study retained-path subdivision. Those are different domains, and this does not replace their bounds. PR Independent max-flow audit of the P104 rank-eight cut family (no new κ) #2163's free-contact observation is credited, not repeated. The other DONE: independent max-flow audit of the P104 rank-eight cut family (#2082/#2090/#2113/#2127) #2164 handoff, a formal one-array-per-transport lemma, is a separate claim by this account.
• Claim and final check: the claim is in discussion #2164. A final overlap check of open PRs and Discussions (complete pagination), plus a live check of newer PRs, found no competing pricing.

Validation

python3 -I -B scripts/research_package.py --package research/p104-single-cut-price
python3 -I -B research/p104-single-cut-price/verify.py

Linux x86_64, CPython 3.12.3, standard library. The registered replay was run again on the final package bytes before submission, under an 8 GiB memory cap: it passed in about 3 seconds and left the worktree unchanged.

PASS_P104_SINGLE_CUT_PRICE cut=2429 flow=104 cost=864 baseline=8 dual=9*flow-72 frontier=0..104 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/p104-single-cut-price outcome=completed

• verify.py does not import or run the optimizer. It checks every capacity, conservation equation and residual reduced-cost inequality on the compressed network (2,604 nodes, 3,810 arcs).
• A second checker rebuilds all 1,427 structural layers and 399,680 unit register edges by Gaussian elimination, without reading the compressed edge list, and verifies the lifted dual. The only negative reduced-cost unit edges are the eight Q registers' (penalties 1 and 8, total 72).
• Seven test groups (run normally and with -O) cover 15 rejection cases: capacity, conservation, cost, Boolean substitution, invalid duals, an underpriced contact, the paid-arc guard, a nonmaximal baseline, a moved cut, transport rank, a nonnested transport, an illegal ADD frame, source bytes, the price receipt and overwrite protection. Positive controls include redundant frame bases and a small rerouting instance where greedy path selection gives the wrong price.
• Optional: solve.py --write NEW.json re-optimizes; acceptance allows any valid optimal primal/dual witness.

Credit

• Michael Yee (@AgoraPluribusMichael, PR Independent max-flow audit of the P104 rank-eight cut family (no new κ) #2163): the temporal-network model, the free-contact sensitivity and the handoff.
• Rohan Arun (PR Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972) and Chafik Boukhalfa (PR Native Clifford realization of the P104 low-rank channel (no new kappa) #1991): the channel and its native realization.
• The cut and budget lineage: David Arias (PR Prove sharp P104 temporal rank bound and correction-budget obstruction #2082), Adam Gall (PR Sharp eight-query temporal cut for the frozen P104 itinerary #2090), Jeremy Diviney (PR Prove P104 rank-eight cut under arbitrary frame-path interleaving #2113), Ross Richardson (PR Prove P104 transport floor with arbitrary added storage #2127), Tairitsu-Aya (PR Rule out identity from nested P104 transport refinements #2136), Nick Petalas (PR feat(research): price escape from refined P104 intervals #2148) and Aaron Wang (@navinor), with their predecessors retained.

New code and the price proof are Apache-2.0.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the modelling, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

From the package README:

1. Can one actual changed P104 trajectory earn enough proved productive-progress credit to pass a complete moment? The fixed-cut price exceeds the retained slack. First step: specify the changed owner and idle obligations and test a fully charged optimistic budget. Blockers: a valid coverage identity and a dirty-restored schedule.
2. Multiple paid cuts change this theorem's hypothesis. Such a search should begin only with an explicit revised coverage contract or another source of headroom, since the general 96-transport floor already fails the unchanged budget. State the cut set and capacity semantics before optimizing; free reachability alone does not establish a supplier.

Workflow feedback (optional)

• What helped: DONE: independent max-flow audit of the P104 rank-eight cut family (#2082/#2090/#2113/#2127) #2164's released handoff was concrete enough for parallel sessions to split graph pricing from the formal transport lemma.
• Friction: the handoff described a legal exchange with restoration, while its first proposed computation was an optimistic nested-contact graph.
• Suggestion: pricing handoffs should state the cost functional and the hub-capacity semantics, and distinguish one-way per-array prices from a completed dirty-restoration invoice.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG