Independent max-flow audit of the P104 rank-eight cut family (no new κ)

[AgoraPluribusMichael]

Change and mathematical scope

This PR adds research/p104-cut-maxflow-audit/, an independent audit (not a construction) of the finite premises behind the P104 side-correction obstructions #2082, #2090, #2113 and #2127, plus the budget line summarized in #2133. Every checked premise reproduces exactly. No discrepancy found. There is no new κ, supplier, scalar saving or selected-frontier change.

Why this matters: #2133 concludes that the interleaved and terminal I-G/2 routes are closed within their premises. It relies on five cut PRs that it did not replay ("I did not replay any cited package"). Each of those PRs was checked only by its own author's tooling.

Method.

• Input. The only input is the hash-pinned PR1972 witness research/p104-native-channel/inputs/LOWRANK-WITNESS.json (0f541bef…580e), authenticated before use. No predecessor checker was read, imported or executed. Author numbers were compared only after the computation, and only from PR descriptions.
• Frames. Frames are canonicalized by GF(2) reduced echelon form. All 4058 events are legality-checked: common chart for add/copy, nested transports with the stated rank, and private lifetimes.
• Frozen chronology (Prove sharp P104 temporal rank bound and correction-budget obstruction #2082/Sharp eight-query temporal cut for the frozen P104 itinerary #2090 model). A time-expanded vertex-capacitated network: one array per register per layer, and one hub per equal-frame class, which encodes complete common-chart mixing exactly. Mixing after the final event is included. The max-flow is certified by weak duality: an explicit feasible integral flow plus an explicit separating vertex cut of the same size. So the flow code itself need not be trusted.
• Prove P104 rank-eight cut under arbitrary frame-path interleaving #2113 model. The union of all interleavings of the fixed frame paths, via BFS.
• Prove P104 transport floor with arbitrary added storage #2127 model. The collapsed frame graph.
• Scalars. An exact Fraction replay of all 280 input columns.

Claim	Author value	This audit
Events / cuts / frames / intervals	4058 / 4059 / 444 / 1698	identical
Equal-frame pairs over all cuts	106,254,468 (#2090)	106,254,468
Cross-register equal-frame interval pairs	71,224 (#2113)	71,224
Frozen-chronology X→Y capacity	8 arrays (#2082, #2090)	max-flow = min vertex cut = 8
Q0..Q7 at cut 2429	separator; cones 104X+32G+8Q / 104Y+32D+8Q (#2090)	identical; each query alone reopens; next event is the first D transport
Union graph without the 8 descents	0 targets; each descent reopens 52 (#2113)	0; 52 each; intact 104
Frame graph without descents	222 / 222 (#2127)	222 / 222, disjoint, covers all 444
X→Y block	G/2, rank 8; minor det 1/256 (#2090)	rank 8, symmetric, unit diagonal, M²=13M, minor 1/256
Residual	(I-G/2)^-1 = I-G/24; 48 off-diagonal nonzeros per row (#2082, #2083)	(I-M)(I-M/12)=I; 48 in every row
Upward / downward transports	1402 / 8 (#2136)	1402 / 8, all private rank-7 descents

New details:

• Extremal minimum cuts.

  • Source side: Q7…Q0 at cuts 1570–1597.
  • Target side: C#0…C#7 just after their descents (cuts 2464–2524).

  So Sharp eight-query temporal cut for the frozen P104 itinerary #2090's query cut and Prove P104 rank-eight cut under arbitrary frame-path interleaving #2113's descent cut are both minimum, and they bracket every minimum cut of the frozen schedule.

• Premise sensitivity. Suppose nested charts (A ⊂ B), not only equal charts, may mix at a single cut. At cut 2429 the max-flow then rises from 8 to the full 104 (registered command). A local run gives the same at cuts 1570, 2000 and 2470. The schedule has no combinatorial bottleneck beyond the equal-chart rule, so the obstruction is entirely the price or crediting of cross-chart transfer.

• Source-bound budget line. docs: P104 side-correction scope map (results by premise, shared 328/9 budget line; no new kappa) #2133's moment fractions used constants copied from PR bodies. Here they come from the witness: boundary coverage 2312 from frame dimensions, descent mass 56, idle coverage 39096. That reproduces 7551/7592, 7659/7592 (+96), 1917/1898 (+104), 7785/7592 (+208) and slack 328/9.

• Private discards. All eight private discards drop nonzero values. This is the inherited private-copy semantics, stated explicitly for reviewers.

Remaining obligations (not established here):

• The reduction from "rank ≤ (8+k)d with k added cross-frame transports" to "≥ 96 transports" (Prove sharp P104 temporal rank bound and correction-budget obstruction #2082, Prove P104 transport floor with arbitrary added storage #2127). Its finite premise is checked; the reduction is the authors' written argument.
• Rule out identity from nested P104 transport refinements #2136's nested-refinement model, and feat(research): price escape from refined P104 intervals #2148, which builds on it. Only Rule out identity from nested P104 transport refinements #2136's interval counts are reproduced.
• Prove a terminal-cut paid-rank floor for the P104 side correction #2083's terminal argument. Only its distinct-hyperplane and row-support premises are checked.
• The conditional coverage contract s=N=9, m=73, W=832, native Clifford/phase recipes, hosting, precision, all-size lifting, and the selected cold replay. This is a finite audit of one pinned witness, not a universal P104 barrier.

Task and overlap

• Selection. choose_research_task.py seed 13c7ea26593c41082f85327d2b361d6b gave track independent-verification, card independent-moment (scope_key c6f2494b197df3bc3591). Scope is refined to the P104 cut implication. The card asks for an independent check of a claim lacking one, and this is the central unreplayed implication on the revision-4 top card. The reason is in TASK.json.

• Canonical thread. Discussion #2164, status DONE. There is also a short pointer in #1166. No advance claim was posted, because Discussion access was unavailable until after the work was done.

• Base. Canonical main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4, re-fetched 2026-10-14 21:23Z with no change.

• Compared heads (all unchanged at 21:13Z): Prove sharp P104 temporal rank bound and correction-budget obstruction #2082 60f4e337, Prove a terminal-cut paid-rank floor for the P104 side correction #2083 7231ab97, Sharp eight-query temporal cut for the frozen P104 itinerary #2090 dff116ef, Prove P104 rank-eight cut under arbitrary frame-path interleaving #2113 fa1a45f9, Prove P104 transport floor with arbitrary added storage #2127 befebf0f, docs: P104 side-correction scope map (results by premise, shared 328/9 budget line; no new kappa) #2133 a79f0f91, Rule out identity from nested P104 transport refinements #2136 9b8689e0.

• Overlap. scripts/find_overlap.py --refresh at 21:23–21:26Z covered 410 open PRs and 834 Discussion records, with complete pagination and no warnings.

  • Max-flow/min-cut/Menger terms found no P104 cut audit.
  • The p104-side-correction card terms found no third-party replay of Prove sharp P104 temporal rank bound and correction-budget obstruction #2082/Sharp eight-query temporal cut for the frozen P104 itinerary #2090/Prove P104 rank-eight cut under arbitrary frame-path interleaving #2113/Prove P104 transport floor with arbitrary added storage #2127.
  • feat(research): price escape from refined P104 intervals #2148 (escape pricing on Rule out identity from nested P104 transport refinements #2136's separator) and Construct complete P104 forward word and conditional hosting invoice #2152 (a complete forward word) appeared during the session. They are complementary, and the README notes them.

  This is not a novelty certificate.

Validation

python3 -I -B research/p104-cut-maxflow-audit/verify.py                              # ~30 s
python3 -I -B research/p104-cut-maxflow-audit/verify.py --premise-sensitivity 2429  # ~50 s
python3 -I -B research/p104-cut-maxflow-audit/test_verify.py                        # 9 tests
python3.12 scripts/research_package.py --package research/p104-cut-maxflow-audit

Run locally on Windows 11:

• Registered replay on CPython 3.12.7: PASS_REGISTERED_RESEARCH_PACKAGE research/p104-cut-maxflow-audit outcome=completed, 3 min 35 s for all three commands.
• verify.py also passes on CPython 3.14.6. Metadata/pins: PASS. Workflow tests test_research_*.py: 65 OK (3 skipped).

Failure controls:

• In-verifier: one added cross-frame contact gives 9; nested mixing at one cut gives at least 9; deleting one private lifetime gives 7; max-flow engine fixtures pass.
• test_verify.py rejects modified source bytes, a wrong transport rank, an add outside a common chart, a non-nested transport, scalar corruption with frames still legal, a corrupted receipt and an overwrite attempt. An equivalent frame-basis presentation reproduces the identical receipt.

Proof versus finite checks. The rank bound follows from the written cut-factorization argument. The code checks its finite premise on this witness and certifies optimality by duality.

Not run: hosted CI, make frontier1094-verify, native Clifford/phase checks.

Credit

The audited results belong to David Arias (#2082), Adam Gall (#2090, #2083, #1940), Jeremy Diviney (#2113), Ross Richardson (#2127), Tairitsu-Aya (#2136) and Aaron Wang/navinor (#2133). The channel is due to Rohan Arun (#1972), and its native realization to Chafik Boukhalfa (#1991). Their geometry and coverage lineage is credited in those packages. Prepared for Michael Yee with substantial Claude (Anthropic) assistance in method design, implementation, verification and prose. There has been no independent human review. Apache-2.0.

Open next questions

These are in the README and TASK.json:

1. Cheapest legal single-cut exchange. Find the cheapest legal single-cut exchange that the nested relaxation exposes. That is the bill a changed coverage interface (docs: P104 side-correction scope map (results by premise, shared 328/9 budget line; no new kappa) #2133 route 2) must credit. First step: a min-cost flow on the layer-701 nested-frame graph.
2. The "+1 array per added transport" step. Machine-check the reduction behind the 96-transport floors.
3. Rule out identity from nested P104 transport refinements #2136's exact capacity. Find the exact capacity of the nested-refinement model; only the bound ≤ 56 is claimed.

Workflow feedback (optional)

• Checkout on Windows. Checkout under a long base path fails without core.longpaths=true. The full tree is about 2.1 GB, of which research/ is 2.06 GB. A non-cone sparse checkout worked well: everything outside research/, plus research/*/README.md and TASK.json, plus the needed packages. A note in CONTRIBUTING would save time.
• Undocumented tool requirements. scripts/research_package.py needs Git ≥ 2.43 (cat-file -z), and find_overlap.py needs an authenticated gh. Neither is mentioned in the package guide.

🤖 Generated with Claude Code

https://claude.ai/code/session_01A2DF8QgsTwrE4t3793qQUx