Research: P104 Y0 needs 48 new helper lifetimes, and 48 suffice

[commdata2338]

Change and mathematical scope

P104's geometric two-hyperplane cover does not supply the source values it needs. In the retained-source-path monotone relay model, completing Y0 requires exactly 48 new helper lifetimes. Each of its 48 correction sources leaves Y0's endpoint subspace on its first frame expansion; contributing reads must occur on 48 distinct, incomparable initial lines. The written causal-path argument permits arbitrary relays, rational cancellation, retiming and relaxed restoration.

A matching chronological word starts with 48 independent dirty helpers, acquires the sources at original cut 0, combines them in the two hyperplanes recorded by PR #2353, and delivers at original cut 3467. Exactly two helpers enter the target endpoint. All helper logical inputs restore. Y0 becomes y0 + x0; the other 103 target rows remain the original G/2 channel.

Only research/p104-y0-source-acquisition/ is added: proof, literal circuit, generator, exact receipt, two separate checkers and package metadata. Added invoice: 52 H1 + 46 H2 + 48 H6, 146 transports, rank mass 432, and 238 scalar ADDs (142 unit, 96 half coefficients). The whole partial word has 4,442 events, 328 independent inputs and rank mass 2,800.

This is a scoped proof and finite common-chart construction. It is not a complete supplier or new κ. The 432 mass is sharp only when every new lifetime must run ZERO to FULL and transports cost their dimension gaps. It is productive boundary progress, not automatically recursive excess cost. The 48-lifetime bound must not be multiplied by 104: acquisition feeders can be shared across targets. Native phases/lowering, global owners, residual charges, precision and all-size admission remain open.

Task and overlap

Contribution commit: 8b3e3a22c9e957990863ef60a561072dfcf08271; main was rechecked unchanged at 07:21 UTC before submission.

Base and final refreshed canonical main: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4, checked again at 2026-10-15 07:07 UTC. No upstream commit, guidance, selected record or scientific input changed.

Exact predecessors:

• PR Audit P104 relay floor with an explicit monotone-lifetime model #2353: 2704eadf6b5205d4b965e9c6fc6dce28b89783cf, reread by anonymous Git fetch and unchanged at close. Model/cover/handoff only; no executable imported.
• PR Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972: fd82d172f46933fd253c2c6de4bd5eaff5429cfd. Runtime word retained by PR Native Clifford realization of the P104 low-rank channel (no new kappa) #1991 on the pinned main base, SHA256 0f541bef7f19fdeffd67157684e9f03d785f42c89973632f508fc1a0452b580e.
• PR Research: chronological P104 schedule attains known 32-chain bound #1940: a9b7dfe2809092a9a4c90784d12bb60061e995e8, schedule lineage. Fixture SHA256 516c06f6ec3cb84be588c546108de897650288576dd5fbf5fe44b9c5546ad8a4. Exact blobs and provenance are in SOURCES.json.

Advance claim in discussion #2245, following PR #2353's one-target handoff.

The local task card is retained in TASK.json:

Task ID / proposed scope key: p104-side-correction / 72dfbf0708467ad34793;
  P104/Y0/two-hyperplane-cover/source-acquisition/monotone-new-helpers.
Base commit; exact predecessor and dependency heads: the pinned commits above.
Concrete object and question: can two new monotone lifetimes acquire Y0's correction sources along the retained X paths, with dirty restoration?
Scope status: refined; the second chooser suggestion matches a live central handoff.
Existing restriction being changed or audited: replace a geometric two-cover with actual source-read availability; retain the ban on original G/Q/D/C and target reads.
Method; domain: exact F2^9 subspaces, arbitrary-length linear causal-path proof, one target, 104 frozen source paths, recorded cover normals 1 and 4; explicit 48-helper sharpness word at cuts 0 and 3467.
Nearest overlap: PR2353's cover/floor; PR2251/2345/2358 pooled-space minima; the 41-contact existing-helper lift; this account's PR2172/2190. Different obligations as described below.
Optimistic full cost budget: the two-lifetime ZERO-to-FULL screen is 18 rank units and is excluded before synthesis. The matching sharpness certificate pays 432 rank units and 238 ADDs for one target, with no favorable global moment claim.
Success evidence; falsification test; cap: explicit lower-bound premises, all 328 endpoint columns, dirty/ordering/frame/source controls; one roughly 60-minute session, standard library, one busy verification process. Stop at this local result.
Predecessor authors; artifact and reproduction: authors credited below; research/p104-y0-source-acquisition; python3 -I -O -B research/p104-y0-source-acquisition/verify.py.

Selection nonce 9aee6e45a250cebc67a06b8bffe3797e, track complex-mechanisms. Passed generic complex-new-helper-inventory and the crowded complex-rank-shape-decoder for this concrete revision-4 handoff; no reroll.

PR #2353 leaves actual acquisition/restoration open. PR #2251/#2345/#2358 and #2367 concern pooled algebraic availability; the active 41-contact lift reads existing D helpers. This account's #2172 prices cut 2429 and #2190 excludes a fixed room's capture order. Parallel private-code access grants information excluded here; the no-relay cover audit concerns a global weighted cover with helper relays forbidden. None supplies this one-target source-acquisition result.

Final cached overlap: 587 open PRs including drafts, fetched 2026-10-15 07:07:38 UTC; 1,116 Discussion records, fetched 07:09:50 UTC; both report complete pagination. Queries: task card, two-hyperplane/monotone-relay/source-reads, and source-acquisition/48-lifetime/Y0-correction. The canonical thread and active claims were reread. Expected offline warnings mean later changes may be absent; metadata search does not certify novelty or cover arbitrary code diffs.

Validation

CPython 3.12.3, standard library, no third-party dependencies:

python3 -I -O -B research/p104-y0-source-acquisition/verify.py
python3 -I -O -B research/p104-y0-source-acquisition/crosscheck.py
python3 scripts/research_package.py --package research/p104-y0-source-acquisition --check-only
python3 scripts/research_package.py --package research/p104-y0-source-acquisition

All pass. Primary expected output:

PASS_P104_Y0_SOURCE_ACQUISITION lifetimes=48 arrivals=2 columns=328 added_rank=432 added_ADD=238 controls=10 no_new_kappa

The primary checker uses sparse rational rows and enumerated binary subspaces. The second uses dense integer rows at exact scale 8, checks every division, and uses binary Gaussian elimination. They independently reconstruct all 328 endpoint columns and agree on the complete endpoint digest. Both are AI-assisted implementations, not separate human reviews.

Nine rejecting controls cover missing dirty compensation, capture, unmerge or cleanup; wrong source frame, transport charge, delivery timing, source-escape premise or cover assignment. A positive changed-source-path example shows one dirty helper can capture two sources when the crucial frame premise changes. Original 280-column replay is a positive baseline.

Circuit regeneration matches the complete parsed JSON. Both checkers pass in a Git-free read-only export; all 11 files remain unchanged. Source-byte drift, false receipt and occupied-output controls reject, preserving existing output. Registered replay takes about 1.2 seconds locally. Staged whitespace check passes. No native phase replay, full 104-target/global supplier replay, selected cold replay, precision/tape or all-size proof was run. Hosted CI and independent human review are not claimed.

The registered replay (with the dense cross-check) was run again on the final package bytes before submission, under an 8 GiB memory cap: it passed in about a second and left the worktree unchanged. The figures are consistent with the P104 geometry: Y0's correction support has 48 source labels, the construction adds 48 columns to the original 280 (328 in all), and 432 = 48 * 9.

Credit

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.

Credit kumagi / PR #2353 for the explicit relay model, checked 208 floor and one-target handoff; Philippe Van Velzen for the original 208 screen and two-hyperplane covers; Rohan Arun / PR #1972 for the scalar/frame channel; Chafik Boukhalfa / PR #1991 for native realization; Adam Gall / PR #1940 for the schedule; Josie Elliston / PR #1071 for the fixture; Ali Romman / PR #1125 and the retained #968/#717/#446/#595/#620 lineage. Dirty compensation is inherited. Gonzalo Mier, Gareth Davidson and AAAmirinu retain credit for the separate pooled-space/prefix results. Existing sources and notices remain unchanged. Apache-2.0.

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

Open next questions

See README and TASK.json. The smallest useful mechanism step is two target covers sharing source-acquisition feeders, with actual common-frame contacts, a complete dirty-restored word and full pricing. The blocker is a compatible relay/owner schedule; summing one-target lower bounds is invalid. Native phase checking of this exact 146-transport word is a separate local obligation and would still leave 103 correction rows plus global admission.

Workflow feedback (optional)

The explicit one-target handoff in #2353 led directly to a bounded, source-bound result. Generic P104 searches are dominated by earlier work; narrow mechanism terms and exact current-status summaries were substantially more useful. A small machine-readable handoff specifying target, cover normals and the unproved acquisition condition would make this boundary faster to find.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG