Prove a terminal-cut paid-rank floor for the P104 side correction

[adamgall]

Research claim

After the complete P104 native low-rank channel finishes, every I-G/2 correction in the stated canonical common-chart/nested-transport model costs at least 104 additional paid ranks, even when fresh owned zero/copy/discard lifetimes are granted for free. Without allocation/disposal the floor is 208. These are arbitrary-length schedule lower bounds, allowing arbitrary rational scalar mixing and arbitrary dirty workspace at the full frame, not finite search conclusions.

Under the retained paired-chain coverage contract, even this favorable 104-rank floor makes the completed moment exceed one at saving zero. This rules out the post-completion route in that model, not earlier interleaving, other chart alphabets or changed coverage laws. No admitted supplier, new kappa, selected-record change or all-size theorem is claimed.

Object, predecessors and changed premise

Rohan Arun's #1972 (fd82d172f46933fd253c2c6de4bd5eaff5429cfd) supplies the 4058-event P104 scalar/frame channel. Chafik Boukhalfa's #1991 (6e222517e182e98499239bc9a4009f540e1c5520) supplies its native realization. After the entire channel, including query cleanup:

• each source is x_s and all 72 arbitrary dirty helpers are restored at the full rank-nine chart;
• target t holds y_t+(Gx/2)_t in its rank-eight hyperplane H_t;
• the 104 target hyperplanes are distinct;
• each row of S=I-G/2 has 48 nonzero off-diagonal entries.

Concurrent claims by 1337isnot1337, 18931708 and 18931718, now collected in discussion #2080, describe a target-up/add/down completion with 208 rank-one uses and the direct-target path floor. Neither that construction nor that restricted floor is claimed new. This contribution allows arbitrary scalar mixing, moving other registers and the more favorable disposable-copy route, proving the 104-rank floor and complete coverage obstruction. The earlier query-live proposal was withdrawn by its author because it leaked independent query dirt; earlier cuts in general are not excluded here. The favorable conditional profile screen in #2080 is not used as a coverage proof: terminal ranks must be added to, not substituted for, required idle coverage.

Ali Romman's #1125 is also distinct: it bounds fixed-term delivery carriers across the P104 decoder family, not a side correction after the complete restored channel with arbitrary scalar mixing.

Proof-critical argument

If no transport touches H_t, every row available there remains in the span of its only original nonzero row z_t=y_t+(Gx/2)_t; new zero rows, same-chart copies, additions and disposal cannot import information. Preserving the independent input y_t forces the final coefficient of z_t to be one. Therefore the nonzero correction (Sx)_t cannot be added without a transport incident to H_t.

Distinct rank-eight hyperplanes are incomparable, so a nested transport can touch at most one of them. Each costs at least one rank: total cost is at least 104. Without allocation/disposal, occupancy conservation forces both arrival and departure at every touched target chart, yielding 208.

A matching 104-rank relaxed word forms each side row in one owned full-frame copy, descends once to the target hyperplane, adds it and disposes of it. The code replays all 280 persistent input columns and all restored endpoints. It independently checks each new drop as a complete native operator with one positive rank-one child, including its global phase.

The owned-copy allocation/disposal grant is deliberately a favorable relaxation. Exact transport operators and a relaxed algebraic witness do not establish that this lifecycle is an admitted physical supplier.

Completed optimistic bill

The inherited local profile is {1:1074,2:58,3:86,4:104,5:48,6:24,7:8,8:8}, rank mass 2368=2312 persistent boundary progress +COPY56. The relaxed correction gives the profile {1:1178,2:58,3:86,4:104,5:48,6:24,7:8,8:8}, rank mass 2472.

The retained paired-chain formulas are s=2p+1, m=16p+9, W=2p*v. Their source is pinned to Dominik Scholz's larger-gain model. Substituting p=4,v=104 is an explicit conditional P104 template premise, not global owner admission: s=N=9,m=73,W=832.

With one unpaid coordinate per live family, unchanged idle coverage is 72*832-9*2312=39096. The terminal correction has zero persistent net frame progress, so its new ranks cannot replace those idle coordinate advances. Favorably charging no scalar, allocation, extra hosting or precision cost still gives:

paid rank mass >= 9*(2312+56+104)+39096 = 61344
capacity       = 73*832                = 60736
M(0)           >= 1917/1898            > 1

For 0<=a<1 and every child 0<r<=m, (r/m)^(1-a)>=r/m, so every such completed moment is also above one. No numerical root calculation is needed. Fully paid auxiliary banks add m to mass and capacity and cannot repair the deficit. Under the same assumptions at any feasible odd stage count, D<=104(s-1)-160s=-56s-104<0.

The coverage context is credited to Sagar Thakrar's #1012 and Douglas Colkitt's #1240. This is not an application of the unchanged selected-word ceiling outside its hypotheses.

Evidence and reproduction

Only research/p104-terminal-correction-floor/ is added: README proof, exact receipt, independent selection, task/source/replay metadata and a standard-library verifier. External proof inputs authenticate before the retained Gaussian interpreter executes; no producer or Clifford compiler is imported. The prior native core is inherited from #1991, while its scalar/frame word is independently replayed here.

Observed with CPython 3.12:

uv run --python 3.12 --no-project python -I -O -B research/p104-terminal-correction-floor/verify.py --write
# PASS_P104_TERMINAL_CORRECTION_FLOOR
uv run --python 3.12 --no-project python scripts/research_package.py --package research/p104-terminal-correction-floor
# PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed

The registered replay regenerates and compares the full canonical JSON receipt. It checks 280 arbitrary input columns, 104 isolated target hyperplanes, 104 exact positive-rank-one native drop identities and 14,400 physical pairings. Eight semantic controls reject: omitted target correction, wrong coefficient, dirty-copy contamination, omitted descent, persistent-target disposal, unrestored source, wrong native global phase and falsely debiting terminal ranks from old idle coverage. Optimized Python does not disable the checks. The card working example also passed PASS_RESEARCH_BASELINE_BINDINGS_ONLY; no selected cold replay, hosted CI or formal-kernel run is claimed.

A separate AI-assisted reviewer read the written proof and code and executed the unchanged registered package successfully, confirming the same exact receipt and all eight controls. This is a separate replay and proof review, not independent human review, a second implementation of every native primitive, or a formal-kernel verification.

Provenance, limitations and next questions

Base: 564148b82ae9df72163f22106f15e73efca8aff2; priority revision 4. One independent selector seed: cfe3b8f6d8ab3b2c86dececdc8d1d726, retained verbatim in SELECTION.json. Git blobs, SHA256 pins, source roles and overlap limits are recorded. Submission overlap covered 353 open PRs including drafts and 738 discussion records with complete pagination and no refresh warnings at 19:02Z, followed by targeted live/local/merged reading including the full current #2080 claim; absence of matches is not a novelty claim.

Submission compatibility was checked against canonical main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. Its AGENTS/autoresearch/priority-log usability update was read; priority revision, catalogue, selected record and scientific inputs are unchanged. Original scientific pins are retained.

The useful next step is an earlier cut: list simultaneously available source/target charts before native completion, then price a correction that reuses unfinished paths. Alternatively, derive a genuinely changed coverage interface before taking extra unpaid-coordinate credit. Neither is constructed here. Unpriced chart changes, physical copy/erase admission, arbitrary-size lifting, precision, chronological hosting and the bit supplier remain outside the result.

Retain credit for Rohan Arun/#1972; Chafik Boukhalfa/#1991; Adam Gall/#1940; Josie Elliston/#1071; Ali Romman/#1125 and the #1862 independent Gaussian interpreter retained through #1931; the #968/#717/#446/#595/#620 geometry lineage; and the template/coverage authors above. Prepared for Adam Gall with substantial OpenAI Codex assistance in proof, implementation and exposition. Apache-2.0. No independent human review is claimed.