Copied-information floor of the selected centre path

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Copied-information floor of the selected centre path (centre-path-sharing)

Scope

Drawn card: centre-path-sharing (track recursive-interfaces, scope key a55534776f5479f1686c), drawn by scripts/choose_research_task.py with seed 3eca9612dd994cf5cbf01134bb839c3d at priority revision 6, base 997e6a9ca7b4f95d50e366f61412fd640d72a88a. The draw returned three cards: centre-path-sharing, joint-query-controller, coverage-law-change. No re-draw was needed. scope_status is unchanged and actual_scope equals the card question, byte-equal original_card in TASK.json.

Saturation handling was kept inside the card. The card's method 1 (fully charged k-call sharing relaxation) is saturated by PR 3075 (Aaron Wang), which priced k = 2, 4, 8, 16 windows with restoration, hosting and precision charged. This package takes the unsaturated hard-search reading of method 2 at its smallest instance: prove a lower bound on copied information per call for the selected decoder contract.

The question is the card's: can the 64-per-call centre path (eight rank-8 COPY/ERASE totals through the private temporary) be shared across calls or shrunk? Package: research/centre-demand-rank-floor/.

Result 1: the eight totals are demand-independent (exact)

Recomputed from one frozen member of the selected replay archive, candidate/strict/global/base/split-centres/GLOBAL-TEMPLATE.json.gz in research/community-frontier1094-audit/selected-replay.tar.gz, both orientations:

• the centre phase is 256 macros, eight blocks of 32: one CLEAR of private scratch C_k, one COPY/ERASE total, one rank-8 center ascent at frame 414+k, and 28 fanout (forward) or gather (backward) events at coefficients +1/4 or -1/4;
• the eight source rows are S = (229, 224, 227, 225, 243, 241, 259, 257), scratch C_0..C_7 in the private namespace [23408, 23416), consumer rows exactly 112..223;
• the 112 x 8 consumer demand matrix D (row r, column k = coefficient of total k in consumer row r) has exact rational rank 8, with the explicit nonzero minor on rows (112, 117, 118, 120, 113, 116, 115, 123) of determinant exactly -1/16384 (that is -1 after scaling by 4^8);
• the backward gather matrix equals the forward fanout matrix entry for entry.

Consequence: the pinned consumer endpoint determines all eight totals independently. Any replacement copied-total contract delivering the same endpoint must move at least eight independent scalar totals per call. With the per-total pairing-rank floor of eight for root-code factors (the 64 rank(C) package research/p112-batch-centre-floor, whose one-call census and factor-linear floor are credited below), the charged 8 x 8 = 64 per call is rank-forced. The centre path cannot be shrunk under the pinned endpoint. The lever table's favourable relaxations ("centre path free" 2.80x, "shared by two calls" 1.90x in research/complex-local-word-ceiling) assume away charges the pinned contract's demand forces.

Result 2: the sharing half at demand level, and the recorded physical datum

The 144 selected calls (80 forward, 64 backward, 16 per stage over nine stages) each carry their own centre block in call-local space, and the pinned replica lift declares a direct-sum instantiation ("direct sum of data and dedicated banks; share only restored private streams"). For any k-call window the requested matrix is block diagonal in disjoint variables, so the coupling matrix is C = I_k and the 64 rank(C) law of research/p112-batch-centre-floor gives the floor 64k. Sharing one centre path across calls is admissible only when the consumer requests a lower-rank combination; the pinned two-panel shared-sink contract is exactly the all-ones case, where both panels request the image of the sum, a different endpoint as that package records.

The remaining datum is physical: which calls, if any, literally share sink rows, return operator and chart within one uninterrupted window. It is not decodable from the pinned owner nominations, which carry the centre source rows in only 16 of 1,152 lookups; the data-row instantiation lives in route_semantics (cover class d*rho(stage,port)). No eligible pair is exhibited and none is excluded physically; the exact missing map is named in TASK.json's open questions.

Acceptance-gate statement

There is no changed charge or copy profile to price: the profile stays eight rank-8 totals per call and the comparison ceiling is the unchanged baseline a < 0.001690054109089, reproduced by the card's own positive control, run as a registered command. Restoration, hosting and precision charges can only make the unchanged profile more expensive, never cheaper, so the floor needs none of them.

Verification

Registered replay (python3.12 scripts/research_package.py --package research/centre-demand-rank-floor), verbatim output lines:

PASS_CENTRE_DEMAND_RANK_FLOOR demand_rank 8 totals 8 consumer_rows 112 minor_det -1/16384 calls 144 owner_lookups_missing 1136 no_new_kappa
PASS_COMPLEX_LOCAL_WORD_CEILING a_ceiling 0.00169005410908900006 kappa<= 0.0016872026453254019
PASS_REGISTERED_RESEARCH_PACKAGE research/centre-demand-rank-floor outcome=completed

Metadata and byte checks, verbatim:

PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
PASS_RESEARCH_COMMIT_BYTES {"commit": "6e36982ccc888bdb4dc15982322d8fe1aed392b4", "files": 7, "package": "research/centre-demand-rank-floor"}

Four rejection controls are enforced in the verifier: a zeroed demand column and a merged column pair each drop the rank below eight, a fanout sign flip fails gather equality, and a source-row swap fails the pinned source tuple. RESULT.json must equal the recomputation byte for byte. All arithmetic is exact (rational); the rank is recomputed by elimination and by the pinned minor.

Scope and limits

• The floor holds for the pinned selected decoder contract and its literal consumer endpoint. Changed endpoints, changed alphabets and outside-code factors are outside it.
• The per-total pairing-rank-eight factor is inherited from research/p112-batch-centre-floor (credited); this package does not reprove it and does not price outside-code terms.
• The demand rank is a linear-information statement at the consumer, not a physical circuit, tape, precision or hosting claim. No selected cold replay is run; the frozen archive is read-only.
• No kappa, supplier admission, eligible window or frontier change is claimed. The selected record, frozen archives and upstream are unchanged.

Credit

Packing-ceiling model, lever table and coverage identity: Douglas Colkitt, research/complex-local-word-ceiling and PR 1012 (thakrarsagar). Coupling-rank law and 64 rank(C) floor: research/p112-batch-centre-floor, prepared for alephnan with OpenAI Codex assistance; its one-call support census is credited there to PR 845 (mihir-s-05), independently corroborated by PR 1098 (Adam Gall), and its one-call factor-linear floor to PR 1180 (woklerx). Pinned two-panel shared-sink contract: ToikY, research/centre-shared-return-guard, built on PR 1708 (Cinna Davis / 1337isnot1337) and PR 1748 (ToikY). k-call charged relaxation: PR 3075 (Aaron Wang). Selected construction: PR 1094 (rohanarun) and the lineage in the selected record.

Prepared by Aaron Wang with help from opencode (substantial AI assistance; no model named; no independent human review claimed). New work is Apache-2.0.