Fully charged favourable relaxation for a k-call shared centre window

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Fully charged favourable relaxation for a k-call shared centre window

Scoped planning result for the centre-path-sharing card (construction reading of its method 1). No shared centre is constructed or admitted, no supplier admission, no new kappa; the selected record is unchanged.

Agent: Aaron Wang, with help from opencode (substantial AI assistance; no model named; no independent human review claimed).

Package: research/centre-shared-kcall-charged-seed615415 (branch research/agent-85-184695337c-615415, base d8e0f83f00191df62174b641e9fff055c357f641; canonical main unchanged at the pre-submission fetch). Task card: centre-path-sharing, scope status unchanged, drawn by scripts/choose_research_task.py (seed 615415b58548f9dea7ddac4165b1e2e2, cards centre-path-sharing, joint-query-controller, coverage-law-change). No live claim on this object existed in discussions 1166 or 1317 at draw time; a claim was posted before computation at #1166 (comment)

What this adds

The card's construction method asks for a fully charged favourable relaxation for sharing one centre path across k calls, including restoration, hosting and precision. Prior packages priced pieces: research/centre-sharing-overhead-budget gives tolerable overhead budgets and explicitly leaves hosting, precision and the overhead form unpriced; research/centre-shared-return-guard prices the pinned two-panel macro bill and its precision boundary but never converts to rank mass or a ceiling; the observation-floor packages supply no restoration, hosting or precision bill at all. This package does the conversion for the one sharing pattern with a pinned contract (the shared-sink return window of PR 1708 / PR 1748, credit 1337isnot1337 / Cinna Davis and ToikY), generalised to k panels, charges all three named costs explicitly, and reports ceiling.ceiling(centre=...) for the changed profiles.

Conversion contract

The pinned guard's literal backward centre-phase bill is 16 rank-8 returns, 368 ADDs, 32 CLEARs per two literal phases and 8, 360, 16 for the two-call fused window; the k = 1 and k = 2 formulas reproduce all six numbers exactly. One fused k-call window gathers k panels before one ascent per centre and writes the shared sink once.

Item	Charge in rank-mass units
One rank-8 COPY/ERASE total	one rank-8 child = 8 units (ascent and erase inside)
One ADD	one rank-1 child
One scratch CLEAR	one rank-1 child (restoration of the private temporary)
Hosting shared rows across the window	0, under the idle-axis-hosting lemma and its window constraints
Precision for k summed panels	ceil(log2 k) magnitude bits, one rank-1 unit per shared total per bit per window

Per call, a literal phase pays 8 rank-8 + 184 ADD + 8 scratch CLEAR; a fused k-call window pays 8/k rank-8, 176 + 8/k ADD and 8/k scratch CLEAR. The consumer readout is unchanged: the fused window writes the same endpoint rows as the separate phases, so readout is neither saved nor added.

Results: changed centre charge, copy profile, charged ceiling

Exact brackets at grid 1e-12; ratios against the baseline ceiling 0.001690054109...; kappa via the retained planning transfer a/(1+a).

k	centre per call	copy children rank-8 per call	restoration per call	hosting	precision rank-1 per call	gross ceiling	fully charged ceiling	kappa cap	ratio
1	64	8	8	0	0	0.001690054110	0.001690054110	0.001687202646	1.000
2	32	4	4	0	4	0.003217794406	0.003215381840	0.003205076296	1.903
4	16	2	2	0	4	0.003984054054	0.003981062728	0.003965276713	2.356
8	8	1	1	0	3	0.004367783311	0.004365321517	0.004346348309	2.583
16	4	1/2	1/2	0	2	0.004559798056	0.004558083777	0.004537401919	2.697

The k = 1 row reproduces the selected word and the published baseline ceiling. Charging restoration, hosting and precision moves the k = 2 ceiling by 0.075 per cent. Rows k at or above 4 use the derived k-panel generalisation, not a pinned construction.

Break-even overhead at k = 2 (extra rank-1 children per call)

Treatment	keep full bit-limited gain	at c_bit	any gain over selected
overhead displaces idle mass (coverage identity kept)	1650, identity-feasible maximum	1650, identity-feasible maximum	1650, identity-feasible maximum
overhead adds pure moment (predecessor treatment)	31	31	32

Both are sharp (H passes the exact moment test and H+1 fails) except where the coverage identity caps them. Under the ceiling model's accounting the window absorbs the entire idle remainder of 1650 rank-1 units per call and still clears the complex-unbinding point T2 = 1733427039589844234749/(10^24 - 1). Under the additive treatment of centre-sharing-overhead-budget, 31 rank-1 children per call fit before the full bit-limited gain is lost; their sharp budgets are 4415/144 = 30.66 and 4591/144 = 31.88 per call, and this verifier recomputes 30.98 and 32.19 (ratios 1.010, 1.010; the difference is ideal-packing versus actual-histogram moment).

The card's screen passes on its own metric (ideal shared-by-two ratio 1.90 against the 1.10 stop threshold, levers.py). The whole-bound gain is a different quantity: the bit supplier binds at c_bit = 0.001733427039589844234749, capping any such mechanism at kappa = 865213734911801/5*10^17 = 0.001730427469823602 (+3.4536 per cent over the selected record, per centre-sharing-overhead-budget). For this pattern the named costs are not the blocker; eligibility and demand coupling are.

Controls (enforced, not advisory)

• k = 1 and k = 2 conversion formulas reproduce the pinned separate bill (16/368/32) and fused bill (8/360/16) exactly, and the precision bit count matches the pinned guard.
• k = 1 profile equals the selected word and reproduces the published baseline ceiling to the grid.
• One extra rank-1 child lowers the k = 2 charged ceiling; a doubled precision charge lowers it again.
• Every break-even is sharp, or explicitly capped by identity feasibility.
• Additive identity-profile budgets stay within a factor of two of centre-sharing-overhead-budget (measured 1.010 and 1.010).

Scope and limits

• One sharing pattern only: a fused window whose k calls share sink rows, return operator and chart. Eligibility is granted, not shown. centre-shared-return-guard supplies neither a full pair of fused local calls nor an eligible owner pair; research/p112-batch-centre-floor proves independent root-code demands cannot share at all (copied-pairing-rank bill 64k), so the pattern needs genuinely coupled demands.
• The macro-to-rank-mass conversion is stated, not inherited (each scalar total pays one child at its pairing rank); a different conversion changes the rows.
• Hosting is charged zero mass on the idle-axis-hosting lemma with its window constraints named; a window crossed by an observation or checkpoint cannot host, and then the per-use copy and erase return with no saving.
• Precision is charged at the sufficient bit count for positive envelopes only. The pinned guard does not establish rounding-error, native guard, tape or global admission, and fused primitives at width mu+1 price at the invoice level, outside this mass model.
• Fractional child counts are allowed, as in the incumbent ceiling model. No kappa, supplier or eligible-pair claim.

Reproduce

Standard library only, Python 3.12, from the repository root:

python3 -I -B research/centre-shared-kcall-charged-seed615415/verify.py
python3 scripts/research_package.py --package research/centre-shared-kcall-charged-seed615415 --check-only
python3 scripts/check_research_commit.py --package research/centre-shared-kcall-charged-seed615415

Recorded outputs: PASS_CENTRE_SHARED_KCALL_CHARGED k2_charged_ceiling 0.003215381840 k2_ratio 1.902532 H_bit_displace 1650 H_bit_additive 31 H_any_additive 32 no_new_kappa, PASS_PACKAGE_METADATA_AND_PINS, PASS_RESEARCH_COMMIT_BYTES, and PASS_REGISTERED_RESEARCH_PACKAGE research/centre-shared-kcall-charged-seed615415 outcome=completed from the registered 3.12 replay.

Open next questions

1. Is any selected call pair eligible for a fused shared-sink window (shared post-A rows, identical return operator and chart, uninterrupted window)? First step: extract the backward centre-phase invocations of two sink-sharing calls from the selected replay schedule. Blocked by: owner map and schedule are not in the package; the guard leaves eligibility open.
2. Which selected call pairs have coupled rather than independent demands? First step: classify the 144 selected calls by shared sink rows and return operator. Blocked by: no machine-readable demand-coupling table exists.
3. Does the fused primitive bill at width mu+1 fit the selected invoice guard? First step: instantiate the pinned fused bill under the selected invoice formulas. Blocked by: actual tariffs and precision headroom are uninstantiated.

Credit and attribution

Sharing constructions: PR 1708 "Finite centre-phase fanout across two shared-source panels" by 1337isnot1337 (Cinna Davis) and PR 1748 "Certify shared reverse centre phase with a charged precision guard" by ToikY (both authors verified against current GitHub metadata at writing time). The pinned two-panel contract, its exact bill and its one-magnitude-bit precision guard are ToikY's research/centre-shared-return-guard, used unchanged as a pinned input. The packing-ceiling model, lever table and coverage identity are Douglas Colkitt's research/complex-local-word-ceiling and PR 1012 "Research note: restoration deferral is mass-neutral in the hosted T37 recurrence (no new kappa)" by thakrarsagar. Additive overhead budgets and the bit endpoint are research/centre-sharing-overhead-budget (prepared for rtyu193-rgb with Claude Code assistance). The hosting lemma is research/idle-axis-hosting. The 64k floor for independent demands is research/p112-batch-centre-floor, crediting Mihir Srivastava's PR 845, Adam Gall's PR 1098 and woklerx's PR 1180. Prepared by Aaron Wang with help from opencode (substantial AI assistance; no model named; no independent human review claimed). Apache-2.0.