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

research/centre-shared-kcall-charged-seed615415/README.md

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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).
+No shared centre is constructed or admitted, and no new kappa is claimed.**
+
+The card asks whether the 64-per-call centre path (eight rank-8 COPY/ERASE totals
+through the private temporary) can be shared across calls or shrunk, and its
+construction method asks for a fully charged favourable relaxation for sharing
+one centre path across k calls, including restoration, hosting and precision.
+This package supplies that relaxation for the one sharing pattern whose contract
+is pinned: the shared-sink return window of
+[centre-shared-return-guard](../centre-shared-return-guard/README.md)
+(PR1708 forward fanout / PR1748 reverse phase lineage), generalised from two
+panels to k panels. The three named costs are charged explicitly instead of
+being left as free overhead, and the ceiling is computed with
+`ceiling.ceiling(centre=...)` from [complex-local-word-ceiling](../complex-local-word-ceiling/ceiling.py).
+
+## Conversion contract
+
+The pinned contract's literal backward centre-phase bill is 16 rank-8 returns,
+368 ADDs and 32 CLEARs for two literal phases, and 8 rank-8 returns, 360 ADDs
+and 16 CLEARs for the two-call fused window (its `RESULT.json` counts; the
+k = 1 and k = 2 formulas below reproduce all six numbers exactly). One fused
+k-call window gathers k panels before one ascent per centre and writes the
+shared sink once. The conversion to the ceiling model's mass units is:
+
+| Macro or obligation | Charge |
+| --- | --- |
+| One rank-8 COPY/ERASE total | one rank-8 child, 8 mass units (ascent and erase inside) |
+| One ADD | one rank-1 child |
+| One scratch CLEAR | one rank-1 child (restoration of the private temporary) |
+| Hosting of shared rows across the window | 0 mass, idle-axis-hosting lemma |
+| Precision: k summed panels | `ceil(log2 k)` magnitude bits on 8 shared totals, each bit one rank-1 unit per total 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 per
+call. The k = 2 row matches levers.py's ideal lever "shared by two calls
+(copies and charge halved)", and the charged profile exceeds it slightly because
+the ADD and scratch-CLEAR savings are credited too.
+
+Restoration is the scratch CLEAR set (charged) plus the contracted zero scratch
+endpoint; scratch dirt is not promised to return to its incoming value. Hosting
+is charged zero mass on the idle-axis-hosting lemma, on the contract's
+constraints: the shared rows, panels and scratch must be untouched across the
+window (no source or panel mutation, observation or required canonical
+checkpoint may intervene). Precision is charged at the derived bit count
+(`additional_magnitude_bits_sufficient = 1` for k = 2 in the pinned guard;
+`ceil(log2 k)` for k panels by envelope summation).
+
+## Results: changed centre charge, copy profile and charged ceiling
+
+All values exact; ceilings are certified brackets at grid `1e-12`; ratios are
+against the baseline ceiling `0.001690054109...`. `kappa` uses the retained
+planning transfer `kappa < a/(1+a)`.
+
+| k | centre | copy children (rank-8/call) | restoration (scratch CLEAR/call) | hosting | precision (rank-1/call) | rank-1/call | gross ceiling | fully charged ceiling | kappa cap | ratio |
+| ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
+| 1 | 64 | 8 | 8 | 0 | 0 | 2412 | 0.001690054110 | 0.001690054110 | 0.001687202646 | 1.000 |
+| 2 | 32 | 4 | 4 | 0 | 4 | 2408 | 0.003217794406 | 0.003215381840 | 0.003205076296 | 1.903 |
+| 4 | 16 | 2 | 2 | 0 | 4 | 2404 | 0.003984054054 | 0.003981062728 | 0.003965276713 | 2.356 |
+| 8 | 8 | 1 | 1 | 0 | 3 | 2401 | 0.004367783311 | 0.004365321517 | 0.004346348309 | 2.583 |
+| 16 | 4 | 1/2 | 1/2 | 0 | 2 | 2399 | 0.004559798056 | 0.004558083777 | 0.004537401919 | 2.697 |
+
+The k = 1 row is the selected word and reproduces the published baseline
+ceiling exactly. The k = 2 fully charged ceiling is `a < 0.003215381840`,
+`kappa < 0.003205076296`; charging restoration, hosting and precision moves the
+gross ceiling by only 0.075 per cent. Rows k >= 4 use the derived k-panel
+generalisation of the pinned two-panel contract, not a pinned construction.
+
+## Break-even overhead at k = 2 (rank-1 children per call)
+
+The named charges are not the binding constraint; how much *additional*
+unmodelled overhead the window can absorb is. Two treatments, both sharp
+(H passes the exact moment test at the threshold and H+1 fails):
+
+| Treatment | keep full bit-limited gain (at T2) | 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 |
+
+Under the ceiling model's own accounting (extra children displace idle mass at
+rank 51), consuming the *entire* idle remainder of 1650 rank-1 units per call
+still leaves the ceiling above the complex-unbinding point
+`T2 = 1733427039589844234749/(10^24 - 1)` and above the selected endpoint: the
+break-even is capped by coverage-identity feasibility, not by cost. Under the
+additive treatment of [centre-sharing-overhead-budget](../centre-sharing-overhead-budget/README.md)
+(overhead as pure moment cost), at most 31 rank-1 children per call fit before
+the full bit-limited gain is lost and 32 before any gain remains; their sharp
+budgets on the identity profile are 4415/144 = 30.66 and 4591/144 = 31.88 per
+call. The verifier recomputes both under its ideal-packing relaxation (30.98 and
+32.19, ratios 1.010 and 1.010) and the residual difference is the ideal-packing
+versus actual-histogram moment, not a shared term.
+
+## Screen and what the relaxation decides
+
+The card's first experiment screens the route on the levers.py scale: the ideal
+shared-by-two ceiling ratio is 1.90 against the 1.10 stop threshold, so the
+route survives the screen. The whole-bound gain is a different quantity: the bit
+supplier binds at `c_bit = 0.001733427039589844234749`, so any mechanism whose
+complex ceiling reaches that point gains at most the bit-limited
+`kappa = 865213734911801/5*10^17 = 0.001730427469823602` (+3.4536 per cent over
+the selected record). The fully charged k = 2 window clears that point with the
+wide margins recorded above, so **for this pattern the cost side of sharing
+(restoration, hosting, precision) is not the blocker**. The blockers are
+eligibility and demand coupling, below.
+
+## Scope and limits
+
+- This is a favourable relaxation for one sharing pattern: a fused window whose
+  k calls share sink rows, return operator and chart. It grants eligibility.
+  [centre-shared-return-guard](../centre-shared-return-guard/README.md) supplies
+  neither a full pair of fused local calls nor an eligible selected owner pair;
+  [p112-batch-centre-floor](../p112-batch-centre-floor/README.md) proves that
+  independent root-code demands cannot share at all (copied-pairing-rank bill
+  `64k`), so the pattern needs genuinely coupled demands.
+- The conversion contract (macro to rank-mass) is stated, not inherited: each
+  scalar total pays one child at its pairing rank. Changing it 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/erase returns (no saving).
+- Precision is charged at the sufficient bit count for positive envelopes only.
+  The pinned guard explicitly does not establish rounding-error, native guard,
+  tape or global admission, and the fused primitives at width `mu+1` price at
+  the invoice level (outside this mass model).
+- The consumer and central readout are unchanged: the fused window writes the
+  same endpoint rows as the separate phases (linearity of the shared-sink
+  write), so the readout count is neither saved nor added.
+- No new kappa, supplier admission or eligible pair is claimed; the selected
+  record is unchanged. Fractional child counts are allowed by the relaxation,
+  as in the incumbent ceiling model.
+
+## Reproduce
+
+Standard library only, from the repository root (Python 3.12):
+
+```sh
+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
+```
+
+Expected first output: `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`.
+`verify.py --write` regenerates `RESULT.json`; the normal run requires the
+committed file to equal the recomputation byte for byte. Controls (each enforced,
+not advisory):
+
+- the 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;
+- the 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, and a doubled
+  precision charge lowers it again;
+- every break-even is sharp, or explicitly capped by identity feasibility;
+- the additive identity-profile budgets stay within a factor of two of
+  centre-sharing-overhead-budget (measured 1.010 and 1.010).
+
+## Open next questions
+
+- **Is any selected call pair eligible for a fused window?** Trace one candidate
+  pair through the selected owner map and call schedule: shared post-A sink
+  rows, identical return operator and chart, and an uninterrupted window with no
+  intervening observation or checkpoint. *Why it matters:* the relaxation shows
+  cost is not the blocker for this pattern; eligibility is. *First step:*
+  extract the backward centre-phase invocations of two calls that share sink
+  rows from the selected replay schedule. *Blocked by:* the owner map and
+  schedule are not in this package; the guard leaves this open explicitly.
+- **Which selected call pairs have coupled rather than independent demands?**
+  *Why it matters:* the `64k` copied-pairing-rank floor of p112-batch-centre-floor
+  blocks sharing for independent root-code demands; only coupled (shared-sink)
+  demands can use this pattern. *First step:* classify the 144 calls of the
+  selected record by shared sink rows and return operator. *Blocked by:* a
+  machine-readable demand-coupling table for the selected record.
+- **Does the fused primitive tariff fit the scalar guard?** The guard prices
+  fused primitives at `mu+1`; centre-sharing-overhead-budget leaves about
+  654,144 extra macros per call of primitive headroom, and the fused window adds
+  none (it removes 32 macros per two calls). *Why it matters:* it would close
+  the invoice-level obligation the mass model cannot see. *First step:*
+  instantiate the two-panel fused bill under the selected invoice formulas.
+  *Blocked by:* the actual tariffs and precision headroom are uninstantiated.
+
+## Credit
+
+The sharing constructions are 1337isnot1337's PR1708 (forward fanout) and
+ToikY's PR1748 (reverse phase). The pinned two-panel contract, its exact bill
+and its one-magnitude-bit precision guard are ToikY's
+[centre-shared-return-guard](../centre-shared-return-guard/README.md); its
+`RESULT.json` counts are used unchanged as a pinned input. The packing-ceiling
+model, the lever table and the coverage identity are Douglas Colkitt's
+[complex-local-word-ceiling](../complex-local-word-ceiling/README.md) and
+PR1012. The additive overhead budgets and the bit endpoint are
+[centre-sharing-overhead-budget](../centre-sharing-overhead-budget/README.md)
+(prepared for rtyu193-rgb). The hosting lemma is
+[idle-axis-hosting](../idle-axis-hosting/PROOF.md). The `64k` floor for
+independent demands is [p112-batch-centre-floor](../p112-batch-centre-floor/README.md).
+Prepared by Aaron Wang with help from opencode (substantial AI assistance; no
+model named; no independent human review claimed). Apache-2.0.