Planning result: exact overhead budgets for centre-path sharing on the actual selected profiles (no new kappa)

research/centre-sharing-overhead-budget/README.md

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+# Centre-path sharing: exact overhead budgets on the actual selected profiles
+
+**Planning result for the `centre-path-sharing` card. No construction, no physical sharing is
+admitted, and no new κ.**
+
+[PR1708](https://github.com/CrocSwap/integer-mult-bounds/pull/1708) and
+[PR1748](https://github.com/CrocSwap/integer-mult-bounds/pull/1748) construct the two halves of a
+two-call shared centre phase at the scalar-macro level. Their open blocker is the overhead ΔH:
+preparation, routing, restoration, ownership and hosting. PR1708 also screened the favorable
+relaxation: sharing lifts the complex cap to about 0.00321, the bit supplier then binds, and the
+whole-bound gain is at most 3.4536%.
+
+This package prices the lever on the **actual** selected #1094 histograms (its real idle
+remainder, not ideal packing). It gives sharp integer budgets for ΔH and states what the route
+needs for the card's 10% target.
+
+## Model
+
+Sharing one centre path across `k` calls keeps every child of the selected complex histogram
+except the shared rank-8 centre children. Their count goes from 1,152 to 1,152/k, and the coverage
+identity becomes `72W + (64/k)N`; this is checked exactly. Overhead is modeled as extra paid
+children of a single rank `r`.
+
+The moment is linear in child counts. Each budget is therefore a **sharp integer**: that many
+extra children pass the exact moment test at the stated point, and one more fails. Exact moment
+enclosures come from Navinor's
+[selected-suppliers-de-diagnostic](../selected-suppliers-de-diagnostic/README.md). κ uses the
+retained seven-level bootstrap, with all 47 outer slacks checked, from Wronskode's
+[seed81 recheck](../frontier1042-invoice-audit-seed81/README.md). Both are loaded unchanged as Git
+blobs at the base.
+
+## Results
+
+**1. With any sharing, the complex side stops binding.** At the bit endpoint
+`c = 0.001733427039589844234749`, the actual-profile residual `f(c) − 1` is:
+
+| k | rank-8 children | centre charge per call | f(c) − 1 | Rank-1 budget (per call) | Rank-8 budget (per call) |
+|---:|---:|---:|---:|---:|---:|
+| 2 | 576 | 32 | −4.25·10⁻³ | 4,415 children (30.66) | 553 children (3.84) |
+| 4 | 288 | 16 | −6.46·10⁻³ | 6,710 (46.60) | 841 (5.84) |
+| 8 | 144 | 8 | −7.57·10⁻³ | 7,858 (54.57) | 985 (6.84) |
+| 16 | 72 | 4 | −8.12·10⁻³ | 8,432 (58.56) | 1,057 (7.34) |
+| 144 | 8 | 4/9 | −8.61·10⁻³ | 8,942 (62.10) | 1,121 (7.78) |
+
+The two budget columns are overhead children (record total, then per call) that keep the full
+bit-limited gain.
+
+With `k = 2` the result is bit-limited, `κ = 865213734911801/5·10¹⁷ = 0.001730427469823602`
+(+3.4536%). This reproduces PR1708's figure on the actual histogram.
+
+**2. ΔH budget for two-call sharing.** Sharing saves 32 rank-units of paid mass per call. Of that:
+
+* **Full +3.4536% kept:** ΔH may consume about **30.5–30.8 units per call** at any overhead rank
+  from 1 to 72. For example: 4,415 rank-1, 553 rank-8 or 92 rank-48 children in total.
+* **Any gain over the selected record:** ΔH must stay below about **31.9–32.0 units per call**.
+* **Exact break-even control:** re-adding 575 rank-8 children still gains; 576, which recreates
+  the selected histogram, does not.
+
+The full table is in [RESULT.json](RESULT.json).
+
+**3. Why the budget is so large.** The complex side currently binds by only **1.23–1.33 paid-mass
+units per call**. Removing 177 rank-1, 23 rank-8 or 4 rank-48 children from the selected record
+makes the bit side bind. Any mechanism netting that much on the complex side reaches the same
++3.45% cap.
+
+**4. The card's stop rule triggers on its own.** Even ideal sharing gains under 10% on the whole
+bound. For **+10%** (κ ≥ 1.1 × selected) with the complex side ample, the **bit endpoint must
+reach `921658892752012102303/5·10²³ ≈ 0.0018433`**, 1.0634× its current value.
+
+On the cited PR589 bit histogram (deficit 10,272, `T = 32`), that needs a net paid-mass reduction
+of at least **649 units**:
+
+* as 649 rank-1 children, about 20.3 per replica; or
+* as 16 rank-42 children, 672 units.
+
+The bad-residue surcharge is recomputed from the reduced child count. So centre sharing is
+worthwhile only jointly with a bit-side mechanism of roughly that size.
+
+**5. Scalar overhead is not the constraint.** Under the selected invoice formulas, each extra
+local macro (or child) adds 16 adapters at the compiler constant, 414,949,376 primitives. The
+binding finite guard is `local_primitive_bound < 2⁴⁸`, currently 10,038,040,553,630. It leaves
+room for at most **654,144 extra macros per call**, before counting the new macros' own
+primitives. PR1748's literal two-call reverse phase is 384 macros.
+
+## Reproduce
+
+Standard library and `git` (history for the base commit), from the repository root:
+
+```sh
+python3 scripts/research_package.py --package research/centre-sharing-overhead-budget --check-only
+python3 -I -B research/centre-sharing-overhead-budget/verify.py
+```
+
+Expected output, after about 60 s:
+
+```text
+PASS_CENTRE_SHARING_OVERHEAD_BUDGET k2_full_gain_budget_rank8=553 bit_limited_kappa=0.001730427469824 bit_endpoint_for_10pct=0.0018433178 no_new_kappa
+```
+
+The run regenerates and compares [RESULT.json](RESULT.json); `--write` rewrites it. Controls:
+
+* the selected complex and bit endpoints are accepted, and their next grid points rejected;
+* the selected κ is recomputed;
+* the bit surcharge reconstruction matches the ledger;
+* the sharpness of every budget is checked;
+* the break-even control above.
+
+The script refuses to run under `-O`.
+
+## Scope and limits
+
+This is a favorable planning relaxation. It does not establish:
+
+* that any selected call pair has common post-A sinks, charts or uninterrupted ownership
+  (PR1748's first blocker);
+* native precision or tape for fused primitives at μ+1;
+* hosting or admission;
+* that ΔH appears only as paid children of one rank.
+
+Overhead that changes the idle histogram, the coverage identity or the outer recipe is outside
+the model. Bit-side numbers are removals from the cited PR589 histogram at fixed `m`, `W` and
+surcharge law, not a bit construction.
+
+Inherited all-size/compiler, precision, analytic and H1–H5 obligations are unchanged. This was run
+on Windows with CPython 3.14; the registered 3.12 replay was not run locally.
+
+## Open next questions
+
+1. **Can a bit-side mechanism net ≥ 649 paid-mass units?** That is about 20.3 per replica, or
+   deficit 10,272 → ≥ 10,921.
+   * *Why it matters:* together with any complex saving of ≥ 1.33 units per call (two-call centre
+     sharing gives up to 32), it would reach the card's +10% target.
+   * *First step:* price the bit-side levers of `bit-revision3-levers-seed03` at the endpoint
+     `0.0018433` with this package's sharp-removal method.
+   * *Blocked by:* no bit construction with that saving is known, and the bit copy charge and
+     full-frame helper floors are scoped obstructions there.
+
+## Credit
+
+* **Sharing constructions:** 1337isnot1337 (PR1708) and ToikY (PR1748). This package prices
+  their lever and admits neither.
+* **Reused code:** Navinor (exact moment enclosure) and Wronskode (bootstrap and outer formulas),
+  both unchanged.
+* **Lever framing and coverage identity:** Douglas Colkitt (`complex-local-word-ceiling`) and
+  PR1012.
+* **Selected construction:** Rohan Arun (#1094), with the credited k8/T16 and supplier lineage.
+
+Prepared for rtyu193-rgb with Claude Code (Anthropic Claude Opus 5.5) in one session, with no
+subagents and no independent human review. Apache-2.0.

research/centre-sharing-overhead-budget/RESULT.json

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+{
+ "complex_saving_to_reach_bit_cap": [
+  {
+   "children_removed": 177,
+   "mass_per_call": 1.2292,
+   "rank": 1
+  },
+  {
+   "children_removed": 23,
+   "mass_per_call": 1.2778,
+   "rank": 8
+  },
+  {
+   "children_removed": 4,
+   "mass_per_call": 1.3333,
+   "rank": 48
+  }
+ ],
+ "controls": [
+  "complex endpoint accepted, next grid rejected",
+  "bit endpoint accepted, next grid rejected",
+  "selected kappa recomputed",
+  "undoing the sharing (+4 rank-8 per call) is exactly break-even"
+ ],
+ "k_sharing": [
+  {
+   "centre_charge_per_call": "32",
+   "complex_above_bit": true,
+   "k": 2,
+   "overhead_budget_full_gain": {
+    "1": {
+     "mass_per_call": 30.65972222222222,
+     "per_call": "4415/144",
+     "total": 4415
+    },
+    "8": {
+     "mass_per_call": 30.72222222222222,
+     "per_call": "553/144",
+     "total": 553
+    }
+   },
+   "rank8_children": 576,
+   "residual_at_bit_endpoint": -0.004250247071287914
+  },
+  {
+   "centre_charge_per_call": "16",
+   "complex_above_bit": true,
+   "k": 4,
+   "overhead_budget_full_gain": {
+    "1": {
+     "mass_per_call": 46.59722222222222,
+     "per_call": "3355/72",
+     "total": 6710
+    },
+    "8": {
+     "mass_per_call": 46.72222222222222,
+     "per_call": "841/144",
+     "total": 841
+    }
+   },
+   "rank8_children": 288,
+   "residual_at_bit_endpoint": -0.006460266623503219
+  },
+  {
+   "centre_charge_per_call": "8",
+   "complex_above_bit": true,
+   "k": 8,
+   "overhead_budget_full_gain": {
+    "1": {
+     "mass_per_call": 54.56944444444444,
+     "per_call": "3929/72",
+     "total": 7858
+    },
+    "8": {
+     "mass_per_call": 54.72222222222222,
+     "per_call": "985/144",
+     "total": 985
+    }
+   },
+   "rank8_children": 144,
+   "residual_at_bit_endpoint": -0.007565276399610871
+  },
+  {
+   "centre_charge_per_call": "4",
+   "complex_above_bit": true,
+   "k": 16,
+   "overhead_budget_full_gain": {
+    "1": {
+     "mass_per_call": 58.55555555555556,
+     "per_call": "527/9",
+     "total": 8432
+    },
+    "8": {
+     "mass_per_call": 58.72222222222222,
+     "per_call": "1057/144",
+     "total": 1057
+    }
+   },
+   "rank8_children": 72,
+   "residual_at_bit_endpoint": -0.008117781287664697
+  },
+  {
+   "centre_charge_per_call": "4/9",
+   "complex_above_bit": true,
+   "k": 144,
+   "overhead_budget_full_gain": {
+    "1": {
+     "mass_per_call": 62.09722222222222,
+     "per_call": "4471/72",
+     "total": 8942
+    },
+    "8": {
+     "mass_per_call": 62.27777777777778,
+     "per_call": "1121/144",
+     "total": 1121
+    }
+   },
+   "rank8_children": 8,
+   "residual_at_bit_endpoint": -0.008608896743712543
+  }
+ ],
+ "scalar_overhead": {
+  "adapter_growth_per_extra_macro": 414949376,
+  "extra_macros_per_call_upper_bound": 654144,
+  "guard": "< 2^48",
+  "local_primitive_bound": 10038040553630,
+  "note": "each extra macro or child adds 16 adapters at the compiler constant; primitive counts of the new macros reduce this further"
+ },
+ "status": "PASS_CENTRE_SHARING_OVERHEAD_BUDGET",
+ "ten_percent_target": {
+  "bit_deficit": 10272,
+  "bit_endpoint_needed": "921658892752012102303/500000000000000000000000",
+  "bit_endpoint_ratio": 1.0633950800376242,
+  "bit_saving_needed": [
+   {
+    "children_removed": 649,
+    "mass": 649,
+    "mass_per_replica": 20.281,
+    "rank": 1
+   },
+   {
+    "children_removed": 16,
+    "mass": 672,
+    "mass_per_replica": 21.0,
+    "rank": 42
+   }
+  ],
+  "kappa_target": "9199631083923049/5000000000000000000",
+  "note": "surcharge recomputed from the reduced child count"
+ },
+ "two_call": {
+  "bit_limited_kappa": "865213734911801/500000000000000000",
+  "bit_limited_kappa_decimal": "0.001730427469823602",
+  "budgets": [
+   {
+    "any_gain_mass_per_call": 31.8819,
+    "any_gain_total": 4591,
+    "full_gain_mass_per_call": 30.6597,
+    "full_gain_total": 4415,
+    "rank": 1
+   },
+   {
+    "any_gain_mass_per_call": 31.9167,
+    "any_gain_total": 2298,
+    "full_gain_mass_per_call": 30.6944,
+    "full_gain_total": 2210,
+    "rank": 2
+   },
+   {
+    "any_gain_mass_per_call": 31.9375,
+    "any_gain_total": 1533,
+    "full_gain_mass_per_call": 30.7083,
+    "full_gain_total": 1474,
+    "rank": 3
+   },
+   {
+    "any_gain_mass_per_call": 31.9444,
+    "any_gain_total": 1150,
+    "full_gain_mass_per_call": 30.7222,
+    "full_gain_total": 1106,
+    "rank": 4
+   },
+   {
+    "any_gain_mass_per_call": 31.9444,
+    "any_gain_total": 920,
+    "full_gain_mass_per_call": 30.7292,
+    "full_gain_total": 885,
+    "rank": 5
+   },
+   {
+    "any_gain_mass_per_call": 31.9444,
+    "any_gain_total": 575,
+    "full_gain_mass_per_call": 30.7222,
+    "full_gain_total": 553,
+    "rank": 8
+   },
+   {
+    "any_gain_mass_per_call": 32.0,
+    "any_gain_total": 288,
+    "full_gain_mass_per_call": 30.7778,
+    "full_gain_total": 277,
+    "rank": 16
+   },
+   {
+    "any_gain_mass_per_call": 32.0,
+    "any_gain_total": 144,
+    "full_gain_mass_per_call": 30.6667,
+    "full_gain_total": 138,
+    "rank": 32
+   },
+   {
+    "any_gain_mass_per_call": 32.0,
+    "any_gain_total": 96,
+    "full_gain_mass_per_call": 30.6667,
+    "full_gain_total": 92,
+    "rank": 48
+   },
+   {
+    "any_gain_mass_per_call": 31.875,
+    "any_gain_total": 90,
+    "full_gain_mass_per_call": 30.8125,
+    "full_gain_total": 87,
+    "rank": 51
+   },
+   {
+    "any_gain_mass_per_call": 32.0,
+    "any_gain_total": 64,
+    "full_gain_mass_per_call": 30.5,
+    "full_gain_total": 61,
+    "rank": 72
+   }
+  ],
+  "gain_percent": 3.4536167505890347
+ }
+}