Independently check #4568's arithmetic (κ = 0.00302016597): outer ledger, gap, absorption; frozen rank 6.9x smaller with #4565 (no new κ)

research/pr4568-arithmetic-check/.gitattributes

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+PR4568-RESULT.json -text

research/pr4568-arithmetic-check/PR4568-RESULT.json

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+{
+  "BIT_admission": {
+    "admitted": "17/5000",
+    "after_cover": "69/20000",
+    "atom": "1/250",
+    "before_cover": "3459/1000000",
+    "ordinary_saving": "8590884599/2500000000000",
+    "prime_rule": "odd q > 2^26*N^5 avoiding all fixed denominators and rank witnesses"
+  },
+  "absorption": {
+    "conclusion": "A*N^theta*(133+logN)^(2/5)<=N^t above this finite cutoff; physical implementation overhead not inferred",
+    "cost_all_larger_N_monotonicity": true,
+    "cost_margin_evaluation_N": "2^100000000000000",
+    "cost_margin_lower": "418/15",
+    "logA_upper": "11588535078387598648766751054204368156120388952774219/850705917302346158658436518579420528640000000000000",
+    "log_threshold": 100000000000000,
+    "margin_evaluation_log_N": 100000000000000,
+    "margin_lower": "62512649175114201706254157498695754664126611047225781/850705917302346158658436518579420528640000000000000",
+    "margin_scope": "margin_lower is evaluated at log(N)=log_threshold; cost_margin_lower is the conservative certified margin at primary_certified_N. The two margins use different cutoffs.",
+    "positive_gap": "1/1000000000000",
+    "primary_certified_N": "2^100000000000000",
+    "secondary_conservative_N_cutoff": "2^200000000000000"
+  },
+  "critical_root": {
+    "beta": "141/250",
+    "conclusion": "Unique crossing on this bracket; critical-root hypotheses hold.",
+    "critical_root_interval": [
+      "9969798340229/10000000000000",
+      "996979834023/1000000000000"
+    ],
+    "derivative_floor": "1",
+    "endpoint_target_minus_paid": [
+      [
+        "-1/1000000",
+        "0"
+      ],
+      [
+        "0",
+        "1/1000000"
+      ]
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+    "funded_mirror_cells": [
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+        "upper_ratio": "141309/500000"
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+        "upper_ratio": "37509/250000"
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+        "upper_ratio": "3127/250000"
+      },
+      {
+        "left": "1/5",
+        "right": "1/4",
+        "upper_ratio": "13237/1000000"
+      }
+    ],
+    "maximizing_cell": [
+      "1/48",
+      "1/41"
+    ],
+    "mirror_upper": "562811/1000000",
+    "scope": "Rank-inventory admission criterion; native and outer obligations inherited.",
+    "source_capacity_lower": "491053/500000",
+    "uniform_mean_gap_lower": "521143/500000"
+  },
+  "finite_envelope": {
+    "G_depth_upper": 20000000000000,
+    "class_envelope": "C_N <= W0*(W0+P0+1)^(G_depth+1) < 2^340000000000031",
+    "divisible_width_bound": "B_N < C_N^N",
+    "full_call_centre_episodes_per_raw_unit": 144,
+    "height_base": 1024,
+    "height_multiplier": 2,
+    "log2_C_N_strict_upper": 340000000000031,
+    "owned_space_scope": "144U and1680U bound separate raw episodes, not total live scratch. Actual compiler/parking allocations require their own O ledger.",
+    "primary_N": "2^100000000000000",
+    "proof_scope": "Height/class recurrence, denominator clearing and positive-surplus occurrence inequalities are inherited proof premises. Only their displayed integer arithmetic is checked; no gigantic word or B_N is materialized.",
+    "raw_occurrence_bound": "U <= B_N*N/752",
+    "raw_paid_jobs": 64620,
+    "raw_productive_fields": 10928,
+    "raw_rank_surplus": 752,
+    "thin_query_episodes_per_raw_unit": 1680
+  },
+  "functional_checks": 4,
+  "kappa": "302016597/100000000000",
+  "minimum_outer_margin_above_kappa": "2996979834027/1000000000000000000000000",
+  "outer_checks": 54,
+  "outer_parameters": {
+    "a": "1510082987/500000000000",
+    "b": "120806639/40000000000",
+    "beta": "999999999997/1000000000000",
+    "bit_saving": "17/5000",
+    "complex_uniform_saving": "377520747/125000000000",
+    "d": "1/1000000000000",
+    "epsilon": "999999999999/1000000000000",
+    "kappa": "302016597/100000000000",
+    "layout": "1/1000000000000",
+    "q": "3020165973/1000000000000",
+    "r0": "1/10000000000000",
+    "stop": "1/1000000000000",
+    "tested_saving": "3020165977/1000000000000"
+  },
+  "physical_bridge": "Conditional proof premise: retained native, movement, precision, stock and fixed-tape closure must apply to the fixed expanded word; this arithmetic check does not establish physical admission.",
+  "rejected_mutations": {
+    "closedrootgap": "closed root gap",
+    "kappa_equals_uniform": "raw kappa lacks uniform slack",
+    "log_absorption_failure": "log absorption derivative",
+    "nativeprofile": "native profile altered"
+  },
+  "retained_root_applications_replayed": [
+    "p7",
+    "p16"
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+}

research/pr4568-arithmetic-check/README.md

@@ -0,0 +1,44 @@
+# Independent arithmetic check of #4568 (conditional κ = 0.00302016597)
+
+**Task:** independent-moment (independent source-bound coverage). **Base:** canonical main `f626db5c`. **No new κ** of my own.
+
+hexapode's #4568 (head `b269a729`) raises #4535's conditional κ by 9.7·10⁻¹⁰, to **0.00302016597**. It uses my #4564's outer slack `d = 10^−12` together with the unchanged #4470 profile, #4535's critical-root bracket and #4532's majorant. The new outer point is `z = .996979834024`, with `q, a, b = κ + 3d, 4d, 5d`, `s_C = κ + 6d` and tested saving `κ + 7d`. The frozen word is at `N = 2^(10^14)`. #4568's `RESULT.json` is vendored (blob `540ffb95`, SHA256-pinned).
+
+## Results
+
+All checked claims are confirmed.
+
+- **Outer ledger.** I replayed the exact source that #4535 and #4568 pin (`pr4359-p7-padding/.../endpoint.py`, SHA256-checked) at #4568's parameters, which are confirmed to be exactly `κ + jd`.
+  - All 54 strict predicates, the 4 functional guards and the 9 margins above κ pass.
+  - The minimum margin is `2996979834027/10^24`, matching #4568's record.
+- **Saving order.** `κ < s_C < tested saving = 1 − θ_hi < 1 − θ`. The tested saving equals the bracket endpoint `1 − θ_hi` exactly. It stays strictly below the root's saving only because the certified root lies strictly inside #4535's open bracket, which the bracket certificate guarantees. The absorption gap is `z − θ_hi = 10^−12` exactly.
+- **Absorption.** At `N = 2^(10^14)`, my certified margin is 42.94 nats, above #4568's conservative 418/15. The least certified ranks at #4568's κ are:
+
+| majorant | least `K` (`N = 2^K`) | vs #4568's `10^14` |
+|---|---|---|
+| #4532 witness, #4568's coarse `ln A` | 37,477,695,815,212 | 2.67× smaller |
+| #4532 witness, `A`'s own definition | 32,523,485,165,724 | 3.07× smaller |
+| **my #4565 retuned witness** (`M = 2^24`, `ℓ0 = 64`, `p = 3/10`, `c = 101`) | **14,562,291,820,122** | **6.87× smaller** |
+
+So #4568's frozen word can be built at a rank exponent at least 6.9× smaller, at the same κ, by combining it with #4565. Each `K` is the least certified one, and the inequality holds monotonically beyond it.
+
+I also replayed #4568's registered package: `PASS_REGISTERED_RESEARCH_PACKAGE research/critical-slack-multiplication outcome=completed` in 8.4 s.
+
+## Verify
+
+```text
+python3 -B research/pr4568-arithmetic-check/verify.py
+```
+
+This runs in about 3 s with the standard library only.
+
+## Limitations
+
+- **Not covered.** The critical-root bracket, mirror and #4532 gates are #4535's and are not recomputed here; my #4536 and #4569 cover them. Native, physical and outer-cutoff premises are out of scope, as in #4568.
+- **Admission.** Whether a κ gain obtained by retuning outer slack against the same raw root counts as a hill-climb step is a maintainer decision. Per #4568 itself, the native paid profile and raw root are unchanged. This package takes no position.
+
+## Open next questions
+
+- None new from the arithmetic. Whether a slack-retuned κ is admitted is for the maintainers.
+
+Credit: hexapode (#4568, #4535), Dillon (#4532), Cinna Davis (#4470), zxv and Jacob Alan Sussman. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

research/pr4568-arithmetic-check/SOURCES.json

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