Independently check #4535's arithmetic (κ = 0.003020165): critical root, gap 377e-12, mirror, absorption, binding outer row (no new κ)

research/pr4535-arithmetic-check/.gitattributes

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+RESULT-4535.json -text

research/pr4535-arithmetic-check/README.md

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+# Independent arithmetic check of #4535 (conditional κ = 0.003020165)
+
+**Claim type:** an independent arithmetic check with certified intervals. **No new κ.** It covers hexapode's
+[#4535](https://github.com/CrocSwap/integer-mult-bounds/pull/4535) (head `cca1112d`). That PR combines
+dillona's #4532 critical-root compiler with #4470's unchanged p16 supplier, and proposes conditional κ
+= 0.003020165, up from 0.00302015. #4535's `RESULT.json` is vendored here (SHA256-pinned, `-text`). The p16
+profile is read from main.
+
+## What is checked
+
+1. **The critical exponent.** `[θ_lo, θ_hi] = [0.9969798340229, 0.996979834023]` brackets the crossing
+   `T_θ = P_θ`, certified on both sides. It equals exactly 1 minus the raw-root bracket
+   `(0.0030201659770, 0.0030201659771)` that my #4478 certified for #4470.
+2. **The saving order.** κ = 604033/200000000 is below the uniform saving 3775207/1250000000, and
+   `t − θ_hi = 377/10¹²` exactly, with `t = 1 − uniform saving`.
+3. **The mirror condition at θ.** `E_J(θ_hi) ≤ β·S_J(θ_lo)` holds on every elementary window, with
+   `β = 141/250`. The worst ratio is **0.56281**, in the window (1/48, 1/41), matching #4535's maximizing cell.
+4. **Absorption.** `A·N^θ(133 + ln N)^{2/5} ≤ N^t`, with `ln A ≤ (1 − θ_lo)·ln(145·1024^{650}) = 13.6223`,
+   which is #4535's `logA_upper`:
+
+   | evaluation point | margin (lower) |
+   | --- | ---: |
+   | `ln N = 10¹²`, #4535's `log_threshold` | 352.33 |
+   | the stated `N = 2^{10¹²}` | 236.79 |
+
+   It is monotone beyond both points, since `t − θ > p/(133 + ln N)`.
+
+   **Labelling note.** #4535's `margin_lower`, 352.33, is computed at `ln N = 10¹²`, while its
+   `primary_certified_N` is `2^{10¹²}`, which has the smaller `ln N ≈ 6.93·10¹¹`. The inequality still holds
+   there, with margin 236.79, so the conclusion is unaffected.
+5. **The outer rows.** Seven of the nine rows exceed κ:
+
+   | Row | Formula |
+   | --- | --- |
+   | main transform | `εq` |
+   | global router | `a` |
+   | fused Gaussian | `1−ε+qβ` |
+   | group bit router | `1−ε+aβ` |
+   | group scans | `1−ε+β` |
+   | polynomial products | `ε` |
+   | micro-box faces | `79−ε−β` |
+
+   The **binding main-transform margin is exactly the reported 29969798347/10²⁰**. The parameters follow the
+   schedule `κ + 7d, 6d, 5d, 4d, 3d` and `1 − d`, `1 − 3d`, with `d = 10⁻¹⁰`. The two ξ rows are not recomputed,
+   because ξ is not printed among #4535's outer parameters.
+
+6. **The finite envelope.**
+   - The height `G = 2⌊log₁₀₂₄⌊N/145⌋⌋` is at most `2·10¹¹` at `N = 2^{10¹²}`.
+   - Since `W₀ = 10,928 < 2¹⁴` and `W₀ + P₀ + 1 = 75,549 < 2¹⁷`, the class envelope is
+     `C_N < 2^{17G+31} ≤ 2^{3400000000031}`.
+   - The episode counts are 144 = 18·8 (centre) and 1,680 = 15·112 (thin queries).
+
+   All of this matches #4535's `finite_envelope`.
+
+## Not covered
+
+- #4532's critical-root theorem: the leaf bound `L(n) ≤ A n^θ(c + log n)^p` and its gate criterion.
+- Owned storage, routing, precision, stock and tape bounds, beyond the displayed envelope arithmetic.
+- The native and physical premises.
+- The two ξ outer rows.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/pr4535-arithmetic-check --check-only
+python3.12 -B scripts/research_package.py --package research/pr4535-arithmetic-check
+```
+
+The verifier prints `PASS_PR4535_ARITHMETIC_CHECK` in about 10 seconds, using only the standard library.
+
+## Open next questions
+
+1. **Print ξ (`cap_xi`) in #4535's outer parameters, so all nine rows can be recomputed.**
+2. **An independent check of #4532's critical-root leaf bound.**
+
+## Credit
+
+- The construction is hexapode's (#4535).
+- The critical-root compiler is dillona's (#4532). The profile comes from #4470.
+
+Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.
+Apache-2.0.

research/pr4535-arithmetic-check/RESULT-4535.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_lower": "646/3",
+    "logA_upper": "11588535078387598648766751054204368156120388952774219/850705917302346158658436518579420528640000000000000",
+    "log_threshold": 1000000000000,
+    "margin_lower": "299725246478781048809002279469549139266283611047225781/850705917302346158658436518579420528640000000000000",
+    "positive_gap": "377/1000000000000",
+    "primary_certified_N": "2^1000000000000",
+    "secondary_conservative_N_cutoff": "2^2000000000000"
+  },
+  "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"
+      ]
+    ],
+    "funded_mirror_cells": [
+      {
+        "left": "1/72",
+        "right": "1/65",
+        "upper_ratio": "141309/500000"
+      },
+      {
+        "left": "1/65",
+        "right": "1/64",
+        "upper_ratio": "37509/250000"
+      },
+      {
+        "left": "1/64",
+        "right": "1/57",
+        "upper_ratio": "477989/1000000"
+      },
+      {
+        "left": "1/57",
+        "right": "1/56",
+        "upper_ratio": "303319/1000000"
+      },
+      {
+        "left": "1/56",
+        "right": "1/49",
+        "upper_ratio": "135741/250000"
+      },
+      {
+        "left": "1/49",
+        "right": "1/48",
+        "upper_ratio": "187981/500000"
+      },
+      {
+        "left": "1/48",
+        "right": "1/41",
+        "upper_ratio": "562811/1000000"
+      },
+      {
+        "left": "1/41",
+        "right": "1/40",
+        "upper_ratio": "25767/62500"
+      },
+      {
+        "left": "1/40",
+        "right": "1/33",
+        "upper_ratio": "140197/250000"
+      },
+      {
+        "left": "1/33",
+        "right": "1/32",
+        "upper_ratio": "430399/1000000"
+      },
+      {
+        "left": "1/32",
+        "right": "1/25",
+        "upper_ratio": "68367/125000"
+      },
+      {
+        "left": "1/25",
+        "right": "1/24",
+        "upper_ratio": "220287/500000"
+      },
+      {
+        "left": "1/24",
+        "right": "1/17",
+        "upper_ratio": "52779/100000"
+      },
+      {
+        "left": "1/17",
+        "right": "1/16",
+        "upper_ratio": "450273/1000000"
+      },
+      {
+        "left": "1/16",
+        "right": "1/9",
+        "upper_ratio": "101799/200000"
+      },
+      {
+        "left": "1/9",
+        "right": "1/8",
+        "upper_ratio": "60647/1000000"
+      },
+      {
+        "left": "1/8",
+        "right": "1/7",
+        "upper_ratio": "9019/500000"
+      },
+      {
+        "left": "1/7",
+        "right": "1/5",
+        "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": 200000000000,
+    "class_envelope": "C_N <= W0*(W0+P0+1)^(G_depth+1) < 2^3400000000031",
+    "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": 3400000000031,
+    "owned_space_scope": "144U and1680U bound separate raw episodes, not total live scratch. Actual compiler/parking allocations require their own O ledger.",
+    "primary_N": "2^1000000000000",
+    "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": "604033/200000000",
+  "minimum_outer_margin_above_kappa": "29969798347/100000000000000000000",
+  "outer_checks": 54,
+  "outer_parameters": {
+    "a": "15100827/5000000000",
+    "b": "6040331/2000000000",
+    "beta": "9999999997/10000000000",
+    "bit_saving": "17/5000",
+    "complex_uniform_saving": "3775207/1250000000",
+    "d": "1/10000000000",
+    "epsilon": "9999999999/10000000000",
+    "kappa": "604033/200000000",
+    "layout": "1/10000000000",
+    "q": "30201653/10000000000",
+    "r0": "1/100000000000",
+    "stop": "1/10000000000",
+    "tested_saving": "30201657/10000000000"
+  },
+  "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/rank-convergence/PROOF.md": "3c971c5670669cb36312815b111e64c83e1490a10ee82f5412bd9aee01e9c376",
+    "research/rank-convergence/README.md": "e77af882d3f02a9d3f2dd19dce889052f63892a1c8f39fdbe0813ff5fab5e8c1",
+    "research/t74-native-fast/vendor/native/new/code/borrow/vendor/clifford/clifford_exact.py": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962"
+  },
+  "status": "PASS_EXACT_CONDITIONAL_PARAMETER_CERTIFICATE",
+  "uniform_saving": "3775207/1250000000"
+}

research/pr4535-arithmetic-check/SOURCES.json

@@ -0,0 +1,24 @@
+{
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+  "base_commit": "f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc",
+  "files": [
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+    {
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+      "usage": "input",
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+}