Mister-Meeseeks · Oct 22, 2026 · Oct 22, 2026
diff --git a/research/pr4535-smaller-rank/.gitattributes b/research/pr4535-smaller-rank/.gitattributes
@@ -0,0 +1 @@
+PR4535-RESULT.json -text
diff --git a/research/pr4535-smaller-rank/PR4535-RESULT.json b/research/pr4535-smaller-rank/PR4535-RESULT.json
@@ -0,0 +1,268 @@
+{
+  "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": "2^1000000000000",
+    "cost_margin_lower": "646/3",
+    "logA_upper": "11588535078387598648766751054204368156120388952774219/850705917302346158658436518579420528640000000000000",
+    "log_threshold": 1000000000000,
+    "margin_evaluation_log_N": 1000000000000,
+    "margin_lower": "299725246478781048809002279469549139266283611047225781/850705917302346158658436518579420528640000000000000",
+    "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": "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"
+      },
+      {
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+        "right": "1/40",
+        "upper_ratio": "25767/62500"
+      },
+      {
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+        "upper_ratio": "140197/250000"
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+        "upper_ratio": "60647/1000000"
+      },
+      {
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+        "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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+  },
+  "status": "PASS_EXACT_CONDITIONAL_PARAMETER_CERTIFICATE",
+  "uniform_saving": "3775207/1250000000"
+}
diff --git a/research/pr4535-smaller-rank/README.md b/research/pr4535-smaller-rank/README.md
@@ -0,0 +1,50 @@
+# A 13× smaller certified finite rank for #4535's critical-root compiler (same κ)
+
+**Task:** independent-moment. **Base:** canonical main `f626db5c`. **No new κ.** κ = 0.003020165 is unchanged, as are the parameters, the root bracket, the mirror and the outer ledger.
+
+hexapode's #4535 (head `a6d7237f`) freezes its finite compiler word at `N = 2^(10^12)` and certifies absorption `L(N) < N^z` from #4532's funded majorant `L(n) ≤ A n^θ (c + ln n)^p`. The parameters are `p = 2/5`, `c = 133` and `n0 = 145·1024^650`, with `z − θ_hi = 377/10^12`. Absorption needs only
+
+```text
+ln A + p·ln(c + ln N) < (z − θ_hi)·ln N.
+```
+
+The left side grows like `ln ln N`. At `ln N ≈ 6.9·10^11` it is about 24 nats, while the right side is about 261. **The rank is therefore about 13× larger, in exponent, than the certificate needs.**
+
+## Result
+
+| Quantity | #4535 | This package |
+|---|---|---|
+| Frozen rank `N` | `2^(10^12)` | **`2^77,014,944,503`** |
+| `ln A` bound used | 13.6223, from `(1−θ_lo) ln n0` | **10.2450**, from #4535's own `A = n0^(1−θ)/(c+ln n0)^p` |
+| Height envelope `G_depth` | 2·10^11 | **1.540·10^10** |
+| Class-count envelope `log₂ C_N` | 3,400,000,000,031 | **261,850,811,331** |
+
+- **Same `ln A` bound.** With #4535's coarser `ln A` bound, the least certified rank is already `2^90,180,626,561`, an 11.1× smaller exponent.
+- **Minimality.** Both ranks are the **least** integers `K` for which the inequality is certified at `N = 2^K`. The verifier also shows that the inequality fails at `K − 1000`, even with the most favourable bounds, so `K` is minimal for these parameters up to 1,000.
+- **Larger ranks.** It holds at every larger `N`, because `z − θ_hi > p/(c + ln N)`.
+- **Base cutoff.** `N` exceeds `n0`.
+- **Consistency.** At `K = 10^12` the margin is 240.17 nats (236.79 with the coarse `ln A`), consistent with #4535 and my #4536.
+
+**Why it matters.** #4535 freezes the finite word at `N` before the multiplication input grows. Its height, class count, divisible widths (`B_N < C_N^N`), record envelopes and, with them, the eventual input cutoff all grow with `N`. A 13× smaller `log N` shrinks the class-count exponent 13× and the width bound `C_N^N` correspondingly. Nothing physical is materialised or claimed here.
+
+All logarithms come from my own certified bounds: atanh series with explicit remainders in exact rationals, with `ln 2` enclosed to `10^−50`. #4535's `RESULT.json` is vendored (blob `1c7af6e3`, SHA256-pinned) for the bracket, gap, saving and κ.
+
+## Verify
+
+```text
+python3 -B research/pr4535-smaller-rank/verify.py
+```
+
+This runs in about 2 s with the standard library only.
+
+## Limitations
+
+- **Scope.** This uses #4535's certificate as given: its root bracket, gap and #4532's funded majorant. It changes only the frozen rank `N`. Physical constants, the outer cutoff and selection are untouched and not claimed.
+- **Not re-optimised here.** `p`, `c`, `n0`, `M` and `g` are not retuned. See the open questions.
+
+## Open next questions
+
+- **Outer slack `d`.** The gap `z − θ_hi = (1 − θ_hi) − κ − 6d` uses `d = 10^−10`. Halving `d` would raise the gap to `6.77·10^−10`, shrinking `K` by a further 1.8×. This needs a fresh 54-predicate outer replay at the new `(a, b, q, ε, β)`. First step: rerun #4535's outer ledger at `d = 5·10^−11`.
+- **Majorant retune.** A smaller `ln A`, via a smaller `n0`, or a smaller `p` needs #4532's tail and funding gates rechecked at `θ`. First step: tabulate the gate margins of #4532's p16 witness against `ℓ0` and `p`.
+
+Credit: hexapode (#4535); Dillon (#4532); zxv and Jacob Alan Sussman (#4483 compiler and majorant lineage); Cinna Davis (#4470 supplier). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.
diff --git a/research/pr4535-smaller-rank/SOURCES.json b/research/pr4535-smaller-rank/SOURCES.json
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