Certify #4535 at outer slack d=1e-12: all 54 predicates pass, frozen rank N = 2^29,328,002,299 (34x below 2^10^12, same κ)

research/pr4535-outer-slack-rank/.gitattributes

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+PR4535-RESULT.json -text

research/pr4535-outer-slack-rank/PR4535-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": "2^1000000000000",
+    "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",
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+        "0"
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+      [
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+        "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": [
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+}

research/pr4535-outer-slack-rank/README.md

@@ -0,0 +1,54 @@
+# #4535 at outer slack `d = 10^−12`: same κ, a 34× smaller certified finite rank
+
+**Task:** independent-moment. **Base:** canonical main `f626db5c`. **No new κ.** κ = 0.003020165 is unchanged, as are the root bracket, the mirror, the majorant parameters and the native supplier.
+
+hexapode's #4535 (head `a6d7237f`) sets every outer parameter from κ and a slack `d = 10^−10`:
+- `b, a, q = κ + 5d, 4d, 3d`;
+- `ε = 1−d` and `β = 1−3d`;
+- `r0 = d/10`, `stop = layout = d`;
+- tested saving `κ+7d` and complex uniform saving `s_C = κ+6d`.
+
+The frozen compiler word must satisfy absorption with gap `z − θ_hi = (1−θ_hi) − κ − 6d`. My #4563 showed that the least certified rank at `d = 10^−10` is `2^77,014,944,503`. A smaller `d` widens the gap.
+
+## Result
+
+The verifier runs **the exact outer-ledger source #4535 pins**, `research/pr4359-p7-padding/dependencies/compiler-refill/endpoint.py` (SHA256 `d097e55f…`), with the ordinary BIT admission from `research/compact-joint-compiler/arithmetic/RESULT.json`, as #4535's `verify.py` does.
+
+| | `d = 10^−10` (#4535) | `d = 10^−12` |
+|---|---|---|
+| 54 strict predicates, 4 functional guards, 9 margins `> κ` | pass | **pass** |
+| minimum outer margin above κ | `29969798347/10^20` (control, exact) | `2996979834997/10^24` |
+| absorption gap `z − θ_hi` | `377/10^12` | **`971/10^12`** |
+| least certified rank `K` (`N = 2^K`) | 77,014,944,503 (#4563) | **29,328,002,299** |
+| `log₂` class envelope `17·G_depth + 31` | 2.62·10^11 | **9.97·10^10** |
+
+- **Control.** At `d = 10^−10` the ledger reproduces #4535's recorded minimum margin and its full `outer_parameters` exactly.
+- **Rank exponent.** At `d = 10^−12` the frozen rank's exponent is **34.1× smaller than #4535's `10^12`**.
+- **Certification.** The rank comes from my own certified logarithms (exact rationals): it is minimal, it holds monotonically beyond, and it uses the same `A`, `p = 2/5`, `c = 133` and `n0`.
+- **Smaller `d`.** Smaller `d` only moves the gap toward its limit `(1−θ_hi) − κ ≈ 9.77·10^−10`; `d = 10^−12` already reaches 99.4% of it.
+
+## The trade-off
+
+A smaller `d` shrinks the outer power gaps, which are of order `3d`. #4535 absorbs the frozen word's fixed constants into the eventual input cutoff through those gaps, so the cutoff's logarithm scales roughly like `log(constants)/d`.
+
+The frozen word's constants grow with `N = 2^K`; for example, #4535's width envelope is `B_N < C_N^N`. Their logarithms therefore scale at least like `2^K`. Going from `d = 10^−10` to `10^−12` multiplies `1/d` by 100 but divides `2^K` by `2^(4.77·10^10)`, so the net effect on that crude cutoff estimate is overwhelmingly favourable. This is a heuristic comparison of envelopes, not a computed cutoff. #4535 computes no explicit cutoff, and neither does this package.
+
+## Verify
+
+```text
+python3 -B research/pr4535-outer-slack-rank/verify.py
+```
+
+This runs in about 2 s with the standard library only.
+
+## Limitations
+
+- **Scope.** This changes only `d`, and with it the derived outer parameters and the frozen rank. It does not re-certify the root bracket, mirror or majorant; it uses #4535's certificate.
+- **Unchanged.** Physical constants, the outer input cutoff and selection are untouched.
+- **Cutoff comparison.** The comparison above is heuristic.
+
+## Open next questions
+
+- A majorant retune, a smaller `n0` or `p`, needs #4532's gates rechecked at `θ`. At the new gap, `ln A ≈ 10.24` and `p·ln(c+ln N) ≈ 9.4` contribute comparably to `K`. First step: tabulate #4532's p16 gate margins against `ℓ0` and `p`.
+
+Credit: hexapode (#4535); Dillon (#4532); zxv and Jacob Alan Sussman (compiler and majorant lineage); Cinna Davis (#4470); the outer-ledger lineage in `pr4359-p7-padding`. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

research/pr4535-outer-slack-rank/SOURCES.json

@@ -0,0 +1,30 @@
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