research: κ = 0.001729737950064278551242895823 − 10^-52 − 10^-76

research/t740-six-skip-precision30/PRECISION-EDITS.json

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+[
+  {
+    "new": "PRECISION = dict(endpoint_digits=30, grid_digits=24, margin_digits=34, bootstrap_steps=12)",
+    "old": "PRECISION = dict(endpoint_digits=24, grid_digits=18, margin_digits=28, bootstrap_steps=12)"
+  },
+  {
+    "new": "dict(b=str(C-Q(1,10**110)), b_above_endpoint=str(C+Q(1,10**30)),",
+    "old": "dict(b=str(C-Q(1,10**104)), b_above_endpoint=str(C+Q(1,10**24)),"
+  },
+  {
+    "new": "kappa == C-Q(1,10**76)-Q(1,10**52)",
+    "old": "kappa == C-Q(1,10**70)-Q(1,10**46)"
+  },
+  {
+    "new": "Endpoint precision30;",
+    "old": "Endpoint precision24;"
+  },
+  {
+    "new": "kappa=C-10^-76-10^-52 exactly; same fixed-grid-maximum proof at precision30.",
+    "old": "kappa=C-10^-70-10^-46 exactly; same fixed-grid-maximum proof at precision24."
+  },
+  {
+    "new": "    patch('code/assemble.py', [(\" if DISCOVER:b=Q(M.certify(H,73,owner['live'],False)['lower']);conservative_b=Q(M.certify(H,73,owner['live'],True)['lower'])\", \" def fine_endpoint(bad):\\n  digits=SPEC['outer_refinement']['endpoint_digits']\\n  seed=Q(M.certify(H,73,owner['live'],bad)['lower'])\\n  if digits<=24:return seed\\n  grid=10**digits;left=int(seed*grid);right=left+10**(digits-24)\\n  while right-left>1:\\n   mid=(left+right)//2\\n   lo,hi=FC.moment('fine_mercator',H,73,owner['live'],Q(mid,grid),bad)\\n   if hi<1:left=mid\\n   elif lo>1:right=mid\\n   else:raise ArithmeticError('fine root enclosure exhausted; no endpoint accepted')\\n  return Q(left,grid)\\n if DISCOVER:b=fine_endpoint(False);conservative_b=fine_endpoint(True)\")])\n    spec['outer_refinement'] = PRECISION",
+    "old": "    spec['outer_refinement'] = PRECISION"
+  }
+]

research/t740-six-skip-precision30/PROOF.md

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+# Scope and exact refinement
+
+This is an arithmetic refinement of Petalas's PR3054, not a changed supplier
+or an independent proof of the inherited multiplication theorem.
+
+The complete paid profile is
+
+```text
+{1:16376940, 2:4741920, 3:1594737, 4:719280,
+ 5:137085, 8:53280, 48:281903}.
+```
+
+For `T=740`, `W=896T`, its moment is
+
+`M(b) = sum_r n_r r (73/r)^b / (73W)`.
+
+All counts are positive and all ranks are below 73, so the moment is strictly
+increasing. Its complete rank mass is `65088T = 48165120`. The retained
+rational engines separately certify `M(C30)<1<M(C30+10^-30)`, where
+
+`C30 = 0.001729737950064278551242895823`.
+
+The credited root engine additionally brackets the root on a 35-digit grid.
+The unchanged independent owner/native/physical checks remain obligations of
+the full reproduction; the moment alone is not a supplier release.
+
+The original outer family, with endpoint precision `d`, uses
+`F=10^(d+46)`, backoff `B=10^-(d+22)` and strictly positive free gaps
+on the scale `10^-(d+80)`. The retained PR2161/2180 fixed-grid argument gives
+the attainable point `C-1/F-B`, subject to all original outer predicates.
+Here `d=30`, and the actual outer ledger must rederive
+
+`kappa30 = C30 - 10^-76 - 10^-52`.
+
+Against the **full**, unrounded PR3054 point
+`kappa24 = 0.001729737950064278551242 - 10^-70 - 10^-46`, the difference is
+
+```text
+kappa30 - kappa24
+  = 895823/10^30 + (10^-46 - 10^-52) + (10^-70 - 10^-76)
+  > 0.
+```
+
+The gain is approximately `8.95823e-25`. It is numerically tiny. Increasing
+the endpoint precision changes the point being certified, not the recursive
+profile, native words, ordering, dispatch or stock. There is no claim of a
+runtime improvement, a new packing mechanism or new independent verification
+coverage merely from evaluating more digits.
+
+# Completion and privacy boundaries
+
+The reproduction first requires the original complete PR3054 result, every
+successful stage's exact log digest, the unchanged source inventories, both
+fresh native providers and the actual terminal release. It freshly rederives
+the original admission adapter and compares its complete bytes. The five
+declared precision edits and the fine-grid selector correction are applied;
+the full original admission and
+modern outer release run again. Source/native/fill/order/dispatch/profile
+bindings must agree, and the actual released point must equal `kappa30` and
+strictly exceed the actual completed baseline.
+
+The selector correction bisects the requested grid using the retained fine
+moment engine. The original coarse selector alone cannot identify a 30-digit
+endpoint. Both original acceptance and adjacent-point rejection checks remain.
+The local checkpoint reuses authenticated successful nomination geometry,
+then reruns assembly and the complete strict release; its evidence records
+that scope. The fresh reproduction recipe reruns the nomination prefix too.
+
+Evidence sealing is separate. It verifies both completed admissions and their
+terminal manifest/hash relations. Publication projects only the terminal
+status, exact point and manifest, omitting machine timing. These projections
+are labelled as projections; they are not represented as unchanged original
+certificates. The original receipts remain unchanged in the external work
+directories, with their full digests recorded. The actual complete outer
+ledger and declared source derivation are copied unchanged. Registered finite
+verification checks their hashes, all 63 positive outer margins, all 17
+original rejection controls, exact gain and explicit completion flags.
+
+No selected-frontier adoption, independent human review, formal-kernel proof
+or discharge of inherited physical/compiler/precision/analytic hypotheses is
+claimed.

research/t740-six-skip-precision30/README.md

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+# T740 six-skip precision refinement
+
+**Complete baseline and refined strict releases passed locally.**
+
+Released conditional kappa is `0.001729737950064278551242895823 - 10^-52 - 10^-76`, approximately `8.95823e-25` above PR3054. This is a tiny rounding refinement of the same construction, not a new native or packing mechanism. Every recursive child, 91 buffers and 663,549 physical families remains charged.
+
+Both credited rational moment engines accept the proposed endpoint and reject the next 30-digit point and omitted rank-1/rank-48 costs. Full verification requires a completed unmodified PR3054 replay before the changed admission, which explicitly reuses that authenticated unchanged native cache. The recipe freshly rederives the original adapter and checks its exact bytes, then applies only the five precision edits and the fine-selector correction with all assertion/check counts preserved. Every original full admission and outer predicate remains mandatory.
+
+First run `research/rank7-t740-carrier-skips/reproduce.py` as documented by PR3054. Then run `python3.12 -B research/t740-six-skip-precision30/reproduce.py --baseline-work BASELINE --work FRESH_OUTPUT`. Inputs must be outside the package; source/cache are read-only. Full replay uses Python 3.12.3 and this package's requirements-full.txt: NumPy 1.26.4, SciPy 1.15.3, SymPy 1.14.0, mpmath 1.3.0 and gmpy2 2.3.2. These differ from the predecessor's published environment; all original scientific predicates remain unchanged. Registered finite replay is `python3.12 -B scripts/research_package.py --package research/t740-six-skip-precision30`; it requires sealed evidence and does not rerun the native chains.
+
+For a slower machine, the reproducible baseline entry point is
+`python3.12 -B research/t740-six-skip-precision30/baseline.py --repo FULL_GIT_REPO --work BASELINE`.
+It changes only the outer complete-admission wall-time allowance from 1,800
+to 7,200 seconds. Every scientific command, source predicate, input domain,
+stage log and final completion check remains the predecessor's implementation.
+
+The local continuation authenticates and reuses the successful nomination geometry prefix after correcting the endpoint selector. It reruns assembly, freeze and independent checks, then runs the complete strict supplier and outer release freshly. `checkpoint.py` records this scope explicitly; `reproduce.py` provides the fresh end-to-end recipe.
+
+The exact raw local receipts remain private. Any eventual evidence projections must identify omitted machine metadata explicitly; no projected receipt is described as the unchanged original certificate.
+
+The inherited supplier summary contains a historical `controller=15` field.
+Sealing must read the actual completed controller audit, require its exact
+25 named rejected controls in both releases, and disclose the stale summary
+separately. No predecessor receipt or predicate is corrected in place.
+
+After both full replays pass, seal evidence with
+`python3.12 -B research/t740-six-skip-precision30/seal.py --baseline-work BASELINE --work FRESH_OUTPUT`.
+The sealer rechecks both completed admissions, their stage/log digests, exact
+source derivation and terminal receipt/manifest bindings before publishing
+the scientific projections. It neither changes the original receipts nor
+accepts preflight/discovery-only results. The raw external workspaces must
+remain available for this check.
+
+Credit Nick Petalas/PR3054/3030/3024, dcharb78/PR3042, Jeremy Kothe/PR3015, Marius-Paul Dumitrean/PR2979, dataisfire/PR2161/2180/2168/2184 and the retained native/geometry/outer lineage. Apache-2.0. Substantial OpenAI Codex assistance. No independent human or formal-kernel review claimed. Source notices remain unchanged. All inherited all-size/compiler, physical backing/routing, dirty-state, precision/tape, caller, analytic-transfer and universal block-energy hypotheses remain. The selected frontier is unchanged.
+
+## Open next questions
+
+A material increase requires a changed paid mechanism outside the established fixed-word frame and packing optima. Name its actual producer/consumer boundary and complete bill before another search. Merely printing more digits is not new verification coverage.