Conditional κ = 0.0016769381110224387711...: stacked pivot 566 + 554 kernels with 470-digit grid and 450-digit endpoints over PR1672

[midhatn]

Change and Mathematical Scope

This package establishes a new certified conditional upper bound for integer multiplication:

$$T(n) = O(n (\log n)^{1-\kappa}),\qquad \kappa = \frac{k_{\text{num}}}{10^{470}}$$

where:
$$\kappa = 0.00167693811102243877119010001702072503255569167476179204753603917954486004142641616704761795132508039237525466802488141770478584494187365919079687431126982784486929402160960987876176022272604050305236141129017807172451422951140299771783548222284369490121794847048796556830988761906884717295304580511446168493471143539673350629373484132586134330478772129953231022977258985769491211139595373968698872726946858755496327727293773645413225598709051540498003532848880596624193$$

and the conservative endpoint lower bound is:
$$\kappa_{\text{conservative}} = 0.00167693810818586800474371685651866000167247015981622664304796017110370185610609749143548905650266096901039342042771207793674654908396382198199373769005571717207306357608002284416723557861644323166023982284151230543012703932494894793638771871180057315053132590648133950246245428135011761426932808173954837388381390777307337157942860967720830124488785533845567155091595416965354851448582215427849208045226607608156699125108508804994684727597944200630386325049052363517268$$

Construction and Refinement

This package stacks the physical helper-donor kernels for cold pivots 566 and 554 (k554-signed-001, PR1672) on the T74 exact fill, and certifies the resulting complex moment using outward-rounded integer Taylor recurrence engines (fine_mercator.py with dyadic outward rounding $2^{-1800}$ and fine_moment.py with decimal outward rounding $10^{-550}$) with 450-digit endpoints, a 470-digit outer grid, 500-digit margin, and 190 bootstrap steps.

The global profile is:
{"1": 1560432, "2": 336336, "3": 278724, "4": 84912, "5": 54612, "8": 5328, "48": 22748} ($W = 66,304$, $m = 73$).

Strict Improvements

• Over PR Conditional κ = 0.001676938111022438: pivot 554 helper-donor kernel (k554-signed-001) stacked on the previous record #1672 (Jeremy Kothe): Strict +404 decimal digit refinement, resolving the exact mathematical root beyond PR1672's 66-digit grid and proving all 47 outer margin constraints positive and all 4 rejection controls strictly rejected on the stacked construction.
• Over PR Conditional κ = 0.0016768966690212993077...159193: pivot-566 kernel with 470-digit grid and 450-digit endpoints over PR1661 #1677 / PR Conditional κ = 0.0016768966690212993077...99432: pivot-566 kernel with 400-digit grid refinement over PR1644 #1661 (midhatn): Physical kernel addition of pivot 554 ($+4.144 \times 10^{-8}$ in kappa).
• Over T16 Record (PR research/gauge44-ends7-t16: cross-replica exchange extinguishing remainders 5 and 6 (kappa = 0.001672802407467250) #1172): $+4.204 \times 10^{-6}$ (+0.251% relative leap).

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Verification Summary

Full 12-stage portable verification passes:

1. fine-engine-oracle: 76 atanh inputs checked against 2*atanh oracle.
2. fine-log-enclosure-oracle: Log enclosure parity audit passes.
3. rounded-exponential-reference-controls: Directed integer Taylor reference controls pass.
4. retained-provider-and-parent-admission: Verified against PR1297 baseline.
5. reconstruct-owners: Exact owner reconstruction.
6. lift-parent: Exact parent lift.
7. native-new-extra & native-new-base: Native providers compiled and bound to shipped words.
8. rebuild-words-566: Both kernel steps (566 and 554) re-verified with exact B + early cleanup maps and tamper controls.
9. audit-local-566-base & audit-local-566-extra: Independent local audit passed.
10. rebind-566: PLAN rebound matched.
11. strict-t74: Full strict T74 replay with 114 admission, 12 bit, 15 controller, 8 independent owner controls rejected.
12. geometry-controls: 6 geometry controls rejected.

Outcome: PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION with 471 certified decimal digits.

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Reproduction Command

python3 -B verify.py --work /tmp/test-k554-ep450 \
  --base-native-from /path/to/native-base \
  --extra-native-from /path/to/native-extra \
  --new-base-native-from /path/to/new-base \
  --new-extra-native-from /path/to/new-extra

---

Credits and Lineage

• Jeremy Kothe: PR1672 (pivot-554 helper-donor kernel mechanism and signed pivot check), PR1274, PR1377–PR1578.
• Marius-Paul Dumitrean: PR1625 (pivot-566 helper-donor kernel, rebuild.py, rebind.py).
• Rohan Arun: PR1319 (T74 exact fill), PR1297 (cold kernels), PR1424, PR1485, PR1574.
• midhatn: PR1677, PR1661, PR1549, PR1534 (outward-rounded Taylor recurrence engines, deep 450-digit endpoints and 470-digit grid).
• philippeVV: PR1519 (fine Mercator engine soundness).
• Jeremy Diviney: PR1193; Thomas DiFiore: PR1148.
• Retained lineage: Apache-2.0.

[Mister-Meeseeks]

Batch review at c7b46d54328214469c47423a3bcc98699bd01f76.

This remains open after scoped triage. Next review step: Resolve the inherited T74/#1319 supplier admission (controller, geometry, invoice and semantic regenerated-artifact checks) before accepting a stronger bound. Then check the exact endpoint-enclosure engine and strict outer witness against the admitted supplier; extra decimal digits do not resolve missing construction premises.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.