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Research: three reciprocal guard bits from directed rounding (no new kappa) - #1638
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Independent audit of PR #1638 at 394b90e. Required Python 3.12 registered replay passes. A separately written integer-significand checker validates 65600 candidate cells and 646656 conservative endpoint inequalities through p 12, retaining six-bit incoming warm-up precision. It reproduces generic-state zero/one/two-guard counterexamples. Evidence: b8d843e / research/pr1638-independent-prefix-audit. Exact source Git-blob/SHA256 pins, checker, deterministic receipt and reproduction commands are retained in the artifact. Exact finite cells corroborate the written quantified induction; they are not the all-precision proof. Seeded two-guard correctness, full root/recovery caller, physical tape and theorem formalization remain open. Next: Derive a reachable-state invariant for the actual warm-up before deciding seeded two guards; additional finite passes alone cannot settle it. Credit: ToikY PR1638; Hanson Sun PR1528; retained OpenAI manuscript and classical Brent reductions; Douglas Colkitt repository guidance. Substantial OpenAI Codex assistance; no independent human review or new kappa claim. Exact committed audit artifact. Commit |
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Batch review at Merged to Reviewed scope: Directed-rounding proof that three reciprocal guard bits suffice for every real normalized input and capped precision, with exact finite real-cell corroboration. Invariant-state lower examples do not establish seeded optimality; root/caller integration remains separate. The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions. Batch review, attribution and evidence. Thank you for the contribution. |
Change and mathematical scope
Section09's reciprocal proof treats all rounding errors symmetrically and uses five fractional guard bits. Preserving the actual floor-rounding signs gives a written uniform theorem with three guards, retaining seed5/8, the six-bit warm-up, exact products and capped doubling. For every real1<=c<2 and integerp>=1, the returned x satisfies |1-cx|<2^-(p+1) and |x-1/c|<2^-(p+1).
Three is sharp for propagating the existing residual invariant from arbitrary admitted dyadic states: exact witnesses reject zero, one and two guards. Those states are not the actual warm-up outputs, so optimality of the seeded two-guard routine remains open.
Only research/reciprocal-directed-guards/ is added: proof, two separately written arithmetic ports, exact receipts, eight semantic/package test groups, source pins and opt-in replay. Explicit side result; no kappa gain. Section09 is a downstream corollary, not a multiplication dependency. Root outer precision, physical tape/head costs, selected suppliers, compiler/analytic transfer, complete root/recovery substitution and proof-kernel checking remain outside scope.
Task and overlap
Task uniform-transfer-audit, priority3; scope refined to its precision branch, explicitly labelled as a side result. Nonce b43a43f25c1b77e3be01ef6fdcf22b71, scope key603f341b99f590f17a76. Claim: Discussion1634. Base0a201997097b8a89b638f3a2afcb35db4184c8b2; publication parent858771bcca3b8dd5d4ef4aa682a13cc9b7e23920 has unchanged guidance, catalog, selected record and Section09 blobs. TASK.json and SOURCES.json pin provenance and the consumed source.
Nearest predecessor: Hanson Sun's PR1528, head f43caa5, checks the retained Newton routines atp1..8 and leaves the uniform argument open. This adds a directed-rounding guard refinement and quantified induction; no pending executable is imported. PR1379 already closes the tempting parking target and was not duplicated.
Final overlap:661 open PR bodies with complete pagination and349 local metadata records, fresh board/handoff reads and a Newton Discussion search. Discussion pagination is incomplete; text coverage is not a novelty guarantee. No matching guard refinement was found in the inspected scope.
Validation
Reproduce: python3.12 -B scripts/research_package.py --package research/reciprocal-directed-guards
Registered CPython3.12.14/macOSarm64 replay passes in13.989s. Both implementations agree on complete traces for16,320 common prefix cells atp1..8; the separate integer checker covers81,920 cells throughp10,387,392 stage cells and774,784 endpoint checks. It preserves all incoming warm-up bits at the smaller first-stage precision. Exact symbolic residual/slack identities corroborate the written proof; finite cells do not prove arbitrary precision by enumeration.
All eight test groups pass:4,995 general-invariant states,120 nondyadic interior executions, wrong sign, lower guards, omitted warm-up, stale prefix, invalid domains, actual source/receipt corruption, occupied output and read-only optimized relocation. Metadata/pins,20 repository research-workflow tests, selected baseline source bindings and staged whitespace checks pass. VALIDATION.json records commands/environment/omissions. No selected cold replay, hosted CI, kernel theorem, literal tape implementation or end-to-end timing is claimed.
Credit
Credit Hanson Sun (@bbMisk)/PR1528, OpenAI's retained manuscript and its classical Brent reductions, and Douglas Colkitt's repository/audit guidance. Source notices remain unchanged. Prepared for @ToikY with substantial OpenAI Codex assistance in derivation, independent implementations, mathematical review, verification and writing. Two bounded subagents; no independent human or kernel review claimed. Apache-2.0.
Open next questions
Package README/TASK.json state the two bounded follow-ups: derive a stronger reachable-state invariant to decide seeded two-guard correctness; formalize the floor-grid induction, then bind/replay the complete nested-root and exact-recovery caller. Additional finite no-failure cells cannot settle the first question.