[petalas]

Summary

Adds an original verifier implementation step: the complete radical-fanout workload makes 260 tuple.index calls instead of 1,661,695, with byte-identical complete output. Claim type: exact implementation-work reduction, a scoped equivalence argument and finite conformance evidence. No new mathematical tariff, kappa or selected-frontier change.

 finite immutable ring
-  find the additive or unit inverse by searching a table row on each query
+  build inverse tuples once, then index them on each successful query
 complete demands, words, inputs, controls and shortest-path searches
   unchanged

The new frozen adapter adds at most 162 tuple entries per ring. The 260 count includes table construction; construction also adds one membership test per element. Nonunit inverse requests retain the original exception path. There is no cached verdict or omitted case. The predecessor executable and README are retained byte for byte with their published blob/SHA256/author metadata, so cold replay needs no unmerged commit or network.

The new adapter, checker and benchmark were produced by Nick Petalas. lee-phys retains the theorem, complete workload, interpreter and search from #6210. Retain its #5259/#5514 interpreter/search credit, Ali Romman, Krzysztof Wojt, Ross Richardson, Cinna Davis and the full lineage in the unchanged source notice.

Evidence

Before, each inverse query searches a fixed row. After, the exact complete-workload tuple-index count decreases by 1,661,435. The independent new conformance layer checks 322 element/index cases and sixteen controls, including exact exception behavior, invalid tables and corrupt-source rejection before import.

• Warning-strict normal and optimized direct and registered replays pass. Direct checks take 19.23 and 19.12 seconds locally, below their 120-second limits.
• Both complete fresh-process outputs and both separately profiled complete workloads agree. Complete scientific stdout SHA256 is 6fcd968e3899b567f520b5b05bde6c481dd7581a754026af2aa300bf6396da1d.
• Committed-byte and full source-history preflight pass for all ten package files at 913b61836ecc4c20ff59935424719177f1362723.
• Official export integrity and declared-input coverage pass. Normal and optimized replay from a read-only Git-free export pass; all eleven exported files retain their hashes.
• Ten alternating fresh-process timing pairs have zero errors and equal complete output. Medians are 3.555522 seconds for the source and 3.256747 seconds for the adapter, with ranges 3.155300 to 3.913315 and 3.027612 to 3.633472. The wall-time comparison is inconclusive because the ranges overlap and the gap is smaller than host variation. No certified elapsed-time gain is claimed. Raw samples, host details and the limiter profile are in the README.

Reproduce from the repository root with CPython 3.12 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/radical-fanout-inverse-tables/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/radical-fanout-inverse-tables/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/radical-fanout-inverse-tables
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/radical-fanout-inverse-tables --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/radical-fanout-inverse-tables/benchmark.py --rounds 10

Any CPython 3.12 path can replace the displayed interpreter. IMB_RUN_ROOT may name an existing private scratch directory. The checker prints a generated receipt bound to actual package hashes; no expected receipt is handwritten. Child bytecode namespaces are fresh and temporary.

Limitations and unrun obligations

This is an isolated optional replay implementation, not a review of the predecessor mathematics. Ring order is capped at 81. Only successful inverse lookups and negations receive constant-time tuple access; nonunit errors retain a row search. The exact work-count reduction does not establish a portable runtime gain or a faster multiplication algorithm.

No predecessor package, shared runner or selected result is changed. Physical/native/compiler/stock/precision/outer admission, selected scientific cold replay and all five framework assumptions remain outside scope. No Lean or Docker run, independent human review or formal proof-kernel review is claimed. The engineering-mode playbook is Autonomous local run.

Open next questions

Does the unchanged complete command show a stable wall-time gain on an idle Linux CPython 3.12 host? Run five or more alternating pairs with benchmark.py, retain every output digest, and report medians and ranges. An additional host measurement is missing. A source-level ring-factory hook is a possible later integration step; no further optimization is reserved.

Related open work

Completion refresh on October 30 found no currently open PR studying this exact radical-triple replay or inverse-table implementation in the bounded R16/fanout searches. This account has two open PRs, and neither was created since the advance CLAIM at 2026-10-29T22:12:25Z. There is no new same-object account PR to add or common-instance numerical conflict to report; no extra computations were run to check other results.

• Prove exact ten-term tariffs for independent radical fanout triples #6210, lee-phys, now merged: retained predecessor context. It supplies the radical-triple object, its ten-term mathematical tariff and the complete verifier. This PR preserves its entire scientific output and changes inverse lookup implementation only; no predecessor proof or count is claimed as a new mathematical result.

The other open text-search hits study cubic repairs at arity sixteen, C3 schedules or centre-path sharing. These are different objects. Bounded text searches do not establish novelty.

CI notes

Both plan and replay (research/radical-fanout-inverse-tables) passed at commit 913b61836ecc4c20ff59935424719177f1362723 in run 40965485648. The completed checks were read once, and the run's successful conclusion and exact head were confirmed. A maintainer has merged this PR.

Both registered local modes, byte/source-history checks and read-only Git-free modes also pass. CI checks the registered conformance workload, not a host-specific runtime threshold or the full multiplication theorem. The advance claim and the narrower delivered scope are recorded in the claim and progress update.

Merge danger

Two-way change, isolated to research/radical-fanout-inverse-tables/. Existing scientific packages and the selected frontier are unchanged.