Mister-Meeseeks · Oct 30, 2026 · Oct 29, 2026
diff --git a/research/radical-fanout-inverse-tables/README.md b/research/radical-fanout-inverse-tables/README.md
@@ -0,0 +1,194 @@
+# Static inverse tables remove 1,661,435 repeated tuple searches
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+The complete radical-fanout command performs **260 `tuple.index` calls instead
+of 1,661,695**, with byte-identical complete output. The change precomputes
+additive inverses and unit inverses once per immutable finite ring. It retains
+every demand, word, literal input, arithmetic check, control and shortest-path
+query in lee-phys's [PR6210](https://github.com/CrocSwap/integer-mult-bounds/pull/6210).
+
+Claim type: an exact implementation-work reduction with a scoped equivalence
+argument and finite conformance evidence. This is a verifier-engineering
+contribution, not a new mathematical tariff, an audit of the predecessor theorem,
+a supplier admission or a new kappa. Wall time is reported separately below.
+
+## Implementation and equivalence
+
+[retained_source.py](retained_source.py) is the exact published verifier at
+`4e08457f4d0b69c00a6ca62eca81c8d0a55f3e60`. [UPSTREAM.json](UPSTREAM.json)
+records its original path, Git blob, SHA256 and author, together with the exact
+source README. The pending commit is not required in the local Git object store.
+All executable bytes required for replay are retained here. [SOURCES.json](SOURCES.json)
+pins only canonical-ancestor task and selected-record provenance.
+
+[fast.py](fast.py) authenticates the retained executable before importing it.
+Its only replacement is the `Ring` class used by the retained ring factory.
+The new frozen subclass adds two tuples and overrides `neg` and `inv`.
+The source's full `main`, constructors, scalar interpreter, checks, Dijkstra
+search and output statements execute unchanged.
+
+For a valid ring table, the cached entry at x is exactly the original
+`add[x].index(0)` or `mul[x].index(one)`. The tuples cannot become stale because
+the original operation tables and the new tables are immutable. A fresh ring
+builds fresh tables. There is no persisted answer or workload cache.
+For a nonunit, the sentinel sends execution through the original failing
+lookup, preserving its `ValueError` and message. Python negative indices and
+out-of-range indices also preserve their original behavior.
+
+It follows by induction over scalar instructions that every intermediate state,
+gate coefficient and endpoint is unchanged. The search's neighbor states, costs,
+heap ties and returned paths are unchanged too. No case is replaced by a cached
+verdict. Constructor table scans are included in the full command and in the
+260-call count. Construction also adds one membership test per ring element;
+the count above is specifically `tuple.index` invocations, not every table read.
+
+The executable accepts ring orders 1 through 81. The theorem's supplied fixtures
+have orders 16, 81 and 64, and its nonlocal rejection fixture has order four.
+The adapter adds at most 162 tuple entries per ring. Successful inverse queries
+and all negations then use constant-time tuple indexing instead of a row search.
+The explicit nonunit failure path retains its row search.
+
+## Evidence
+
+The new checker compares every element at both its nonnegative and negative
+Python index in all three rings, 322 element/index cases. It also checks the
+out-of-range exception type and message, rejects invalid orders and nonsquare
+tables, and rejects changed executable source before import. Sixteen controls
+run under normal and optimized Python.
+
+Two fresh complete commands have equal stdout. Separate untimed profiles run
+both complete workloads again and count exactly 1,661,695 and 260 tuple searches.
+All four outputs agree, with SHA256
+`6fcd968e3899b567f520b5b05bde6c481dd7581a754026af2aa300bf6396da1d`.
+The generated receipt is printed to stdout and binds the actual local package
+hashes. No receipt is handwritten or stored as an expected scientific answer.
+
+The preserved source output enumerates 22,152 demands, 221,520 words,
+1,772,160 formal forward/inverse evaluations, 1,310,720 literal evaluations,
+four oracle queries and 8,465 settled entries. These are the predecessor's
+workload and counters, not new mathematical findings claimed here.
+
+## Wall-time measurement
+
+The benchmark times fresh complete scientific commands, including imports,
+table initialization, every check and the four shortest-path searches. It
+compares every complete stdout and fails on a nonzero exit or stderr. Profiling
+is separate from the reported timing sweep. The generic package runner and
+export tools are outside the measured boundary.
+
+The wall-time comparison is **inconclusive on this shared host**. Ten alternating
+pairs have zero errors and equal complete output. The optimized median is lower,
+but the ranges overlap and the difference is smaller than run-to-run variation.
+No portable elapsed-time improvement is claimed.
+
+| Complete command | Median seconds | Range seconds |
+| --- | ---: | ---: |
+| Retained source | 3.555522 | 3.155300 to 3.913315 |
+| Inverse-table adapter | 3.256747 | 3.027612 to 3.633472 |
+
+Raw alternating pairs, in seconds, are:
+
+| Pair | Retained | Adapter |
+| ---: | ---: | ---: |
+| 1 | 3.762916 | 3.384731 |
+| 2 | 3.551804 | 3.531334 |
+| 3 | 3.913315 | 3.633472 |
+| 4 | 3.899865 | 3.448595 |
+| 5 | 3.621823 | 3.143365 |
+| 6 | 3.559239 | 3.291734 |
+| 7 | 3.329549 | 3.221761 |
+| 8 | 3.421450 | 3.030692 |
+| 9 | 3.202639 | 3.033553 |
+| 10 | 3.155300 | 3.027612 |
+
+Environment: CPython 3.12.15, macOS 27.0.1 arm64, 18 logical CPUs. The final
+load averages were 4.43, 4.00 and 3.12. Other research processes shared the host;
+the measured commands themselves ran serially. Both engines used identical
+warning-strict flags and fresh private bytecode namespaces. The adapter digest
+was `a556f916c57b6fb25d1cbcb54abbaf92d1cc32dc96e4c5fc886b461328e31f32`.
+
+The separate baseline profile records 0.238 seconds of tuple-index self-time
+within 6.546 profiled seconds. Endpoint interpretation, count bookkeeping and
+shortest-path expansion remain substantial work. Tracing changes the timing
+distribution, so this profile explains the limited scope of the optimization
+rather than establishing an untraced speed bound. The exact lookup-count
+reduction is the result being shipped; timing does not certify a faster multiplier.
+
+## Reproduce
+
+From the repository root, with CPython 3.12 and the standard library:
+
+```sh
+/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
+```
+
+Another CPython 3.12 path may replace the displayed interpreter. Set
+`IMB_RUN_ROOT` to an existing private scratch directory to collect all temporary
+files there; otherwise the system temporary directory is used. Direct replay
+needs no Git, network, foreign commit or
+third-party package. Child interpreters use fresh private bytecode namespaces
+and `-B`; all temporary directories are removed. The optional benchmark prints
+all raw samples, complete-output digests, environment and summary statistics.
+CI registers conformance, not a host-specific runtime threshold.
+
+## Limits and engineering choices
+
+This changes only an isolated optional replay. It does not patch the predecessor
+package, shared runner or selected result. The predecessor mathematics remains
+a cited result with its own review obligations. No selected scientific cold
+replay, physical/native/compiler/stock/precision/outer proof, Lean or Docker run
+is claimed. All five selected framework assumptions remain. Independent human
+and formal proof-kernel review are unclaimed.
+
+New functions fit within sixty lines. The new table loops are bounded by 81;
+conformance uses three fixed rings; timing accepts five through ten rounds;
+each child has a 120-second timeout. The retained source already has explicit
+search resource guards and is unchanged. Its long functions and search loops
+are retained dependencies, not newly refactored code. Managed Python allocation
+is a documented deviation from fixed-allocation guidance. New table capacity
+is bounded as above. The four inverse-method targets in the conformance helper
+and the single explicit ring-factory replacement are bounded uses of dispatch.
+Warning-strict normal and optimized execution are registered. No external linter
+or proof assistant is required. Local compilation is checked without writing
+bytecode files.
+
+Build the Lever led to the rerunnable checker and benchmark. Prove It Works led
+to complete-command equality, actual profile counts and the corrupt-source
+control. Explain the Number keeps the wall-time uncertainty separate from the
+exact lookup reduction. The engineering-mode playbook is Autonomous local run.
+
+## Open next questions
+
+Does this table adapter produce a stable complete wall-time gain on an idle
+Linux CPython 3.12 host? Run five or more alternating complete commands with
+`benchmark.py`, retain all output digests, and report medians and ranges. The
+workload is supplied; an additional host measurement is missing. A source-level
+factory hook could later integrate the adapter without altering the mathematical
+workload, but no integration or additional optimization is reserved here.
+
+## Credit and overlap
+
+The new inverse-table adapter, equivalence explanation, checker and measurement
+script were produced by Nick Petalas. The radical-triple theorem, complete
+enumerated workload, interpreter and search are lee-phys's. Retain the source's
+credit to #5259/#5514, Ali Romman #5266/#5310, lee-phys #5139/#5163, Nick Petalas
+#5208/#4916, Krzysztof Wojt #5045, Ross Richardson #3063/#3121, Cinna Davis,
+Anthony, Oscar Fennell, Tekalign and the retained lineage. The two retained
+source files preserve their original credit verbatim.
+
+The scientific base is canonical main
+`a5bff3bf133e482541c09eba4e773884bdb2b1b7`, priority revision 8. The selected
+binding-only orientation passed and resolves to PR6030 at kappa .003125804;
+the older value in current-priority prose is not used as the selected value.
+The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19279673)
+reserves this optimization. Live open and account PRs, active Discussions,
+recent handoffs, local locks, merged term searches and main README searches
+found no matching optimization in their stated coverage. The overlap helper
+covered 338 open PRs and 5,957 Discussion records with complete remote pagination,
+but only 91 of 8,623 local note records. Separate Git README searches cover the
+omitted main files for the distinctive terms; neither search proves novelty.