Classify every intact-region minimizer: exact 43-bit cube (no new κ)

research/intact-region-minimizer-count/README.md

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+# Exactly 2^43 intact-region minimizers in each orientation
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+Independent human and formal proof-kernel review are unclaimed.
+
+The fixed-boundary intact-region model has exactly
+**8,796,093,022,208 = 2^43 minimizing chart assignments in each orientation**.
+The minimizing set is a complete Boolean cube. Of the predecessor's 203
+residual binary region choices, 160 are fixed and 43 vary independently.
+The free region IDs and all fixed bits are identical forward and backward.
+Every minimum has the same rank histogram throughout `0<t<1`.
+
+This is a scoped equality-class proof with an exact finite certificate. It
+answers the uniqueness gap in Krzysztof Wojt's
+[PR #5024](https://github.com/CrocSwap/integer-mult-bounds/pull/5024), whose
+README explicitly makes no uniqueness claim. His optimum, graph reduction,
+source correctness and elimination certificate are **cited premises**. This
+package neither reruns his native word nor re-establishes that optimum.
+The new result identifies every equality case and shows that their set does
+not depend on the exponent inside the open interval.
+
+Chris R / Clear1nsight's open
+[PR #5161](https://github.com/CrocSwap/integer-mult-bounds/pull/5161) already
+proves that all `2^43` subsets of its 43 declared zero moves preserve the
+paid and helper invoices, following Cinna Davis's #5124 census. That
+construction retains credit. The new step here is the **converse completeness
+statement** for #5024's entire intact-region domain, including every one of
+its 203 residual binary choices. The numerical count agrees; this package
+does not compute a mapping between the two packages' region/group IDs or
+assert equivalence of their different declared domains.
+
+## Why the equality class matters
+
+An unchanged minimum recursive moment leaves a substantial choice of charts.
+A later contributor can restrict a finite routing, storage or precision
+objective to these 43 independent bits, instead of searching all 203 bits
+again. This result supplies that exact domain. It does not claim any reduction
+in those additional costs, legal absolute-phase lowering, physical execution,
+supplier admission, improved kappa or selected-frontier change.
+
+The fixed scalar word, native maps, start/final charts, owned-center interfaces
+and intact source zero-edge regions are the hypotheses of #5024. Changing a
+region boundary or splitting a region remains outside this theorem. The
+43 choices count assignments of contracted-region subspaces, not physical
+circuits, bases for a subspace, labelled owners or independent forward/backward
+circuits that may be combined without another compatibility proof.
+
+## Equality extraction theorem
+
+Let a certified binary bucket-elimination trace have the meaning established
+in #5024. At each step its two candidate histograms are the exact conditional
+minima for the eliminated bit. Keep the recorded choice when the comparison
+is strict, both choices when the histograms coincide, and no choice when
+both candidates are infeasible. This gives one Boolean relation on the
+eliminated bit and its later separator bits.
+
+**The complete global minimizing set is exactly the conjunction of these
+relations.** To prove this, reverse the elimination order. Given a minimizing
+assignment of later variables, a strict loser cannot extend it to a minimum;
+each equal candidate can. Each retained candidate has a minimizing extension
+by the predecessor's elimination invariant. Repeating this step includes
+every minimum and excludes every nonminimum. Disconnected terminal buckets
+combine independently because their eliminated variables are disjoint.
+
+In this source, every unequal certified comparison is strict for all
+`0<t<1`. For majorization, strict concavity of `r^t` on the nonnegative reals
+makes equality possible only for identical sorted rank lists. The histograms
+already distinguish these lists, with equal edge counts including rank zero.
+The five exceptional power inequalities are strictly positive in #5024's
+written proof. Forced choices have no feasible alternative. Thus no isolated
+exponent inside the interval adds an equality branch. This is a consequence
+of the cited inequalities, not a new audit of them.
+
+[verify.py](verify.py) performs exact sum-product elimination on the new
+Boolean relations. It counts the assignments and each bit's two marginals.
+Each orientation has 43 varying bits, each with `2^42` solutions per value;
+the other 160 bits are forced. The total is `2^43`. Since the solution set
+is a subset of the cube on those 43 bits and has that cube's full cardinality,
+it equals the whole cube. The receipt contains all 203 marginals and the
+complete fixed/free classification.
+
+[independent.py](independent.py) proves the same cube by a different method,
+with no primary-counter or predecessor-code import. It removes unsupported
+bit values until the domains stabilize. Every removal is necessary for any
+solution. It then explicitly enumerates each local Cartesian product and
+checks that **every** tuple in it is allowed. This second step proves
+sufficiency; arc consistency alone would not. Four passes leave 160 singleton
+domains and 43 two-point domains, and 270 local Cartesian tuples per
+orientation certify the full product. The source relations have 552 allowed
+rows before propagation in each orientation.
+
+## Explicit enumeration and reproduction
+
+[RESULT.json](RESULT.json) is generated by the primary verifier and binds the
+source witness and package verifier/document bytes. Its `free_regions` list
+is sorted. For each integer `0 <= j < 2^43`, assign free region number `k`
+the bit `(j >> k) & 1`, and use `fixed_bits` at every other binary region.
+Choose the lower or upper basis in the predecessor witness's `bounds` row
+according to that bit; all nonbinary regions keep their unique bound.
+This is a bijection and an explicit enumeration without storing trillions
+of assignments. Region IDs refer to #4997's contracted graph, not word offsets.
+
+Use CPython 3.12 and the standard library, from the worktree root:
+
+```sh
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/intact-region-minimizer-count
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/intact-region-minimizer-count --check-source-history
+~/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/intact-region-minimizer-count/verify.py
+~/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/intact-region-minimizer-count/independent.py
+```
+
+The manifest registers both direct commands in normal and optimized modes.
+The main success marker is `PASS_INTACT_REGION_MINIMIZER_COUNT`.
+`verify.py --write` explicitly regenerates only RESULT.json. Ordinary replay
+compares its freshly computed receipt and writes no package files.
+Controls include a coupled equality pair whose two varying marginals do not
+form a cube, independent factors, contradictory relations, an unreachable
+conditional tie, duplicate variables and propagated implications.
+
+## Scope, provenance and implementation limits
+
+Canonical base is `a5df48c55ea806d7952b65acc9c76d878ff09b13`, priority revision
+7. TASK.json preserves the original `complex-rank-shape-decoder` card and
+records its changed scope to this equality question. The card's decoder
+synthesis experiment is replaced by equality extraction from #5024's first
+working certificate, because this contribution does not propose a new paid
+profile. The selected baseline metadata orientation passed separately.
+
+SOURCES.json pins the live compressed witness and preserves the predecessor
+proof/README as Git provenance. No predecessor executable is an input.
+The universal equality-extraction argument and strictness above use #5024's
+proved trace semantics. The finite verifier establishes this pinned trace's
+new equality class. It is not independent validation of the native graph,
+absolute phases, arbitrary dirty endpoints or the predecessor optimum.
+No native/all-column, compiler, stock, precision/tape, complete selected cold
+replay, Lean or outer multiplication proof is run or claimed.
+
+All new loops have finite input or explicit iteration bounds; the source is
+capped at 203 variables, separators at three bits and the decompressed witness
+at one million bytes. Both algorithms are iterative. Python's managed
+allocation is a Power of Ten deviation: tables have at most 16 entries,
+integer counts at most 204 bits, and at most 406 elimination factors. Package
+metadata checks and warning-strict normal/optimized replays supply the
+available diagnostics. Assertions are not verification gates.
+
+## Open next questions
+
+Can one of the 43 independent minimum-moment chart choices reduce a concrete
+routing, storage or precision cost? First attach that actual cost to the free
+region IDs, retain all incident operations and absolute phases, and compare
+fully charged alternatives. A physical implementation and a complete consumer
+bill are the blockers. This package assigns no such improvement.
+
+Can splitting one source region exploit a minimum-moment choice to remove an
+actual paid transition? Coordinate with the broader
+[PR #4993](https://github.com/CrocSwap/integer-mult-bounds/pull/4993) before
+selecting one literal split. The first test needs its exact operator, dirty
+restoration, all new edges and full cost. The intact-region theorem does not
+answer that changed-boundary question.
+
+## Credit and overlap
+
+Chris R / Clear1nsight #5161 and Cinna Davis #5124 retain the existing
+43-zero-move family and its `2^43` invoice-preserving subsets. This submission
+adds exhaustive equality classification in the intact-region model.
+Krzysztof Wojt #5024 supplies the global optimum, complete elimination witness
+and five strict all-exponent comparison arguments, with the first three
+inequalities credited there to Cinna Davis #4997. Cinna Davis supplies the
+intact-region graph/model. Dugongue #4971 supplies carried-frame swaps and the
+retained native source. Cao Yifan, zxv, Jacob Alan Sussman, James Reed and all
+contributors named in those predecessor packages retain their attribution.
+No predecessor file or credit line is changed.
+
+The broader numerical domain of #5016, Clear1nsight's small-component
+fixed-price result #5068 and Cinna Davis's continuing #4993 search have
+different restrictions. Their minima and search results are cited, not
+recomputed here. Live PR bodies, this account's recent work, handoffs, active
+Discussions, merged/main-tree terms and question reservations were checked.
+The overlap cache covered 331 open PRs and 4,359 Discussion records; sparse
+local notes were supplemented with all-main README searches. This is bounded
+overlap evidence, not a novelty certificate. Unpublished work and arbitrary
+code diffs remain outside coverage.