[petalas]

Summary

Every minimizing assignment in #5024's complete intact-region domain lies in the same 43-bit cube, in each orientation, throughout 0<t<1.

203 residual binary region choices
= 160 fixed bits + 43 independent bits
= exactly 8,796,093,022,208 minimizing assignments

This is a scoped equality-class proof with a finite exact certificate. #5024's optimum, native correctness, graph reduction and certified elimination trace are cited premises, not replay targets. The new step retains every equality branch and proves converse completeness. #5161 already constructs all 2^43 subsets of its declared 43-zero-move family; that construction retains credit. No mapping between those distinct packages' region/group IDs is computed here.

The result provides an exact domain for a later routing, storage or precision optimization at unchanged recursive moment. No such physical saving, new supplier, kappa or selected-frontier change is claimed.

Evidence

Before: #5024 explicitly leaves uniqueness unclaimed. After: exact sum-product counting and all 203 bit marginals prove the full cube. A separate implementation removes unsupported values, then checks all 270 local Cartesian tuples per orientation, proving sufficiency beyond arc consistency. Seven primary controls and three independent controls include a coupled pair with varying marginals that is not a cube, contradictory constraints and an unreachable conditional tie.

CPython 3.12.15, standard library only. Four registered warning-strict normal/optimized commands pass. Locally timed counting commands take about 1.3 seconds each; independent commands about 0.02 seconds each. Metadata, exact committed bytes and source history pass for all eight package/input files. Official export integrity and all four read-only Git-free replays pass; all nine exported files, including the receipt, retain their bytes and modes.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/intact-region-minimizer-count
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/intact-region-minimizer-count --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/intact-region-minimizer-count/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/intact-region-minimizer-count/independent.py

VERIFY.json registers both direct commands with and without -O. Expected output includes fixed=160 free=43 count=8796093022208 and local_cube_rows=270 in each orientation. RESULT.json is generated and binds the witness and verifier/document hashes. The README gives a bijection from integers 0 <= j < 2^43 to every minimizing chart assignment.

Canonical base a5df48c55ea806d7952b65acc9c76d878ff09b13; contribution head e5d7e4dc31331f19fb9ee795f180e6058ccf779c. Source #5024 is retained at 390d7255f0e128be54e007962d7febb47697a7c5. TASK records the changed complex-rank-shape-decoder scope. Live PRs, claims, account history, main/merged terms and the overlap helper were checked. Coverage excludes unpublished work and arbitrary diffs.

Limitations

The theorem fixes the scalar word, native maps, boundaries, owned-center interfaces and intact zero-edge regions. It counts contracted-region subspace assignments. It does not count physical circuits or establish compatibility of independently chosen forward/backward implementations. The predecessor's optimum and strict comparisons remain cited mathematical premises. Native/absolute-phase, dirty-endpoint, all-column, physical precision/tape, stock, compiler, complete selected cold replay, outer multiplication and proof-kernel obligations are unrun.

Related open work

• Verify three bounded joint-Clifford chart certificates (no new κ) #5161 proves a 2^43-subset invoice-preserving family from 43 declared zero moves. The count agrees; this contribution adds converse completeness for the entire Prove the global fixed-boundary intact-region optimum #5024 intact-region domain without asserting a cross-domain ID mapping.
• Explore joint frame retiming after carried swaps (ongoing; no new κ) #4993 explores broader variable boundaries and split regions. Those changed hypotheses are outside this classification.
• Screen paid nonmonotone owned contacts with shared late native maps (no new κ) #5144 permits nonmonotone persistent contacts and shared late native maps. This result keeps those maps and intact regions fixed.
• Extend owned-wire floor to free native/helper boundaries and nonnested C moves #5140 proves an owned-wire floor with freer native/helper boundaries. This classification concerns the complete moment at the fixed boundaries of Prove the global fixed-boundary intact-region optimum #5024.

Open next questions

Can one of these 43 independent choices reduce an actual routing, storage or precision cost? First bind that cost to the named regions and price every incident operation and phase. A physical implementation and complete consumer bill are missing.

Can one literal region split exploit an equality-case chart to remove paid work? Coordinate with #4993 and supply its exact operator, dirty restoration and full boundary bill before extending the search.

Credit

This contribution was produced by Nick Petalas. Krzysztof Wojt #5024 supplies the optimum and trace; Cinna Davis #4997 supplies the intact-region graph/model and the first three comparison inequalities. Chris R / Clear1nsight #5161 and Cinna Davis #5124 retain the existing zero-move family. Credit Dugongue #4971, Cao Yifan, zxv, Jacob Alan Sussman, James Reed and the retained native/controller lineage. Independent human and formal proof-kernel review are unclaimed.

CI notes

CI pending at e5d7e4dc31331f19fb9ee795f180e6058ccf779c. All four registered commands, metadata, committed-byte/source-history and read-only Git-free export checks pass locally. Hosted results have not been checked.

Merge danger

Reversible; isolated research package. Existing sources and the selected frontier are unchanged.