[petalas]

Summary

Prove a scoped classification of exact carried-frame rounding. For every integer d >= 1, refinement m >= 2 and group G <= GL_d(Z),

uniformly bounded G-equivariant q : (1/m)Z^d -> Z^d
                         exists iff G is finite

The d seed errors form an integer matrix congruent to -I modulo m; its invertibility bounds every frame matrix. For finite G, a construction on all of R^d rounds into the lattice fixed by each canonical orbit point's stabilizer. It fixes integer inputs and needs no continuity or coordinatewise rule.

This extends Sricharan Vagvala's merged #1631 shear obstruction and addresses the compatibility concern in woklerx's merged #1892. Sagar Thakrar #1537, Ross Richardson #1287 and the retained frame/dirty-echo lineage keep their credit. The new contribution is produced by Nick Petalas. No predecessor verifier is imported or replayed.

Evidence

Before: one shear obstruction and separate orbit-window optima. After: a dimension-uniform equivalence and a compatible positive construction.

Local CPython 3.12.15 normal and isolated optimized replays pass, with 900 finite group/input fixtures, 5,100 equivariance comparisons and 63,424 seed matrices. Three deliberate mutations reject: omitted stabilizer filtering, omitted invariant norm and omitted inverse transport. A Git-free read-only standalone replay also passes. Each registered command takes about five seconds; the complete registered runner takes about ten seconds. AST parsing and staged whitespace checks pass.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/equivariant-grid-frame-classification/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -O -B research/equivariant-grid-frame-classification/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/equivariant-grid-frame-classification
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/equivariant-grid-frame-classification --check-source-history

An equivalent CPython 3.12 interpreter also works. Expected markers are PASS_EQUIVARIANT_GRID_FRAME_CLASSIFICATION no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed and PASS_RESEARCH_COMMIT_BYTES. The publication preflight passes for all eight package/input files and source ancestry at 43f558c1b8fc14fb0ed33a4512fbc976beea6b0f. Optional receipts are computed and bound to package SHA256 hashes.

Limitations

This is a scoped proof with finite corroboration, not a complete multiplication theorem. Necessity requires the whole refined grid, or the specified d seed orbits. Smaller reachable domains, changing grids, frame-dependent rounding and nonuniform error bounds can escape it. Translation-equivariance is not promised. No optimal radius, efficient physical rounder, actual supplier reachability, dirty-restoration programme, bit/tape bill, selected cold replay, Lean proof or independent human review is supplied. No #4100 caller obligation or selected-frontier change is claimed.

Open next questions

• Determine the optimal uniform radius for the displayed nonorthogonal order-two reflection, beginning with its stabilizer-fixed nearest points.
• Identify an actual indivisible reader whose dirty-input image contains the seed orbits; source-bound reachability and physical arithmetic prices remain missing.

Related open work

No open PR studying this integer-frame-group rounding classification was found in the refreshed open-PR and Discussion searches. #1631 and #1892 are merged predecessors; unpublished work and arbitrary diffs remain outside search coverage.

CI notes

CI pending for 43f558c1b8fc14fb0ed33a4512fbc976beea6b0f. All declared commands passed locally; no hosted result is yet claimed. The registered workflow is the requested hosted replay. No historical suite, native cold replay, Docker or Lean run is requested by this isolated package.

Merge danger

Door: two-way. Blast radius: isolated. This adds only research/equivariant-grid-frame-classification/, with no shared-code or selected-record changes.

Coordination: #1166 (comment)