[apoursh]

Summary

New package research/mode-fiber-radius-three-halves. It continues the merged #6320, which partly answered #6312's open question.

Theorem D: R ≤ 3/2. With #6312's lower bound, the least uniform ℓ∞ radius of exact mode-preserving rounding for the PR4933 mode matrix M now satisfies 1 ≤ R ≤ 3/2. This replaces the R ≤ 2 of #6320 (Theorem C). The bound holds for every x, with no genericity assumption.

Argument. It is shorter than #6320's and needs neither its genericity reduction nor its Hall lemma:

1. Centre x at its nearest integer vector.
2. Take a vertex of the fiber polytope in [−½,½]^112. All but at most 8 rows land exactly at ±½. Snapping the rest leaves a demand u ∈ L with ‖u‖∞ ≤ 4.
3. Balancing lemma. Each block of 8 rows has one pinned coordinate and three free ones, so it splits into four sign-free pairs worth ±2e_j each. With a rainbow matching in every block, each coordinate gets 7 pairs. Making one or two blocks straight fixes the parity class of u in H8. That leaves 6–10 pairs per coordinate against a demand of at most 4.
4. Set eᵢ = (sᵢ + κᵢ)/2 ∈ {−1,0,1}. Then the error is at most 1 + ½.

Verification

• verify.py uses the standard library and exact arithmetic, and runs in about 5 s, both normally and with -O. It checks:
  • the 14 pinned blocks and MᵀM = 56·I;
  • H8 closure and that complements of supports are supports;
  • column sums divisible by 8;
  • the rainbow and straight matchings with every pair identity;
  • the 16-case balancing table;
  • 150 exact runs of rounder.py, covering all three parity cases (worst error 503/336).
• Seven rejection controls fail closed.
• A separate exact stress run of 12,000 inputs found no failure. Its worst error was 1259/840 ≈ 1.4988, so 3/2 is the limit of this construction.
• research_package.py --check-only prints PASS_PACKAGE_METADATA_AND_PINS. The Python 3.12 replay is left to CI.
• The overlap search found only Prove exact mode preservation needs a full rounding unit (no κ) #6312 and Mode-fiber rounding radius: 1 ≤ R ≤ 2; R = 1 reduces to zonotope filling (no κ) #6320. Mode-fiber rounding radius: 1 ≤ R ≤ 2; R = 1 reduces to zonotope filling (no κ) #6320's package is unchanged.

Scope

• This is a property of the exact-mode contract only. Complete κ and the selected frontier are unchanged.
• It is not a claim that any caller needs the contract.
• It supplies no physical rounder, error or tape bill.
• R = 1 stays open; TASK.json records the next step and its blocker.

AI assistance: Anthropic Claude (Claude Code) found the argument, wrote the code and ran the checks under apoursh's direction (disclosed in TASK.json).

🤖 Generated with Claude Code