[apoursh]

Summary

A scoped partial answer to open question 1 of #6312 (research/mode-preserving-rounding, head 46593234). The question: what is the least uniform ℓ∞ radius R for exact mode-preserving integer rounding on all nonempty fibers of the PR4933 mode map M (112×8)? #6312 proves R ≥ 1 and gives no upper bound. This adds research/mode-fiber-radius-bounds:

• Structure (exact). L = Mᵀℤ¹¹² = {x ∈ ℤ⁸ : x mod 2 ∈ H8}, the Hamming construction of √2·E8, with index 16. The 14 row supports are the weight-four words of [8,4,4], with 8 rows each. MᵀM = 56·I. The rows are not closed under negation.
• Theorem A (finite reduction). R ≤ 1 if and only if every lattice point of L in the open zonotope Mᵀ(0,1)¹¹² is a 0/1 subset sum of rows. One interior lattice point that is not a subset sum forces R > 1. So R = 1 is a finite decision problem, and either answer has a finite exact certificate.
• Theorem C. R ≤ 2, so 1 ≤ R ≤ 2. Each support block is one pinned coordinate plus every ±1 pattern on the other three, so each block holds four disjoint row pairs equal to ±2eⱼ for each of its coordinates. Rounding at a vertex of the fiber polytope leaves a lattice residual with |r′ⱼ| ≤ 4. One or two rows fix its parity, and a Hall-theorem assignment of lightly damaged blocks pays the rest with intact pairs. Every coordinate moves by less than 2. rounder.py executes the construction in exact arithmetic.
• Theorem B (weaker, independent): R ≤ 6, from an explicit 8-row basis of L with column ℓ₁ norm 6 plus Beck–Fiala rounding.
• Evidence for R = 1 (not a proof). 1,864 sampled interior lattice points near vertices and faces of rank 0–7 are all 0/1 subset sums. Each witness is rechecked exactly; interiority came from a floating-point LP and is not certified.

No κ, frontier, caller or framework-assumption change. Like #6312, this is a property of the exact-mode contract only.

Verification (macOS, Python 3.14.8, base 2485805a6)

• python3 -I -W error -B research/mode-fiber-radius-bounds/verify.py and the same with -O both print PASS_MODE_FIBER_RADIUS_BOUNDS in about 5 s. Exact integer/rational checks cover:

  • the structure, the HNF lattice identity and the rank condition Theorem A uses;
  • the Prove exact mode preservation needs a full rounding unit (no κ) #6312 witness relation;
  • the τ = 6 basis;
  • the 14 pinned blocks and every 2eⱼ pair;
  • the Hall lemma, exhaustively over every set of at most three damaged blocks (minimum slack 3);
  • 120 exact Theorem C rounder runs, worst error 279/140;
  • all 1,864 witnesses.

  Nine rejection controls fail closed.

• A separate exact stress run of the rounder on 12,000 random and sparse inputs found no failure; the worst error was 839/420 ≈ 1.998. So 2 is the limit of this construction, not evidence about R.

• CI (plan and the 3.12 package replay) passed on the first push.

• scripts/research_package.py --check-only: PASS_PACKAGE_METADATA_AND_PINS. The registered replay pins Python 3.12, which isn't installed locally, so it is left to CI.

• The non-certifying search scripts (search/, numpy/scipy) are included for regeneration.

Remaining obligations / open next questions

1. Decide R = 1: prove or refute that every interior lattice point is a subset sum. The suggested route is symmetry reduction plus a face recursion. Brute force over about 10¹² points is out.
2. Close the gap between 1 and 2. Theorem C loses a full unit at the vertex step. A suggested route is to pick a fiber point near y blockwise instead, then reuse the pair-and-Hall repair.

Theorems A, B and C are written proofs (PROOF.md); no proof-assistant or independent human review is claimed.

Credit

Dillon Amburgey (PR4933) supplies M. Nick Petalas (#6312) supplies the contract, the radius-one witness and the question, and his predecessor credits carry over. Beck–Fiala rounding, Hall's theorem and the Hamming E8 construction are classical.

Prepared with substantial AI assistance (Claude Code, Anthropic). Apache-2.0.

🤖 Generated with Claude Code