[pinterbanget]

Summary

This adds research/delta-cross-cell-transfer/. It answers the open question of #4452 (and of my #4460): do jwr404's within-cell δ exponents 11, 24, 26, 26 and 6 survive on the banded reference across cells?

Claim type: a conditional transfer lemma plus a robustness lemma, with exact rational checks. No new κ. The ideal gaps 1/(8p²) stay hypotheses.

Results

• Lemma A (downward reading, LOG-A + Prove the downward rounding contract for across-cell Gaussian entries #4449's 0 ≤ H ≤ n). This is Apply the smaller halo exponents across a cell, under the downward reading (no new κ) #4442's column split with δ in place of 1/(16p²). It gives off(H) ≤ T_in/(1−δ) + T_out ≤ (1−g)/(1−δ). The global inverse ceiling is therefore exactly Lower the within-cell halo exponents to 11, 24, 26, 26 and 6 (no new κ) #4452's U, and all five exponents transfer verbatim.
• Lemma B (any reading with excess ρ ≤ 1/(32p³) per row or column). We get ‖H^{−1}‖₂ < 10p², so 11, 24, 26, 26 and 6 all hold. Per entry, an absolute error up to 1/(64p⁴) in either direction suffices.
• What Lemma B changes. Apply the smaller halo exponents across a cell, under the downward reading (no new κ) #4442 found that under 16p²−1, a 1/64 overshoot broke p = 2, so the downward-vs-"rounded" (FINE-PRIME) wording mattered. Under δ it no longer matters for these exponents. FINE-PRIME's poly(p)2^{−(C−o(1))p} perturbation works eventually.
• Near-tightness. At p = 2 the excess tolerance on this route lies in (0.0091, 0.0092).

Evidence

python3.12 -B scripts/research_package.py --package research/delta-cross-cell-transfer --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/delta-cross-cell-transfer
  PASS_REGISTERED_RESEARCH_PACKAGE research/delta-cross-cell-transfer outcome=completed   (~45 s, stdlib only)

The checks are exact for each p from 2 to 64, plus all-p inequalities. The split identity is proved as a polynomial identity on a 3×3×3 grid. The LOG-A, FINE-PRIME and downstream phrases are pinned. Head: c8fd7f81696d5b8b838c996cc384e76ab378dae3. Base: canonical main 2a13eb18.

Not covered

• The ideal gaps.
• Page-generator implementation.
• The Schur complement and the core exponent 400.
• Any new κ.

Credit: James Reed (jwr404: #4442, #4451, #4452) and mihir-s-05 (#4449). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code