[Matarener]

Requested by Mark

Change and mathematical scope

Optimistic model screen. No new κ, no supplier, no native word. Package research/window-break-even-4470.

Under the productive-helper accounting of the selected #4470 word, a shorter label window h (fresh block d = h − 1) lowers the cost a line system may pay per line. It also raises the lines per dimension the system needs. For dense units with one deleted frame (the case of every built complex supplier), h = 3, 4, 5 and 7 need more lines per dimension than the proven Theorem B cap v/T ≤ 2^k′ + 1 of line-system-saturation allows, and h = 6 misses by about 2%. So "fewer fresh axes" does not pay by itself. The live cells are the even windows h = 6 and h = 8, and deleting more than one frame.

• verify.py rebuilds the Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 targets and paid histograms exactly from joint-helper-lattice/RESULT.json. The root is 0.003020165977.
• It generalises the invoice to a line system (v, T, h, D, k deleted lines): ports v − k, centre D − k paid as T jobs (the E8 floor 72 − k), the same Conditional κ = 0.00296 via query-thinned productive controllers (+10.18%) #4440 controller, and the Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 local word scaled by V/112 in counts and d/8 in ranks. The surplus is exactly S = (p+2)(v − D) − V (checked in exact arithmetic).
• Named systems as a multiple of the record: E7 0.710 (0.825 with two frames deleted), E6/D4/D5/D6/Z8 0, E8+A3 0.238, D16⁺ 0.565 (Delsarte cap 0.649), D24⁺ 0.130.
• At the Theorem B cap: h = 6 gives 0.980 / 1.166 / 1.435 for k = T / 2T / 3T, h = 7 gives 0.710 / 0.825 / 0.983, and h = 8 gives 1.460 or more. At k′ ≤ 2 a saturated dense unit has D = v and no saving at all.
• Break-even v/T for k = T, needed vs cap: h = 7: 10.06 vs 9, h = 8: 12.36 vs 16, h = 11: 21.7 vs 33 (best known 12.7). Nothing ties at h ≥ 21.

Remaining obligations, all outside this package: a native word with free central shears for any new unit, its real local profile (the scaled word is a proxy), sparser pairings, and the BIT side.

Task and overlap

Base ff3cc58e3. Card complex-new-helper-inventory, scope refined: the earlier line-system screens (#2244 line-system-saturation x*(h), #2873, discussion #3622) used the pre-#4359 break-even fit. This package re-prices the window question under the #4470 invoice. find_overlap.py (open PRs and discussions refreshed 2026-10-23, closed PRs from a 2026-10-18 cache) found no such re-pricing.

Validation

python3.12 scripts/research_package.py --package research/window-break-even-4470 --check-only
python3.12 scripts/research_package.py --package research/window-break-even-4470
python3.12 -B scripts/check_research_commit.py --package research/window-break-even-4470 --check-source-history

All three PASS locally (Windows, CPython 3.12, standard library only). The verifier takes about 3 minutes and replays RESULT.json. test_verify.py has rejecting controls (a changed helper count, a dropped centre, a wrong pair count) and positive controls (a tie unit has no saving, monotone in v/T, the Theorem B caps, the root near S/G). This is a model screen, not a lower bound.

Credit

It uses the #4470 invoice of joint-helper-lattice (zxv's native supplier, the PR4440/PR4441/PR4447 contributions and the other authors credited there), the cost table of char2-threshold, and the Theorem B cap and constructions of line-system-saturation. All are referenced unchanged. Prepared with substantial Claude Code assistance.

Open next questions

1. Does a saturated even-window system exist at h = 6 (v = 8T) or h = 8 (v = 16T)? These are the only short windows whose cap clears the record here.
2. Is the centre floor D − k attainable beyond one deleted frame? The h = 6 and h = 7 verdicts depend on it.

🤖 Generated with Claude Code

[jwr404]

Independent review (James Reed, with Claude Code). Head 0a66b2f9da7acbf85e8142b392f0c41e55207ff8.

Verdict: the screen's arithmetic and model reductions check out. It is correctly scoped as an optimistic model screen, with no κ.

Checks

• Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 calibration, independently.
  • I took the targets and paid histograms straight from joint-helper-lattice/RESULT.json, without the package's invoice builder.
  • A 50-digit Decimal bisection of Σ_T m r^(1−c) − Σ_P m r^(1−c) gives root 0.00302016597701. That matches the recorded calibration_root and lies above the selected 60403/20000000.
  • The rank-mass surplus is exactly 752, which is (p+2)(v−D) − V = 18·48 − 112 as the README states.
• "T = 8 without loss of generality" holds. In the generalised invoice every count scales linearly with T, and every rank (1 + d·…, r·d/8, centre rank (D−k)/T = h − k/T) is independent of T at fixed v/T and k/T. Numerically, best() returns identical roots to 12 digits for T = 3, 5, 7, 8 and 13 in three cells:
  • v/T = 9, h = 7, k = T: 0.002143944044, the E7 row, 0.710 of the record;
  • v/T = 8, h = 6, k = T: 0.002958698456, the 0.980 cell;
  • v/T = 9, h = 7, k = 2T: 0.002492505900, 0.825 of the record.
• Theorem B caps. cap(h) matches line-system-saturation §4: (i) v ≤ T(2^k′+1), and (ii) v ≤ T·2^k′ when 1 ∈ col(Ā), the even window h = 2k′. Taking the k′ given by h is the largest admissible cap, so this direction is optimistic, as a screen should be.
• Replay. scripts/research_package.py passes locally on Linux in 20 s (outcome=completed), so cross-platform float rounding didn't bite.

Notes (non-blocking)

1. Fractional proxy ranks. Scaling local ranks by d/8 gives non-integer ranks for d ≠ 8; at h = 7 the helpers have ranks 3, 5.25, 6 and 6.75. That is fine for a mass-neutral moment proxy. A sentence saying the proxy ranks need not be realisable would help readers, since the per-row C values depend on that proxy rather than on any word.

2. Even windows (corrected after posting). I first flagged the hypothesis 1 ∈ col(Ā) behind the even-h cap. It is automatic. The window is h = rank(J + Ā) with Ā alternating of rank 2k′:

   • if 1 ∉ col(Ā), then h = 2k′ + 1;
   • if 1 = Āx, then h = 2k′ − 1 would need xᵀĀx = 1, which is impossible for an alternating form over F2, so h = 2k′.

   Even h is therefore equivalent to the hypothesis, and cap(h) is exactly Theorem B. No change is needed.

3. The break-even bisection compares against the rounded kappa_decimal 0.00302015, not the reconstructed root 0.003020166. The difference (~1.6·10⁻⁸) is negligible at the reported precision, but worth one line.

This review covers the model and arithmetic. It does not cover native realisability (open question 1) or the BIT side, which the PR also leaves out of scope.