[pinterbanget]

Summary

This adds research/depth4-count-vectors-check/, an independent check of #4279 (Nick Petalas). #4279 classifies the call-count vectors of the unsigned depth-four dirty chain connector at m = 2..5: a vector is attained, for any nonzero c, exactly when its entries are multiples of m and it dominates a member of P(m). The bills follow:

m	cheapest bill
2	2S + 2(p2+p3)
3	3S + 6·min(p2,p3)
4	4S + 4·min(p2,p3)
5	5S + 5·min(p)

#4279's completeness step is a finite reduction checked only by its own search.

Claim type: an independent exhaustive finite computation in the box of entries ≤ 3m. No new κ.

Result

For every m = 2..5 and every residue c, within the box:

• the minimal attained vectors are exactly Classify depth-four chain connector count vectors and unequal-price tariffs for m = 2..5 (no new κ) #4279's P(m);
• the attained set is exactly the multiples of m dominating P(m): 36, 54, 72 and 80 vectors per residue;
• P(m) does not depend on c;
• the four bill formulas equal min over P(m) at 2,000 random rational prices, giving 12, 18, 20 and 25 at unit prices.

Method and independence

The method is different from #4279's middle-letter reduction:

• Reachable sets. R(n), the products of words with exact counts n, satisfies R(n) = ⋃_l g_l·R(n − e_l) in the lower unitriangular group mod m.
• Coset compression. The adjacent entries of R(n) are n_l mod m, so each set lives on six coordinates.
• Implementations. Python, standard library, reruns m = 2, 3 in about 19 seconds. countdp.c covers m = 2..5; m = 5 takes 21 s and about 128 MB. Its recorded runs are bound to the source by SHA256, agree with Python at m = 2, 3, and are local evidence that CI does not recompile.

Nothing is imported from #4279.

python3.12 -B scripts/research_package.py --package research/depth4-count-vectors-check --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/depth4-count-vectors-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/depth4-count-vectors-check outcome=completed

Head: 2dce07ef435e776889e23f1f1f58afaa989611fc.

Not covered

• Vectors outside the box. Classify depth-four chain connector count vectors and unequal-price tariffs for m = 2..5 (no new κ) #4279's reduction covers every length and is not re-derived here.
• m ≥ 6, where Classify depth-four chain connector count vectors and unequal-price tariffs for m = 2..5 (no new κ) #4279 cites Attain the depth-four chain connector floor 4m for every 6 ≤ m ≤ 32 (no new κ) #4263; Independently check the depth-three and depth-four chain-connector tables of #4249, #4263 and #4280: every word and record holds (no new κ) #4343 checks Attain the depth-four chain connector floor 4m for every 6 ≤ m ≤ 32 (no new κ) #4263's words separately.

Task and overlap

• Card: connector-alphabet-change, refined, priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244.
• Classify depth-four chain connector count vectors and unequal-price tariffs for m = 2..5 (no new κ) #4279 has no comments or outside check.

Credit

The model, the theorem, P(m) and the bills are Nick Petalas's (#4279). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code