[petalas]

Result

New package research/cubic-three-query-arities. It answers part of Open next question 1 of CubeJerry's PR6250: which remaining arities admit a three-query dyadic repair of the fixed cubic reader outside directions 0,d,2d.

Claim type: scoped written proof with exact finite certificates. At each of the 17 arities

6, 7, 12, 13, 21, 25, 42, 51, 52, 76, 85, 93, 97, 114, 156, 157, 160

three integer-direction cube queries with dyadic weights, plus integer source shears, retain all r children and restore arbitrary incoming y, a, b to the eager endpoint y + r*A^3. None of these arities is in an earlier three-query class: r = 2 mod 3, r = 3, 4 (apoursh, PR6212); r mod 9 in {0,1} (apoursh, PR6215); r = 2^s (AAAmirinu, PR6246); r = 2^s+1 (CubeJerry, PR6250). PR6212 proves that three queries are necessary for r > 2, so the exact dyadic minimum at each of these arities is three.

The new step is a node lemma. Distinct directions c1,c2,c3 represent the defect exactly when e2(c) = m*e1(c) - S1, where m = (2r-1)/3 and S1 = r(r-1)/2. The weights are then unique: w_j = S1(m - c_k - c_l)/((c_j-c_k)(c_j-c_l)). Example at r = 6, from the deferred source a = A+6B: a -= 8b; y += a^3; a += 4b; y += 10a^3; a += b; y -= 11a^3; a -= 3b.

No new κ. The selected result is unchanged.

Evidence and reproduction

Interpreter: CPython 3.12 (~/git/integer-mult-bounds/.venv/bin/python3.12 locally), standard library only.

python3.12 -B scripts/research_package.py --package research/cubic-three-query-arities
python3.12 -I -W error -B research/cubic-three-query-arities/verify.py
python3.12 -I -O -W error -B research/cubic-three-query-arities/verify.py
python3.12 -I -W error -B research/cubic-three-query-arities/search.py --check research/cubic-three-query-arities/CERTIFICATES.json
python3.12 -B scripts/check_research_commit.py --package research/cubic-three-query-arities --check-source-history

Local results, CPython 3.12 on macOS arm64:

• The registered replay printed PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=partial in 11.2 s total. The search check takes about 9 s; each verifier run takes under 0.1 s.
• The commit check printed PASS_RESEARCH_COMMIT_BYTES at 3ea9965e1.

verify.py accepts a certificate only by its own checks, not by the closed weight formula:

• It solves the first three coefficient equations by exact elimination and requires a zero B^3 residual.
• It requires every weight to be nonzero and dyadic.
• It runs each literal program on 12 dirty integer and rational inputs.
• It corroborates the node-lemma residual identity on random rational triples.
• It reproduces PR6250's r=33 weights and anchors PR6215's residue table and PR6212's two-query statement in the pinned README bytes.
• Six rejection controls must fail.

search.py regenerates CERTIFICATES.json byte for byte.

Limitations

• Search failure only: 50 of the 67 uncovered arities r <= 160 have no integer-direction certificate with c1,c2 in [-160, r+160). They are listed in the JSON. No four-query obstruction is claimed. Non-integer dyadic directions and general dyadic frames were not searched.
• This is a local coefficient and restoration result. There is no query or shear price, physical backend, complete invoice or supplier admission. The lower bound of three is cited from PR6212, not re-proved.
• No independent human or formal proof-kernel review.

Open next questions

1. Do r = 15, 22, 24, 30, 31, ... admit three queries with non-integer dyadic directions or general frames, or is there a four-query obstruction? A suggested first step is a valuation test on e2(v) = (9-3(2r-1)^2)/2 with v_j = 6c_j-(2r-1).
2. Do the weight shape (-r, r/2, r/2) seen at r = 7 and r = 97 and a Pell-type recurrence give an infinite new family?
3. Does the r = 6 or r = 7 repair beat r eager inverse shears under one pinned backend?

Related open work

No open PR studies the cubic-reader repair or its query count. The open PRs touching cubic objects (#6410 central UT4 tariffs, #6617 affine seven-cube decoding) concern different objects. This account's PRs opened since the claim (#6649-#6652, #6654-#6662) study other objects; none computes a cubic-repair query count.

Credit

Produced by Nick Petalas. The question is CubeJerry's PR6250. The interface and lower bound are apoursh's PR6212. The original-b classification is apoursh's PR6215, and the power-of-two branch is AAAmirinu's PR6246. The fixed reader comes from PR6200. No predecessor file is modified.

CI notes

Both hosted checks passed at head 3ea9965e13457fdafc70b784305d026ebfa8faa2 in run 41661232737: plan (4m57s) and replay (research/cubic-three-query-arities) (4m8s).