KB3d: (11,6)/(6,11) remainder rows (5,1),(6,0) of the 17-point cover question impossible by an exact Fourier/Parseval identity (no new kappa)

[navinor]

KB3d: the (11,6)/(6,11) remainder rows (5,1),(6,0) of the 17-point cover question — impossible by an exact Fourier/Parseval identity (no new kappa)

Contributor: Aaron Wang, with help from opencode (substantial AI assistance in scoping, implementation, verification and writing; no model named; no independent human review claimed). No kappa, supplier, certificate or selected-frontier change.

Change and mathematical scope

In the PR943 prefix-walk model (PR 943 by Ross Richardson), universality of a restored seven-source singleton word is the covering condition P + P = F2^7 on its prefix set. PR 2542 by Aaron Wang asserts "No 17-point set covers F2^7". PR 2773 by Aaron Wang audited that claim at the set level and recorded the whole remainder: the rows (5,1),(6,0) of the (11,6)/(6,11) family (pair-sum excess 5 and 6 at 11 points, shared = 1, c = 1), mattering only at L = 22. This package decides those two rows.

Result (set level; no words enumerated; labels in CERTIFICATE.json):

• PROVEN: rows (5,1) and (6,0) are impossible. An exact Fourier/Parseval identity on the forced odd-coverage profile excludes the entire (11,6)/(6,11) family: for a set A in the even coset and B in the odd coset of F2^7 with sizes 11 and 6 (either order), the 66 cross sums cannot cover the 64 odd masks. Exact obstruction in both orders: Parseval remainder 696 versus allowed 504, deficit 192.
• PROVEN (jointly with PR 2542 and PR 2773): "No 17-point set covers F2^7" is complete at the set level — all six counting-feasible splits are excluded. Consequently at L = 22 every universal word needs at least 18 distinct prefixes.
• CHECKED: the recorded predecessor anchors are reproduced before any new number (table below; commands and outputs in ANCHOR-REPRO.json).
• DEMOTED: the construction route (an 11-set of pair-sum excess 5 or 6 covering F2^7) is refuted; the planned row-restricted joint (A, B) fallback search was not needed and is not run.

The identity (exact hypotheses)

Write S = A u B with A in the even-weight coset E of F2^7 and B in the odd-weight coset O. Sums of two points of S land in O exactly when the two points differ in parity, so the odd half of S + S = F2^7 requires every odd mask to be a cross sum a + b. Let r be the cross-sum multiplicity of A x B, 1_O the indicator of O, and rho = r - 1_O, which is nonnegative on O with total $E_o = s t - 64$.

For characters $\chi_u(x) = (-1)^{u \cdot x}$ put $\hat f(u) = \sum_{a \in A} \chi_u(a)$ and $\hat g(u) = \sum_{b \in B} \chi_u(b)$, so $\hat r(u) = \hat f(u) \hat g(u)$ exactly, while $\mathrm{FT}(1_O)$ is supported on $\lbrace 0, \mathbf{1} \rbrace$ (the zero vector and the all-ones vector). Then:

1. for every $u \notin \lbrace 0, \mathbf{1} \rbrace$ one has $\lvert \hat f(u) \hat g(u) \rvert = \lvert \hat\rho(u) \rvert \le E_o$;
2. $\hat f(u) \equiv s$ and $\hat g(u) \equiv t \pmod 2$ for every $u$, so the odd-cardinality side's character is nonzero, and the even-cardinality side's character $\hat c(u)$ satisfies $\lvert \hat c(u) \rvert \le E_o$ with $\hat c(u) \equiv 0 \pmod 2$ off $\lbrace 0, \mathbf{1} \rbrace$;
3. Parseval on that side, with $\hat c(0) = \mathrm{card}$ and $\hat c(1) = -\mathrm{card}$ (all its points sit in one coset), gives $\sum_{u \notin \lbrace 0, \mathbf{1} \rbrace} \hat c(u)^2 = 2^{n} \mathrm{card} - 2 \mathrm{card}^{2}$.

If that remainder exceeds $(2^{n} - 2)$ times the square of the largest even integer up to $E_o$, the hypothesis is contradictory. For $(s, t) = (11, 6)$ at $n = 7$: $E_o = 66 - 64 = 2$ forces $\hat c(u) \in \lbrace -2, 0, 2 \rbrace$, yet $128 \cdot 6 - 72 = 696 \gt 126 \cdot 4 = 504$ — contradiction, deficit 192. The mirror $(6, 11)$ is identical with the roles exchanged (an odd translation swaps E and O and preserves S + S). Only the odd half of the covering condition is used; even coverage is not needed. All arithmetic is exact Python integers; no floats are used.

Machine results (exact integers)

Split	E_o	Bounded side	Characters off the hyperplane	Parseval remainder	Allowed	Verdict
(11,6)	2	B (6 points)	-2, 0, 2	696	504	impossible (PROVEN)
(6,11)	2	A (6 points)	-2, 0, 2	696	504	impossible (PROVEN)
(10,7)	6	A (10 points)	-6 ... 6	1080	4536	no Fourier contradiction
(7,10)	6	B (10 points)	-6 ... 6	1080	4536	no Fourier contradiction
(9,8)	8	B (8 points)	-8 ... 8	896	8064	no Fourier contradiction
(8,9)	8	A (8 points)	-8 ... 8	896	8064	no Fourier contradiction

The checker is not vacuous: it fires exactly on the (11,6)/(6,11) rows among the 17-point splits (the four others are excluded by PR 2542's rows and PR 2773's census route), and a real 10-point cover of F2^5 ({0,1,2,3,4,5,6,8,16,31}, split (4,6), odd excess 8) passes every check. Independent controls: the convolution identity is exhaustively verified over all 65,536 set pairs of F2^3; the bound on $\hat\rho$ is exhaustively verified over all 2,080 total-2 multiplicity shapes on the 64 odd masks (worst value 2); Parseval is re-checked by direct pair double counting (768 = 768 and 1408 = 1408).

Reproduce

From the repository root, standard library only, CPython 3.12:

python3 -I -B research/f7-17point-fourier-obstruction-seedb5e318fe/test_engine.py
python3 -I -B research/f7-17point-fourier-obstruction-seedb5e318fe/independent.py
python3 -I -B research/f7-17point-fourier-obstruction-seedb5e318fe/verify.py
python3 scripts/research_package.py --package research/f7-17point-fourier-obstruction-seedb5e318fe --check-only

Observed markers: Ran 13 tests ... OK; PASS_FOURIER_INDEPENDENT {...}; PASS_FOURIER_VERIFY seconds=0.0 printing OBSTRUCTION (11, 6) odd_excess=2 bounded=B parseval=696 allowed=504 deficit=192 contradiction=True, OBSTRUCTION (6, 11) odd_excess=2 bounded=A parseval=696 allowed=504 deficit=192 contradiction=True, REPRO rows remainder (5,1),(6,0) shared=1 c=1 (both orders; census floor e_ex >= 5), CONTROL other 17-point splits (10,7),(7,10),(9,8),(8,9): no Fourier contradiction, CONTROL real 10-point cover of F2^5 passes every check, CONTROL vendored engine small-group 420/840/168 ok; PASS_PACKAGE_METADATA_AND_PINS; PASS_REGISTERED_RESEARCH_PACKAGE research/f7-17point-fourier-obstruction-seedb5e318fe outcome=completed.

The row remainder is reproduced from the vendored PR 2773 row engine: sharp_rows(7, 11, 6) yields exactly the seven rows (e, 6-e, 1) with shared = 1, c = 1, and the recorded k = 11 census floor e_ex >= 5 leaves exactly (5,1),(6,0) — the object decided here, in both parity orders.

Recorded anchors (reproduced before any new number)

In-session against byte-exact predecessor extractions (agent_tools.py pr-tree-extract; commands and observed outputs in ANCHOR-REPRO.json):

Anchor	Value	Status here
PR 2542/2634 REPRO tiers	945/31, 10395/49, 945945/59 (records SHA-256 match)	recomputed
PR 2542 length-18 census	15,775,992 nodes, 312,064 cycles, 0 universal	recomputed
PR 2773 zero shards (k10 budgets 0/1/2, k11 budget 2)	all 0, match receipt	recomputed (verify.py --full, 909.6 s)
PR 2773 row table	remainder (5,1),(6,0)	recomputed
n = 5 set controls	0 nine-point covers (max 29/32), 99,200 ten-point covers	recomputed
PR 2773 k = 10 budget-3	149,990,400	recomputed in full (PASS_ANCHORS_K10B3 count=149990400, 62/62 sub-shards)
PR 2773 k = 11 levels 3-4	0, 0 (62/62 sub-shards)	recomputed in full (PASS_ANCHORS_K11B4 count=0, 62/62 sub-shards)

anchors.py recomputed both heavy anchors in-session by the established raw-domain walk (complete GL(6,2)-transitive domain through {0,3}, sub-shards by the smallest free element, per-sub-shard receipts in ANCHOR-PROGRESS.json, vendored engine byte-pinned): every k = 10 leaf sits at exact excess 3, and the k = 11 walk has zero leaves at excess <= 4. Neither heavy anchor is load-bearing for the verdict: the rows (5,1),(6,0) are excluded by the identity above, which uses only exact integer arithmetic.

Pair-collision disclosure and concurrency

Entry, refined (before heavy computation) and pre-submission pair-collision checks — discussions 1317 and 1166 with nested replies via agent_tools.py disc-comments, plus scripts/find_overlap.py --refresh with the object's proper names ("17-point", "pair-sum excess", "Sidon", "11-set", "(11,6)", "cover F2^7", "Fourier", "Parseval", "rows (5,1)") — found no claim on these rows and no Fourier/Parseval treatment of the 17-point question. The concurrent claim of Nicholas Stone (discussion 1166 comment 18954844) is the repeated-state L = 20 word census; this package is set-level only and enumerates no words. Handoff: the 17-point object is now closed at every split, so no |P| = 17 word family survives at any length (its (10,7) split was already empty a fortiori by PR 2773); the claim's remaining splits (10,9),(10,8),(9,9) are 18/19-point objects this identity does not reach ($E_o \ge 8$ there) and are untouched.

Scope and limits

A finite set-level result inside PR943's abstract interface. Not checked: every word-level question, the 18-/19-point cover question r((C2)^7), and any kappa, supplier, tariff or selected-frontier implication. The recorded gap 20 <= f(7) <= 24 is unchanged. Base commit 544a7e82810e30cab61c297cb2e2ea86127fe944 is retained per the assignment pin; canonical main was fetched at 734ef056148cea45daf9af0bd089a7b5c46fea8f at the pre-submission refresh (2026-10-16), where certificates/selected-result.json differs from the pin only in one scripts/verify_hpe_lean_binding.py manifest hash (tooling, not the frontier) and docs/research/agent-task-catalog.json is unchanged.

Credit

• PR 2542 by Aaron Wang (navinor): the audited claim and the pair-representation identity.
• PR 2773 by Aaron Wang (navinor): the 17-point cover audit, the sharp collision-multiplicity rows, the census floors and the exact remainder rows decided here. Its cover17.py and search17.py are vendored byte-exact (SHA-256 pins 082bc838134cba97... and 9eb1bf79ffb99b0e..., enforced by test).
• PR 2634 by Aaron Wang (navinor): the recorded KB3b state reproduced here before new work.
• PR 2691 by Nicholas Stone (NicholasStone, Kimi agent): the completed (10,10) simple-20-cycle census (recorded state, not re-run).
• PR 943 by Ross Richardson (RossRichardson): the prefix-walk model.
• PR 1185 by Doug Colkitt (Mister-Meeseeks): the pinned f(5) = 12, f(6) = 16 record and the frozen tier EXPECTED.json reproduced here.

All authors verified against current ground truth with gh pr view NNN --json author,title at writing time. Claim thread: discussion 1166 comment 18961235.