Research: seven-source singleton universal path — length 18 excluded, f(7) >= 20 (no new kappa)

[navinor]

Seven-source singleton universal path: f(7) >= 20 (no new kappa)

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

Change and mathematical scope

In the PR 943 prefix-walk model (#943, research/universal-connector-boundary), a restored source word w over n singleton labels is universal when paired readouts realize every mask, D(w) = {q_s xor q_t} = F2^n; f(n) is the minimum universal word length. The pinned record (#1185, research/universal-connector-six-source) leaves 18 <= f(7) <= 24 with f(7) explicitly open. This package removes 18:

No universal seven-source word of length 18 exists. Hence f(7) >= 20 and the gap is 20 <= f(7) <= 24.

The alphabet is singletons only. The charged DUO pair-source family (#2396, #2128, #2156) and the 63-mask weight-1/2/3 sparse family (#2229, #2236, #2397) are distinct scopes and are recorded as concurrent, not overlapping.

Method

Smallest falsifying instance first: the engine rebuilds the pinned research/universal-connector-six-source/EXPECTED.json byte-for-byte (945 words max 31/32; 10,395 words max 49/64; 945,945 words max 59/64; records SHA-256 3dcc5689…, 491b295f…, 3d711ffa…), plus the counting lower bounds 10/14/18 and the Gray constructions. The pinned record is f(5) = 12, f(6) = 16; the assignment's placeholder values were corrected by reading the pinned packages.

Then the length-18 census at n = 7, via the prefix vertex set P = E union O (even/odd weight):

1. Position parity gives |E|, |O| <= 9; counting gives |E||O| >= 64, C(|E|,2) + C(|O|,2) >= 63.
2. A pair-representation identity (machine-checked rows in CERTIFICATE.json, independently re-derived by collision-multiset enumeration) contradicts the 17-point splits (9,8) and (8,9): full odd coverage forces sum_x A_x B_x >= 4, full even coverage bounds the same sum by 1. So |P| = 18 and every universal 18-word would walk a simple 18-cycle with the forced (9,9) split and collision budget e_ex + b_ex <= 9.
3. engine.cyclic_census(7, 18, budget=9, simple=True) exhausts the restricted-growth direction words (complete under coordinate relabeling, a hypercube automorphism) with those proved-necessary prunes: 15,775,992 nodes, 312,064 simple 18-cycles, 0 universal words. Shards partition the tree; leaf counts agree with an unsharded run exactly.

Controls and non-vacuity: the cyclic census's leaf set equals the reference canonical word set exactly at n = 5 length 10 and n = 6 length 12; zero universal words at n = 5 length 10 and n = 6 length 14 reproduces f(5) = 12, f(6) = 16 from the other side; planted hits are found at n = 5 length 12 and n = 6 length 16 and re-checked; the set-level engine agrees with a brute-force census of all 20,160,075 ten-point sets of F2^5 (99,200 covers, none cycle-realisable) and reproduces the community nine-point result (7,888,725 sets, max 29, zero covers, #1166 (comment)). independent.py re-checks coverage with the dual predicate, realizes cycles with subset dynamic programming, and re-derives the identity rows without the closed-form bound.

Reproduction and checks

From the repository root:

python3 -I -B research/universal-f7-singleton-seedb859517b/verify.py
python3 -I -B research/universal-f7-singleton-seedb859517b/independent.py
python3 -I -B research/universal-f7-singleton-seedb859517b/test_engine.py
python3 scripts/research_package.py --package research/universal-f7-singleton-seedb859517b --check-only

Observed markers:

PASS_F7_SINGLETON_GAP_VERIFY f(7) >= 20 (length 18 excluded) cycles=312064 universal_words=0 nodes=15775992 seconds=72.2
PASS_F7_INDEPENDENT {"control_sets": "ok", "lemma_rows": {...}, "witnesses_replayed": 0}
Ran 22 tests in 154.694s — OK
PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)

The decisive census is search.py (about 15 minutes, resumable from per-shard checkpoints): PASS_F7_SINGLETON_GAP f(7)>=20 L18-excluded.

Scope and open questions

A finite exclusion inside PR 943's abstract interface; no instantiation in the selected supplier and no kappa claim. Open: whether a universal seven-source singleton word of length 20 or 22 exists or the Gray bound 24 is sharp; and whether any 18-point cover of F2^7 exists at all (the set-level restricted addition basis value r((C2)^7), whose six-source analogue the community records as 14 after Gavrylkiv). Structured next questions are in TASK.json and the package README.

[NicholasStone]

Correction to the WORD-census evidence: #2973 (head 91bde9fedb6a1872906876e60235af8b8a45f732) confirms that cyclic_census uses bit-0 groups with fixed collision budgets derived for weight parity. A concrete simple balanced n4 L8 universal counterexample is 01021323 (bit0 split6,2, excess8 > wrongly substituted budget5). Old traversal agreement is therefore insufficient to justify the word exclusion.

The corrected weight-parity engine independently restores L18 (9,9) with 951,416 nodes/zero hits and completes all five admissible L20 splits with zero hits; every run uncapped, exit0, complete=true, and then freshly replayed from the packaged source. With exact split classification, parity-swap and padding, the new finite result is 22 <= f(7) <= 24 inside PR943's abstract singleton word model. No selected-frontier/kappa change. Independent validation covers 164 small complete comparisons and 36 capped n7 comparisons, plus proof controls. Frozen predecessor packages are preserved. Actual-size bit0 SET-level enumeration remains algebraically valid, so unrelated later set-level proofs are not revoked. See PR2973 for the proof, exact receipts, source pins, limitations and predecessor attribution. Substantial OpenAI Codex assistance; no independent human review claimed.

[navinor]

Corrective review (sweep #8): the word-census justification, reproduced independently

Replying to Nicholas Stone's correction on this PR (#2542 (comment)). I reproduced the finding before writing this reply.

1. Reproduced: the counterexample. A fresh standard-library CPython 3.12 script, written against the definitions in PR 2973's MATH.md section 4 (#2973, fetched at head 91bde9fedb6a1872906876e60235af8b8a45f732), confirms every stated number for the balanced n=4 L=8 word 01021323:

• each of the 4 labels occurs exactly twice; all eight nonclosing prefix states are distinct; q_8 = 0 (restored, simple, balanced);
• prefix set {0,1,2,3,4,6,8,12}; pair-XOR coverage is all 16 masks, so the word is universal;
• weight split (4,4); bit-0 split (6,2);
• wrongly substituted budget C(4,2)+C(4,2)-7 = 5; bit-0 collision excess 8; correct bit-0 budget C(6,2)+C(2,2)-7 = 9.

Replaying this package's own engine.py::cyclic_census accounting on that word (states grouped by nq & 1, single caller budget) rejects the walk at cumulative excess 8 with the substituted budget 5, and admits it with the correct budget 9. Script output: PASS bit0_collision_excess=8 old_prune_rejects=True (8 > 5) and PASS corrected_budget_admits=True (8 <= 9). As PR 2973 notes, this establishes unsoundness of the prune; it is not by itself a seven-label counterexample at length 18 or 20.

2. Reproduced: the budget substitution. This package's decisive invocation is engine.cyclic_census(7, 18, budget=9, simple=True). The engine groups prefix states by bit 0 (elif nq & 1 in cyclic_census), while budget=9 = C(9,2)+C(9,2)-63 is the pair-identity budget derived for the (9,9) weight-parity split. Time parity fixes the sizes of the weight classes; it does not fix the sizes of the bit-0 classes (PR 2973 MATH.md section 4: for bit-0 sizes 10+d,10-d the valid budget is 27+d^2, not invariably 27). PR 2973's corrected engine groups by Hamming-weight parity ("groups":"hamming_weight_parity" in its receipts) and uses each target split's own budgets even_budget = C(S,2)+C(T,2)-63 and odd_budget = S*T-64. The substitution is the error; the number 9 itself is correct for a weight-parity (9,9) census.

3. Affected sentences in our records (word-level claims resting on the old traversal-agreement justification):

• PR body, blockquote: "No universal seven-source word of length 18 exists. Hence f(7) >= 20 and the gap is 20 <= f(7) <= 24."
• PR body, method step 3: "with those proved-necessary prunes: 15,775,992 nodes, 312,064 simple 18-cycles, 0 universal words." The bit-0 fixed-budget prune is not proved necessary.
• PR body, controls: "the cyclic census's leaf set equals the reference canonical word set exactly at n = 5 length 10 and n = 6 length 12" and "Shards partition the tree; leaf counts agree with an unsharded run exactly." Traversal agreement between implementations that share a prune proves replay fidelity, not the necessity of the shared prune; PR 2973 states this directly ("Reproducing node counts between the old Python and C implementations proves traversal agreement, not the necessity of their shared prune").
• Receipt markers PASS_F7_SINGLETON_GAP_VERIFY f(7) >= 20 (length 18 excluded) cycles=312064 universal_words=0 nodes=15775992 and PASS_F7_SINGLETON_GAP f(7)>=20 L18-excluded (search.py, EXPECTED.json, CERTIFICATE.json) are historical outputs of the unsound filter, not self-standing proof.
• Package wording carrying the same claim: research/universal-f7-singleton-seedb859517b/README.md ("the proved-necessary prunes above", the length-18 blockquote), CLAIM.md ("each prune a proved necessary condition"), TASK.json ("Delivered: length 18 excluded, so f(7) >= 20 ..."), CERTIFICATE.json and EXPECTED.json ("f(7) >= 20 (length 18 excluded)").

4. What stands, and on whose evidence. The conclusion "no universal seven-source word of length 18 exists" stands. It now stands on PR 2973's corrected weight-parity engine by Nicholas Stone (NicholasStone) (#2973), which independently restores the L18 (9,9) census at 951,416 nodes and zero hits, uncapped, exit 0, complete=true, and completes all five admissible L20 splits with zero hits; corroborated by PR 3005 by Nick Petalas (petalas) (#3005, open at writing time). What falls is our justification and gap wording: the census receipts above no longer support the exclusion on their own, "proved-necessary prunes" is withdrawn as applied to the bit-0 fixed-budget prune, and the corrected finite result 22 <= f(7) <= 24 (PR 2973) supersedes our "20 <= f(7) <= 24" gap phrasing.

5. SET-level claims are unaffected and are not demoted. PR 2973 scopes this explicitly ("Actual-size bit0 SET-level enumeration remains algebraically valid"): the pair-representation identity rows, the identity contradiction of the 17-point splits (9,8)/(8,9), the reduction of L=18 to simple (9,9) cycles, the counting lower bounds and the verified Gray constructions are not revoked. The 17-point cover chain remains completed at set level by PR 2773 (#2773) and PR 2870 (#2870). PR 2973's MATH.md section 7 records this scoping.

6. Record state in main. PR 2973 is merged and its package research/f7-weight-parity-census-correction/ records the correction in main's tree, with an affected-material inventory in MATH.md section 8 that prescribes preserving frozen predecessor bytes rather than silently changing pinned sources. I therefore opened no separate docs PR: the correction is already recorded in main, and the demotion above applies to how our merged records read against it.

Correction comment on this PR: #2542 (comment)
Correction comment on the sibling PR: #2634 (comment)
PR 2973 (correction, merged): #2973
PR 3005 (independent corroboration, open at writing time): #3005

Contributor: Aaron Wang, with help from opencode (no model named; substantial AI assistance in scoping, verification and writing; no independent human review claimed). All cited PRs were verified with gh pr view --json author,title at writing time: PR 2542 and PR 2634 by Aaron Wang (navinor), PR 2973 by Nicholas Stone (NicholasStone), PR 3005 by Nick Petalas (petalas).