[petalas]

Result

For the unsigned dirty connector of #4231 in characteristic c, let N(c) be the largest source count at which the floor c(k + 2^k − 1) is attained. #4248 proved N(c) ≥ 2c and asked how N(c) grows (open question 2). This PR adds research/connector-five-halves-growth:

• Theorem T (scoped proof, finite core checked): one explicit 30-event base walk with per-source cut positions expands to a five-source depth-two block in every characteristic c ≥ 5. Replace the (a_j+1)-th event of source j by a run of c − 5 copies (deleted at c = 5). Each expanded prefix is congruent mod c to an integer "lifted" prefix, and every nonzero mask is an integer sum of two lifted prefixes, so the identity holds in every c ≥ 5.
• Corollary (proof): with Reach the unsigned connector floor for k ≤ 2c in every characteristic; 9 ≤ N(4) ≤ 21 (no new κ) #4248's concatenation and deletion lemmas, N(c) ≥ ⌊5c/2⌋ for every c ≥ 5, for example N(5) ≥ 12, N(6) ≥ 15, N(7) ≥ 17 and N(8) ≥ 20 (previously 2c).

Characteristic four is untouched. #4271 excludes five-source depth-two blocks at c = 4, and the template needs c − 5 ≥ 0. There is no new κ, no consumer, no selected-frontier change and no multiplication theorem.

Evidence and reproduction

verify.py uses only the standard library and takes about 2 s. It checks:

• the integer template identity for all 31 masks (the only finite step the proof needs);
• the expanded block for every 5 ≤ c ≤ 64, by direct search modulo c and through the lifted readouts;
• literal floor words on 12, 15 and 17 sources for c = 5, 6, 7 (20,535, 196,692 and 917,616 calls), simulated with generic coefficients over Z/c;
• five rejection controls.

~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/connector-five-halves-growth/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/connector-five-halves-growth/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/connector-five-halves-growth
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/connector-five-halves-growth --check-source-history

All four passed locally (CPython 3.12, macOS). Marker: PASS_CONNECTOR_FIVE_HALVES depth_two_k5_every_c>=5 N(c)>=floor(5c/2). Both registered VERIFY.json commands take about 1.5 s each.

Limitations

• Only the unsigned, immutable-source, complete-call interface of Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231. The prefix criterion, concatenation lemma and deletion lemma are cited from Reach the unsigned connector floor for k ≤ 2c in every characteristic; 9 ≤ N(4) ≤ 21 (no new κ) #4248/Bound the unsigned F3 connector floor for every source count: attained for n ≤ 6, impossible for n ≥ 14 (no new κ) #4232 and not re-derived.
• Theorem T is proved for every c ≥ 5. The direct block check covers c ≤ 64 and the literal word simulation c ≤ 7.
• N(c) for c ≥ 5 remains open between ⌊5c/2⌋ and the counting bound. No physical consumer or call pricing is supplied.
• Proof-kernel checks and independent human review are not claimed.

Open next questions

1. Does any characteristic admit a depth-two block on six or more sources? One valid for all large c would give N(c) ≥ 3c.
2. Is there a depth-three block on eight or more sources valid for every large c (density 8/3 > 5/2)?
3. Is N(c)/c unbounded? Counting gives O(c log c).
4. Does any concrete consumer use a characteristic-c batch at a favorable complete invoice?

Related open work

• Research: seven-source weight-at-most-four connector boundary (L<=18 excluded, 216-call floor) (no new kappa) #4079, Add five-source arity-four connector cost bound #3909 and research: audit minimal joint-output connector charge #3897 study other connector variants (weight-at-most-four outputs, arity-four calls, joint-output charges). None studies N(c) or depth blocks; no shared number.

• Certify that balanced depth-three chain multiplicities break every unit symmetry for m ≤ 24 (no new κ) #6037 studies balanced multiplicities of the depth-three chain connector (Classify all depth-three connector count vectors and unequal-price optima #4257), a different connector object from the unsigned all-mask connector here; no shared number.

No open PR studies N(c) or depth-two blocks.

Credit

Builds on #4231 (interface, floor), #4248 (prefix criterion, depth, concatenation and deletion lemmas, N(c) ≥ 2c, open question), #4232 (deletion lemma), #4255 (counting bound) and #4271 (c = 4 depth-two obstruction). Claim: #1166 (comment)

CI notes

Both hosted checks passed on head 119de1d7fca5124fa8d01ea49d258560aca618e6: plan and replay (research/connector-five-halves-growth), in run 40608712659. The run head matches the PR head.