[JosieElliston]

Change and mathematical scope

Adds only research/connector-bounded-depth-cap/. It is a scoped upper bound for the unchanged unsigned dirty connector of #4231/#4248. No operation is added, and there is no new κ or selected-frontier change.

Theorem. A depth-d block (#4248's definition) on k sources in characteristic c ≥ 2 satisfies

2^k − 1 ≤ k · min(k, c) · binom(kc + d − 2, d − 1).

Proof: boundary t has coordinate sum t, so the read boundaries of a mask m sum to |m| mod c. Fix the first d−1 reads. The last read is then one of k candidates for each weight, and it determines m.

Consequences:

• Depth two: 2^k − 1 ≤ k²c·min(k, c) ≤ k³c, so k ≤ 1 + log₂ c + 3 log₂ k. Pair counting allowed about 2 log₂ c. Prove unsigned connector floor capacity N(c) = Θ(c log c) via Gray-walk blocks #6071's reflected-Gray blocks (k = 1 + ⌊log₂ c⌋) are within O(log log c) of the largest possible depth-two block. The depth-two search "near k ≈ 2 log₂ c" suggested in Prove unsigned connector floor capacity N(c) = Θ(c log c) via Gray-walk blocks #6071 open question 1 cannot succeed for large c.
• Sharp constant for depth ≤ 2 concatenations: Reach the unsigned connector floor for k ≤ 2c in every characteristic; 9 ≤ N(4) ≤ 21 (no new κ) #4248-style concatenations of blocks of depth ≤ 2 have n ≤ c · max(2, K₂(c)/2) = (1/2 + o(1)) c log₂ c, which Prove unsigned connector floor capacity N(c) = Θ(c log c) via Gray-walk blocks #6071's lower bound attains.
• Depth ≤ D: such concatenations are capped at (1 − 1/D + o(1)) c log₂ c. A lower constant above 1/2 therefore needs depth ≥ 3 blocks; reaching 1 needs unbounded depth or reads that cross block boundaries.
• A routine sharpening of Reach the unsigned connector floor for k ≤ 2c in every characteristic; 9 ≤ N(4) ≤ 21 (no new κ) #4248's count puts the general constant in [1/2, 1]. That question stays open, and no new lower bound is claimed.

At practical c the cap is loose (21 against Gray's 9 at c = 2^8). Convergence to 1/2 is only logarithmic in log c, and the README table shows this.

Task and overlap

Base 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8. Chooser seed 9aa6e642b3ad1c5695053de603870c91 → connector-alphabet-change. Scope changed: the alphabet stays fixed, following #6071's released open question 1 (the leading constant). The reason is in TASK.json. This is a local interface side result and does not pass any supplier budget gate.

Canonical claim. scripts/find_overlap.py --task connector-alphabet-change, live handoffs, open PRs and searches for "leading constant" and Gray found no earlier depth-two cap beyond pair counting. That is not a novelty certificate.

Nearest work: #4231/#4248 (interface, prefix criterion, concatenation/deletion, Lemma C, counting; this package is the depth-d analogue of Lemma C, and its README is the pinned input), #6035/#6057/#6061 (finite depth-two blocks), and #6071 at a0f08278dd755e0bf228855e45c5efaa3b8ad97a (Gray blocks, Θ(c log c), the open question). #6071 is pending and used for comparison only. Its block is rebuilt in the verifier from its README statement.

Validation

CPython 3.12, macOS arm64, standard library only, about 2 s:

python3.12 -I -W error -B research/connector-bounded-depth-cap/verify.py
python3.12 -I -O -W error -B research/connector-bounded-depth-cap/verify.py
python3.12 -B scripts/research_package.py --package research/connector-bounded-depth-cap
python3.12 -B scripts/check_research_commit.py --package research/connector-bounded-depth-cap --check-source-history --check-modes

All pass. The verifier covers:

• An exhaustive census of all 39,288 floor words for (k, c) ∈ {(2,2..5), (3,2..4), (4,2)}, with the weight identity and the per-weight last-read count checked on every representing multiset at depths 1 and 2 (and 3 when kc ≤ 9). That is 1,031,591 checks.
• 21 rebuilt Gray blocks, each checked for depth two on the literal word.
• The exact capacity table up to c = 2^4096, with Fraction ratio checks.
• 4 rejected controls.

Finite checks support the written proof; they do not replace it. No selected replay, Lean, Docker or native supplier was run, and none is claimed.

Review. A separate OpenAI Codex read-only adversarial review found no error in the theorem or its corollaries. It confirmed the boundary-kc reduction, the zero padding, the multiset count, the cases c = 2, k ≥ c and repeated reads, and that the global d = c version is weaker. It raised four points, all fixed at the current head:

• The exact d = 2 form is k²c·min(k,c); k³c is only a relaxation.
• K_d needed a definition.
• The ratio assertions should compare exact Fractions, not rounded values.
• One control was circular and has been dropped.

This is an AI review, not human or proof-kernel review.

Open next questions

1. A depth-three block family with k ≈ 2 log₂ c would give lower constant 2/3, the most depth three allows.
2. Is the general constant above 1/2, and is it 1?
3. What is the exact depth-two capacity at small c?

Details are in the README and TASK.json.

Credit

Prepared by JosieElliston with substantial Anthropic Claude (Claude Code) assistance. Credit Nick Petalas (#4231/#4248/#6035) and the retained Ross Richardson / Riquelme Soares lineage, apoursh (#6071, Gray construction and the question), and zxv (#6057). Reflected and balanced Gray codes are established combinatorics. Apache-2.0; no predecessor files changed.

🤖 Generated with Claude Code