[petalas]

Summary

- 9 ≤ N(4) ≤ 20
+ 9 ≤ N(4) ≤ 17

A scoped written proof excludes every eighteen-source floor walk. Cube intersections, unions of cubes with the same parity, and colexicographic triple-sum classes exclude every repeat-count case except three. Axis-deletable prefixes and a conservative nonattainment bound close both branches of those cases, with every final count below 262,144. The deletion lemma excludes every larger source count.

A complete finite census also proves that a balanced five-source depth-two block covers at most 28 of 31 nonzero masks. Exactly four words attain 28 up to source relabelling; all four and their missing masks are retained. This excludes concatenating two five-source blocks of depth two, while leaving general ten-source walks open.

The package includes an eight-inventory all-ones normal form for future searches and a finite witness attaining the cube cap H(18,0)=22. That witness claims only intersection sharpness.

Claim types: scoped proof, exhaustive finite classification, structural lemma and finite sharpness certificate. No new κ or selected-frontier change.

Evidence

• Before: Lower the characteristic-four connector floor bound to N(4) ≤ 20; the known nine-source walk does not extend (no new κ) #4255 supplies 9 ≤ N(4) ≤ 20 and excludes insertions into one nine-source walk.
• After: the written triple-class argument excludes eighteen sources. All 21 initial cases, the parity tail and six residual branches replay; the earlier 76-case cube arithmetic remains a control. The depth-two censuses account for all 2,546,168,625 canonical words and find zero words covering at least 30 masks including zero, and exactly four covering at least 29.
• Standard-library normal/optimized replays pass, including separate literal component and catalog checks, exhaustive small cube controls, future-cut table controls and unpruned small-case calibration. A separately written C engine matches both sharp censuses and accounts for every four-source canonical word with negative feasibility pruning disabled.

From the repository root:

~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/connector-char4-floor-obstructions/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/connector-char4-floor-obstructions/independent_check.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/connector-char4-floor-obstructions
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/connector-char4-floor-obstructions --check-source-history

Optional independent native commands are in research/connector-char4-floor-obstructions/DEPTH2.md. Generated receipts bind all fifteen package files.

Limitations and unrun obligations

N(4) is not settled exactly. Ten through seventeen sources remain open; no ten-source floor word is supplied. The depth-two classification does not exclude coupled depth-four words. The earlier single-cube cap is attained at the eighteen-source, zero-duplicate corner. The new proof controls collections of cubes to close that case; the finite sharpness witness itself makes no floor-attainment claim.

Independent human and formal proof-kernel review are unclaimed. No physical caller, complete tariff, admission, dispatch or precision obligation is instantiated. The optional native cross-check passed locally and is not a registered CI obligation. Every registered command passed locally.

Open next questions

• Find a ten-source floor walk in the eight-inventory normal form, or prove ten sources fail. The four maximizing depth-two blocks give a complete small seed catalog for coupled searches.
• Apply or improve the triple-class/cube-union count at seventeen sources.
• Identify and completely price a characteristic-four caller before turning abstract batches into a recursive supplier.

Related open work

These are arithmetic consistency checks on cited results, not independent audits of their proofs or searches.

• Lower the characteristic-four connector floor bound to N(4) ≤ 20; the known nine-source walk does not extend (no new κ) #4255: its interval [9,20] is narrowed to [9,17]; the nine-source lower bound and its one-family nonextension result are retained.
• Reach the unsigned connector floor for k ≤ 2c in every characteristic; 9 ≤ N(4) ≤ 21 (no new κ) #4248: source of the nine-source construction and deletion lemma. Its k=9 characteristic-four floor is 4(9+511)=2080, unchanged here; [9,17] refines its [9,21].
• Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231: source of the interface and floor. At c=4,k=5, 4(5+31)=144 agrees with its optimum.
• Bound the unsigned F3 connector floor for every source count: attained for n ≤ 6, impossible for n ≥ 14 (no new κ) #4232: characteristic-three source-count bounds. The common floor at k=6 is 3(6+63)=207, agreeing with its attained six-source case.
• Prove the seven-source F3 connector floor fails: optimum 405, floor attained exactly for n ≤ 6 (no new κ) #4250: characteristic-three seven-source exclusion. Its floor 3(7+127)=402 agrees with the common model; its 405-call optimum and N(3)=6 are cited, not rerun.
• Prove the F3 connector excess grows with n: exact 795-call eight-source optimum (no new κ) #4264: characteristic-three excess. At k=8 the common floor is 3(8+255)=789, and its cited optimum 795 has excess 6, agreeing with its statement. No c=3 bound changes here.
• Prove the exact 108-call unsigned F3 five-source connector optimum #4225: characteristic-three five-source case; 3(5+31)=108 agrees with its optimum.
• Research: exact F3 minimum for the one-source additive connector #4189: characteristic-three one-source case; 3(1+1)=6 agrees with its optimum.
• Count optimal unsigned one-source connector words in every characteristic #4236: unsigned one-source counts in every characteristic; at c=4 its minimum 2c=8 agrees with 4(1+1)=8. Word multiplicities are not used here.

Merge Danger

Door: Two-way.

Blast Radius: Package.

Adds one isolated research package. Shared code and the selected construction are unchanged.

CI notes

All four registered commands and the optional native cross-check passed locally. Committed-byte and source-history preflight passed on e68a4999f9a13f0aa86807b7b951bf697611ae98; all fifteen files match the commit. Both hosted checks passed on the same commit: plan (2m45s) and package replay (4m38s). CI run 39518247635 completed successfully. The only change since 0f07447 is the wording of scope_change_reason in research/connector-char4-floor-obstructions/TASK.json.

This package was produced by Nick Petalas. It builds on #4231, #4248, #4255 and #4232, and Ross Richardson's #943/#934 connector lineage. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.