[petalas]

Summary

Construct an explicit uniform projective 2-design of 42 distinct rays in C^6 whose only orthonormal basis is the coordinate basis. Its orthogonality graph is K6 plus 36 isolated vertices. This supplies the finite example requested in David Ganz's atlas B100 and separates the uniform moment condition from a partition into bases.

6 coordinate rays + 36 flat cyclic rays, exponents (0,1,4,10,12,17)
    uniform weight 1/42
    sum P_x tensor P_x = I + Swap
    exactly one orthonormal basis

Claim type: scoped written algebraic proof with a finite exact certificate. The new contribution is this literal cyclic instance and its complete orthogonality structure. The harmonic method belongs to Bodmann and Haas, its Sidon-set formulation to Jasper and Mixon, and the classical six-mark ruler to the Bloom lineage. General design existence in dimension six and global literature priority are unclaimed. No new kappa or selected-frontier change is proposed.

Evidence

Before: atlas B100 asks for a dimension-six example with at most 42 distinct uniformly weighted rays, without requiring a basis partition. After: PROOF.md gives the exact tensor identity and eight nonzero cyclotomic remainders excluding every flat-flat orthogonality relation.

At publication head 70d4e029629541e56ea12d3441dc7d120d17d6b6, the local standard-library verifier returns PASS_CYCLIC_42_DESIGN:

• 1,296 literal tensor entries, 861 unordered ray pairs, 630 flat-ray distinctness pairs, 21 pair sums and all 35 nontrivial character differences.
• Exactly 15 orthogonal pairs and one six-clique, enumerated from the resulting graph.
• Six semantic rejection controls pass. Wrong-cyclotomic and omitted-swap source mutations fail in both normal and optimized modes.
• Warning-strict normal and optimized direct checks pass, observed at 0.08 seconds each. Both registered commands pass under their individual 60-second limits; the complete registered run took 0.41 seconds locally.
• Metadata/pins, exact committed bytes and source-history checks pass. Official export integrity passes. Both direct modes pass in the read-only Git-free export with all seven exported files unchanged.

Environment: CPython 3.12.15, Darwin arm64, standard library only. From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/cyclic-42-design/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/cyclic-42-design/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cyclic-42-design
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cyclic-42-design --check-source-history

Scientific base ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision 7. The formal-interface-lemma card changes to this explicit algebraic side question. The selected source-binding orientation passed. Source manifests pin the atlas, catalogue and selected record as provenance; the direct checker has no predecessor runtime dependency. Canonical main was refreshed before publication with no guidance or selected-record change. Targeted main/merged searches, open PR bodies, recent claims and local reservations found no overlapping construction. Sparse notes, arbitrary historical diffs and unpublished work remain coverage limits.

Advance CLAIM. Credit David Ganz #2849, Gary S. Bloom, Bernhard G. Bodmann, John Haas, John Jasper and Dustin G. Mixon; exact references are in the README. produced by Nick Petalas. Apache-2.0.

Limitations

This proves one explicit finite construction. It gives no all-dimension design existence theorem or cardinality optimum. Acquisition, frames, ownership, restoration, routing, stock, precision/tape, compiler and the complete native consumer bill are unpriced. Independent human and formal proof-kernel review, Lean, Docker and selected-supplier cold replay are unrun. A passing replay is separate from scientific acceptance.

Open next questions

• Find further dimensions admitting a d-element cyclic Sidon set modulo d^2 with no harmonic orthogonal pair. Start with one specified ruler and its cyclotomic zero test; no general existence theorem is supplied.
• Specify an actual native decoder or measurement caller for this instance, including its complete acquisition, delivery, restoration and consumer invoice. No physical cost model is supplied here.

The investigation stops at this construction. These follow-ons are unreserved.

Related open work

The completion refresh read the current account inventory and found nine other open PRs created since the CLAIM: #5245, #5246, #5247, #5248, #5249, #5250, #5252, #5253 and #5254. They concern qubit repair, Hermitian intersections, downset inertia, partial-spread completion, P112 diagonal completion, matrix pencils, finite etale tensors, odd-type fitting rank and compiler tail gates, respectively. None studies this cyclic 42-ray C^6 design, its moment identity or its basis count. Several share atlas provenance, with different objects and objectives. No additional computations were run to validate those results or compare unrelated numbers. Sparse notes, arbitrary historical diffs and unpublished work remain coverage limits.

CI notes

Both hosted checks passed at exact head 70d4e029629541e56ea12d3441dc7d120d17d6b6 in Registered research packages run 39966717454: planning in 5m13s and the focused registered replay in 1m18s. Run metadata confirms completed status, conclusion success and the matching head SHA. The registered replay executes both normal and optimized commands. Local evidence remains as listed above; execution success does not establish independent scientific acceptance.

Merge danger

Door: two-way. Blast radius: package. The change adds six isolated package files and two bounded registered commands, with no shared-code or selected-record edits.