[petalas]

Result

Scoped proof with finite exact exclusion certificates. No κ is claimed. This answers the open minimum question of Ali Romman's #4729 within two explicit Gram families. For #4729's pinned 8×112 source matrix, both raw responses are dyadic units exactly when det(52I−G) and det(4I−G) are signed powers of two, where G is the selected Gram.

• Two-block family: the minimum is 38. Here G_B = 52I − 32(I+V/4)(I+Vᵀ/4) with V = [[0,B],[0,0]] and B integral. No selected set with fewer than 38 ports has a Gram equal to a signed coordinate conjugate of G_B and passes both unit conditions. Construct selected response units and reduce two normalization costs #4729's 38-port set lies in this family, so 38 is sharp.
• One three-coordinate block: the minimum is 39. Here the Gram is H ⊕ 20I₅ with H integral symmetric 3×3. No such selected set with at most 38 ports passes both conditions. A 39-port set exists, with H = [[16,−8,−8],[−8,20,0],[−8,0,20]]; it is the norm-two-row construction already in Construct selected response units and reduce two normalization costs #4729's proof.
• Reflections. Both exclusions still hold after conjugating by any word in the reflections O_r = I − rrᵀ/2 of the 112 columns plus the 8 axis vectors 2eᵢ.
• Still open: the minimum over arbitrary selected sets below 38.

Proof outline

• Two-block reduction. Positive semidefiniteness forces 50I − 13BBᵀ ≻ 0, so every entry of B is 0 or ±1. Enumerating 107,240 row multisets and canonicalising signs leaves 11 classes.
• Three-block reduction. The trace identity bounds the trace of H. The two unit conditions make det(52I−H) and det(4I−H) powers of two. Of 8,568 parameter tuples, 20 characteristic polynomials pass the exact filters; of 33,350 integral candidates, 24 signed-permutation classes remain.
• Placements. The source supports are exactly the 14 affine planes of F₂³ with all sign patterns, so the coordinate symmetries form AGL(3,2) (order 1344). That leaves 30 placement classes, and all 40,320 permutations are checked.
• Exclusions.
  • All 900 two-block cases have exact integer box-separating certificates.
  • For the three-block cases:
    • 600 of the 720 fail a root-pair congruence modulo 4.
    • The other 120 have box certificates.
  • No solver output is used in the proof.

Evidence and reproduction

research/selected-unit-family-minima/. From the repository root:

python3 scripts/research_package.py --package research/selected-unit-family-minima --check-only   # PASS_PACKAGE_METADATA_AND_PINS
python3.12 scripts/research_package.py --package research/selected-unit-family-minima             # PASS_REGISTERED_RESEARCH_PACKAGE, ~4 s
python3 -B scripts/check_research_commit.py --package research/selected-unit-family-minima --check-source-history   # PASS_RESEARCH_COMMIT_BYTES

verify.py uses only the Python 3.12 standard library and runs in under 2 s. It rebuilds both reductions, the placement coverage and the reflection closure. It checks all 1,620 exclusions and rebuilds the 38- and 39-port witnesses from the literal columns. It passes both normally and with -O.

SOURCE-INTERFACE.json is a byte copy of #4729's interface file at head b556092c. #4729 is not merged, so SOURCES.json cannot pin that commit; the verifier checks the copy's SHA256 instead. The README records the exact path and blob.

Limitations

• Only the two named families are covered. General upper shears, non-integral coefficients, several coupled blocks and other determinant families are outside the proof.
• Searches outside the proved families found nothing below 38: annealing for sets of size 36 and 37, and small-block targets. That is a search failure, not an exclusion.
• No native, routed-access, restoration, precision or selected-construction replay was run.

Open next questions

• Is there a selected set below 38 outside the integral two-block family?
• Can a target coupling four or more coordinates, or several interacting blocks, reach 38 or fewer?
• Does any new scalar witness meet Construct selected response units and reduce two normalization costs #4729's native access, endpoint, restoration, cost and precision contracts?

Related open work

• Construct selected response units and reduce two normalization costs #4729 (Ali Romman): the source matrix, the two-unit criterion, the 38-port witness and the open question this answers in part. No other open PR claims these minima; the open-PR and Discussion check was refreshed at about 18:24 UTC today.

Credit

This package was produced by Nick Petalas. Ali Romman's #4729 supplies the source matrix, the unit criterion and both sharpness constructions. Its lineage includes dcharb78's #3133 and Jeremy Diviney's #3148. Its retained notices credit Rohan Arun, Cinna Davis (#2226) and Jeremy Kothe (#2588).