[pinterbanget]

This answers #4600's open question 2 (@petalas) within #4280's row model. Three short rows realize every unit coefficient for every m ≥ 6, and every nonzero c for every m ≥ 7. For m ≤ 5, no nonzero c is realized with three short rows, so the threshold is sharp. No new κ.

Construction (m ≥ 14)

Take λ = (m−1, m−2, 3) and ρ = (m, 3t, 2t, padding ≤ 2t).

• Choice of t. t = 1, unless c = −1. Then t is a unit with 2 ≤ t < m/3: t = 2 for odd m, t = 3 when 3 ∤ m, and otherwise the smallest prime not dividing m.
• Choice of β. β = (balanced block of sum X + 2m, balanced middle of sum S, X, 0), where X ≡ c·t⁻¹ lies in [1, m−2] and S ≡ −2(X + 2m) with S just above (m−5)X.
• Result. B(3) ≡ X, B(m−2) ≡ −X and B(m−1) ≡ B(m) ≡ 0, so the coefficient is tX = c.

Checks

• Construction: fully written, including the monotonicity step (with X = m − k, the excess (−3k mod m) fits under (m−5)·⌊2k/3⌋). It is also checked for every nonzero c and every 14 ≤ m ≤ 200.
• Exact enumeration of the whole model for m ≤ 13. It also gives the m ≤ 5 negative result.
• Fixed-shape dynamic programme for m ≤ 30, as a cross-check.

Together with #4622 (two short rows exactly when gcd(c, m) ≥ 3), the short-row complexity of every coefficient is now determined in the row model.

research/three-short-row-units/verify.py uses the standard library only and takes about 20 s.

Limitations

• Inherited model. The row model and its converse come from Attain the depth-four chain connector floor 4m for every 33 ≤ m ≤ 64, via an exact row model (no new κ) #4280.
• Not claimed. Call counts beyond Prove the two-short-row connector obstruction for every modulus #4600's.

Credit: petalas (Nick Petalas) for #4600 and #4280. Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance.

Head: 7e1b489

🤖 Generated with Claude Code