The spectral-gap constant of the pentachoric transfer operator: exact traces, two-sided bounds, and the obstruction to a closed form

The one-dimensional reduction of the pentachoric transfer operator on the Kuhn triangulation of Z⁴ 1 is the compact, symmetrizable operator on L² (0, 1) with kernel G (x, y) w (y), where G (x, y) = exp (- (15/2) (x − y) ²) erf (√6 (1 − μ) ) + erf (√6 μ), μ = (x + y) /2, and w (y) = erf (√30 min (y, 1 − y) ). Its parameter-free spectral-gap constant δ* = ln (λ₁/λ₂) = 0. 934082. . . has no known closed form. We make the trace side exact: Tr T closes in error functions plus three Owen T-functions with algebraic arguments whose angles sum exactly to π/2 and whose exponential scales are the integers 9, 9, 6 T1; the anti-diagonal trace is elementary, J = √ (π/30) erf (√6/2) erf² (√15/2), so both parity-resolved traces Tr T± close T1. The second trace reduces exactly to one-dimensional integrals whose infinite-window kernel is a single Owen term; the finite-window remainder is trivariate normal, for which no closed form exists T1/T2. Trace-power inequalities on the squared parity sectors bracket δ* two-sidedly to width 6. 3 × 10⁻⁵² T2, and the uniform N-level alphabets of 3 converge to δ* at rate 1/N T2. We further exclude the natural local Schrödinger reading: the midpoint effective Hamiltonian mispredicts the gap by +24% with growing level defects, and the effective potential has logarithmic walls of coefficient exactly one, outside every shape-invariant solvable family, and the log-spectrum grows sub-quadratically, against the Weyl law of any local operator T1/T2. The gap of the unconfined kernel vanishes: δ* is a pure boundary constant. Both negative results are documented as such, blockers and defects made explicit. Every claim is verified by the companion script (71 tests, 0 failures).