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).