Pre-registered statement on the dark-energy equation of state from a cost-asymmetric lattice geometry: w0 = −1 + 13/324, wa ≥ 0

This note pre-registers, with dated custody of every link, the dark-energy statement of the cost-asymmetric lattice programme (the CAL model; the note itself refers to it under its earlier name, the pi-square-equals-one discrete geometry programme): at z = 0, w0 = −1 + ξ with ξ = 13/324 exactly, i.e. w0 = −0.95988…, and, in the CPL parameterisation, wa ≥ 0, with wa = 0 exactly under the constant-invariant reading. The invariant ξ = 13/324 is a theorem with an exact closed form. The identification of −1 + ξ with the equation of state is a conjecture, marked as such at its source; this note does not upgrade it. The registered point (−0.960, wa ≥ 0) currently sits between ΛCDM and the DESI DR2 central preferences, and stakes a sign on wa opposite to the present quintom-B central fits. The statement is designed to be cleanly refutable by the converged DESI, Euclid and Roman analyses. As evidence of good faith, we record that the same corrigendum which fixed ξ at 13/324 destroyed this programme's most celebrated numerical agreement: the fine-structure coincidence 1/α = 137.036 does not survive it, the corrected chain yielding 138.09. This is externally checkable rather than merely asserted: the earlier deposits of this programme, published in June 2026 and still online, display the superseded 37/1296 and the 137.036 agreement; the corrected value is stated here, and the correction is being propagated to those deposits. The numbers follow the mathematics here, not the reverse. Epistemic status: theorem for ξ; conjecture for the identification; pre-registration for the dated claim.