Toward Open Problem 3: The Annulus Liouville Correlator

This working note addresses Open Problem 3 of the reconstruction programme — the Liouville two-point functions on the annulus with FZZT boundary conditions — not by solving it, but by showing that its single bottleneck separates into a modular component, which is resolved, and a boundary component, which reduces to a small set of sharply identified residuals; one of those is then shown to be irreducibly external. Status markers: [E] established, [C] conditional, [N] numerical, [O] open. Open Problem 3 had been the common lock for several programme components at once: the value of the elementary scale κ (OP1), the OPE factorisation hypothesis (F) together with the modular Hamiltonian of the KMS/modular article, and the Bohr–Sommerfeld route to the action quantum. The note establishes that it splits into two independent sectors — modular and boundary — and treats each. Modular sector (resolved in the scale-invariant bulk). On the flat cylinder [0, Lρ] × S¹, the physical dilation flow is generated by Hdil = −i∂ρ = L0 + L̄0 of radial quantisation, and the optimal reconstruction state is KMS for it. By Tomita–Takesaki and the conformal invariance of bulk Liouville, the normalised modular flow is the dilation rescaled by the temperature, with modular operator Δ = exp(−2π Hdil) and modular Hamiltonian K = 2π Hdil. The factor 2π is the Bisognano–Wichmann temperature, verified numerically to coincide exactly with the inverse temperature β = 2π obtained earlier [E] (modulo the standard radial-quantisation and Tomita–Takesaki hypotheses); the FZZT boundaries deform this only within collar neighbourhoods, negligible over Lρ ≈ 140. The cylinder is moreover the logarithmic image of the punctured plane, which supplies rigorously the periodicity and single-valuedness previously assumed under hypothesis (F). [E] Boundary sector (reduced, not closed). The bulk two-point function is assembled in spectral form from the established FZZT boundary Liouville data, conditional on the standard boundary bootstrap [C]. By SO(2) invariance the boundary velocity potential is constant, so the Bohr–Sommerfeld integral collapses to a single number S0 = ℏ fixed by the FZZT saddle momentum P* — exactly the reconstruction-area problem already isolated, A* = (2πP*)² Tmin², with no new gap; the residual is the value of P*. [O] The calibration sector, and the no-go for γ. The remaining question is whether the cosmological calibration selects the Liouville coupling b. It reduces to a single boundary number, the cutoff-scaling exponent γ of the boundary cosmological constant: a physical b ∈ (0, 1.3] exists iff γ ≤ 0.03 [N]. The note then closes this question with a no-go. The exponent γ — equivalently the asymmetry of the two FZZT branes, equivalently the residual Δn in the absolute lepton-mass exponent κ = mP exp(−n Lρ), with n = 1/3 + Δn and 1/3 = Vol(S³/Z3)/Vol(S³) the holomorphic Z3 section weight, c = Δn · Lρ ≈ 0.285 — is fixed by no internal object [E]. Stationarity symmetrises the brane moduli; the modular/dilation flow carries the boundary modulus at dilation weight exactly zero, so it can neither generate nor differentiate an asymmetry; channel duality underdetermines it; the curvature dictionary μB = K∂ overshoots by some five orders; and the Z3 boundary flux φℓ = 2/9 (the η-invariant of the lens space) organises only the relative spectrum — the closure cost factorises exactly as c = cα + cflux = −1.640 + 1.925, the flux contributing the relative-mass structure (+1.925), not the observed residual (0.285), which lives in the absolute on-site scale. Resolution. The exponent γ (equivalently Δn) is therefore the external absolute-scale datum — the lepton on-site energy α² — a third non-local calibration alongside the unit ℏ and the span Lρ (the closure constant, itself shown external in The Closure Constant Is Not Determined by the FZZT Sector). Structurally κ = mP exp(−n Lρ) is dimensional transmutation, which always leaves the exponent free, so fixing the absolute scale is the flavour/hierarchy problem, not an internal boundary datum; the calibration is non-selective in the only internally accessible sense, and b decouples from Lρ. [C] The note also retracts an earlier suggestion of a cutoff-independent boundary ratio, which had conflated the fixed Fisher conformal factor with the dynamical Liouville field. Significance and status. The achievement is structural: a problem previously viewed as a single bottleneck is decomposed into a resolved modular component and a small set of sharply defined residual boundary computations, one of which is then proved external. [E] the modular resolution (Δ = exp(−2π Hdil), the 2π Bisognano–Wichmann temperature, the periodicity from the logarithmic map) and the five falsification arguments of the no-go · [N] the numerical 2π coincidence and the γ ≤ 0.03 window · [C] the boundary spectral assembly (conditional on the boundary bootstrap) and the external-datum identification · [O] the residual boundary computations — the FZZT saddle P*, the reconstruction normalisation, the collar corrections — and the value of Δn itself, which is external and remains open.