The Reconstruction Area Problem

The Reconstruction Area Problem This article addresses one of the central calibration problems of the reconstruction programme: how to determine the elementary area scale associated with the Fisher–Rao geometry without assuming Planck’s constant as an input. Purpose Article 12 showed that an action scale is structurally required by the canonical symplectic completion of the programme, and that its non-vanishing follows from the rigidity of the Fisher–Rao structure. However, the numerical value of this action scale was not derived. Article 13 reformulates this issue as the reconstruction area problem: compute an area quantum A* from Fisher–FZZT geometric data without using ℏ, the Planck length, or any quantity already defined in terms of ℏ. Main Problem The target is to determine A* independently. If this can be done, then the relation ℏ = c³A* / G would become a calibration theorem rather than a dimensional restatement. The Planck area is therefore not assumed; it is the target value to be recovered. Main Results Flat-cylinder obstruction. On the flat cylinder, the area scale cannot be determined spectrally. All relevant eigenvalues scale with the conformal factor A, so the flat geometry contains no intrinsic mechanism for fixing A*. Curvature as symmetry breaking. The Liouville sector introduces non-zero curvature and therefore breaks the conformal freedom of the flat cylinder. This provides a possible geometric scale independent of ℏ. FZZT route to the area quantum. Under the boundary-action hypothesis from Article 10, the reconstruction area quantum is related to the FZZT saddle momentum P* by: A* = (2πP*)²Tmin². Self-consistency condition. The circularity of using a Planck-scale cutoff is reduced to a dimensionless fixed-point condition: P* = 1/(2π). The remaining task is to solve the FZZT saddle equation and determine whether this condition is satisfied. Numerical evidence. The paper reports a three-level numerical test showing that the condition P* = 1/(2π) admits real, finite, non-pathological FZZT boundary parameters for the tested values of the Liouville coupling. Open Computation The remaining problem is explicit: compute the FZZT saddle using the full Barnes double-gamma function Υb, solve the resulting equation for the cylinder length L, and then evaluate whether the induced action scale a₀ = c³Tmin² / G matches the experimentally measured value of ℏ. Conclusion Article 13 does not claim to derive Planck’s constant. Its contribution is to isolate the exact mathematical problem whose solution would make such a derivation possible. It shows that the flat-cylinder route cannot fix the area scale, that the Liouville–FZZT sector can in principle break the conformal freedom, and that the calibration of ℏ reduces to a sharply defined FZZT saddle computation.