PAPER-Ξ: Finite-Depth Readability and Scale-Readout Cosmology

This paper records a conditional bridge between the thimble-depth parameter D, finitedepth information readability, and an effective scale-factor readout in the Fracture–Berry–Tension framework. The central point is not that a thimble flow by itself proves a cosmological constant. Rather, the paper isolates a precise sequence of conditional statements. First, readable matter sectors require neither complete thimble fusion nor complete thimble isolation, but a finite-depth window in which sectors are distinguishable while still dynamically comparable. Second, a three-sector density-matrix toy model shows that inter-thimblecoherence is exponentially suppressed as the depth increases; this model does not produce a local minimum of von Neumann entropy at D = 0, but it motivates a separate readability functional balancing sector separation and sector coherence. Third, decomposing the thimble depth as Di = Dcom +δi separates relative generation data from a common scale mode. Relative depth differences control mass or overlap ratios, while the common mode may enter as a microscopic readout scale. Finally, if the common depth mode is read as aeff (teff ) = a0eγDcom(teff ), under the effective Lorentzian time teff of Paper-Ω, then the corresponding flat FLRW effective equation of state is wD(teff ) = −1 −2Dc¨om3γDc˙om2 . The de Sitter value wD = −1 is therefore obtained conditionally when the common thimbledepth flow is approximately inertial in the force-free common-mode direction. In canonical depth coordinates this means ¨QD ≃ 0, and, in a local window where the depth metric is approximately constant, it becomes Dc¨om ≃ 0. The golden weak-resonance branch then selects the stable dimensionless rate or slope of this inertial branch, rather than being the primary source of linearity itself. The resulting structure is best read as a scale-readout hypothesis, not as a completed derivation of dark energy. Revision notes (v0.7). This version incorporates a Torricelli-type finite-capacity interpretation of thimble convergence. The compact six-dimensional phase carrier has finite Liouville volume, while asymptotically deep thimble channels may still possess unbounded effective depth or interface complexity through shrinking transverse readout scale. This provides a geometric interpretation of negative-pressure readout: microscopic thimble contraction is read macroscopically as expansion under the scale-readout map. This version also clarifies the relation between inertial linearity, Diophantine stability, and global torus monodromy. The approximate linearity Dc¨om ≃ 0 is attributed to quadratic-carrier inertia in a force-free common-mode window. The Diophantine golden branch selects the stable dimensionless slope or rate of that inertial flow. FBT0B supplies a compatible hyperbolic SL(2,Z) monodromy branch whose contracting eigenvalue is also φ−2gr . The agreement is read as structural compatibility, not as a derivation of the Diophantine selection from monodromy alone.