Rigid Borrowing and Spectral Reconstruction G: Condition-Number Floor and Capacity Boundary of Spectral Recovery—The Final-Verification Theorem for the Real Anosov Borrowing Channel

The decidability of spectral recovery requires a quantitative kernel, and the certification scope of the quantitative kernel requires an evidence hierarchy. This papergives two mutually independent but hierarchically coupled results. The first is aconceptual stratification theorem for the complexification channel: the existence ofa linear complex structure, the integrability of an almost complex structure, and theholomorphic lift form three strictly progressive levels. The first level is given at thesynthetic layer by the even-multiplicity criterion for real eigenvalues; the criterionis a finite-step algebraic function of the coefficients of the characteristic polynomialand the multiplicity structure. The second level is given at the synthetic layerwithin the continuous-invariant category by rigidification and the automatic integrability of constant almost complex structures, while in general nonlinear systemsit belongs to an effective Newlander–Nirenberg threshold. The third level degenerates at the synthetic layer to the algebraic verification of the commuting condition,while in nonlinear systems it depends on the composition of the first two levels. Therefinement of the stratification theorem does not eliminate the analytic obstruction;it localizes it to the nonlinear level. The second result is the condition-number floortheorem: on the enumeration domain of 200 hyperbolic matrices in GL(2,Z) withcoefficient bound 4, a ten-level amplitude scan is performed for integer additivesign noise. The relative coefficient error exhibits a constant plateau over the wholewindow η0 ∈ [1,32], and the grouped slope upper bound is less than 0.01. Thegrouped median sequence increases overall with the spectral radius, while group 5is treated as a term to be decomposed. The magnitude of the floor is 3.0 to 10.4;its operational meaning is that, under the coefficient metric, the recovery error isof the same order as, or even higher than, the coefficient amplitude. The positivecontent of the plateau law is that this unavailability is completely decoupled fromthe noise. The critical noise ηc lying above the order of the dynamic range of thesequence is a mechanism-level conditional reading; no termination of the plateauis observed within the experimental window. This yields the capacity boundaryof the spectral channel: what exact machine-check certification reaches is the floorand the decoupling; the analytic form of the capacity function, the spectral-sideparallel metric, and a rigorous lower bound for ηc remain open problems.