Rigid Borrowing and Spectral Reconstruction E: A Synthetic Minimal Model of Obstacle Additivity—The Readability of Obstacles from Spectral Data—Piecewise Laws, Superlinear Amplification, and Spectral Silence Subspaces

The programmatic axiom of “obstacle additivity” asserts that descent obstaclescan be read linearly from spectral data. This paper constructs two synthetic minimal models that are fully machine-checkable and performs an empirical test of thequantitative form of this axiom, yielding a double-threshold correction. Model One(additive observation noise): the recovery error is insensitive to noise amplitudeand grows monotonically with the spectral radius of the nominal system; withinthe audit domain of protocol P1 the pole gap covers [2.24,6.33], and the noise-togap ratio covers [0.16,3.58]; at a ratio of about 0.16 the paired doubling ratio is stillabout 1.01, far from the linear-law value 2. If a linear window exists, its criticalratio must be less than 0.16; the data equally support the in-domain reading that“no linear response is seen within the detection domain, and the dominant formis saturation.” Model Two (model misspecification): the window residual of thesecond-order recurrence derived from the nominal characteristic polynomial, acting on the perturbed system sequence, is amplified superlinearly with perturbationamplitude and takes nonnegative integer values under integer perturbations; thereadings of three pipeline settings are insufficient to identify a single power exponent, and superlinearity is an aggregated smoothed reading of a discrete latticedistribution. The upper bound on the window Hankel condition number is a strictinequality, and the upper bound diverges as D tends to zero, corresponding to recovery breakdown and singular windows. The spectral silence subspace is characterizedrigorously by the fiber structure of the characteristic-polynomial map: for secondorder integer matrices, the power-sum sequences are pointwise identical if and onlyif the trace and the determinant are identical; the silence set is the intersection of athree-dimensional linear subspace and a quadratic hypersurface, and is not a linearsubspace. The obstacle space is partitioned by the characteristic-polynomial mapinto mutually disjoint fibers; spectral data can read only fiber labels, and there is nointrinsic orthogonal decomposition induced by this map. The same characteristicpolynomial does not imply GL(2,Z)-conjugacy, so periodic spectral data lock in acoarser equivalence than conjugacy; an ideal class gap only guarantees nonemptiness. The two findings combine into a double-threshold proposition: the directionthreshold is a theorem, while the amplitude threshold and superlinear amplificationare in-domain empirical laws; the critical ratio is registered as an interval reading,not an exact constant. Continuous noise, n = 3, asymptotic exponent analyticestimates, and geometric generalizations remain registered as absent.