Persistent Particle Identity from Finite Spectral Response: Self-Certification, Robust Reconstruction, and a Pythagorean Regulator Budget

The full spectral response describes a particle through sources, dynamics, and phase-sensitive feature correlations. We prove that on an atom of dimension d, for a bank of rank r_src, the finite word Gram matrix up to depth d − r_src + 1 and the resolvent block at a single nonreal point determine the entire visible package up to a single unitary. The rank plateau determines the required depth, a bounded resolvent defect recognizes a single energy, and a positive Gram margin gives linear robustness of the reconstruction on a constant-rank stratum. The full resolvent moment at a single point on a spanning bank also recovers the generator of a split multiplet. A Pythagorean decomposition of the leakage carries the carrier, the source, and the features along the regulator. The centered defect gives a quadratic cost of the energy change; a summable cost preserves the invertible source profile, the positive margin, and the entire complex signature. The actual leakage gives a necessary and sufficient non-collapse criterion, readable through Gram determinants. For a two-sided power law of the actual leakage, the non-collapse threshold of the profile is exactly the exponent 1/2. The operator law also carries a split island, and a positive block construction produces its isolation, centering, and budget. We derive checkable rank conditions for data with controlled error, as well as the reconstruction of exact PVMs. An explicit q³D realization arises from local excitations, positive exchange, and co-occupation energy. Local pulses prepare the source and the phase-sensitive features. Eight Walsh modes give a common anchor and seven Fano directions; seven anchored three-block windows reconstruct the full Gram matrix with a sharp contraction of the error and a Pythagorean balance of the inconsistency. The number seven is minimal in this three-block architecture, and positivity of all local Gram matrices does not replace positivity of their global gluing. We compute exactly the ranks 1, 8, 8, the unit margin, and a family of nonzero regulators with tail O(4⁻ⁿ). Counterexamples show the necessity of controlling the tail and of preserving the cycle phases. Moreover, an exact Schur condition decides when an external amplitude family actually belongs to the same reconstructed carrier and when it contributes a new orthogonal sector. A uniform tail is the exact condition for the norm limit of the full first-exit record with preservation of the persistent leg; the q³D model produces it from a constant splitting of the generator.