88. A complete perturbed-operator construction
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The input to the new proposer contains only two perturbed polynomial operators and the initial vector. A separate fixture generator creates those inputs from the preceding continuous model by adding \(\pm\delta E(u)\), with opposite signs for the two actions and
where
The perturbation preserves integrated mass. It is not a statistical sample. The proposer does not receive the predecessor metric or a hidden generator.
A noisy operator ratio supplies a center and a least-squares shape. A finite observation-anchor list supplies a scale proposal. Rationalizing the center and a triangular metric factor with denominator limit 48 gives
and hence
This is not the exact predecessor metric. The transformed family is not assumed conformal. The raw fitted family actually fails the proposed domain: for the direct action at \(u=-1\) and Lorentz input \((1,1,0)\), its output mass is \(106321/82000>0\), while the output quadratic form is
This is a genuine rejected input state, not a negative Bernstein coefficient of uncertain meaning. The normalized output lies outside the ellipse. No positivity certificate is claimed for the raw fit.
The construction next mixes each operator with the center-reset reference using \(\eta=1/8\). The exact checker proves a margin of at least \(1/16\) for the resulting family. It verifies eight polynomial inequalities across both actions and both semicircle charts. The witness contains 116 subdivision leaves and 1,780 exactly recomputed nonnegative Bernstein coefficients. These are arithmetic certificate sizes, not separate theorems.
The exported family admits mass and spatial perturbation budgets each equal to \(1/128\), leaving certified slack \(3/64\). In particular, it covers every degree-two transformed coefficient perturbation with each entry bounded by \(1/4096\), subject to the exact integrated mass constraints. Proposition 76 verifies that the whole coefficient box lies inside the declared output budgets. No uncertainty vertices or future trajectories are sampled to establish that inclusion.
The certificate is replayed independently from its JSON record. The record contains rational operators, initial state, transform, margin, uncertainty definition and complete subdivision trees. Floating-point proposal diagnostics are stored elsewhere and are not trusted by the replay routine.