The exact error calculation passes

All 16 software controls pass, including both 292-epoch prefixes. Eight small cases match independent exact matrix calculations. Each longest prefix takes about 5.1 seconds. No aircraft observations or model fits were used.

The first version stopped while printing a large rational number. Its failure remains preserved. This version changes only the bounded output capacity; the mathematics, generated inputs and acceptance tests are unchanged.

What the bound means

Write the training observations as y = mean + scale × Gaussian error, with a fixed positive-definite covariance shape C. Choose a fixed reference space X with p columns. Any fitted coefficients are allowed: their residual quadratic form Q is at least the minimum over that reference space.

If its actual rank is r <= p, the minimum divided by the true squared scale has a noncentral chi-square law with n-r degrees. Noncentrality moves mass away from zero; reducing the degrees to d = n-p is conservative for the lower tail. Thus a certified lower-tail floor q gives the scale upper bound sqrt(Q/q). Poor reference means can make this bound larger; they cannot invalidate it by requiring an Earth candidate to supply its own noise level.

For an integer k < d/2, the inverse chi-square moment is 1 / product(d-2j, j=1..k). Markov's inequality therefore bounds the probability below q by q^k / product(d-2j). The calculation selects a dyadic q using integer comparisons and checks the final floating-point square-root endpoint by exact rational squaring. No numerical rank cutoff or floating-point chi-square quantile determines that guarantee.

The covariance combines a discrete OU state, constant offset, linear drift and positive white noise. Exact state-space innovations avoid a dense whole-record matrix. The OU transition is explicitly rationalized, rather than presented as an exact continuous-time exponential.

Next gate

Prepare a separate assessment using only the first 292 receiver epochs of the already studied recording. Report every predeclared reference/covariance case. Its scale bounds remain valid only under their declared Gaussian error law; they do not establish actual receiver accuracy or calibrated aircraft coverage.

This result also leaves the broad nonlinear prediction ranges unresolved. No observed Earth-model fits or protected ESA outcomes follow from this pass.

Results and frozen limits · Training-only source plan · Roadmap.