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Alternating values bound every critical value - #2523
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`Polynomial.one_le_abs_eval_critical_of_alternation`: if a degree-`N` polynomial with `N` simple roots in `(-1, 1)` takes values of alternating sign and absolute value at least one at `-1 = ξ 0 < ⋯ < ξ N = 1`, then `|q c| ≥ 1` at every critical point `c ∈ (-1, 1)`. Used for the critical-value propagation and the degree-five start in #2513. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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Polynomial.one_le_abs_eval_critical_of_alternationtoRealRooted/Mathlib/Analysis/Polynomial/InverseBernstein.lean.Suppose a degree-
Npolynomial hasNsimple roots in(-1, 1), and takes values of alternating sign with absolute value at least one at points-1 = ξ 0 < ⋯ < ξ N = 1. Then|q c| ≥ 1at every critical pointc ∈ (-1, 1).Proof: each gap holds exactly one root. Each critical point is the unique one between two consecutive roots, and the alternation point between those roots shares the monotone pieces, which gives the comparison.
This is the ingredient for the critical-value propagation and the degree-five start in #2513 (step 3/4).
Checks: build, runLinter, lint-style and the private-import check pass, as do root imports, module system, import architecture and proof status.
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