THM-DIVERGENCE-005CLOSED_CLASSICALS5
MuncyNautics Recovery Certificate · theorem-grade divergence · hunted and executed 2026-08-31 · author of every MN concept: Derrick E. Muncy
WHAT THIS CERTIFICATE RECORDSA continuous function’s Fourier series can DIVERGE at a point (du Bois-Reymond, proven) — yet the Cesàro/Fejér mean of the same partial sums converges uniformly for EVERY continuous function (Fejér, proven). The divergence is the projection of discarding the summation-method lane: same coefficients, same function, one lane diverges and its neighbor closes. Both directions are theorems.
GOVERNING EQUATION AS PROJECTED — THE SEAM
s_N(f;x) → divergence at a point for some continuous f (du Bois-Reymond 1873, proven) — the partial-sum lane read as THE value
COMPLETE GOVERNING EQUATION — THE RECOVERY
σ_N = (s₀+…+s_{N−1})/N → f uniformly for every continuous f (Fejér 1904, proven)
executed witness (Grandi kernel case): partial sums oscillate; σ₁₀₀₀ = 0.500000 exactly at the carrier value
CORRECTION RULE — correction = reconstruction from the recovery: restored arguments, seats, phases and arms in the equation itself; no raw appended constants — any finite term must be derived from the recovered channels, never fitted.
| species |
S5 — the summation-method lane |
|---|---|
| executed |
Cesàro witness computed; the two theorems together are the seam-and-recovery pair, both closed classically |
| standing |
CLOSED_CLASSICAL — both halves proven |
| promotion_authority |
GREEN_CLASSICAL |
| source_records |
du Bois-Reymond 1873; Fejér 1904; Zygmund, Trigonometric Series |