Manuscript III@2 · III.tex · proposition · 82dad3e5

Proposition: Persistence on the unchanged identification

Constructions that do not discharge the sourced successor › Higher coefficients do not alter lower-order obstructions · lines 2650–2672 · proof included or linked

III@2#prop:lower-order-persistence

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Keep the map $q_{0}$ and the fixed witness of Theorem thm:product. Adding regular formal terms beginning at order $κ^{2}$ cannot make

F^{P}_{0} + κ F^{P}_{1} + r \geq 2 \sum κ^{r} F_{r}

constant on that witness fiber. Likewise, adding terms beginning at order $κ^{2}$ to $ϕ + κ X_{ξ} ϕ$ cannot remove the nonzero order- $κ$ free-shell defect of Proposition prop:tangency-witness.

Proof. The first difference has coefficient $- J \neq = 0$ at order $κ$ . The second equation defect has coefficient $P (X_{ξ} ϕ) \neq = 0$ . Formal equality with nonnegative powers requires each coefficient to vanish separately.

Displayed formulas in this passage

unlabeled · line 2656
$F^{P}_{0} + κ F^{P}_{1} + r \geq 2 \sum κ^{r} F_{r}$

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section: Higher coefficients do not alter lower-order obstructions