Manuscript X-G@1 · X-G.tex · theorem · 01566e14

Theorem: Conditional fixed-background-input identity

What background covariance follows conditionally › The restricted graph statement · lines 1706–1731 · proof included or linked

X-G@1#thm:conditional, X-G@1#eq:PhiG

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Assume that such an admitted coefficient $ℓ$ has been constructed. Define

Φ^{G} (f) = Φ^{F} (f) + κ^{2} ℏ ϕ (ℓ) (mod κ^{3}, ℏ^{2}) .

If the inherited X-F polynomial identity holds in the corrected carrier, then under (eq:twojet), with the free fields held fixed,

Φ^{G} (f) [h_{cl}^{'}, k_{cl}^{'}] - Φ^{G} (f) [h_{cl}, k_{cl}] = 0 (mod J, κ^{3}, ℏ^{2}) .

Proof. The tree difference belongs to $J$ by the inherited identity. The added coefficient has no dependence on the two classical backgrounds, so its difference is identically zero.

Displayed formulas in this passage

X-G@1#eq:PhiG · line 1710
$Φ^{G} (f) = Φ^{F} (f) + κ^{2} ℏ ϕ (ℓ) (mod κ^{3}, ℏ^{2}) .$
unlabeled · line 1719
$Φ^{G} (f) [h_{cl}^{'}, k_{cl}^{'}] - Φ^{G} (f) [h_{cl}, k_{cl}] = 0 (mod J, κ^{3}, ℏ^{2}) .$

Theorem: Conditional fixed-background-input identity

Displayed formulas in this passage

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