Manuscript Hawk05-C@1 · Hawk05-C.tex · proposition · ba0719f1

Proposition: The ξ 1 compatibility obstruction

Replacing the clock: compatibility, crossover, and cutoff › An inverse-square example · lines 1056–1083 · proof included or linked

Hawk05-C@1#prop:xi

The inverse-square clock itself can reparametrize a fixed tensor-release proxy. The assignments

lo g_{2} r = A (M), lo g_{2} b = B (M)

cannot both be exact assignments in that proxy over a nontrivial mass interval when $ξ \neq = 1$ . More quantitatively, they must satisfy

∣ ξ - 1∣ [B (M_{0}) - B (M)] \leq w_{0} + w_{M}, w = e_{R} + e_{B} .

Proof. Here

A (M) + B (M) = ξ B (M_{0}) + (1 - ξ) B (M) .

It is constant on a nontrivial interval exactly when $ξ = 1$ . Subtract its initial value and apply (eq:sumdifference).

Displayed formulas in this passage

unlabeled · line 1060
$lo g_{2} r = A (M), lo g_{2} b = B (M)$
unlabeled · line 1066
$∣ ξ - 1∣ [B (M_{0}) - B (M)] \leq w_{0} + w_{M}, w = e_{R} + e_{B} .$
unlabeled · line 1076
$A (M) + B (M) = ξ B (M_{0}) + (1 - ξ) B (M) .$