Manuscript En01-C@1 · En01-C.tex · theorem · 00a2161c

Theorem: Result A: preparation-aware cancellation

Gaussian preparation and outcome-conditioned splitting › Gaussian kernel and point-pair coherence · lines 1032–1073 · proof included or linked

En01-C@1#thm:prep, En01-C@1#eq:apre

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For the kernel (eq:kernel),

V_{pair} (d) = \frac{∣ ρ ( d /2 , - d /2 ) ∣}{ρ ( d /2 , d /2 ) ρ ( - d /2 , - d /2 )} = e^{- a_{pre} d^{2}},

where

a_{pre} = \frac{1}{ξ ^{2}} - \frac{1}{8 v _{x}} = \frac{4 det V - ℏ ^{2}}{8 ℏ ^{2} v _{x}} \geq 0.

Consequently, every weight vector satisfying $\sum_{i} w_{i} d_{i}^{2} = 0$ obeys

i \sum w_{i} [- lo g V_{pair, h} (d_{i})] = 0

for each hypothesis $h$ .

In particular, $w = (5, - 8, 3)$ at $d_{i} = R (1/4, 1/2, 3/4)$ cancels all hypothesis changes encoded only in $a_{pre, h}$ , including gravitational changes. It does not cancel a separately justified nonquadratic post-split response merely because that response is gravitational.

Proof. The off-diagonal numerator is $(2 π v_{x})^{- 1/2} e^{- d^{2} / ξ^{2}}$ . Both diagonal entries are $(2 π v_{x})^{- 1/2} e^{- d^{2} / (8 v_{x})}$ . Their ratio gives (eq:apre). The uncertainty relation makes its coefficient nonnegative. The weighted logarithm is $a_{pre, h} \sum_{i} w_{i} d_{i}^{2}$ .

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En01-C@1#eq:apre · line 1043
$a_{pre} = \frac{1}{ξ ^{2}} - \frac{1}{8 v _{x}} = \frac{4 det V - ℏ ^{2}}{8 ℏ ^{2} v _{x}} \geq 0.$
unlabeled · line 1035
$V_{pair} (d) = \frac{∣ ρ ( d /2 , - d /2 ) ∣}{ρ ( d /2 , d /2 ) ρ ( - d /2 , - d /2 )} = e^{- a_{pre} d^{2}},$
unlabeled · line 1052
$i \sum w_{i} [- lo g V_{pair, h} (d_{i})] = 0$

Theorem: Result A: preparation-aware cancellation

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