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

Theorem: Result B: exact complete-record mean signature

Structured-force identifiability from four free times › Settings and declared competing sources · lines 1499–1536 · proof included or linked

En01-C@1#thm:fourtime, En01-C@1#eq:fourtime

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Under the preceding specifications and moment-domain assumptions,

T_{h} := i = 0 \sum^{3} a_{i} E_{h} Z_{i} = R (D_{f} + hλ A_{r_{c}}), R = \frac{4 ℏ ^{2} Δ ^{3}}{M ^{2}} r_{x} > 0.

This identity holds for the full regularized kick model, not just its Gaussian quadratic approximation.

All shared preparation second moments cancel, including their gravitational and support-noise changes. The sensor-source offset also cancels. The retained gravitational contribution is its diffusion during the flat interval.

Proof. Insert (eq:free) into (eq:recordlinear). Every contribution from $v_{x, B}, c_{B}, v_{p, B}$ is a polynomial in $t_{i}$ of degree at most two. The sensor-source offset is constant. The $r_{c} c (t_{i})$ diffusion term is quadratic, and the $r_{p} v_{p} (t_{i})$ diffusion term is linear. Only the cubic position term survives:

r_{x} \frac{2 ℏ ^{2} D _{f, h}}{3 M ^{2}} i \sum a_{i} t_{i}^{3} = \frac{4 ℏ ^{2} Δ ^{3} r _{x}}{M ^{2}} D_{f, h} .

Lemma lem:momentsG proves that the full kick model has the same second-moment evolution. Lemma lem:rx proves positivity.

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En01-C@1#eq:fourtime · line 1502
$T_{h} := i = 0 \sum^{3} a_{i} E_{h} Z_{i} = R (D_{f} + hλ A_{r_{c}}), R = \frac{4 ℏ ^{2} Δ ^{3}}{M ^{2}} r_{x} > 0.$
unlabeled · line 1527
$r_{x} \frac{2 ℏ ^{2} D _{f, h}}{3 M ^{2}} i \sum a_{i} t_{i}^{3} = \frac{4 ℏ ^{2} Δ ^{3} r _{x}}{M ^{2}} D_{f, h} .$

Theorem: Result B: exact complete-record mean signature

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