Manuscript Psy03-B@1 · Psy03-B-v1.tex · theorem · 8ca3caf9

Theorem: Exact reduced operational memory

An exact one-edge structural-memory pilot › A carrier-only causal-break witness · lines 2286–2329 · proof included or linked

Psy03-B@1#thm:operationalmemory, Psy03-B@1#eq:memorygap

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In the two-probe experiment, first-outcome probabilities are

P (r = 0) = \frac{1}{2} + w C, P (r = 1) = \frac{1}{2} - w C .

The posterior unscarred probabilities are

w_{0} = \frac{w ( 1/2 + C )}{1/2 + w C}, w_{1} = \frac{w ( 1/2 - C )}{1/2 - w C} .

After the identical carrier re-preparation, the difference in future $P_{0}$ probabilities is

J (w, C) = C (w_{0} - w_{1}) = \frac{w ( 1 - w ) C ^{2}}{1/4 - w ^{2} C ^{2}} > 0.

Consequently, the reduced carrier process is operationally non-Markovian under this carrier-only experiment.

For the noiseless quarter-rotation pilot,

J (w, 1/2) = \frac{w}{1 + w} .

Proof. Sector-zero and sector-one first-outcome probabilities for $r = 0$ are $1/2 + C$ and $1/2$ , respectively. Bayes' rule gives $w_{0}, w_{1}$ . The reset makes the carrier input identical in both histories while leaving these conditional scar distributions unchanged. The future $P_{0}$ probability is $1/2 + w_{r} C$ . Subtract and simplify.

Both first outcomes have positive probability because $0 < w < 1$ and $C \leq 1/2$ . The denominator is positive. The nonzero conditional future difference satisfies Proposition prop:memory and the causal-break criterion PollockMarkov.

Displayed formulas in this passage

Psy03-B@1#eq:memorygap · line 2300
$J (w, C) = C (w_{0} - w_{1}) = \frac{w ( 1 - w ) C ^{2}}{1/4 - w ^{2} C ^{2}} > 0.$
unlabeled · line 2289
$P (r = 0) = \frac{1}{2} + w C, P (r = 1) = \frac{1}{2} - w C .$
unlabeled · line 2294
$w_{0} = \frac{w ( 1/2 + C )}{1/2 + w C}, w_{1} = \frac{w ( 1/2 - C )}{1/2 - w C} .$
unlabeled · line 2309
$J (w, 1/2) = \frac{w}{1 + w} .$

Theorem: Exact reduced operational memory

Displayed formulas in this passage

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