Manuscript Psy03-C@1 · Psy03-C-v1.tex · proposition · d3380b90

Proposition: Carrier forgetting versus structural memory

A separate input-driven quantum model › Persistent structural labels prevent an automatic global forgetting theorem · lines 1853–1890 · proof included or linked

Psy03-C@1#prop:register, Psy03-C@1#eq:registerbound

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For common label distribution $π$ ,

d_{tr} (ω_{T}, ω_{T}^{'}) = s \sum π_{s} d_{tr} (ρ_{T, s}, ρ_{T, s}^{'}) \leq s \sum π_{s} κ_{s}^{T} d_{tr} (ρ_{s}, ρ_{s}^{'}) .

For different label distributions,

d_{tr} (ω_{T}, ω_{T}^{'}) \geq d_{TV} (π, π^{'}) .

The reduced carrier satisfies the upper bound

d_{tr} (ρ_{Q, T}, ρ_{Q, T}^{'}) \leq d_{TV} (π, π^{'}) + s \sum min {π_{s}, π_{s}^{'}} κ_{s}^{T} d_{tr} (ρ_{s}, ρ_{s}^{'}) .

Thus carrier contraction in each sector does not erase a different persistent label distribution.

Proof. Orthogonal classical blocks have additive trace norm. Tracing out the carrier preserves the label distributions and contracts distance, proving the lower bound. For the reduced upper bound, match mass $min {π_{s}, π_{s}^{'}}$ in each sector. The unmatched mass totals $d_{TV} (π, π^{'})$ on each side; its trace-distance contribution is at most that number. Apply sector contraction to the matched pairs.

Displayed formulas in this passage

unlabeled · line 1856
$d_{tr} (ω_{T}, ω_{T}^{'}) = s \sum π_{s} d_{tr} (ρ_{T, s}, ρ_{T, s}^{'}) \leq s \sum π_{s} κ_{s}^{T} d_{tr} (ρ_{s}, ρ_{s}^{'}) .$
unlabeled · line 1865
$d_{tr} (ω_{T}, ω_{T}^{'}) \geq d_{TV} (π, π^{'}) .$
Psy03-C@1#eq:registerbound · line 1869
$d_{tr} (ρ_{Q, T}, ρ_{Q, T}^{'}) \leq d_{TV} (π, π^{'}) + s \sum min {π_{s}, π_{s}^{'}} κ_{s}^{T} d_{tr} (ρ_{s}, ρ_{s}^{'}) .$

Proposition: Carrier forgetting versus structural memory

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