BornJoinGleason · section of the QIQT-H book

QIQTH.BornJoinGleason

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BornJoinGleason · entries 4–4 of 1000

Theorem 4 (finite_noCollapseBorn_fromNoncontextuality).  source ↗

No-collapse Born representation with the single-trial law DERIVED (not assumed). Given the prize ensemble PLUS non-contextuality of the single-trial statistics (the law p is the value of a non-contextual effect assignment M on a measurement {Pₐ}), there is a density matrix ρ such that: (i) every world has a UNIQUE actual pointer-value history (capacity + selector, no collapse); (ii) the single-trial law is the Born weight Re tr(ρ Pₐ) — FORCED by effect-Gleason; (iii) the world-mass of each history is the Born PRODUCT law; (iv) atypical-frequency histories carry vanishing world-mass. The Born weights are no longer a free parameter — only NON-CONTEXTUALITY + independence (+ the world measure) are assumed.

(∀(a:Fin m),IsEffect (P a))→(∀(a:Fin m),M.μ (P a)=E.p a)→∀(k:Fin m){ε:R},0<ε→0<n→∃ρ,ρ.PosSemidef∧ρ.trace=1∧(∀(ω:E.Ω),∃!h,∀(t:Fin n),∃r∈(E.V ω t).config.active,(E.V ω t).ctx.valueOf r=h t)∧(∀(a:Fin m),E.p a=(ρ⋅P a).trace.re)∧(∀(h:Fin n→Fin m),E.P.massSet {ω∣E.actualHist ω=h}=w E.p h)∧E.P.massSet {ω∣(n⋅ε)2≤(count k (E.actualHist ω)−n⋅E.p k)2}≤E.p k⋅(1−E.p k)/(n⋅ε2)(\forall (a : \mathrm{Fin}\,m), \href{/browser/qiqth-effectgleason#d-qiqth-effectgleason-iseffect}{\mathrm{IsEffect}}\,(P\,a)) \to (\forall (a : \mathrm{Fin}\,m), M.\mu\,(P\,a) = E.p\,a) \to \forall (k : \mathrm{Fin}\,m) \{\varepsilon : \mathbb{R}\}, 0 < \varepsilon \to 0 < n \to \exists \rho, \rho.\mathrm{PosSemidef} \wedge \rho.\mathrm{trace} = 1 \wedge (\forall (\omega : E.\Omega), \exists !h, \forall (t : \mathrm{Fin}\,n), \exists r\in (E.V\,\omega\,t).\mathrm{config}.\mathrm{active}, (E.V\,\omega\,t).\mathrm{ctx}.\mathrm{valueOf}\,r = h\,t) \wedge (\forall (a : \mathrm{Fin}\,m), E.p\,a = (\rho \cdot P\,a).\mathrm{trace}.\mathrm{re}) \wedge (\forall (h : \mathrm{Fin}\,n \to \mathrm{Fin}\,m), E.P.\mathrm{massSet}\,\{\omega|E.\mathrm{actualHist}\,\omega = h\} = w\,E.p\,h) \wedge E.P.\mathrm{massSet}\,\{\omega|{(n \cdot \varepsilon)}^{2} \le {(\mathrm{count}\,k\,(E.\mathrm{actualHist}\,\omega) - n \cdot E.p\,k)}^{2}\} \le E.p\,k \cdot (1 - E.p\,k) / (n \cdot {\varepsilon}^{2})

Proof. Immediate from the definitions. □\square

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