Target 1 — the Born rule: reductions and a no-go

Born rule

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Target 1 — the Born rule: reductions and a no-go

finite_noCollapseBorn_fromNoncontextuality

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.

finite_noCollapseBorn_fromNoncontextuality · capstone — there is ρ\rho such that all of:

  1. ρ.PosSemidef\rho.\mathrm{PosSemidef}
  2. ρ.trace=1\rho.\mathrm{trace} = 1
  3. (∀(ω:E.Ω),∃!h,∀(t:Fin n),∃r∈(E.V ω t).config.active,(E.V ω t).ctx.valueOf r=h t)(\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)
  4. (∀(a:Fin m),E.p a=(ρ⋅P a).trace.re)(\forall (a : \mathrm{Fin}\,m), E.p\,a = (\rho \cdot P\,a).\mathrm{trace}.\mathrm{re})
  5. (∀(h:Fin n→Fin m),E.P.massSet {ω∣E.actualHist ω=h}=w E.p h)(\forall (h : \mathrm{Fin}\,n \to \mathrm{Fin}\,m), E.P.\mathrm{massSet}\,\{\omega|E.\mathrm{actualHist}\,\omega = h\} = w\,E.p\,h)
  6. 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)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})

assuming

plus 1 routine conditions (1 bridge) — full list in the per-track PDF.

finite_effect_gleason

finite_effect_gleason · spine — there is ρ\rho such that all of:

  1. ρ.PosSemidef\rho.\mathrm{PosSemidef}
  2. ρ.trace=1\rho.\mathrm{trace} = 1
  3. ∀(E:Mat (Fin d) (Fin d) C),IsEffect E→(m.μ E)=(ρ⋅E).trace\forall (E : \mathrm{Mat}\,(\mathrm{Fin}\,d)\,(\mathrm{Fin}\,d)\,\mathbb{C}), \href{/browser/qiqth-effectgleason#d-qiqth-effectgleason-iseffect}{\mathrm{IsEffect}}\,E \to (m.\mu\,E) = (\rho \cdot E).\mathrm{trace}

positive_ray_certain_forces_born

positive_ray_certain_forces_born · spine — we have

w E=born ψ Ew\,E = \mathrm{born}\,\psi\,E

assuming

plus 1 routine conditions (1 bridge) — full list in the per-track PDF.

continuous_additive_fMeasure_eq_born

continuous_additive_fMeasure_eq_born · spine — we have

fMeasure (f) w k=w k\mathrm{fMeasure}\,(f)\,w\,k = w\,k

assuming

plus 1 routine conditions (1 setup) — full list in the per-track PDF.

decoherent_partition_additive

decoherent_partition_additive · spine — we have

born ψ ((∑aSC a).conjTranspose⋅∑aSC a)=∑aSborn ψ ((C a).conjTranspose⋅C a)\mathrm{born}\,\psi\,((\sum_{a S} C\,a).\mathrm{conjTranspose} \cdot \sum_{a S} C\,a) = \sum_{a S} \mathrm{born}\,\psi\,((C\,a).\mathrm{conjTranspose} \cdot C\,a)

assuming

finite_noCollapseBornRepresentation

finite_noCollapseBornRepresentation · spine — we have all of:

  1. (∀(ω:E.Ω),∃!h,∀(t:Fin n),∃r∈(E.V ω t).config.active,(E.V ω t).ctx.valueOf r=h t)(\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)
  2. (∀(h:Fin n→Fin m),E.P.massSet {ω∣E.actualHist ω=h}=w E.p h)(\forall (h : \mathrm{Fin}\,n \to \mathrm{Fin}\,m), E.P.\mathrm{massSet}\,\{\omega|E.\mathrm{actualHist}\,\omega = h\} = w\,E.p\,h)
  3. 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)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})

assuming

product_born_measure_unique

product_born_measure_unique · spine — we have

μ S=((kronN λx↦ρ)⋅eventEffect E S).trace.re\mu\,S = ((\mathrm{kronN}\,\lambda x \mapsto \rho) \cdot \mathrm{eventEffect}\,E\,S).\mathrm{trace}.\mathrm{re}

assuming

chebyshev_freq

chebyshev_freq · spine — we have

∑ωw p ω≤p k⋅(1−p k)/(N⋅ε2)\sum_{\omega} w\,p\,\omega \le p\,k \cdot (1 - p\,k) / (N \cdot {\varepsilon}^{2})

assuming

qiqth_born_typicality_conditional

qiqth_born_typicality_conditional · spine — we have

expectedIndicator outcome M.μ k=c k2\mathrm{expectedIndicator}\,\mathrm{outcome}\,M.\mu\,k = {c\,k}^{2}

born_distribution_realizable_conditional

born_distribution_realizable_conditional · nogo — there is μ\mu such that all of:

  1. (∀(γ:Γ),0≤μ γ)(\forall (\gamma : \Gamma), 0 \le \mu\,\gamma)
  2. ∑γμ γ=1\sum_{\gamma} \mu\,\gamma = 1
  3. ∀(k:Outcome),outcomeMarginal outcome μ k=c k2\forall (k : \mathrm{Outcome}), \mathrm{outcomeMarginal}\,\mathrm{outcome}\,\mu\,k = {c\,k}^{2}

assuming

decoherence_does_not_concentrate

decoherence_does_not_concentrate · nogo — we have all of:

  1. 0<branchWeight c 00 < \mathrm{branchWeight}\,c\,0
  2. 0<branchWeight c 10 < \mathrm{branchWeight}\,c\,1

assuming

support_preservation_does_not_imply_measure_preservation

support_preservation_does_not_imply_measure_preservation · nogo — there is T, μT,\ \mu such that all of:

  1. Bijective T\mathrm{Bijective}\,T
  2. SupportPreserving T\mathrm{SupportPreserving}\,T
  3. ¬MeasurePreserving T μ\neg \mathrm{MeasurePreserving}\,T\,\mu

operational_data_insufficient

operational_data_insufficient · nogo — there is outcome, ν1, ν2\mathrm{outcome},\ \nu_{1},\ \nu_{2} such that all of:

  1. (∀(k:Fin 2),marginal3to2 ν1 outcome k=marginal3to2 ν2 outcome k)(\forall (k : \mathrm{Fin}\,2), \mathrm{marginal3to2}\,\nu_{1}\,\mathrm{outcome}\,k = \mathrm{marginal3to2}\,\nu_{2}\,\mathrm{outcome}\,k)
  2. ν1≠ν2\nu_{1} \ne \nu_{2}