Manuscript I@2 · I.tex · section · 85631fa7

The finite two-laboratory calculation

Context refinement and the Frauchiger–Renner example › The finite two-laboratory calculation · lines 2578–2659

Take

Ψ = \frac{1}{3} (∣ h ˉ ⟩ ∣ ⇓ ⟩ + ∣ t ˉ ⟩ ∣ ⇓ ⟩ + ∣ t ˉ ⟩ ∣ ⇑ ⟩),

and define

∣ ok ⟩ = \frac{∣ h ˉ ⟩ - ∣ t ˉ ⟩}{2}, ∣ ok ⟩ = \frac{∣ ⇓ ⟩ - ∣ ⇑ ⟩}{2} .

The fail vectors use plus signs.

Let

C_{rec} : C_{W ˉ} : C_{W} : {h ˉ, t ˉ} \otimes {⇓, ⇑}, {ok, fail} \otimes {⇓, ⇑}, {h ˉ, t ˉ} \otimes {ok, fail} .

The corresponding zero-probability events give

C_{rec} : C_{W ˉ} : C_{W} : ⇑\Rightarrow t ˉ, ok \Rightarrow⇑, t ˉ \Rightarrow fail .

Indeed,

⟨ h ˉ ∣ ⟨ ⇑ ∣ Ψ = 0, ⟨ ok ∣ ⟨ ⇓ ∣ Ψ = 0, ⟨ t ˉ ∣ ⟨ ok ∣ Ψ = 0.

However,

[∣ t ˉ ⟩ ⟨ t ˉ ∣, ∣ ok ⟩ ⟨ ok ∣] \neq = 0.

The necessary contexts therefore have no common sharp refinement. In the outer context measuring both interference bases,

Pr (ok, ok) = \frac{1}{2 3}^{2} = \frac{1}{12} .

Transporting the three implications into one outer classical chain is not justified by the stated refinement discipline.

This is an observational-context obstruction. It is not a theorem that linear ownership alone rejects all cross-observer inference. Sequential operations on the same laboratory can be well typed and can disturb later tests.

No interior observable algebra or gravitational dynamics is specified here. The calculation consequently establishes no claim about smooth horizons or firewalls. This boundary agrees with the explicit finite-mode non-claims of Em03-C CheungEm03C; the physical firewall question has a different hypothesis package AMPS.

An internal syntax for observer judgments, certificates, and allowed transports remains P3 below.

Displayed formulas in this passage

unlabeled · line 2581
$Ψ = \frac{1}{3} (∣ h ˉ ⟩ ∣ ⇓ ⟩ + ∣ t ˉ ⟩ ∣ ⇓ ⟩ + ∣ t ˉ ⟩ ∣ ⇑ ⟩),$
unlabeled · line 2590
$∣ ok ⟩ = \frac{∣ h ˉ ⟩ - ∣ t ˉ ⟩}{2}, ∣ ok ⟩ = \frac{∣ ⇓ ⟩ - ∣ ⇑ ⟩}{2} .$
unlabeled · line 2600
$C_{rec} : C_{W ˉ} : C_{W} : {h ˉ, t ˉ} \otimes {⇓, ⇑}, {ok, fail} \otimes {⇓, ⇑}, {h ˉ, t ˉ} \otimes {ok, fail} .$
unlabeled · line 2612
$C_{rec} : C_{W ˉ} : C_{W} : ⇑\Rightarrow t ˉ, ok \Rightarrow⇑, t ˉ \Rightarrow fail .$
unlabeled · line 2620
$⟨ h ˉ ∣ ⟨ ⇑ ∣ Ψ = 0, ⟨ ok ∣ ⟨ ⇓ ∣ Ψ = 0, ⟨ t ˉ ∣ ⟨ ok ∣ Ψ = 0.$
unlabeled · line 2629
$[∣ t ˉ ⟩ ⟨ t ˉ ∣, ∣ ok ⟩ ⟨ ok ∣] \neq = 0.$
unlabeled · line 2636
$Pr (ok, ok) = \frac{1}{2 3}^{2} = \frac{1}{12} .$