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Example observable types and scoped meanings — Examples

1 · Prerequisites

2 · Summary

The companion page works through the finite two-level transition and the one-mode time-ordered correlator at a negative time. The squared first-order coefficient is not treated as a probability, and the oscillator correlation is distinguished from its retarded response. These are exact calculations in the stated constructed models, with no simulated or reported data.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Exact two-level transition versus its first coefficient

Physical scope

One exactly constructed two-level finite Hamiltonian and its explicitly supplied projective measurement. The first-order squared coefficient is not called a probability, and no continuum decay or experimental data is claimed.

Example

Under the explicit pure-state preparation and unitary-evolution assumption of Exact regulated transition probabilities and controlled coefficients, take $H = C^{2}$ , $H_{0} = 0$ , $V (t) = g_{E} σ_{x}$ with $g_{E} > 0$ joules, dimensionless $λ = 1$ , initial vector $e_{1} = (1, 0)$ , and the two-outcome projective measurement $P = ∣ e_{2} ⟩ ⟨ e_{2} ∣$ , $I - P$ , where $e_{2} = (0, 1)$ . At $τ = π ℏ / (2 g_{E})$ seconds the exact model transition probability is $1$ , whereas the first-order squared transition coefficient is $π^{2} / 4 > 1$ . The latter is a coefficient-derived square, not a probability. This is an exact constructed-model calculation, not a simulated or reported experiment.

Facts & Assumptions

Given: ${A C}_{ω}$ , $g_{E} > 0$ in joules, $ℏ > 0$ in $J s$ , the two-level Hamiltonian and projective measurement in the Example, and the conditional theorem of Exact regulated transition probabilities and controlled coefficients.

[F1]

Exact regulated transition probabilities and controlled coefficients: under its stated pure-state preparation $ρ_{0} = T_{e_{1}, e_{1}}$ and model update by the constructed unitary $U (τ)$ , for $H = g_{E} σ_{x}$ and $z = g_{E} τ / ℏ$ the exact transition probability is $\sin^{2} z$ and the first-order squared coefficient is $z^{2}$ ; the coefficient-derived square need not be a probability.

[F2]

QFT example units, metric, gauge, and pairing conventions: $g_{E}$ has energy unit joule and $ℏ$ has unit $J s$ , so $g_{E} τ / ℏ$ is dimensionless when $τ$ is in seconds.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume only ${A C}_{ω}$ , as required by the conditional supplier theorem.

Verification

1.1givenF1F2A1algebra

At the selected time, $z = g_{E} τ / ℏ = π / 2$ by the definition of $τ$ [F2]. The exact transition probability from [F1] is therefore $\sin^{2} (π / 2) = 1$ , while the first-order squared coefficient is $(π / 2)^{2} = π^{2} / 4$ .

2.1step 1.1F1algebra

Since $π > 2$ , $π^{2} / 4 > 1$ . Thus this squared coefficient cannot be a probability for the specified outcome, while the exact model probability remains $1$ under the stipulated constructed model and measurement.

3.1givenF1step 1.1algebra∎

The theorem's quartic error bound specializes at this time to $∣ 1 - π^{2} / 4 ∣ \leq π^{4} / 48$ . It is a valid absolute bound, but it exceeds $1$ since $π > 3$ ; it gives no sub-unit error control at this time. No empirical uncertainty or continuum statement is inferred.

ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

One-mode time-ordered correlator versus retarded response

Physical scope

One exact oscillator mode with the stated core, vacuum, and conventions. The negative-time correlation differs from a retarded response; this makes no spatial-locality, signalling, interacting-field, or empirical claim.

Example

For the exact single oscillator mode of One-mode oscillator, Feynman boundary prescription, and retarded contrast, let $ω > 0$ in $s^{- 1}$ and choose $t = - π / (2 ω)$ . The normalized time-ordered vacuum correlator is $- i / (2 ω)$ , while the retarded response kernel is $0$ . The correlator at negative time is not a response to a future source. This fixed-mode calculation does not establish a spatial-causality or superluminal-signalling result.

Facts & Assumptions

Given: ${A C}_{ω}$ , $m_{o} > 0$ in kilograms, $ω > 0$ in $s^{- 1}$ , $ℏ > 0$ in $J s$ , and the one-mode oscillator result of One-mode oscillator, Feynman boundary prescription, and retarded contrast.

[F1]

One-mode oscillator, Feynman boundary prescription, and retarded contrast: for real $t$ , $C_{F} (t) = e^{- i ω ∣ t ∣} / (2 ω)$ and $G_{R} (t) = 1_{[0, \infty)} (t) \sin (ω t) / ω$ ; the statement assumes ${A C}_{ω}$ and the exact oscillator basis and domain.

[F2]

QFT example units, metric, gauge, and pairing conventions: the frequency $ω$ is in $s^{- 1}$ , time is in seconds, and the Hilbert pairing convention is the stated first-variable-linear one.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ is assumed only because the exact oscillator supplier invokes it.

[A2]

Physical interpretation of a supplied constructed quantum model: a physical reading is limited to an actually constructed state and dynamics; the postulate supplies no spatial-locality or interacting-field conclusion.

Verification

1.1givenF1F2A1algebra

The chosen time has units seconds because $[ω] = s^{- 1}$ , and $ω ∣ t ∣ = π / 2$ [F2]. Therefore [F1] gives $C_{F} (t) = e^{- i π / 2} / (2 ω) = - i / (2 ω)$ .

2.1F1A2step 1.1algebra∎

Since $t < 0$ , the indicator in [F1] is zero, so $G_{R} (t) = 0$ . Thus the two values differ, and the nonzero time-ordered correlation is not the retarded response at this negative time. The physical interpretation remains within the supplied one-mode construction described by [A2].

Example observable types and scoped meanings — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Exact two-level transition versus its first coefficient

Example

Facts & Assumptions

Verification

One-mode time-ordered correlator versus retarded response

Example

Facts & Assumptions

Verification

Sources