Pipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite Pauli dimer spectrum and exact separability threshold — Examples

1 · Prerequisites

2 · Summary

These examples specialize the global-sector model at $N = 2$ , compare the dimer's separability threshold with its analytic finite partition function, and evaluate the onsite covariance and occupation-step formulas at symmetric parameters. Each is a conditional calculation within the linked finite model; none records an observation or upgrades a local model to a lattice or material claim.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Finite analyticity versus the global-model cusp and shell inequivalence

Physical scope

The N=2 member and positive transition parameter of the explicitly defined finite global three-sector quantum model. This is not an example of a local or short-range quantum lattice.

Example

For the global three-sector model of Commuting quantum Ising and nonadditive phase countermodel, take $N = 2$ and $β_{c} = (\log 2) / ϵ$ . Its finite Gibbs probability of the one-dimensional middle sector is exactly $1 / 9$ , while a uniform spectral shell with half-width $0 \leq Δ < 2 ϵ$ assigns that sector probability $1$ . The finite partition function is still analytic at $β_{c}$ ; the cusp belongs to the limiting normalized log partition as $N \to \infty$ .

Facts & Assumptions

Given: The three-sector Hamiltonian and finite canonical preparation of Commuting quantum Ising and nonadditive phase countermodel, with $ϵ > 0$ joules.

[F1]

Commuting quantum Ising and nonadditive phase countermodel: for each $N \geq 1$ , $Z_{N} = 1 + e^{- β N ϵ} + 4^{N} e^{- 2 β N ϵ}$ , and a shell with $0 \leq Δ < N ϵ$ contains only the middle sector.

Verification

1.1givenF1algebra

At $N = 2$ and $β = β_{c}$ , $e^{- β N ϵ} = e^{- 2 \log 2} = 1 / 4$ and $4^{N} e^{- 2 β N ϵ} = 16 e^{- 4 \log 2} = 1$ . Hence $Z_{2} = 9 / 4$ and the middle-sector Gibbs probability is $(1 / 4) / (9 / 4) = 1 / 9$ . The finite partition is a sum of positive exponentials, so it is analytic at this parameter.

2.1givenF1step 1.1algebra∎

Since $2 ϵ$ is the distance from the middle energy to either outer energy at $N = 2$ , every shell with $0 \leq Δ < 2 ϵ$ selects exactly the one-dimensional middle sector and gives it uniform probability $1$ . Thus the two specified finite preparations disagree on this observable, while the analytic finite sum and limiting cusp remain distinct claims.

ExampleConstruction: AI-adaptedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Entanglement threshold versus thermodynamic singularity

Physical scope

The zero-field, antiferromagnetic finite Pauli dimer at its exact thermal separability boundary. It distinguishes a state-property threshold from a thermodynamic singularity and makes no experimental claim.

Example

For the zero-field dimer of Interacting Heisenberg dimer Gibbs model with $J > 0$ , the equality point $β J = (\log 3) / 4$ gives a separable Gibbs density matrix, and every larger $β J$ gives an entangled one. Nevertheless $Z = e^{3 β J} + 3 e^{- β J}$ and the free energy remain analytic at this threshold. The threshold changes the separability classification without producing a finite-temperature thermodynamic singularity.

Facts & Assumptions

Given: $J > 0$ , $h = 0$ , $β > 0$ , and the exact finite dimer result.

[F1]

Interacting Heisenberg dimer Gibbs model: at zero field the Gibbs state is separable for $β J \leq (\log 3) / 4$ and entangled above, while its finite partition and free energy are analytic for positive $β$ .

Verification

1.1givenF1algebra

Substituting $β J = (\log 3) / 4$ into [F1] gives the separable equality state; taking any $β J > (\log 3) / 4$ gives the entangled branch. These are conclusions about the defined two-qubit density matrix.

2.1givenF1step 1.1algebra∎

At $h = 0$ , $Z = e^{3 β J} + 3 e^{- β J}$ is a finite positive sum of exponentials, so $- β^{- 1} \log Z$ is analytic for $β > 0$ , including at the separability threshold. Therefore this state-property boundary is not a thermodynamic singularity of the finite dimer.

ExampleConstruction: AI-adaptedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Repulsive Fermi covariance and zero-temperature Bose occupation step

Physical scope

A symmetric-parameter instance of the finite Fermi atom and a single Bose zero-temperature boundary in the isolated onsite models. No lattice or material-specific conclusion is drawn.

Example

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the inherited Bose occupation-space construction.

For the Fermi atom of Exact onsite Bose and Fermi interacting atoms, set $h = 0$ , $μ = ϵ$ , and $U > 0$ . Then $Ξ_{F} = 3 + e^{- β U}$ and $Cov (n_{↑}, n_{↓}) = (e^{- β U} - 1) / (3 + e^{- β U})^{2} < 0$ . For the Bose site, set $μ = ϵ + U$ with $U > 0$ ; the two zero-temperature minimizing occupations are $1$ and $2$ , each with limiting Gibbs probability $1 / 2$ , and the entropy tends to $k_{B} \log 2$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $β > 0$ , $U > 0$ , and the two exact onsite model statements.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every at most countable family of nonempty sets has a choice function; the Bose conclusion retains this premise from the theorem.

[F1]

Exact onsite Bose and Fermi interacting atoms: at $h = 0$ , $μ = ϵ$ , the Fermi covariance has the displayed formula; for the Bose site the boundary $μ = ϵ + U r$ has exactly the adjacent minimizing occupations $r, r + 1$ with equal zero-temperature limiting probabilities.

Verification

1.1givenA1F1algebra

Substituting $h = 0$ and $μ = ϵ$ into the Fermi weights gives $w_{↑} = w_{↓} = 1$ , $w_{2} = e^{- β U}$ , and $Ξ_{F} = 3 + e^{- β U}$ . Since $U, β > 0$ , $e^{- β U} < 1$ , so [F1] gives the strictly negative covariance stated in the Example.

2.1givenA1F1algebra∎

In the Bose result of [F1], set $r = 1$ and $μ = ϵ + U$ . The adjacent minimizing occupations are $1, 2$ , their limiting probabilities are $1 / 2, 1 / 2$ , and the binary entropy is $- k_{B} (2) (1 / 2) \log (1 / 2) = k_{B} \log 2$ .

Finite Pauli dimer spectrum and exact separability threshold — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Finite analyticity versus the global-model cusp and shell inequivalence

Example

Facts & Assumptions

Verification

Entanglement threshold versus thermodynamic singularity

Example

Facts & Assumptions

Verification

Repulsive Fermi covariance and zero-temperature Bose occupation step

Example

Facts & Assumptions

Verification

Sources