The Ising gap gives a conditional local parent Hamiltonian
The square-root Gibbs transform of C124 is a positive, three-site local, frustration-free Hermitian operator with unique ground state \(\sqrt{\pi_{N,b}}\) and exact gap \(a[1-\tanh(2b)]\). Multiplication by a supplied action constant gives an energy-unit Hamiltonian. This construction preserves the chosen stochastic clock and does not select physical dynamics or action units. It resolves G04’s finite-volume transfer decision.
1. Object, domain and transform
Use C124’s model: \(N\ge3\) periodic spins, \(0\le b<\infty\), per-site refresh rate \(a>0\) and stationary law \(\pi(\sigma)=Z^{-1}\exp(b\sum_i\sigma_i\sigma_{i+1})\). The generator \(Q=a\sum_i(E_i-I)\) acts on functions in complex \(L^2(\pi)\); \(E_i\) is conditional expectation over spin \(i\). Its time parameter is the auxiliary sampling clock and \(a\) has inverse-time units.
Let \(D=\operatorname{diag}\pi\) and \(U:L^2(\pi)\to\mathcal H_N=(\mathbb C^2)^{\otimes N}\) be \((Uf)_\sigma=\sqrt{\pi(\sigma)}f(\sigma)\). The codomain has the standard counting inner product; \(U\) is unitary between these two inner-product spaces. All operators below have the entire finite-dimensional space as domain. Set
\[ A_N=-UQU^{-1}=-D^{1/2}QD^{-1/2}. \]
Detailed balance makes this matrix Hermitian. Its quadratic form is
\[ \langle\psi,A_N\psi\rangle =\frac12\sum_{\sigma,\eta}\pi(\sigma)Q_{\sigma\eta} \left|\frac{\psi_\eta}{\sqrt{\pi(\eta)}}- \frac{\psi_\sigma}{\sqrt{\pi(\sigma)}}\right|^2\ge0, \]
where diagonal terms vanish. Irreducibility implies a one-dimensional kernel, spanned by the normalized vector
\[ |\Omega\rangle=Z^{-1/2}\sum_\sigma e^{(b/2)\sum_i\sigma_i\sigma_{i+1}}|\sigma\rangle. \]
Its squared basis amplitudes reproduce the Gibbs probabilities. Regarding those amplitudes and other vectors as physical quantum states requires a state/measurement interpretation in addition to this Hilbert-space map.
2. Explicit local terms and their positivity
Write \(Z_i|\sigma\rangle=\sigma_i|\sigma\rangle\) and let \(X_i\) flip spin \(i\). Put \(\theta=\tanh(2b)\), \(s=\operatorname{sech}(2b)\), \(c_+=(1+s)/2\), \(c_-=(s-1)/2\). Then
\[ A_N=\sum_i h_i,\qquad h_i=\frac a2\left[I-\frac\theta2Z_i(Z_{i-1}+Z_{i+1}) -(c_+I+c_-Z_{i-1}Z_{i+1})X_i\right]. \]
Derivation. For fixed neighbours with sum \(S=\sigma_{i-1}+\sigma_{i+1}\), the original flip rate is \(r_i(\sigma)=a[1-\sigma_i\tanh(bS)]/2\). Detailed balance gives transformed off-diagonal entry \(-\sqrt{r_i(\sigma)r_i(\sigma^i)}=-a/[2\cosh(bS)]\). The diagonal entry is unchanged. Since \(S=0,\pm2\), the off-diagonal coefficient equals \(c_++c_-\sigma_{i-1}\sigma_{i+1}\), giving the formula. The two neighbours are distinct even for \(N=3\); all indices are cyclic.
Alternatively \(h_i=a(I-UE_iU^{-1})\). Each \(E_i\) is an orthogonal projection, so \(h_i\ge0\), \(h_i^2=ah_i\) and \(h_i|\Omega\rangle=0\). Thus the sum is frustration-free, although its overlapping terms need not commute. Each term has norm \(a\) and support in the three sites \(i-1,i,i+1\); the global density transform has not introduced interactions of growing range. For a fixed neighbour block the local ground vector has components proportional to \(e^{bS/2},e^{-bS/2}\), which also checks the square-root and sign conventions. At \(b=0\), \(h_i=a(I-X_i)/2\) and the ground state is the product of plus-\(X\) vectors. These statements are direct finite-dimensional derivations.
3. Exact gap, normalization and limits (C126)
Unitary equivalence and C124 give
\[ \operatorname{gap}(A_N)=a[1-\tanh(2b)]. \]
The corresponding nonzero eigenvector is \(U M\) with \(M(\sigma)=\sum_i\sigma_i\), orthogonal to \(|\Omega\rangle\) by spin-flip symmetry. This is an all-mode spectral statement, inherited from C124 rather than inferred from locality or the ground state alone.
For a supplied constant \(K>0\) with action units, define \(H_N=K A_N\). Its ground energy is zero and its excitation gap in energy units is
\[ \Delta_N=Ka[1-\tanh(2b)]. \]
For \(K_N\ge K_0>0\), \(a_N\ge a_0>0\), \(0\le b_N\le B<\infty\), \(\Delta_N\ge K_0a_0[1-\tanh(2B)]\) uniformly in \(N\). These are bounds on a sequence of finite-volume Hamiltonians. They do not construct an infinite-volume Hilbert representation or establish a continuum gap. Fixed total refresh rate replaces \(a\) by \(a/N\) and closes the gap. Growing \(b_N\) or vanishing \(K_N\) can close it as well when the other parameters stay fixed.
4. Which physical premise has been supplied?
The exact semigroup identity is
\[ e^{-tH_N/K}=Ue^{tQ}U^{-1}. \]
It identifies the transformed sampling semigroup with imaginary-time evolution of the constructed operator, in the same time units. Defining the unitary \(e^{-itH_N/K}\) is mathematically possible but is an additional choice of real-time evolution; it is not the original stochastic transition law.
| Input | Use | What remains to identify physically |
|---|---|---|
| Positive Gibbs law and detailed balance | Unitary weighted-space transform and Hermiticity | Physical state and observable interpretation |
| Single-site heat-bath generator | Three-site terms and exact inherited gap | Why this generator governs physical imaginary time |
| Per-site rate \(a\) | Overall spectral rate scale | Conversion from sampling time to physical time |
| Supplied action constant \(K\) | Converts inverse time to energy and sets phase units | Positive universal action normalization |
| Uniform \(K,a,b\) bounds | Finite-volume energy-gap lower bound | Their physical origin and any further limit construction |
The classical energy function \(-J\sum_i\sigma_i\sigma_{i+1}\) used to prepare \(\pi\) is not the constructed \(H_N\): the latter has off-diagonal spin flips and coefficients depending on \(b=\beta J\) and \(Ka\). Knowledge of \(J\) and \(\beta\) fixes neither the refresh clock nor a universal action constant. For the same \(\pi\), replacing \(Q\) by \(\lambda Q\), \(\lambda>0\), preserves the ground state and locality while multiplying every excitation gap by \(\lambda\). Holding \(Q\) fixed and varying \(K\) independently gives the other normalization freedom. Thus the map yields a conditional stochastic-parent Hamiltonian; an independent physical identification is still a premise.
5. Evidence and strategic consequence
Written proof and literature status are recorded separately in the ledger. The B71 audit matches the established detailed-balance parent mapping in Henley and Castelnovo et al.; the explicit Ising term is a derived specialization. The written review checks its coefficients and inherited gap. No novelty is asserted.
G04 stops at the explicit finite-volume operator. Additional Ising spectra would not select its clock or action scale. A further operator task requires an independently specified physical dynamics whose imaginary-time generator can be compared with this one. The quantum track’s reversible-descent premise is selected next: test whether a specified interacting mechanical orientation flow closes on C125’s moment coordinates. STATE owns that bounded decision.