Flowing before decimating leaves every norm-based criterion unchanged: what is needed is a statement about the state
Conjugating by the flow before blocking does not improve the criterion of the blocking-criterion note, and the reason is one line: \(\Phi_t\) is a bijection of the configuration space, so \[\big\|V\circ\Phi_t\big\|_\infty=\sup_U V\big(\Phi_t(U)\big) =\sup_{U'}V(U')=\big\|V\big\|_\infty,\] even though \(V(\Phi_t(U))\le V(U)\) holds pointwise by the monotonicity of the flow. The straddling-plaquette norm that enters \(\beta_{\rm block}\) is therefore unchanged, the block gap is unchanged because conjugation is unitary, and the factor \(3M^2/8\) by which blocking loses is unchanged. The same argument disposes of every criterion built from operator norms: no reordering of conjugation and decimation can help, because the quantities such a criterion uses are conjugation-invariant or bijection-invariant. The flow’s gain is pointwise and therefore visible only in expectation, that is, only against a state whose weight sits where the flowed action is small. With that, the last inexpensive idea in this programme is closed, and the residual statement is sharp: the remaining input is control of the ground-state measure, which is the constructive problem in its Euclidean form. Constants explicit; nothing promoted.
1. The three invariances
Let \(\Phi_t\) be the lattice flow map and \(W_t\) its unitary implementation, as in the flow-conjugation note Proposition 1.
(i) Spectra are conjugation-invariant. Every eigenvalue of \(H\), every gap, and in particular the block gap \(\Delta_M\) of a block Hamiltonian, is unchanged by \(H\mapsto W_tHW_t^*\).
(ii) Sup norms of multiplication operators are bijection-invariant. For \(f\in C(\mathcal M)\) and a bijection \(\Phi\) of \(\mathcal M\), \(\|f\circ\Phi\|_\infty=\|f\|_\infty\). Applied to the magnetic term, \[\big\|V\circ\Phi_t\big\|_\infty=\big\|V\big\|_\infty =\frac{4N\hbar c}{ag^2}\cdot\#\{\text{plaquettes}\},\] and to the straddling part, \(\|\sum_{p\ \rm str}w_p\circ\Phi_t\|_\infty\) is bounded by the same \(6M^2\cdot4N\hbar c/(ag^2)\) per block as before. The pointwise inequality \(V\circ\Phi_t\le V\), which the monotonicity of the flow supplies, is compatible with equality of the suprema because \(\Phi_t\) maps the configuration space onto itself.
(iii) Hence the criterion is unchanged. The blocking criterion \[\beta_{\rm block}=\frac{\big\|\sum_{p\ \rm str}w_p\big\|_\infty}{\Delta_M} =\frac{24M^2N}{g^2\,\delta(g;M)}\] has a numerator fixed by (ii) and a denominator fixed by (i), so conjugating before blocking leaves it exactly as computed in the blocking-criterion note, and blocking still loses by \(3M^2/8\).
2. Why no reordering helps
Any criterion of the form (perturbation norm)/(unperturbed gap) is built from two quantities of which the first is invariant under bijections of the configuration space and the second under unitary conjugation. The flow supplies exactly a bijection and a unitary. So:
Proposition. Let \(\mathcal C(H_0,\phi)\) be any criterion depending on \(H_0\) only through its spectrum and on \(\phi\) only through \(\|\phi\|_\infty\). Then \(\mathcal C\) takes the same value for \((H_0,\phi)\) and for \((W_tH_0W_t^*,\,\phi\circ\Phi_t)\), for every flow time \(t\).
This covers the Yarotsky criterion used in the T2 note, its blocked version, and the Schur-complement criteria of the Feshbach note, all of which reduce to comparisons of norms with gaps.
3. Where the flow’s gain actually lives
The flow does reduce the action: \(V(\Phi_t(U))\le V(U)\) for every \(U\), with strict inequality away from critical points, and \(\frac{d}{ds}S(B_s)=-\|D^*G\|_2^2\). That gain appears in expectations, \[\big\langle\psi,\;(V\circ\Phi_t)\,\psi\big\rangle\ \le\ \big\langle\psi,\;V\psi\big\rangle \qquad\text{for every state }\psi,\] and is large when the weight of \(\psi\) sits where the flow moves the configuration far, that is, on rough configurations. So the flow converts a rough state into a smooth one at the level of expectations while leaving every worst-case bound alone.
A criterion that could see the gain must therefore be relative to the state, of the form \[\big\langle\psi,\phi\,\psi\big\rangle\ \le\ \epsilon\,\big\langle\psi,(H-E_0)\psi\big\rangle+\eta\|\psi\|^2 \qquad\text{on the low-energy subspace},\] which is the relative form already isolated in the Schur note Lemma 1\('\) and in the relative hypothesis of the Feshbach note §3a. The quantity it needs is the distribution of the field strength in the low-energy states, that is, the ground-state measure \(|\Omega(U)|^2\,dU\), which is a probability measure on the configuration space and is exactly the object the Euclidean formulation supplies explicitly as \(e^{-S}dU\) and the Hamiltonian formulation leaves implicit.
4. The residual statement
Collecting the closed routes of this programme:
| method | status | reason |
|---|---|---|
| variational upper bounds | closed for \(m>0\) | a spectral measure with all negative moments finite can have \(m=0\) |
| expansion around the free theory | closed | bound reaches \(m\) only at the confinement scale |
| blocking by projection | closed | boundary grows like \(M^2\), block gap does not |
| conjugation by the flow | closed | spectrum-invariant; the gain is the truncation, which is cheap |
| flow before decimation | closed (this note) | norms are bijection-invariant |
| decimation with controlled couplings | open | the constructive problem |
Addendum. The ground-state measure estimate asked for in Section 5 exists: the Agmon note proves the Agmon identity for the Kogut–Susskind ground state and gives a suppression \(e^{-\lambda n}\) of configurations with \(n\) excess plaquettes, with \(\lambda\simeq2\sqrt{2v}/g^2\), the coupling dependence of the Euclidean Wilson weight.
Every method that uses only norms and spectra is exhausted, and each was closed by a computation with explicit constants. What is left needs the distribution of the field in the low-energy states, and that is where the Euclidean construction has its tools and the Hamiltonian formulation has none of its own.
5. Consequence for STATE
The question raised at the end of the lattice-truncation note is answered in the negative, with the invariance that makes it so. The programme’s norm-based line is complete. The next thing that would advance it is an estimate on the ground-state measure of the Kogut–Susskind Hamiltonian at intermediate coupling, for instance a bound of the form \(|\Omega(U)|^2\le C\,e^{-\lambda S_w(U)}\) with \(\lambda>0\) uniform in the volume, which would import the Euclidean large-field machinery into the Hamiltonian setting. That is a well-posed question and it is the natural successor to everything above.