Blocking and Schur bounds: the surface-to-gap obstruction
A block criterion based on the norm of straddling plaquettes divided by an intra-block gap has a surface cost. For \(M^3\)-site blocks of a three-dimensional spatial lattice the counted sufficient bound is \[\beta_{\rm block}=\frac{24M^2N}{g^2\delta(g;M)},\qquad \Delta_M=\frac{\hbar c}{a}\delta(g;M).\] Its strong-coupling asymptotic exceeds the direct criterion by \(3M^2/8\), and a small-volume zero-mode model gives \(M^3\) scaling. These are regime-dependent comparisons, not a proof of monotonicity at every coupling. The inequality \(\beta_{\rm block}(M)\ge\beta_{\rm block}(1)\) follows if \(\delta(g;M)\le M^2\delta(g;1)\); a separate upper bound on \(\delta(g;M)\) does not establish this comparison. The original all-coupling claim is withdrawn. No uniform physical mass gap or continuum construction follows.
1. Blocking, norms and the Schur issue
Partition a periodic lattice of \(N_s=MK\) sites per side into \(K^3\) blocks, assign links by midpoint and write \[H=\sum_\alpha H_\alpha+\sum_{p\ {\rm straddling}}w_p,\qquad H_\alpha=\frac{\hbar cg^2}{2a}\sum_{\ell\in\alpha}(-\Delta_\ell) +\sum_{p\subset\alpha}w_p.\] The earlier calculation used \(6M^2\) straddling plaquettes per block; face-crossing counting may require a factor of two unless terms are half-attributed. The displayed numerical criterion is conditional on that assignment. For each plaquette, \(\|w_p\|\le4N\hbar c/(ag^2)\). Edge counting and interaction assignment must be fixed before applying a particular polymer theorem. The block Hamiltonians have finite-volume ground states and gap \(\Delta_M>0\). A product projection \(P\) onto block low-energy subspaces gives a compression \(A=PHP\); a Schur reduction additionally requires a lower bound on the complement \(D=\bar PH\bar P\), including the straddling interactions. Such a bound does not follow merely from the uncoupled block gap.
If \(B=PH\bar P\) and a complementary resolvent bound is supplied, a naive per-block estimate is \[\frac{\|B\|^2}{\Delta_M}\lesssim 36M^4\frac{\hbar c}{a}\frac{16N^2}{g^4\delta(g;M)}.\] This measures a large boundary contribution as well as level-dependent variation. The relative Schur lemmas in the Feshbach note show why a common energy shift can cancel from a gap estimate, while its energy dependence and residual variation still need control. A proposed bound \[\|R(E)\|\le C\sum_{p\ {\rm straddling}} \frac{\|w_p\|^2}{\Delta_M}\] would replace the naive \(M^4\) counting by \(M^2\), but is a hypothesis: polymer connectedness by itself does not prove this particular Schur operator inequality or its volume-uniform constants. The earlier one-step threshold and induction based on that unproved inference are withdrawn.
2. Direct block criterion and its limitations
Normalize a block Hamiltonian by its gap, \(h_\alpha=(H_\alpha-E_0^\alpha)/\Delta_M\), and bound the normalized inter-block perturbation by \(\beta_{\rm block}\) above if that count and assignment hold. Yarotsky’s bounded-perturbation theorem gives a sufficient criterion of the form \(\beta_{\rm block}\le\beta_*\), subject to its product-space, unique block-ground-state, finite-range and interaction-assignment hypotheses. Its polymer proof (Commun. Math. Phys. 261 (2006), §2, passage-level reading) has connected excited-region weights; it can be applied only after those hypotheses are checked, and it is not the asserted Schur variation estimate. See the group-specific application and its scope in strong coupling.
At strong coupling, in a block containing an elementary loop and in the gauge-invariant sector, the electric excitation is a fundamental plaquette loop with energy \(2C_2(R_{\min})g^2\hbar c/a\). For fixed block geometry as \(g\to\infty\) this gives \[\beta_{\rm block}\sim\frac{12M^2N}{C_2g^4},\qquad \frac{\beta_{\rm block}}{\beta_{\rm direct}}\sim\frac{3M^2}{8},\qquad \beta_{\rm direct}=\frac{32N}{C_2g^4}.\] Thus the displayed sufficient estimates compare unfavourably at \(M=2\) in that asymptotic regime; they are bounds, not exact optimal thresholds. Boundary conditions and the sector of the block operator matter.
For \(SU(3)\), \(C_2=4/3\) gives \[\beta_{\rm block}=\frac{72M^2}{g^2\delta(g;M)},\qquad \beta_{\rm direct}=\frac{72}{g^4}.\] The same asymptotic ratio is \(3M^2/8\), formally \(3/2\) at \(M=2\). These constants retain the stated half-attribution and block-geometry conditions; they do not establish an all-coupling comparison.
For a periodic small-volume zero-mode model, the conditional asymptotic \(\Delta_M\sim\delta_1g^{2/3}\hbar c/(Ma)\) would give \[\beta_{\rm block}\sim\frac{24M^3N}{\delta_1g^{8/3}}.\] It must not be imported without proof into the free-boundary blocks above or treated as a uniform fixed-cutoff theorem; see the Feshbach reduction. Algebraically the criterion is no better than its \(M=1\) analogue if \(\delta(g;M)\le M^2\delta(g;1)\), assuming both are defined with compatible boundary conventions. This condition has not been established for all couplings. A one-sided upper bound such as the Polyakov variational bound cannot supply the missing lower comparison with \(\delta(g;1)\).
3. Running coupling and generated interactions
The one-loop count for successive doublings, conditional on the running law \(g_{n+1}^{-2}=g_n^{-2}-2b_0\log2\) with \(b_0=11N/(48\pi^2)\), is \[n\simeq\frac{g_{\rm UV}^{-2}-g_{\rm thr}^{-2}}{2b_0\log2}.\] For \(SU(2)\), \(2b_0\log2=11\log2/(12\pi^2)\approx0.064\); the illustrative inputs \(g_{\rm UV}^2=1/2\) and \(g_{\rm thr}^2\approx20\) yield about thirty doublings. This perturbative extrapolation is not a certified arrival at a strong-coupling domain. The number increases with the ultraviolet cutoff along an asymptotically free trajectory. A projection generates longer-range and other interactions, and a coupling label alone does not prove an iterated effective-action or gap estimate. An RG construction must control those generated terms and observable matching at each scale.
4. Consequence for STATE
Keep the boundary count, conditional comparison, asymptotic strong-coupling calibration and Schur distinction. Neither a general no-improvement theorem nor a connected-Schur estimate is established. Control of the blocked action and gap through the weak-to-intermediate regime remains open; the physical energy unit is \(\hbar c/a\) and the volume and continuum limits are separate obligations.