One blocking step: the obstruction is the variation, not the size, of the inter-block coupling
Partition the lattice into blocks of \(M^3\) sites and let \(P\) project onto the tensor product of the low-energy subspaces of the blocks. The transfer lemma then reduces the gap of the whole system to the gap of the blocked Hamiltonian \(PHP\) minus the Schur error of the straddling plaquettes. Counting gives, per block, \(6M^2\) straddling plaquettes of norm \(\|w_p\|\le4N\hbar c/(ag^2)\) each, against an intra-block gap \(\Delta_M=(\hbar c/a)\delta(g;M)\), so the naive operator-norm estimate gives a Schur error of order \[\frac{\|B\|^2}{\Delta_M}\ \sim\ 36M^4\,\frac{\|w\|^2}{\Delta_M} =36M^4\,\frac{\hbar c}{a}\,\frac{16N^2}{g^4\,\delta(g;M)},\] which exceeds the gap it is meant to preserve by a factor of order \(M^4N^2/(g^4\delta^2)\). Lemma 1\('\) of the Schur note shows where that estimate is wasteful: a part of the Schur term proportional to the identity shifts all levels equally and costs nothing, and the boundary energy is exactly of that kind. What the induction needs is the variation of the Schur term over the low-energy subspace, for which a connected estimate replaces \(\|B\|^2=\big(\sum_p\|w_p\|\big)^2\) by \(\sum_p\|w_p\|^2\) and so replaces \(M^4\) by \(M^2\). Under that replacement one blocking step closes when \[\frac{\|w\|}{\Delta_M}\ \lesssim\ \frac1{M\sqrt6}\qquad\text{i.e.}\qquad g^4\ \gtrsim\ \frac{8\sqrt6\,MN}{\gamma\,C_2(R_{\min})}.\] Correction, recorded after this note was written: the connected estimate is supplied by Yarotsky’s polymer expansion, and the resulting criterion is worse than the direct one by a factor \(3M^2/8\), so the induction sketched in Sections 4–5 does not close; the blocking-criterion note carries the corrected statement. What survives is the counting: the number of renormalization-group doublings needed to reach a strong-coupling threshold from the cutoff is \[n\ \simeq\ \frac{1}{2b_0\log2}\Big(\frac1{g_{\rm UV}^2}-\frac1{g_{\rm thr}^2}\Big), \qquad b_0=\frac{11N}{48\pi^2},\] finite and of the order of tens for realistic couplings. The whole difficulty of the mass gap is therefore concentrated in a finite number of blocking steps in the intermediate coupling regime, each of which generates couplings beyond the Kogut–Susskind form that must be carried along. Constants explicit; the connected estimate is stated as the required input and is not proved here; nothing promoted.
1. The block decomposition and its counting
Fix a block factor \(M\ge2\) and partition the periodic lattice of \(N_s^3\) sites, \(N_s=MK\), into \(K^3\) blocks of \(M^3\) sites. Assign each link to the block containing its midpoint, so the links are partitioned. A plaquette is interior when all four of its links lie in one block and straddling otherwise. Write \[H=\sum_{\alpha}H_\alpha+\sum_{p\ \rm straddling}w_p,\qquad H_\alpha=\frac{\hbar cg^2}{2a}\sum_{\ell\in\alpha}(-\Delta_\ell) +\sum_{p\subset\alpha}w_p .\]
Counting. Each block has six faces of \(M^2\) plaquettes, so \[\#\{\text{straddling plaquettes touching one block}\}=6M^2, \qquad \|w_p\|\le\frac{4N\hbar c}{a\,g^2},\] with \(N\) replaced by \(\dim_{\rm f}\) for a general compact group, as in the Lieb–Robinson note §1. Each \(H_\alpha\) is a Kogut–Susskind Hamiltonian on a block with free boundary, so T1 applies to it: discrete spectrum, unique positive ground state \(\Omega_\alpha\), gap \(\Delta_M=(\hbar c/a)\delta(g;M)\).
Let \(\Pi_\alpha\) project onto the spectral subspace of \(H_\alpha\) below a cutoff \(E_c\) chosen in the gap region, and \[P=\bigotimes_\alpha\Pi_\alpha,\qquad A=PHP,\qquad D=\bar PH\bar P\ge\textstyle\sum_\alpha E_0^\alpha+E_c .\] The compression \(A\) is the blocked Hamiltonian: a Hamiltonian on the product of block low-energy spaces, whose gap the induction must track.
2. Why the naive estimate fails
The coupling operator is \(B=PH\bar P\), bounded by the sum of the straddling terms touching the blocks involved, \[\|B\|\ \le\ \sum_{p\ \rm straddling}\|w_p\|\ \sim\ 6M^2\cdot\frac{4N\hbar c}{a g^2} \quad\text{per block},\] and the transfer lemma of the Feshbach note charges \(\|B\|^2/(E_c)\) against the gap. With \(E_c\sim\Delta_M\), \[\frac{\|B\|^2}{\Delta_M}\Big/\Delta_M \ \sim\ 36M^4\Big(\frac{\|w\|}{\Delta_M}\Big)^2 =36M^4\Big(\frac{8N}{\gamma C_2\,g^4}\Big)^2,\] using \(\Delta_M\ge\gamma(g^2/2)C_2\hbar c/a\) from the T2 note at strong coupling. The ratio exceeds one unless \(g^4\gtrsim M^2N\), and at weak coupling it is enormous. Taken at face value, real-space blocking is useless.
3. What the estimate is actually measuring
The quantity \(\|B\|^2/\Delta_M\) is dominated by the boundary energy: the second-order effect of the straddling plaquettes lowers the ground state of the coupled system by an amount proportional to the total boundary area, which is extensive in \(M^2\) per block and has nothing to do with the gap. Lemma 1\('\) of the Schur note isolates that contribution: if \[S(E)=B(D-E)^{-1}B^*=\eta_0(E)\,\mathbb 1+R(E),\qquad 0\le R(E)\le\epsilon\,(A-a_0)+\eta_1,\] then the constant \(\eta_0\) drops out of the gap entirely and only \(\epsilon\), \(\eta_1\) and the slope \(\eta_0'\) enter. The boundary energy is the constant; what survives is the dependence of the boundary energy on which low-energy state of the blocks the system occupies.
The required input. For each straddling plaquette separately, \(w_p(D-E)^{-1}w_p\) has norm at most \(\|w_p\|^2/E_c\), and its variation across \(\operatorname{ran}P\) is at most twice that. Cross terms between two straddling plaquettes \(p\neq p'\) contribute to the constant when they are far apart and to the variation only through connected configurations. A connected estimate of the form \[\big\|R(E)\big\|\ \le\ C\sum_{p\ \rm straddling}\frac{\|w_p\|^2}{E_c} \qquad\text{in place of}\qquad \frac{\big(\sum_p\|w_p\|\big)^2}{E_c}\] therefore replaces \(M^4\) by \(M^2\). This is exactly what a cluster expansion supplies at strong coupling, and it is the one analytic input the blocking step needs. It is stated here as a hypothesis.
4. The one-step inequality under that input
Superseded. Sections 4 and 5 are corrected in the blocking-criterion note: the connected estimate is available from Yarotsky’s polymer expansion, and the resulting threshold is worse than the direct one by a factor \(3M^2/8\), so the induction below does not close. The counting of Section 5 survives as a count of renormalization-group steps.
Proposition (conditional). Assume the connected estimate of Section 3 with constant \(C\), and take \(E_c=\Delta_M\). Then the transfer lemma gives \[\operatorname{gap}(H)\ \ge\ \operatorname{gap}(A)\Big(1-\epsilon\Big)-\eta_1, \qquad \epsilon,\ \frac{\eta_1}{\Delta_M}\ \le\ 6CM^2\Big(\frac{\|w\|}{\Delta_M}\Big)^2,\] so one blocking step preserves a positive gap whenever \[\frac{\|w\|}{\Delta_M}\ \le\ \frac1{M\sqrt{6C}}, \qquad\text{that is}\qquad g^4\ \ge\ \frac{8\sqrt{6C}\;M\,N}{\gamma\,C_2(R_{\min})} .\]
The threshold depends on the block factor \(M\) and on the group, and on no other scale: it is a fixed coupling threshold once \(M\) is fixed, say \(M=2\).
5. The induction, and where the finite difficulty sits
Under blocking, the effective coupling of an asymptotically free theory grows toward the infrared. If the blocked Hamiltonian \(A\) were again of Kogut–Susskind form with a coupling \(g_{n+1}>g_n\), then the threshold of Section 4, once satisfied at some step, would be satisfied at every later step, and the induction would give a gap at all larger scales, uniform in the volume. Two obligations remain.
Reaching the threshold. At the cutoff the coupling is small. With the one-loop running written for a doubling of scale, \[\frac1{g_{n+1}^2}=\frac1{g_n^2}-2b_0\log2,\qquad b_0=\frac{11N}{48\pi^2},\] the number of doublings from \(g_{\rm UV}\) to \(g_{\rm thr}\) is \[n\simeq\frac{1}{2b_0\log2}\Big(\frac1{g_{\rm UV}^2}-\frac1{g_{\rm thr}^2}\Big),\] finite and logarithmic in the ratio of scales. For \(SU(2)\), \(2b_0\log2=\tfrac{11}{12\pi^2}\log2\approx0.064\), so a bare coupling \(g_{\rm UV}^2=1/2\) takes of the order of thirty doublings, a scale ratio of about \(10^9\), to reach a threshold at \(g_{\rm thr}^2\approx20\). The mass gap therefore requires control of a finite, explicitly bounded number of blocking steps, all of them in the intermediate regime where neither the perturbative expansion of the Feshbach note nor the cluster estimate of Section 3 applies.
Staying in the family. The blocked Hamiltonian \(A\) is a Hamiltonian on the product of block low-energy spaces, and it contains, besides a nearest-block plaquette term, every operator generated by the compression: longer-range terms, terms of higher order in the block observables, and terms without a Kogut–Susskind counterpart. Carrying a controlled family of such Hamiltonians through the steps is the constructive renormalization-group problem that Balaban’s programme solves on the Euclidean side in finite volume with an ultraviolet cutoff, and which remains open beyond that.
6. What this step establishes
The blocking route survives the first objection: the enormous inter-block coupling enters the gap only through its variation, and the boundary energy, which is what makes \(\|B\|\) large, is free by Lemma 1\('\). The route then needs one analytic input, a connected estimate for the Schur variation, and one structural input, a stable family of effective Hamiltonians. The first is a cluster expansion and is available at strong coupling; the second is the open problem. The quantitative shape of the difficulty is now explicit: a fixed coupling threshold \(g_{\rm thr}^4\simeq8\sqrt{6C}MN/(\gamma C_2)\), and a finite number of steps, logarithmic in the scale ratio, to reach it.
7. Consequence for STATE
Step 2 of the foreseen route is written out with its constants. The two inputs it needs are named and separated: a connected (cluster) estimate for the variation of the Schur term, which would make the one-step inequality unconditional at strong coupling, and a stable family of blocked Hamiltonians, which is the constructive renormalization-group problem. The next tractable item is the first of these at strong coupling, where Yarotsky’s expansion already produces connected estimates of the required kind and may be readable as such directly.