2026-10-15 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controlthe four rows that do not close, proposals, dependence · Kiichi, Shiori and Rin

The four rows that do not close, and proposals for them

• The part that does not close can be written in four rows: (S), (T-Obs), the absorption of $\eta$ and the multi-step induction, and the seam of re-fixing (§01).
• (U2) of row 1 (the rarity of large fields) is a corollary of Chebyshev's inequality once the group version of property 3 (iii_G) holds, and needs no independent estimate (§02, [Proved here], assuming (iii_G)).
• If the source of row 2 is taken to be a function of the block variables, it commutes with the integration at the fine steps and produces no loss (§03, [Proved here], trivial).
• For row 4, the reading is that the direct-product decomposition of the tree gauge leaves no field-dependent Jacobian of re-fixing, and the cost moves to the ghost determinant and the coupling at the seam (§05).
• The four rows gather at row 3, and the irreducible core is (S_small) (the two-loop remainder uniform over small-field backgrounds, which we read as the content of Bałaban's unpublished Theorem 2) and the multi-step (iv_G) (§06).

The notation is that of 15-00 §01 and 15-07: the effective coupling $g_k$ at step $k$, $\beta_k=4/g_k^2$, the small-field threshold $\varepsilon_k=g_kp(g_k)$, the weight $G_k$, the measure of the fluctuations $\mu_{k,W}$, the set of blocks $B(X)$. The "one line" of Table 1 is a conditional sentence: if it were shown, that row would close Assumption.

The boundary between the parts that close and the parts that do not can be drawn by two questions Reading: "does the law of the fluctuations depend on the background?" and "is it at tree level, or at one loop or beyond?". The closed components lie either in fluctuations independent of the background (Lemma H) or at tree level (Theorem F, the bookkeeping of Lemma F).

Proposal 1-a: (U2) is a Chebyshev corollary of (iii_G). If on $X$ the group version of property 3 (iii_G) holds with $s=tc/h_k^2$, that is, if $\mathbb E_{\mu_{k,W}}\exp\bigl(s\sum_{l\in\bar X}\|A_l\|^2\bigr)\le2^{|B(X)|}$, then by Chebyshev's inequality

holds Proved here (a one-liner assuming (iii_G)). If the threshold is taken of type $\varepsilon=h_k\,p(g_k)$, the right-hand side is $2^{|B(X)|}e^{-tc\,p(g_k)^2|\bar X|}$, which is the form of (U2). The rarity of large fields needs no independent estimate: the exponential moment of (iii_G) is itself the rarity. That the weight of large fields plays this role in the appendix for flat fields is a known structure Cited (15-02 §04); all we added on the group is the single phrase "if (iii_G) holds". For the product of heat kernels in Lemma H, $P\le m^{|\bar X|}e^{-s\varepsilon^2|\bar X|}$ follows at once from $M=m^{|\bar X|}$ Proved here. Caveat: (iii_G) itself depends on row 3, so (U2) of row 1 is subordinate to row 3.

Proposal 1-b: split (S) into (S_small) and (S_large). If the read-off map $b$ (the map that reads $\beta_{k+1}$ off the effective density; a candidate is the quadratic coefficient of the single coarse plaquette) is Lipschitz with respect to total-variation distance with constant $L_b\le C\beta^2$ ($db/dE\approx\frac43\beta^2$ from $\mathbb E[\frac12\operatorname{tr}U]\approx1-3/(4\beta)$ for one link), and the difference in total variation between the density with one large-field region inserted and the density without it is $e^{-c\,p(g)^2}$, then the remainder due to large fields is $|r_k^{\rm large}|\le C\beta_k^2e^{-c\,p(g_k)^2}$, and if $p(g)=c_0(1+\log g^{-1})^{p_0}$ ($p_0\ge\frac12$, with $cc_0^2$ large enough) it is $\le C'g_k^2$ Assumption (arithmetic under two assumptions). The remaining core is (S_small) = the two-loop remainder uniform over small-field backgrounds, which we read as corresponding to the content of the unpublished Theorem 2 Assumption (a reading from one sentence of the preface, 15-02 §02). Caveat: if the reading is taken to be the quadratic coefficient, two coordinates are needed at the quadratic level (15-06), so the Lipschitz property has to be stated on the pair.

Proposal 1-c: send the background dependence to the coefficient of $\theta^4$ by the parity of the critical point. If around the true critical point under the constraint the action of the fluctuations is even in the fluctuations (a fact checked with two plaquettes, 15-06 §02), there is no contribution of $O(\theta)$ to the reading of $\beta'$, and the background dependence goes entirely into the coefficient of $\theta^2$ (the reading of $F^2$ itself) and into $\theta^4$ and beyond (the source). The requirement "uniform over the background" in (S_small) then weakens to "the coefficient of $\theta^4$ is bounded", and that part is paid for by the transverse contraction (15-03 §04) Assumption (parity is a fact checked with two plaquettes, trivial with three, not yet with four).

Proposal 1-d: put large fields into the class as the parameter $z$ of the distribution over the centre. This proposal splits the weight $f$ of one plaquette into $f=f_e+f_o$ by the parity of the characters of the centre $\mathbb Z_2$, and calls the relative weight $z:=f(\pi)/f(0)$ of the far hemisphere the large-field parameter ($z=e^{-2\beta}$ for Wilson; of order $e^{-\beta\pi^2/2}$ for the heat kernel, from the leading term of the sum over images). If the large-field region is counted not as outside the class but as the contribution of $z^{|\Omega^c|}$, (U2) becomes a statement inside the class Assumption. Caveat: this is about class functions of a single plaquette; with many links the shift of the centre becomes a centre vortex (a surface of signs of $\mathbb Z_2$ of the plaquettes) and cannot be fully counted by the $z$ of a single plaquette (the caveat of 15-04 §01).

Proposal 2-a: take the source to be a function of the block variables. If the observable $J$ is defined as a function of the block variables $V$ of step $k_J$ (a Wilson loop of the coarse lattice, a smooth functional of the blocked field), then for $k\lt k_J$ the integration over the fluctuations of step $k$ is carried out with $V$ held fixed, so $\int\rho_ke^{sJ(V)}d\mu=e^{sJ(V)}\int\rho_k\,d\mu$ holds exactly and the constants of (U1)(U2) do not change (loss 0) Proved here (trivial). For $k\ge k_J$, $J$ becomes a local function of the coarse lattice (a term for each loop shape), and $e^{sJ}$ shifts the couplings by loop shape of the action by $s$. The line is that (T-Obs) reduces to "the coordinates of (Cls) include the couplings by loop shape, and the one-step map is Lipschitz in those coordinates" Assumption (the parameter coordinates of 15-06). Caveat: physical observables are loops of the fine lattice, and the difference from the blocked loops is the renormalisation of observables. Restating (T-Obs) with blocked observables is a re-choice of the statement, not (T-Obs) for fine loops. We have not checked in the text which of the two the "expectation values of loops" in Bałaban's preface refers to.

Proposal 2-b: the class of position-dependent couplings $\beta(x)$. A source of plaquette-density type $J=\sum_pf(p)(1-\frac12\operatorname{tr}U_p)$ (with $f$ slowly varying on the scale $L_0/M$) is the Wilson action with $\beta$ replaced by $\beta(x)=\beta+sf(x)$. If the one-step map is the map of $\beta$ applied pointwise plus a gradient correction $O(|\nabla\beta|\,a_k/L_0)$, the gradient terms form a geometric series in $a_k/L_0=2^{-(K-k)}$ and the sum is finite Assumption. For the abelian Villain action of $U(1)$ this is a form that can be checked on paper (whether after one step it becomes a local average of $\beta'(X)=\sum_qf_q^2\beta(q)$). It cannot be used for loop-type observables (the source of the area law) and is suited to local observables.

Rejected: peeling off the source with Hölder alone. The road that splits the estimate into $(\mathbb EG^2)^{1/2}(\mathbb Ee^{2sJ})^{1/2}$ does not close: at the fine steps the support of $J$ becomes a non-local activity spread over blocks of size loop length / lattice spacing, the one-step loss $2^{|B(X_J)|}$ cannot be paid for by a small factor per block, and it accumulates over successive steps (Table 2).

The content of row 3 is in 15-07 §06 and §07. We write only the main points. For the Wilson action of $SU(2)$, the conditional law of one link is a member of a one-parameter family independent of the background, and the background enters only through the local coupling and the direction of the centre Cited (a fact of the heat-bath method; bibliography not checked). With many links, a quaternion identity splits the action into "local coupling × a form independent of the background" + a first-order term + a conjugate shift Proved here, and if the score of the density is sorted by order in $\theta$, the prospect is that $\theta^0$ vanishes by re-centring, $\theta^1$ is absorbed as a coupling derivative, and $\theta^2$ and beyond remain at $O(g^3p^3)$ Assumption.

Putting it on the bookkeeping of the transverse contraction. What remains after absorption ($\theta^2$ and beyond, and the expansion of $b_p=\beta\cos(\theta_p/2)$ at $\theta^4$ and beyond = the source) is a small shift of the transverse coordinates, and with the bookkeeping of 15-03 §04, if $\lambda\lt1$, the deviation is bounded not by a sum but by the supremum $\sup_k\varepsilon(\beta_k)/(1-\lambda)$ Proved here (a geometric series, assuming the one-step inequality). The multi-step (iv_G) is needed in the form "the fine angles of the image of the background field are $L^{-2}$ times the coarse angles + $O(g^2)$ of the transverse coordinates" Assumption. The transverse contraction was measured at the abelian tree level (15-05); its value for non-abelian configurations and at one loop is not yet known.

Proposal 4-a: there is no Jacobian of re-fixing — the direct-product decomposition of the tree gauge. Take a spanning tree $T_B$ (with root $r_B$) in each block $B$, and split the gauge group $G^\Lambda=\prod_xG$ into the factor $G_{\rm int}:=\prod_{x\ne\text{root}}G$ of the non-root points and the factor $G_{\rm coarse}:=\prod_BG$ of the roots (it is pointwise a direct product, so trivially a direct product). The map fixing the tree links to 1 is a section of the action of $G_{\rm int}$; since each point has exactly one tree link heading towards the root, it is solved uniquely, the axial (tree) gauge is complete with no Gribov multiplicity, and the Faddeev–Popov determinant is constant. If block averaging is gauge covariant, $G_{\rm coarse}$ is isomorphic to the gauge group of the coarse lattice, and the tree gauge of step $k+1$ fixes it. That is, the gauge fixings of step $k$ and step $k+1$ each fix one of the two factors of a direct product of the same group, and there is no field-dependent Jacobian of re-fixing Proved here (a restatement of a known structure; that the Faddeev–Popov determinant of the axial gauge is constant is a standard fact).

With this, "the Jacobian (the Faddeev–Popov determinant) enters the effective action" of 15-01 §03 becomes empty for the axial gauge. The cost moves to two other places Reading: (i) the degradation $\|U_l-1\|\le\ell\varepsilon$ of the small-field condition due to the tree length $\ell$ (6-05 §05), and (ii) if a background-covariant gauge is used for the fluctuations, its ghost determinant. We have not checked in the text which gauge Bałaban used in the four-dimensional papers (in three dimensions, an axial gauge per block is introduced by the Faddeev–Popov method).

Proposal 4-b: absorb the ghost determinant in the coordinates of the local couplings. For the ghost operator $\Delta_W^{\rm gh}=d_W^*d_W$ in the background $W$ (the covariant Laplacian of the lattice, in the adjoint representation), the prospect is that $\log\det\Delta_W^{\rm gh}-\log\det\Delta_1^{\rm gh}$ can be expanded into local functionals on small-field backgrounds, with leading term $\sum_pc_{\rm gh}\|F_p\|^2$ (the coefficient of $F^2$ = the ghost contribution to the one-loop coefficient), the rest being $\theta^4$ and beyond (the source) and exponentially decaying non-local terms, with constants independent of $k$ Assumption (of the type of the heat-kernel expansion on the lattice; uniform convergence of the local expansion has not been shown). If so, the $F^2$ part merges into the shift of local couplings of row 3, $\theta^4$ and beyond go into the transverse coordinates, and the seam becomes not a new kind of loss but part of the one-loop coefficient. As a negative control, for $U(1)$ the ghosts do not depend on the background.

Proposal 4-c: make the seam a single matrix. In the tree-level network that fixes the angles of all coarse plaquettes simultaneously (Theorem G of 15-05), the coarse action is a quadratic form $\theta^{\rm T}B\theta$, and the coupling between neighbouring coarse plaquettes and the mixing of the quartic map correspond to the coupling at the seam. The reading is that the question of row 4, "are the terms at the seam a loss with finite sum?", becomes the single number "does the norm of the matrix of the quartic map fall below 1?" Reading. Under Theorem G the map measured in $\ell^4$ exceeds 1/16 in the coarse direction (the largest value found in four dimensions is about 0.15), and we have no upper bound uniform in the volume Reading (15-05).

The wall for a road using class functions alone. On a road whose only coordinates are the class angles of the holonomy of each coarse plaquette (Migdal–Kadanoff is one), the coarse Bianchi identity (the product of the holonomies of the six faces of a coarse cube = 1) needed to reconstruct the coarse gauge field on the four-dimensional hypercubic lattice cannot be written. In the non-abelian case it can only be written once there is a common base point and connecting paths, and with class angles alone the relative orientation of neighbouring faces is lost. We read this as: a road using class functions alone cannot go beyond the hierarchical-lattice approximation Reading.

Can Haar invariance replace re-fixing? In the setting of Lemma H, if the law of the fluctuations is conjugation invariant on each link, the decomposition $U=WY$ is gauge covariant and no gauge fixing at all is needed. But this is limited to renormalisation of the type that "gives" the law of the fluctuations by convolution with heat kernels (a Polchinski flow on the group), and convolution with heat kernels does not reduce the degrees of freedom, so it cannot replace the re-fixing of four-dimensional blocking, which produces block variables Reading.

The four rows are not independent. Written in the direction "when the left closes, the right becomes lighter", they give the following diagram Reading (an estimate of the logic between the proposals).

Row 3 (absorption of η: background dependence = shift of local couplings)
  ├─→ (iii_G), (A.6.5_G) hold with constants independent of the step
  │      └─→ (U2) of row 1 (proposal 1-a: Chebyshev corollary)
  │             └─→ (S_large) of row 1 (proposal 1-b: Lipschitz of the reading × rarity)
  ├─→ coordinates of (Cls) (β, couplings by loop shape, shape coordinates)
  │      └─→ row 2 (proposal 2-a: the source is a shift of coordinates)
  └←── ghost determinant of row 4 (proposal 4-b: merges into local couplings)
         ↑
       row 4 has no Jacobian (proposal 4-a: direct-product decomposition of the tree gauge)

Irreducible core:
  (S_small) = two-loop remainder uniform over small-field backgrounds (read as the content of Theorem 2)
                → may weaken to "boundedness of the θ⁴ coefficient" via proposal 1-c
  multi-step (iv_G) = regularity of the background field (fine curvature ≤ L⁻² × coarse curvature + small terms)

What to read from the diagram Reading: (U2) and (S_large) of row 1, row 2, and the ghosts of row 4 all have a form that can be written, once the coordinates of the absorption of row 3 are fixed, as a corollary of it or as a shift of those coordinates. What remains independent of row 3 is two things, (S_small) and the multi-step (iv_G): the former is the unpublished part that Bałaban's series needed in order to remove assumption (U3), the latter is the multi-step version of the regularity of the background field.

Proposals of the type "with enough computing resources the constants become explicit" (making the constants explicit with interval arithmetic) are not the core of the four rows, so they are not in this table.

Table 3 sets out in ten rows the directions being tried towards ① (moved from Table 3 of 13-11, with the components added in this part written into "What can be said now"). "Smallest setting" is the smallest object on which the direction can first be tried, "What can be said now" is the current state with labels, and the last column is the risk that the direction ends up as a restatement of something already known. We do not give numbers for the prospects. No row closes ①.

What to read from Table 3 Reading (a physical estimate). The directions with components closed on paper are directions 3 and 4 (Lemma R, Lemma H) and direction 6 (Lemma F), and all lie outside the real issue (property 2, background-independent fluctuations, the bookkeeping part). The real issue — the group version of property 3 for the real $\mu_{k,W}$, and the one-step remainder (S) including large fields — is the single point towards which directions 3 and 4 and direction 6 each head, and they meet where the dependence of the measure of the fluctuations on the background takes effect. This is the same place as "row 3 is the core" in §06.

Questions arising from this chapter. Table 4 places, one for each of the four rows, the smallest question that can be tried next.

(a) ① ultraviolet control does not close. None of the four rows closes. Every proposal in this chapter is a proposal, not a result.

(b) The "one line" of Table 1 is a conditional statement — if it were shown, the row would close — and none of them has been shown. That (S_small) corresponds to the content of Theorem 2 is a reading from one sentence of the preface.

(c) Proposal 1-a is a one-liner assuming (iii_G), and (iii_G) has not been shown. The Lipschitz constant and the difference in total variation of proposal 1-b are assumptions and have not been estimated.

(d) The commutativity of proposal 2-a is a trivial fact limited to functions of the block variables, and says nothing about observables given by loops of the fine lattice.

(e) "There is no Jacobian" of proposal 4-a is a restatement of a structure limited to the axial (tree) gauge, and we have not checked the choice of gauge in Bałaban's four-dimensional papers. The local expansion of the ghost determinant is a prospect.

(f) "Row 3 is the core" is an estimate of the logic between the proposals, and none of its implications has been shown. The directions of Table 3 are candidates, and neither their consistency nor their prospects have been assessed.

Dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] ASM ROWS   one line of each of the four rows that do not close  (constants independent of k, a)
[02] RDG EDGE   where the closed components lie ⟸ [01]
[03] PRV CHEB   (U2) ⟸ Chebyshev for (iii_G)  (assuming (iii_G))
[04] ASM SPLIT  (S) = (S_small) + (S_large) ⟸ [01][03]  (two assumptions)
[05] ASM EVEN   to θ⁴ by parity, parameter z ⟸ [04]  (four plaquettes not yet)
[06] PRV BLOCK  a source that is a function of block variables commutes  (blocked observables)
[07] ASM SHIFT  the source is a shift of coordinates ⟸ [06]
[08] PRV TREE   no Jacobian for the tree gauge  (axial gauge)
[09] ASM GHOST  ghost determinant into local couplings ⟸ [08]
[10] RDG CLASS  wall for class functions alone, limit of Haar ⟸ [08]
[11] RDG CORE   row 3 is the core ⟸ [03][04][07][09]
[12] PRV DOM    Wilson is not dominated by the heat kernel  (one link)
[13] RDG DIR    the ten directions ⟸ [11]

The way to check it is to see by eye that every number to the right of a "⟸" is smaller than the number on the left. There are no cycles. [01], [04], [05], [07] and [09] are assumption rows, and all are in the form of questions that split ①. Among the nodes of this chapter there is no node saying that ① has been shown.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the four rows that do not close, (S_small), (S_large), the read-off map, the large-field parameter $z$, a source that is a function of the block variables, position-dependent couplings, the direct-product decomposition of the tree gauge, the ghost determinant, the coupling at the seam, row 3 is the core. Their meanings are in A-02 Glossary.

Next chapter: 15-09 What cannot be said, the index of questions, the dependency diagram — the questions F-UV1–15 of this part and a summary of the five grades.

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