2026-10-15 · chapter Yang–Mills and the mass gapPart 12 The second roadthe insertion ratio, the tanh form, second moments · Kiichi, Shiori and Rin

Vortex-insertion identities

1. In the sector-summed ensemble (12-03), whether a vortex passes through a region is expressed by a sign $\sigma=\pm1$. A map $T$ that inserts (or removes) one vortex flips $\sigma$.
2. We call the ratio of weights when a vortex is inserted in a configuration $x$ the insertion ratio $\rho(x)$. The smaller $\rho$ is than 1, the more expensive it is to insert a vortex.
3. The mean of the insertion ratio is always exactly 1 (Identity 1). This is nothing but a balance equation: "relabelling configurations by a map does not change the total sum of weights". From the same balance also follows $\mathbb E[\rho\sigma]=-\mathbb E[\sigma]$ Proved here.
4. Combining the two, the bias of the sector $|\mathbb E[\sigma]|$ is at most half the square root of the fluctuation (variance) of the insertion ratio (Theorem 2). If one inserts at many positions and averages the insertion ratio, the fluctuation can become small. Averaging over positions plays the role of entropy Proved here.
5. The insertion ratio can take large values (heavy tails), so we also prepare a bounded form that is easy to handle. $\tanh(\Delta S/2)$ always lies in $[-1,1]$, and $\mathbb E[H]=\mathbb E[H\tanh(\Delta S/2)]$ holds (Identity 3). $1-\mathbb E[\tanh]$ is twice the mean acceptance rate of a Monte Carlo move that inserts a vortex (Lemma 4) Proved here.
6. The form using $\sqrt\rho$ has a second moment that is always at most 1, so there is no worry about tails (Identity 6, Theorem 7) Proved here.
7. Taking the flip of a thin vortex sheet (12-04) as the insertion, the slab-confinement theorem comes out as a special case of this chapter (§06).
8. These are not a proof of confinement. An area law needs the bias to be small in many regions at once (large deviations) and a bound on the cost of insertion uniform in the lattice spacing (the ultraviolet), and both are outside this chapter.

Throughout this chapter we place the following three.

• (F) A finite configuration space $X$ (a finite set).
• (W) Positive weights $w:X\to(0,\infty)$. The expectation is $\mathbb E[f]=\sum_xw(x)f(x)/\sum_xw(x)$.
• (T) A bijection that flips the sector: a sign of the sector $\sigma:X\to\{\pm1\}$ and a bijection $T:X\to X$ with $\sigma(Tx)=-\sigma(x)$.

(T) is a finite version of "a map that preserves the Haar measure and flips the sign of the sector of some region" in the sector-summed ensemble (for example, the composite of flipping the $z$ of a vortex sheet with a smooth left multiplication supported inside the region). Being an involution, commuting with one another, and worst-case bounds are not used in §03. To pass from (F) to the continuous configuration space ($SU(2)^{\text{links}}$ and coclosed $z$), one must replace the sums by Haar and counting integrals, and $T$ must preserve the measure (if not, the Jacobian must be put into the weight). The proofs go through with the same relabelling of variables, but that version is outside Lean.

We set the insertion ratio to be

The orientation is chosen so that a vortex is inserted by $T$ into configurations with $\sigma=+1$, and configurations with $\sigma=-1$ are taken back by $T^{-1}$. In terms of the action $S=-\ln w$, $\rho=e^{-\Delta S}$.

Proved here Proof. Set $X_\pm=\{\sigma=\pm1\}$. $T$ maps $X_+$ to $X_-$ and $X_-$ to $X_+$, and since it is a bijection, $T(X_+)=X_-$ and $T^{-1}(X_-)=X_+$. Hence

Adding gives $\sum w\rho=\sum w$, and multiplying by $\sigma$ and adding gives $\sum w\rho\sigma=\sum_{X_-}w-\sum_{X_+}w=-\sum w\sigma$. ∎

Lean InsertionIdentity (Insertion.mean_one, Insertion.mean_rho_sigma) (hypotheses: $X$ is finite, $w\gt0$, $T$ is a bijection of $X$, $\sigma(Tx)=-\sigma(x)$, $\sigma=\pm1$. Conclusion: $\sum w\rho=\sum w$, $\sum w\rho\sigma=-\sum w\sigma$). No involution, commutativity or measure structure is used.

Proved here Proof. Applying Identity 1 to each $T_k$ and averaging gives $\mathbb E[\bar\rho]=1$, $\mathbb E[\bar\rho\sigma]=-\mathbb E[\sigma]$. Hence $\mathbb E[(\bar\rho-1)\sigma]=-2\mathbb E[\sigma]$. From Cauchy–Schwarz and $\sigma^2=1$, $4\mathbb E[\sigma]^2\le\mathbb E[(\bar\rho-1)^2]\mathbb E[\sigma^2]=\mathrm{Var}(\bar\rho)$. ∎

Lean InsertionIdentity (Insertion.sector_bias_le) (hypotheses: $X$ is finite, the index set of the family is finite and non-empty, $w\gt0$, each $T_k$ is a bijection with $\sigma(T_kx)=-\sigma(x)$, $\sigma=\pm1$. Conclusion: $4(\sum w\sigma)^2\le(\sum w)\sum w(\bar\rho-1)^2$).

To apply it to containers (12-03), use Theorem 2 on the conditional measure with the frame fixed. A bijection that moves only the inside of a container does not change the frame, so $\theta_j(\Phi)=|\mathbb E[\sigma_j\,|\,\Phi]|\le\tfrac12\sqrt{\mathrm{Var}(\bar\rho\,|\,\Phi)}$ Proved here. The reading is that the factor $\tanh(|\Delta F_j|/2)$ of Lemma 5 is bounded from above by the fluctuation of the mean insertion ratio of insertions at many positions inside the container.

The insertion ratio $\rho=e^{-\Delta S}$ can take large values, so $\mathrm{Var}(\bar\rho)$ can be large. We therefore switch to a bounded factor. From the $T$ of (T) we build the map oriented by the sector

and $\tau$ is an involution ($\tau\circ\tau=\mathrm{id}$) (if $\sigma(x)=+1$, then $\tau x=Tx\in X_-$, so $\tau(\tau x)=T^{-1}Tx=x$; the converse is the same) Proved here. We set $\Delta S=S\circ\tau-S$, $\rho=w\circ\tau/w=e^{-\Delta S}$, and

Proved here Proof. Since $1-t=2w\circ\tau/(w+w\circ\tau)$, $\mathbb E[H]-\mathbb E[Ht]$ is proportional to $2\sum_xg(x)/\sum w$ with $g(x)=w(x)H(x)w(\tau x)/(w(x)+w(\tau x))$. $g(\tau x)=-g(x)$ ($H$ is odd and the rest is symmetric under $x\leftrightarrow\tau x$), so relabelling the sum by $\tau$ gives $\sum g=-\sum g$, hence 0. ∎

Lean BoundedInsertion (Bounded.single) (hypotheses: $X$ is finite, $\tau$ is a bijection with $\tau\circ\tau=\mathrm{id}$, $w\gt0$, $H\circ\tau=-H$. Conclusion: $\sum wHt=\sum wH$).

Proved here Proof. Relabelling gives $\mathbb E[g]=\mathbb E[(g\circ\tau)\,\rho]$ for any $g$. Since $\Delta S\circ\tau=-\Delta S$, taking $g=\mathbf 1\{\Delta S=s\}$ gives $g\circ\tau=\mathbf 1\{\Delta S=-s\}$, and on the set $\{\Delta S=-s\}$ we have $\rho=e^{-\Delta S}=e^{s}$, which gives the first formula. Pairing $s$ with $-s$ gives $\mathbb P(s)\tanh\frac s2+\mathbb P(-s)\tanh\frac{-s}2=\mathbb P(-s)(e^{s}-1)\tanh\frac s2\ge0$, so $\mathbb E[t]\ge0$. Finally $1-\tanh(\Delta S/2)=2/(1+e^{\Delta S})$. ∎ This lemma has no Lean check.

How to read (6). A Barker-type move is a Monte Carlo move that "accepts the configuration with a vortex inserted with probability $w(\tau x)/(w(x)+w(\tau x))$". If insertions are almost always refused (the cost is high), $\mathbb E[t]\approx1$; if they pass often, $\mathbb E[t]$ moves away from 1. The first formula has the same form as what non-equilibrium statistical mechanics calls Crooks' fluctuation relation (comparison with the original source, and the version driven by dynamics, are in 12-06).

Many cells and the average over positions. For each of mutually separated cells $c$, take insertion involutions $\tau_{c,p}$ at positions $p\in P$ in the cell ($P$ a non-empty finite set), build $t_{c,p}$ by (5), and set $\bar t_c=\frac1{|P|}\sum_pt_{c,p}$.

Proved here Proof. First, for $\prod_ct_{c,p_c}$ with one position chosen from each cell, an induction adding cells one at a time gives $\mathbb E[H\prod_ct_{c,p_c}]=\mathbb E[H]$: at the step adding cell $d$, the product so far $G=\prod_{c\ne d}t_{c,p_c}$ does not move under $\tau_{d,p_d}$, so $HG$ is odd and Identity 3 applies. Expanding $\prod_c\bar t_c$ gives the average over the $|P|^{\#\text{cells}}$ choices, hence the first formula. Taking absolute values with $|H|\le1$ gives the second. ∎ Each factor lies in $[-1,1]$, and no moment assumptions are needed.

Lean BoundedInsertion (Bounded.averaged, Bounded.averaged_bound) (hypotheses: $X$, the set of cells and the set of positions $P$ are finite, $P$ is non-empty, every $\tau_{c,p}$ is an involution, $w\gt0$, $H$ is odd under every $\tau_{c,p}$, $t_{d,q}\circ\tau_{c,p}=t_{d,q}$ if $c\ne d$, and for the latter also $|H|\le1$. Conclusion: (7)). One step of the induction is BoundedInsertion (Bounded.multi).

The tanh form is bounded, but it does not capture the exponential average (the free energy) as it is. Another bounded form is $\sqrt\rho$.

Proved here Proof. $w\sqrt\rho=\sqrt{w\cdot w\circ\tau}$ is symmetric under $x\leftrightarrow\tau x$ and $H$ is antisymmetric, so $\sum_xw(x)H(x)\sqrt{\rho(x)}$ changes sign when relabelled by $\tau$, and is 0. ∎

Lean SecondMoment (SecondMoment.sqrt_identity) (hypotheses: $X$ is finite, $\tau$ is a bijection with $\tau\circ\tau=\mathrm{id}$, $w\gt0$, $H\circ\tau=-H$. Conclusion: $\sum wH\sqrt{w\circ\tau/w}=0$).

Proved here Proof. Averaging Identity 6 gives $\mathbb E[HY]=0$. Setting $C=\mathbb E[Y^2]$, $B=\mathbb E[Y]$ gives $\mathbb E[H(C-BY)]=C\,\mathbb E[H]$. From Cauchy–Schwarz and $H^2\le1$, $C^2\mathbb E[H]^2\le\mathbb E[(C-BY)^2]=C^2-2CB^2+B^2C=C(C-B^2)$. Dividing by $C\gt0$ ($\rho_p\gt0$) gives the first formula. The last inequality follows from $Y^2\le\frac1{|P|}\sum_p\rho_p$ (Cauchy–Schwarz) and, for involutions, $\mathbb E[\rho_p]=\sum_xw(\tau_px)/\sum w=1$. ∎

Lean SecondMoment (SecondMoment.pz_bound) (hypotheses: $X$ and $P$ are finite, $P$ is non-empty, every $\tau_p$ is an involution, $w\gt0$, $H$ is odd under every $\tau_p$, $|H|\le1$. Conclusion: $(\sum wH)^2(\sum wY^2)\le(\sum w)\bigl((\sum wY^2)(\sum w)-(\sum wY)^2\bigr)$, which normalised is the first formula of (8)). $\mathbb E[Y^2]\le1$ has no Lean check.

(8) says that the more concentrated $Y$ is in the sense of second moments (the closer $\mathbb E[Y^2]$ is to $\mathbb E[Y]^2$), the smaller the bias. The second moment is always at most 1, so there is no worry about heavy tails as in the variance form of $\rho$.

Many cells. We also place a form that handles products of insertion ratios over cells.

Proved here Proof. Expand $\prod_c(1-r_c)=\sum_{S\subseteq C}(-1)^{|S|}\prod_{c\in S}r_c$; using the assumption each term becomes $\mathbb E[H]$, and there are $2^{|C|}$ terms. Cauchy–Schwarz and $H^2\le1$ give the second formula. ∎

Lean MultiCell (MultiCell.bound) (hypotheses: $X$ and $C$ are finite, $w\gt0$, $|H|\le1$, the equalities of "oddness under insertion" above for every $S\subseteq C$. Conclusion: the second formula of (9)) and MultiCell (MultiCell.expand_identity) (the first formula). Within Lean, "oddness under insertion" is an equality placed as a hypothesis. For a single bijection it follows from the pushforward equality $\sum w\,(w\circ T/w)\,F=\sum w\,F\circ T^{-1}$ (MultiCell (MultiCell.pushforward), hypotheses: finite, $w\gt0$), but that the equality holds when insertions in separate cells are composed depends on the commutativity of the maps and the invariance of the ratios, and is outside Lean.

Take the thin insertion of 12-04. Let the cells be the plaquettes $p\in S(C)$ of the plane inside the loop, with one position ($|P|=1$), and let $\tau_p$ be the flip of $z$ on the vortex sheet $S_p$. $\tau_p$ is itself an involution, and since it does not change the links it preserves the Haar × counting measure. $H=W(C)$ is odd under each $\tau_p$, and $t_p=(1-r_p)/(1+r_p)$ is determined only by the values on $S_p$, so it does not move under $\tau_q$ for $q\ne p$. From Theorem 5,

($|t_p|\le\tanh(\beta N^2)$ is the one-sheet inequality of 12-04) Proved here. This is exactly (4) of step 4 of 12-04, and the projection to a fixed sector is the same as step 5 of 12-04. The slab-confinement theorem is the special case "thin insertion, one position" of Theorem 5. (10) on the continuous configuration space is the same argument with sums replaced by integrals, and is outside Lean.

The average over positions is entropy. The $\mathrm{Var}(\bar\rho)$ of Theorem 2 is expected to be of order $\approx VK/n$, where $K$ is the number of positions, among the $n$, within reach of each other's correlations and $V$ is the variance of one insertion ratio Reading (physics). Increasing the number of positions $n$ plays the role of entropy, and the decay of correlations between positions (the mass gap) plays the role of keeping $K$ small. For insertions of the smooth fractional-instanton type, $V\sim e^{S_{\rm inst}}$ is expected, the area of the region needed is proportional to $V$, and the string tension is of order $\sigma\sim e^{-S_{\rm inst}}/L^2$, the same form as the formula for a dilute vortex gas Reading (physics, a semiclassical estimate; not a proof).

Collapsing onto a chessboard. If there is reflection positivity in the directions of the plane, chessboard estimates collapse things into a single translation-invariant quantity, $\mathbb E[\prod_{c\in\text{inside}}|\bar t_c|]\le\Lambda^{n}$ with $\Lambda:=(\mathbb E[\prod_{\text{all cells}}|\bar t_c|])^{1/\#\text{cells}}$, and if $\Lambda\lt1$ an area law follows; this is a strand Assumption (where the evidence is: the treatment of reflection positivity is in 13-04 and 13-05. The same structure is in (13)(21) of Kovács–Tomboulis Cited).

What is missing. (i) The large-deviation stage: $\Lambda\lt1$ needs the event that insertions are often refused in all cells at once to be exponentially rare in the number of cells. (ii) The ultraviolet stage: the lower bound on the acceptance rate must be uniform in the lattice spacing, and with thin flips (12-04) it gets worse with $N$. With smooth insertions, it is known that fixed maps pay an extra cost in the ultraviolet (12-07). (iii) An upper bound on second moments: an upper bound on the $V$ of Theorem 2. This book shows none of the three.

A negative control. Four-dimensional $U(1)$ does not confine at weak coupling, so if the bounds of this chapter are applied there, they must stop working somewhere (for example, $V$ becoming large). This check has not yet been done in this book.

(a) Confinement. None of the identities and inequalities of this chapter implies confinement. (i)–(iii) of §07 remain.

(b) The continuous configuration space. All the theorems are finite versions under (F)(W)(T); the version on the Haar measure of $SU(2)$ is the same argument with the measure preservation of the maps (or the Jacobian) added, but it is outside Lean.

(c) The hypothesis of Theorem 8. The equalities of "oddness under insertion" are not shown in this chapter for maps composed of insertions in separate cells (in Lean too they are hypotheses).

(d) Novelty. All are elementary combinations of relabelling of variables and Cauchy–Schwarz, and comparison with earlier literature is not done. The summed ensemble and the container identity lie within the framework of Mack–Petkova, the form bounding Wilson loops by twisted partition functions within Tomboulis–Yaffe, and products on a chessboard within Kovács–Tomboulis. The combination of insertion averaged over positions with Cauchy–Schwarz is recorded as a standard technique in proofs of Mermin–Wagner type, and is said to appear in gauge theory in Chatterjee (arXiv:2602.00436) as well (not yet checked). No novelty is claimed for the results of this chapter.

(e) The estimates of §07. $\mathrm{Var}(\bar\rho)\approx VK/n$, $V\sim e^{S_{\rm inst}}$ and $\sigma\sim e^{-S_{\rm inst}}/L^2$ are physical estimates, and none is a proof.

(f) The negative control. That the bounds stop working for four-dimensional $U(1)$ has not yet been checked.

The bundles in the machine-checking column are those confirmed, by re-checking the main text, to contain no sorry and to use axioms forming a subset of the three standard ones. What is guaranteed is only the algebra of sums over finite sets (10-13). In every chapter Lean names are written in the form "file (namespace.theorem)". The theorems of the namespace Insertion are in the file InsertionIdentity, and those of Bounded in the file BoundedInsertion.

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

[01] PRV MEAN   E[ρ] = 1, E[ρσ] = −E[σ]                          (F)(W)(T)
[02] PRV BIAS   4E[σ]² ≤ Var(ρ̄) ⟸ [01]                          Lean sector_bias_le
[03] PRV INV    the oriented τ is an involution                  (T)
[04] PRV TANH   E[H] = E[H t] ⟸ [03]                             Lean Bounded.single
[05] PRV CROOKS fluctuation relation, acceptance rate ⟸ [03]
[06] PRV AVG    average over positions (7) ⟸ [04]                Lean averaged_bound
[07] PRV SQRT   E[H√ρ] = 0 ⟸ [03]                                Lean sqrt_identity
[08] PRV PZ     Paley–Zygmund (8) ⟸ [07]                         Lean pz_bound
[09] PRV Y2     E[Y²] ≤ 1 ⟸ [01][03]
[10] PRV MULTI  many cells (9) (oddness under insertion assumed) Lean MultiCell.bound
[11] PRV SLAB   slab confinement via thin insertion ⟸ [06]       slab, N fixed
[12] RDG HEUR   VK/n, e^(−S_inst)/L² (physics) ⟸ [02]
[13] CIT KT     product structure of the chessboard
[14] ASM CHESS  collapse to Λ; area law if Λ < 1 ⟸ [06][13]     under reflection positivity

The way to check is to see by eye that every number to the right of "⟸" is smaller than the number on the left. There are no circles. The only assumption row is [14], and it enters as a premise of no row. The "oddness under insertion" of [10] is a hypothesis within the statement of the theorem, and does not come from another row. This chapter has neither a node of confinement nor a node of the mass gap.

What this chapter can say

Sources and reproduction

Words used in this chapter — insertion map, the insertion ratio $\rho$, the involution oriented by the sector $\tau$, the tanh factor $t$, acceptance rate, the average over positions, the $\sqrt\rho$ form, the Paley–Zygmund form, oddness under insertion. Their meanings are in A-02 Glossary (the words of Part 12 are to be added).

Next chapter to read: 12-06 Driven insertion and the flow lemma — if, instead of applying the map at once, the vortex is inserted slowly by heat-bath dynamics, Identity 3 takes the form of the work $W$.

← 12-04 Slab confinementContents12-06 Driven insertion and the flow lemma →