2026-10-01 · chapter Yang–Mills and the mass gapAppendicesindex of assumptions

Index of assumptions — name, one-line meaning, chapter where it is set and status, in one table

Part 10 proceeds in this order: divide the problem into statements, write the statements with assumptions and quantities, and confirm in Lean the algebra by which conclusions follow from assumptions. Along the way nearly twenty named assumptions appear. This appendix is an index that lines them up in one place. It does not give derivations of formulas or the grounds for numbers; it links to the master chapter (the chapter that defined the symbol).

The rows of the tables are divided into four kinds.

• Assumption: a statement set down without a proof. Assumption H, the chains, $T(c,1)$, (P) and so on.
• Statement: a part into which the problem is divided, a statement whose truth is in question. (S1)–(S8).
• Quantity: a number fixed by a definition. $\kappa$, $\delta(b)$, $\theta$, $\eta_{\rm dyn}$ and so on. A quantity itself is not an assumption, but statements such as "this quantity is 0" or "this sum is finite" are assumptions.
• Convention: a choice of symbols, such as the normalisation of $\mathcal E_\vartheta$ and $\mathcal E_A$. It has no truth value, but it can change the truth of other statements.

The status column uses the labels of this book. Known is a known theorem or coefficient; Lean means the algebra "a conclusion follows from this assumption" is machine-checked; Paper is a proof on paper; Computed is a number confirmed on this machine; Physics is something accepted at the level of physics but not a theorem. Even if an assumption row carries a Lean label, it does not mean the assumption itself has been shown. The label shows that there is a theorem that uses the assumption of that row.

Example. The row "H↑" reads "Assumption" and "Lean (hup_is_gap_positivity)". The reading is: H↑ ($a^2C_P\le C$, that is, $C_P\lesssim\xi_{\rm RG}^2$) is not proved, but the algebra that, assuming H↑ and the equality of chain (i) $C_P=1/m^2$, gives a lower bound on the mass in physical units $m_{\rm phys}=m\,\xi_{\rm RG}$ has been checked.

The master copy of Assumption H is 10-01 The line of construction and Assumption H. Its content can be said in one sentence: for the measure of lattice Yang–Mills, the time over which fluctuations are smoothed out (the Poincaré constant $C_P$) grows with the same power as the square of the correlation length $\xi$. The metaphor is the "smearing time ≍ ruler²" of 10-00 Entrance — the staircase, and the random walk, the ruler and the clock. Precisely, it is a statement about the limit of the exponent of the power.

Here $\beta=4/g^2$ (SU(2); $\beta=2N/g^2$ in the table of notation), $C_P$ is the Poincaré constant of the Wilson measure as defined in 1-10 §07, and $\xi$ is the correlation length. The power statement does not change when $C_P$ is multiplied by a power of $\beta$ (under $\ln\beta/\ln\xi\to0$; with the one-loop $\xi\approx e^{3\pi^2\beta/11}$ this holds Physics). On the other hand, the stronger statement that they agree within a band of constant multiples changes its truth depending on the graduation in which $C_P$ is measured. So this book calls the band statement by a different name, with the normalisation named.

$A\asymp B$ means that the ratio is bounded above and below by positive constants. (2) is the same as $C_P\asymp\xi^2$. The band statement $a^2C_P\asymp1$ with the bookkeeping length $\xi_{\rm RG}=1/a$ as the length of comparison (the last row of Table 1), split into its upper and lower halves, gives H↑ and H↓ (10-01 The line of construction and Assumption H). The halves of (2), written with $\xi$, are distinguished by adding the normalisation: H↑[$\mathcal E$], H↓[$\mathcal E$].

The distinction between H↑ and H↓ is also a distinction of quantifiers. $C_P$ is a supremum of a ratio, so for H↓ it is enough to show one test function, while H↑ is an inequality over all functions (10-01 The line of construction and Assumption H). That a proof of the mass gap is a proof of a lower bound is reflected on the side of H↑.

Example. For the free field in the field normalisation, $C_P=\xi^2$, and H, $\text{H}_{\rm band}[\mathcal E_A]$, H↑[$\mathcal E_A$] and H↓[$\mathcal E_A$] all hold. Measuring the same free field in link angles gives $C_P=(g^2/4)\xi^2$; H holds, but the lower half of $\text{H}_{\rm band}[\mathcal E_\vartheta]$ breaks (8-05 §03).

The statement "$C_P$ agrees with the square of a length" involves three choices: whether to say it by powers or by a band; whether the length of comparison is the correlation length $\xi$ or the bookkeeping length $\xi_{\rm RG}$; and whether $C_P$ is measured with the angle $\mathcal E_\vartheta$ or the field $\mathcal E_A=\mathcal E_\vartheta/\beta$. With $2\times2\times2$ there are eight readings. The following table is the master copy.

The derivation is in 10-01 The line of construction and Assumption H and 4-06 §05; here only the sum is shown (this passage may be skipped). The fractions in the right-hand column are sums of three logarithms: the $-1$ of the normalisation ($C_P^{(\vartheta)}=C_P^{(A)}/\beta$), the $+13/22$ of the renormalisation of stochastic time (10-01 The line of construction and Assumption H), and, entering only the rows with $\xi_{\rm RG}$, the difference between one-loop bookkeeping and two-loop running, $-2\cdot\tfrac{51}{121}=-\tfrac{102}{121}$ ($51/121=b_1/(2b_0^2)$, independent of $N$ for pure SU($N$); 4-06 §05). All are accounts under the renormalisation group of one-loop perturbation theory, and carry the Physics label.

We confirmed this sum on this machine in exact rationals Computed. The following figure draws, under this account, how the ratio of the band readings $C_P/\xi^2$ and the exponent of the power readings $\ln C_P/\ln\xi$ move. $\beta$ and $\xi$ are tied by the one-loop formula $\xi\approx e^{3\pi^2\beta/11}$ (up to a constant factor, 4-07).

When JavaScript is disabled, it can be read in Table 3.

There are two things to read. In the ratio columns, all four rows keep moving away from 1, and no band reading closes with a uniform constant. In the exponent columns, all four rows approach 2, and the power reading does not move. But the approach is slow: even at $\xi=10^{48}$ the exponent is 0.008–0.042 away from 2. The last row is the same formula as Table 9 of A-01; the exponents agree, and the ratios agree when the values of A-01 are divided by the value at $\xi=10^2$. This table does not measure the $C_P$ of the Yang–Mills measure; it is an example that lines up the consequences of the account. The difference between the two kinds of column in this table is the reason this book placed H at the level of powers.

As names multiply, one wonders which is stronger than which. We list four relations that follow from the definitions by short arguments. All presuppose that $\xi\to\infty$ as $\beta\to\infty$ Paper.

One: a band contains the power. If $c\xi^2\le C_P\le C\xi^2$, then taking logarithms and dividing by $\ln\xi$ gives

and H follows. H follows both from $\text{H}_{\rm band}[\mathcal E_\vartheta]$ and from $\text{H}_{\rm band}[\mathcal E_A]$. The converse does not. $C_P^{(A)}=\xi^2\sqrt{\ln\xi}$ satisfies H but lies in the band of neither normalisation. Under the one-loop $\ln\xi\asymp\beta$, $C_P^{(A)}/\xi^2\to\infty$ and $C_P^{(\vartheta)}/\xi^2=C_P^{(A)}/(\beta\xi^2)\propto1/\sqrt\beta\to0$. As for $\xi^2\ln\xi$, it leaves the band of that normalisation, but lies in the band of the other normalisation (for breaking of the band in the same normalisation, see 8-05 §04).

Two: the two bands are incompatible. From the relation between normalisations $C_P^{(A)}=\beta\,C_P^{(\vartheta)}$, if both held, then $\beta=C_P^{(A)}/C_P^{(\vartheta)}\le C_A/c_\vartheta$ and $\beta$ would be bounded. This contradicts $\beta\to\infty$. Writing "$C_P\asymp\xi^2$" without naming the normalisation leaves undecided which of two mutually exclusive statements is meant.

Three: the directions of the halves. If $\beta\ge1$, then $C_P^{(\vartheta)}\le C_P^{(A)}$. Hence

$C_P$ is smaller when measured in angles, so an upper bound is harder to state in the field normalisation, and a lower bound harder in the angle normalisation. In the one-loop column of Table 2, that H↓[$\mathcal E_\vartheta$] breaks for the angle band and H↑[$\mathcal E_A$] for the field band is consistent with these directions.

Four: bands and the bookkeeping length. $a^2C_P\asymp1$ is a statement that follows from $\text{H}_{\rm band}$ and chains (ii)(iii) ($a\xi\asymp1$). It splits as $a^2C_P=(a\xi)^2\cdot C_P/\xi^2$, so it follows when both hold (10-01 The line of construction and Assumption H). The converse does not follow. If the factors cancel each other, the product can lie in the band even if one of them breaks (7-12 Figure 2).

A note on reading: 10-02 §06. The H in the bracket "under H and D2" of (8) in 10-02 Roads tried on the four-dimensional lattice, and walls refers to the band statement $a^2C_P\asymp1$ (in the link-angle normalisation) set out in the same paragraph. Under this book's naming convention (A-06 Index of assumptions), we read it as the statement combining $\text{H}_{\rm band}[\mathcal E_\vartheta]$ and chains (ii)(iii) (the row band, $\xi_{\rm RG}$, $\mathcal E_\vartheta$ of Table 2). The power form of H is, by item one, a weaker assumption that follows from $\text{H}_{\rm band}$.

Example. For the free field in the field normalisation, $\text{H}_{\rm band}[\mathcal E_A]$, H↑[$\mathcal E_A$] and H↓[$\mathcal E_A$] hold with equality; in the angle normalisation, by item one, only H remains. The band statement written with $\xi_{\rm RG}$ is decided by the order of the bookkeeping even in the free approximation, so it is not included here (Table 2).

Detour: assumptions are arguments of theorems

Splitting $a^2C_P$ into a product of three ratios gives three chains, each saying that a factor is bounded by constant multiples (the master copy is 7-12 §05; the split into halves is in 10-01 The line of construction and Assumption H).

$\sigma$ is the string tension, $m$ the lightest mass in lattice units, and $\lambda_1=1/C_P$ the gap of Langevin dynamics. H↑ needs three lower bounds (relaxation not too slow, glueballs not too light, confinement not too weak), and the record moved what lies beyond into two further assumptions: the upper half of (ii) into the tube inequality $T(c,1)$, and H↑ into the projection assumption (P) (10-02).

Every arrow in Table 4 points "assumption ⟹ conclusion", and there is no row that shows a chain, $T(c,1)$ or (P) themselves. In the words of the record, directions 15 and 15′ stopped by naming "where the difficulty moved" (10-02 Roads tried on the four-dimensional lattice, and walls).

Example. The upper half of (ii) is the same as $m\ge2\sqrt{c\sigma}$. Inside the leading strong-coupling term $m=4\sigma$, $\sigma$ is large and this inequality actually holds. Extending that leading-term relation down to $\sigma\to0$ gives $m/\sqrt\sigma=4\sqrt\sigma\to0$, and no positive $c$ uniform in $\beta$ can be taken. However, this extension lies outside the region of convergence of the expansion (7-12 §08).

(S1)–(S8) are not assumptions but the parts into which the requirements of the problem statement are divided along the road entering from the lattice. The master copy is 10-01 The line of construction and Assumption H, and the connections by arrows are in 10-01 The line of construction and Assumption H. Only (S5) is the record's formulation, and differs in character from the other seven.

(7) is an overall picture, not a theorem. (S5) does not enter (7); it is placed as a candidate restatement of (S4).

The "non-trivial" of (S8) and H↓ are different statements. H↓ is that the mass in physical units stays finite, and it is satisfied even by a free massive field (10-01 The line of construction and Assumption H). Also, the arrow from (S5) to (S4) needs $\text{H}_{\rm band}$. H↓ can be read as the finiteness of the mass in physical units because H↓ is a statement written with $\xi_{\rm RG}$ ($m_{\rm phys}=m\,\xi_{\rm RG}$). For the free field the two coincide (10-01 The line of construction and Assumption H), but in an interacting theory they are statements about different quantities.

Example. "The gap in lattice units is at least a fixed value for every $\beta$" is not (S4). In physical units it is $\Delta\ge c/a\to\infty$, the opposite side of the continuum limit (10-01 The line of construction and Assumption H). This direction is known, and is not counted as a result.

The record kept the books on one step that coarsens the lattice by a factor $b$ with the following map (10-02 Roads tried on the four-dimensional lattice, and walls)).

The two numbers appearing in the left formula and the two exponents appearing in the right formula are the quantities of this section. The quantities are not assumptions, but statements such as "$\sum\delta\lt\infty$" and "$\eta_{\rm dyn}=0$" are assumptions, and the record's "(S4) ⟺ $\sum\delta\lt\infty$ and $\eta=0$" is the algebra of $p=0$, presupposing the power form and the model of the one-loop kernel.

The symbol $\eta$ needs care. The record's $\eta$ originally referred to two quantities. This book fixes the $\eta$ of the power form and of the equivalence with (S4) as $\eta_{\rm dyn}$, and uses $\eta_s$ only for lower bounds from test functions. An $\eta$ without qualification is $\eta_{\rm dyn}$.

Example. Fixing $b=2$ gives $\delta(2)=0.1335$ (weight $w$), and $\theta=\delta(2)/\ln2\approx0.19$ (10-02 Roads tried on the four-dimensional lattice, and walls). Even with $\eta_{\rm dyn}=0$, $p=2\theta\approx0.39\gt0$, and it falls on the side of $a^2C_P\to0$. Making it larger step by step, as $b_k=k+2$, gives $\sum c_2/(k+2)^2\lt\infty$, and $\theta=0$ can be restored.

There are rows that have no truth value yet change the truth of other statements: the normalisation of $C_P$. The table of notation A-01 §08 and 10-01 The line of construction and Assumption H are the master copies.

$\vartheta$ is the link angle ($U=e^{i\vartheta\cdot\sigma}$), and $A$ the field. Multiplying the Dirichlet form by $1/\beta$ multiplies the best constant by $\beta$. The power statement does not move; the band statement does (§03, §04).

For the free field, $C_P^{(\beta)}=4\,(C_P^{(\alpha)})^2$, and since Assumption H is a statement about (β), it can be reread as "the $C_P$ of the path agrees with the square of the $C_P$ of the vacuum" Computed (10-01 The line of construction and Assumption H). In gauge theory the measure at one time lives on the space of gauge orbits, and this rereading is an analogy from the form of the free field.

Example. The $C_P=1/m^2$ assumed by Lean's hup_is_gap_positivity is the value for the free field in the field normalisation. So this theorem is naturally read in the graduation of $\mathcal E_A$ (10-13 Lean statements and the limits of machine checking).

The rows of Tables 1–7 are not isolated. Finally we give a diagram linking rows by arrows. An arrow points in the direction in which the source is used (together with the sources of other arrows entering the same target) to obtain the target. The target does not necessarily follow from the source alone; what is needed together is written in each row of Table 8. The kinds are distinguished by the form of the line. A solid line is algebra checked in Lean, a thin line a relation that follows from definitions, a dashed line an arrow that stands only under another assumption or model, a dash-dot line a premise (not an implication, but something needed to state the target), and a dotted line an incompatible pair. Every arrow in the diagram is "assumption ⟹ conclusion", and no box in the diagram is proved, except (S2) and (S1) for small $\beta$.

When JavaScript is disabled, it can be read in Table 8.

Looking over the diagram, the solid Lean lines all run from bottom to top, from assumptions towards H↑ and (S5), and not a single solid line enters the bottom boxes (the chains, $T(c,1)$, (P)). The arrows entering (S4) at the very top are also only conditional ones or premises. This is what "the algebra closed in Lean tidies up the form of the assumptions, and does not reach the side of the mass gap" looks like on the diagram. An estimate of the distance to a theorem is in 10-15, and the list of what this book does not claim is in 10-16.

The assumptions set by the assembly of the first step of Part 10 (the double well in 0+1 dimensions) are listed, numbered, as rows among the 117 rows of the integrated table (Table 1 of 10-12), classified by the five grades (proved here, cited, computer-assisted, reading, assumption). Of these, 30 rows are "assumption"; which of them are needed for the road of the core on the box side (A) and which for the road of the core on the sector side (B), and the work needed to close them, are in the table of what remains (Table 1 of 10-15). Here we do not copy the rows, but list only the main ones, one line each. The numbers are those of the integrated table.

The table of assumptions in Part 10 makes no "standard" category, and Lean is a column outside the classification. Each proposition comes with a dependency graph, and by the rule "premise numbers are smaller than the conclusion" readers can check by eye that there is no circle (10-05 §07, 10-12).

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