2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircaselattice directions G0–G15′, lattice walls GA1–GE4 · Kiichi, Shiori and Rin

Roads tried on the four-dimensional lattice, and walls — they reach from below, and not from above

The roads and walls of this chapter are attempts to push one of the arrows of the line of construction in 10-01 §08. Rewritten with the band statement split into its upper and lower halves, it has the following form.

$\delta(b)$ is the shortfall of one step (§02), (S5) is $a^2C_P\asymp1$, (S4) the gap in physical units, and the properties of the limit are (S1)(S2)(S3)(S6)(S7)(S8) (10-01 Table 1). H↓ is $C_P\gtrsim\xi_{\rm RG}^2$ (one test function suffices), H↑ is $C_P\lesssim\xi_{\rm RG}^2$ (an inequality over all test functions). None of the arrows of (1) is proved.

A road (a lattice direction) is the pair of one question and one set of tools applied to it. The symbols are G0–G15′; G9′ and G15′ take up the original question again. A wall (a lattice wall) is a one-sentence statement of why a certain road, with certain tools, stopped short of a certain arrow; in the symbols GA1–GE4 the letter gives the type (§06). They are the numbers of the "directions 0–15′" and "walls A1–E4" of the earlier version of Part 10, with G added.

Example: G12 (a test function giving H↓) pushed H↓ of (1), and stopped short of H↑ at GA1 (test functions bound $C_P$ only from below). G0 (strong-coupling curvature) is not on (1); it sits outside to the left. Uniform estimates at strong coupling are $C_P=O(1)$, the opposite side of (S5) (10-01 §06).

One step is the map that coarsens the lattice spacing from $a$ to $ba$, integrates out some of the fine variables, and makes the action of the coarse lattice (6-05). The bookkeeping here takes three moves: (i) choose links that can be integrated in closed form, (ii) write the remaining action with characters, (iii) read the coefficient of the fundamental representation of the coarse plaquette as $\beta'$. It is of the "decimation" type, keeping some of the original links (Migdal–Kadanoff is an approximation of the same type). The area law, dimension counting and the one-loop coefficient are known rewritings, not counted as results.

Complete families. If the links to be integrated never lie on the same plaquette, then with the rest held fixed the weight of each link separates independently and can be integrated. A family that removes exactly one link from every plaquette is called a complete family.

Proved here An edge lies on $2(d-1)$ plaquettes, so counting the pairs $(P,\ell\in P\cap F)$ in two ways gives the density of the family $|F|/|E|=1/4$. Two perpendicular edges at the same vertex are two adjacent edges of one plaquette, so the number $f_x$ of edges of the family at a vertex $x$ is at most the two along one axis, and comparing with the average $d/2$ gives $d\le4$. The classification including existence is Table 1; the statements are LeanPerfectFamily.perfectK_one_iff (assumption: $2\le d$. Conclusion: the box $(\mathbb Z/L)^d$ of period $L$ has a complete family of edges $\iff d\le4$ and $L$ is even) and no_perfectZ_of_five (assumption: $d\ge5$. There is none on $\mathbb Z^d$). No prior record was found in what we searched, but it may be known. This 4 is a combinatorial number of the lattice; its agreement with the four dimensions of spacetime is a coincidence with no relation.

One-link integration. The 6 plaquettes on which an edge $\ell$ of the family lies contain no other edges of the family, so $U_\ell$ pairs only with the sum $M$ of staples (the products of the three edges of a plaquette other than $\ell$), and the integral is determined by $|M|$ alone.

The left is the standard formula of the $SU(2)$ heat-bath method Cited (5-07); the two-sided estimate on the right is Leanlog_F0_le, log_F0_ge (assumption: only that the argument is non-negative). The conditional density of the remaining edges is a product of six sources and does not close into the one-link type, so this trick works only once Assumption (there is a paper argument, but this book gives no proof).

The deficit of bookkeeping. After integrating a complete family, not a single "intact" plaquette with all 4 edges remaining is left, and the coupling can only be reborn from broken plaquettes. For the fraction $e$ of remaining edges and the fraction $q$ of intact plaquettes, counting pairs gives

Proved here (double counting) LeanIncidence.intact_frac_le (assumption: the incidence is regular. Conclusion: $q\le e$). Stacking steps does not reach it: in a sequence of complete steps the remaining edges are $(2^k+1)/2^{k+1}\gt1/2$, no_blocking (assumption: a sequence of complete steps), and in any sequence removing at most 1/4 at each step, reducing the edges to $1/16$ leaves at most $1/65536$ of the 4-loops, blocking_shortfall (assumption: fraction $\le1/4$). The reading of this subsection is that the wall of bookkeeping was neither curvature nor packing, but the way of bookkeeping that counts only intact plaquettes.

Discs and a weak-coupling step. Integrating the interior of a disc-shaped set of $A$ plaquettes raises the coefficients of every channel of the character expansion to the power $A$: $u(\beta')=u(\beta)^A$, and for a $2\times2$ disc, $\beta'=\beta^4/64+O(\beta^6)$ Cited (a rewriting of the strong-coupling area law and two-dimensional decimation, 5-02, 6-02). It is a semigroup in the area, with $\beta'\lt\beta$ everywhere LeandiscStep_semigroup, step_lowers_coupling (assumption: surfaces of disc shape). In the Gaussian approximation at weak coupling, the $b^2$ plaquettes of a disc act in series and the $b^{d-2}$ transverse layers in parallel, giving

$\beta b^{d-4}$ is the dimension count of the coupling, marginal at $d=4$ Cited (4-04). The contribution of surfaces sharing no edges does not reach 1 LeankappaDisj_lt_one (assumption: $b\ge2$), and the bookkeeping of surfaces has the multiplicative form $\beta'=\kappa(b)\beta$, not the subtractive form of one loop $\beta'=\beta-\kappa\ln b$.

The one-loop slope and the shortfall. On top of the tree-level $\beta'=\beta$ sits the one-loop correction from the fast modes (momenta between $\pi/(ba)$ and $\pi/a$). Measured by the integral over a lattice window, it takes the form

$\kappa=4Nb_0=11/(3\pi^2)=0.3715110$ is the known one-loop coefficient Cited (4-06). The slope read from the difference of windows is 1.000028 at $b=32\to64$; multiplied by $\kappa$ it is 0.371521, agreeing with $11/(3\pi^2)$ to a relative $2.8\times10^{-5}$ Reading (an agreement inside the model of the one-loop kernel; not a non-perturbative check). The coefficients of the shortfall are $c_2=0.0197(5)$ (weight $w$) and $0.4200(5)$ (weight $w^2$), and at $b=2$, $\delta(2)=0.1335$ and $0.2284$ Reading (fits in the same model). On the other side, the effective slope $\Delta\beta/\ln(a_1/a_2)$ read from the actual lattice $a\sqrt\sigma$ (Table 2 of 10-01) is 0.304 over $\beta=2.25\to2.60$, reaching neither the one-loop 0.3715 nor the two-loop 0.397 Reading. Assuming the step-by-step form $\xi(\beta-\kappa\ln b)=\xi(\beta)/b$ for every $b\ge1$ gives $\xi(\beta)=\xi(0)e^{\beta/\kappa}$ Proved here Leanxi_eq_exp (assumption: that functional equation; neither continuity nor monotonicity is assumed). The physical content lies on the side of the assumed functional equation (the one-loop running put down as an equation).

The form of the answer. Stacking steps with $b$ fixed, the shortfall accumulates by the same amount each step, and if $\delta(b)\ne0$ the sum diverges. For a sequence with growing step scales $b_k=k+2$, $\sum_kc_2/(k+2)^2\lt\infty$ and it stays bounded Proved here Leanfixed_step_bounded_iff (assumption: a constant error $\varepsilon_j=d\ge0$. Conclusion: the partial sums are bounded $\iff d=0$). For example, with the provisional value $c_2=1$, fixing $b=2$ gives a total of 250 after 1000 steps, while with $b_j=j+2$ the total stays within $\pi^2/6-1=0.645$ (values of closed formulas). What ties this shortfall to the first arrow of (1) is the algebra of two powers.

$\theta$ is the exponent of accumulation of the shortfall (with $b$ fixed, $\delta(b)/\ln b$), and $\eta$ is the dynamical power $\eta_{\rm dyn}$. If $0\le\eta\le2$ and $\theta\ge0$, then $p=0\iff\eta=0\wedge\theta=0$ Proved here Leandrain_exponent, drainExp_eq_zero_iff (assumptions: the power form of (7) and the signs). In words, "under Assumption H ($\eta=0$), (S4) ⟺ $\sum_k\delta(b_k)\lt\infty$", which is the "form of the answer" of the earlier version. Besides H, this statement stands on the power form of (7), a model in which the errors of the steps are additive and bounded, and $\delta$ being the value of the model of the one-loop kernel Assumption. The algebra that if the partial sums of the errors of the steps are bounded then $a_k^2C_P^{(k)}$ stays in a positive band is also closed LeanS4_two_sided (assumptions: partial sums $|\sum_{j\lt k}\varepsilon_j|\le M$, positivity of coefficients. Reading this as $a_k^2C_P^{(k)}$ is on the side of the text).

A road that changes the kernel to make $\delta(2)=0$ exists in the one-parameter family $c_1(q)=\alpha+(1-\alpha)\cos q$ at $\alpha_0=1.4400$, but the neighbouring weight is negative Reading. For kernels with positive weights, $c_2\gt0$ Leanc2_ge_of_nonneg_weights (assumption: $c_2$ is an increasing affine function $A+\kappa S$ of the kernel quantity $S\ge0$, with $A,\kappa\gt0$, a measured dependence. Not an unconditional claim). Negative weights can collide with reflection positivity, so within the families examined, removing the shortfall and the condition for returning to a quantum theory are not compatible. Outside the families it is undecided.

Table 2 lines up the seventeen roads one row each. The symbols in the "Where it stopped" column are the walls of §06, and the "What remained" column is algebra closed in Lean (assumptions in brackets) or labelled statements.

Seen arrow by arrow, there is a bias. Stopped roads gather at H↓ and outside (1). Only two roads aimed at H↑, G10 and G15′; G10 stopped at the cross terms, and G15′ sandwiched only under (P). No road reached the recovery of rotational symmetry (S6) or non-triviality (S8). (S3)(S6) are claims of a kind that do not come out however many times the map of one step is composed (GB7).

G6, G12, G13 and G14 all aimed at estimates from below, and stopped at a place of the same form. $C_P$ is a supremum over test functions, and the gap is an infimum over states, so what one test function or one operator gives is in either case only one side.

Cited The variational principle (1-10). $\Omega$ is the vacuum, $H$ the Hamiltonian of the transfer matrix, and $m_{\rm eff}(k)=\ln(C(k)/C(k+1))$ the effective mass at time $k$. H↓ is a statement about "some $f$", the side test functions can reach; H↑ is about "every $f$", the side they cannot.

Local operators (G12). In the family made by translating one operator $O_x$ and summing with weights, the freedom of the weights collapses to a single best momentum LeanprofileQuotient_isGreatest (assumptions: finitely many momenta, the Fourier transform of the denominator is positive). The numerator is a susceptibility, and estimating it by the integral up to radius $\xi$,

(assuming a power form of the correlations). $\eta_s$ is the static power of each operator, a different quantity from the dynamical $\eta_{\rm dyn}$ (8-05). The lowest dimension of gauge-invariant local operators is the 4 of $\operatorname{tr}F^2$ Cited, so the ratio does not grow with $\xi$ Leaneta_ge_two_of_bounded_susc (assumption: the susceptibility is uniformly bounded; the boundedness comes from the physical assumptions of the form of correlations and $\Delta=4$). For bare link angles, under the measure without gauge fixing each link angle is uniform and independent, and the ratio is 1 independently of $\beta$ and the volume Proved here (a direct calculation from the invariance of the Haar measure; the same mechanism as Elitzur's theorem).

Gauge fixing (G13). Gauge-fixed long-wavelength modes are non-local and avoid the wall of dimension, but the choice of representative must be single-valued. Taking a weighted average at temperature $T$ over the orbit makes it single-valued and continuous LeanorbitAvg_gauge_invariant, orbitAvg_continuous. In the smallest example, the $U(1)$ ring ($L$ edges), the amplitude of the averaged mode vanishes exponentially with $L$ as $(I_1/I_0)^{L-1}$ for blunt fixing, and jumps at the total angle $\Phi=\pi$ for sharp fixing Proved here (closed formulas from Fourier series). The same functional serves both as "a tool for choosing among copies" and "a tool for evening out long wavelengths", so the two demands sit at the two ends of a single knob. The numbers measured for four-dimensional compact $U(1)$ (the scatter between copies being 3.25 standard deviations, and so on) are Reading; they were not measured for $SU(2)$.

Spectroscopy (G6, G14). Smoothing in the three spatial directions only stays on one time slice, and correlations keep the form $C(k)=\sum_nw_ne^{-E_nk}$ ($w_n\ge0$) Cited (5-03). With this form the effective mass decreases monotonically Leanmeff_antitone (assumption: a representation with non-negative weights), and at every $k$ it is an upper bound on the lowest energy meff_ge_of_corr. A lower bound needs a different input.

Proved here Leangap_ge_of_overlap (assumptions: a representation with non-negative weights and a lower bound $q$ on the overlap). The price of the lower bound is exactly one lower bound on the overlap; without $q$, a light state of weight $\varepsilon$ changes the correlation by only $\varepsilon$ at every $k$ light_admixture_invisible (a concrete two-level example). That no lower bound on the gap follows from finitely many measurements is a restatement of the known framework of the equivalence of a gap and exponential clustering Cited (4-11). On the numerical side, the effective mass of a smoothed operator at $\beta=2.6$, $12^4$ was $\le0.501\pm0.071$, touching the $am(0^{++})=0.514(8)$ of the original source from below Reading (Monte Carlo, reading of an upper bound). A four-dimensional flow that also smooths in the time direction does not stay on one slice, and breaks the upper-bound property with an effective mass that increases monotonically Proved here (closed formulas for the free field). The cost of measuring is $\ge cL^{d+2}$, from the relaxation time $\propto L^2$ of the softest mode in a box of side $L$ cost_ge (assumptions: a lower bound on the relaxation time, and the number of sweeps needed for relative precision).

The three roads reached the same wall. Each needs "the overlap of gauge-invariant operators with the lightest fluctuation is bounded from below", which includes "no other light state is hidden", a statement about all states. It has the same form as H↑.

H↑ is an upper bound on $C_P$. The types of tools that give upper bounds are limited; there are two representative ones.

Cited The left comes from a lower bound on the curvature of the measure $e^{-U}$ (1-10); the right sandwiches $C_P$ within a factor of 4 by the maximum $B$ of "the mass before a point × the resistance beyond it", which can be computed only in one dimension.

Curvature scale by scale (G0, G10). The known places where inequalities of the type of H↑ stand as theorems are only curvature, diameter, convexity, strong coupling, and the multiscale Bakry–Émery uniform in the lattice regularisation for the high-temperature phase of continuum scalar fields in two and three dimensions Cited (Theorem 1.1 of Bauerschmidt–Dagallier; the criterion is Theorem 2.5 of Bauerschmidt–Bodineau). For four-dimensional gauge fields, the curvature after one-link integration is positive only at strong coupling (G0). Examining the curvature on the Gribov region in a finite-dimensional model, it was positive only on the strong-coupling side, and the non-convexity came from the zero-momentum mode Reading (on samples with $L=2,3,4$). The ultraviolet fluctuations remaining after removing the zero modes cannot be overcome by one step of gradient flow: the flow is a bijection commuting with the gauge action, so it does not shrink the space of functions Leancomp_class_eq_of_equivariant_bijection; the only gain is in changing the metric, and the curvature of the pushed-forward measure became more deeply negative with flow time Reading (within the range measured: open box $\ell=3$, typical points at $\beta=2.3,4$, $t\le0.5$).

The deficit of tubes (G15). We moved the lower half of chain (ii), $m^2\gtrsim\sigma$, into an inequality between the transfer operator $\mathcal T$ of closed tubes and the perimeter operator $\hat\ell$.

Proved here (the right is the AM–GM inequality) Leanmass_sq_ge_of_tube (assumption: the norm of the transfer operator is at most $e^{-2\sqrt{c\sigma}}$. Conclusion: $m^2\ge4c\sigma$). The power of $1/\ell$ is the only power that makes the coefficient $c$ dimensionless. The content is on the side of $T(c,1)$, which itself has no type of proof. In the region where the strong-coupling surface expansion converges, the lower half of a single scale is itself false linear_regime_breaks_onescale (assumption: a sandwich in the region of convergence); in counting with the pillar model the deficit can have either sign, the Wulff shape is a rhombus, and $T(c,1)$ stands only after the anisotropy is subtracted linear_deficit_breaks_inv (negative control). The final form is three assumptions, "localised $T$ on short loops, a Casimir floor on large loops, and a seam between them" mass_bound_localized; the floor alone gives no lower bound casimir_only_no_bound.

A one-dimensional chain of scales (G15′). In holographic models the gap is an eigenvalue problem in a single radial variable, and on that variable the band statement $C_P\asymp\xi^2$ becomes an identity needing no proof Proved here (ground-state transformation $H=P^\dagger P$ and the variational principle). To bring this onto the lattice, we put a one-dimensional chain on the scale $s=\ln\ell$ and sandwiched $C_P$ with the discrete version of the right of (11).

Leanconditional_scale_bound (assumptions: projection (P), shortfalls non-negative with bounded partial sums, geometric-series weights; the Hardy sandwich for finite chains is a reproof of a known result). As a negative control, with a constant shortfall the boundedness breaks const_defect_scale_unbounded. The content is on the side of (P), and (P) splits into four, "the variance can be split scale by scale", "the conditional Poincaré constants of each stage are uniform", "the conductance across stages is bounded", "the state space is a single line of scales", of which the last three have not been shown Assumption. (P) is not weaker than H. The one-dimensional chain did not show H↑; it restated H↑ as (P).

Boundaries and the topological charge (G11, G15′). In a periodic box the state space splits into sheets by topological charge $Q$, connected only at the ultraviolet end. For reversible dynamics, for every observable

holds Cited Leantau_int_le, rayleigh_le_tau (finite reversible chains; a reproof of known facts), and every measured $\tau_{\rm int}$ is a lower bound on $C_P$. The original source reports for HMC-type dynamics an effective dynamical exponent of the square of the topological charge of "about 5" Reading (Schaefer–Sommer–Virotta 2011); combining this with the conjecture that HMC and Langevin are in the same universality class Assumption and (14), in a periodic box $C_P\gtrsim a^{-5}$ and H↑ breaks. It is a conditional reading on top of a conjecture. "About 5" is a value for $SU(3)$; the same measurement for $SU(2)$ was found in one source (HMC, periodic boundaries, small boxes, three points), reported to be close to $a^{-2}$ Reading (Abe–Morikawa–Suzuki 2025; the note in 10-01 §05). Open boundaries remove this barrier only inside a finite box with the physical side held fixed (a count using the charge of a subregion as test function Assumption; in the original source too, the autocorrelation time of $Q^2$ with open boundaries grows roughly in proportion to the length in the time direction Reading). In this book's order, which makes the volume infinite first, this is expected not to help, and the candidate direction is to rewrite the statement of Assumption H itself as a sentence sector by sector, or as a local sentence with support limited to a few correlation lengths (the note in 10-01 §05; how to rewrite it is dealt with in Part 11).

Thus every road on the upper side stopped by naming the place to which the difficulty had moved.

The arrows go only one way, from right to left, and every right-hand side is an assumption.

A wall is a one-sentence statement of why a certain road, with certain tools, stopped short of a certain arrow of (1). There are five types, given by letters: principle (GA), which comes from the direction of an inequality and does not depend on the details of the tool; structure (GB), where demands are not compatible within the chosen family, bookkeeping or series (outside the family it is undecided); tool (GC), where the type of inequality needed is known but there is no tool to show it; assumption (GD), the footing of conditional statements; measurement and literature (GE), what was not settled by numbers, boxes or primary sources. No wall is a proof that it "cannot be crossed".

Walls of principle (GA). GA1 and GA2 are the two one-sided halves of (8), and they do not move however many measurements are stacked. For example, for the standard normal distribution ($C_P=1$, with equality at $f=x$), the maximum ratio over a family of odd polynomials not containing $x$ approaches 1 as the family is enlarged, but the numbers inside the family alone do not tell whether 1 is the final value Reading (lower bound over a family). One knows it is 1 because of an input from outside the family ($C_P\le1$ from the convexity $V''=1$), and an input of this kind is what H↑ needs.

Walls of structure (GB). Incompatibilities inside the chosen family; they say nothing about outside it. GB1–GB3 are the ways the three roads of §04 stopped; GB4 is the tension between reflection positivity and removing the shortfall, the same tension, in the opposite direction, as what the problem statement says about continuum regularisations, "once reflection positivity is lost there are few ways to recover it". GB5–GB7 are limits of the form of bookkeeping and series.

Walls of tools (GC). They gather in the column of H↑. GC1–GC3 are the three places of §05 to which the difficulty moved; GC4 alone is not a mathematical obstacle but a lack of tools for checking. The $\beta\lt0.2722$ where the curvature theorem exists and the $\beta=2.60$ used by spectroscopy (correlation length 1.95 lattice spacings) are about 9.6 times apart, and an estimate crossing that interval uniformly is the content of GC1.

Walls of assumption (GD). The band statement $a^2C_P\asymp1$ splits into the product of three ratios $(a^2/\sigma)(\sigma/m^2)(m^2/\lambda_1)$, and the central equivalence also stands on footing other than H. Each assumption corresponds to a concept of Parts 1–8: H and chains (ii)(iii) are candidates for restating "a single scale" (4-07) in terms of the Poincaré constant; chain (i) is the dynamical exponent $z=2$ (8-03); chain (ii) is a single scale (7-12); chain (iii) is the specialisation of asymptotic scaling (5-05) to the bookkeeping length. These are analogies, not mathematical equivalences.

Walls of measurement and literature (GE). They move by computation or by obtaining sources, but even if they move, the logic of (1) does not change. The representative is the size of the box: $12\times0.15896=1.91$ ($\beta=2.55$) and $12\times0.13395=1.61$ ($\beta=2.60$), both small boxes that do not contain 3 physical lengths $1/\sqrt\sigma$ Reading.

The walls in the column of H↑ are GA1, GC1, GC3 and GD3, with no walls of measurement and literature (GE). That is, short of H↑ there is no wall that moves by measuring. Where the setters wrote that "new ideas are needed" is also the side of the gap uniform in the volume (Jaffe–Witten §6.5) Cited, the same place where the walls that measurement does not move gather.

Most of what remained from the seventeen roads is finite inequalities and algebra with the assumptions made explicit as arguments. Table 4 lines up the theorems cited in the text by what their statements are about. Inside Lean every theorem is a sentence about real numbers, sequences, or sums over finite sets; that they are quantities of the Wilson measure is a reading attached by the text. There are no theorems about expectation values, correlations or gaps of the Wilson measure (10-13).

The content of the conditional rows lies on the side of the assumptions, and the theorems do not take a single step towards showing those assumptions. Only the two rows of lattice theorems (the classification of complete families and the lower bound on the Hessian form of the effective action) are theorems on the lattice that assume only the shape of the lattice. The former is a combinatorial fact about the lattice; the latter is an estimate that works on the strong-coupling side, where the correlation length is about 1/10 of a lattice spacing. Both concern places far from the continuum limit.

What remained on the numerical side is readings Reading: the agreement of the slope in the model of the one-loop kernel with the known coefficient (relative $2.8\times10^{-5}$, inside the model), the values of $c_2$ (two weights and two gauges), the effective slope 0.304 read from the lattice $a\sqrt\sigma$ of the original source, the upper bound $0.501\pm0.071$ on the effective mass of smoothed operators ($\beta=2.6$, $12^4$), the scatter of gauge-fixed copies for compact $U(1)$, the strong-coupling range where the curvature of the Gribov region is positive (samples of a finite-dimensional model), and the curvature of the pushed-forward measure of the gradient flow (at the typical points measured). None of them is used in a proof of an upper or lower bound.

Summing up §04–§06: there is no tool to attack the upper side of Assumption H (H↑, every $f$) while staying on the four-dimensional lattice. The tools of the lower side (test functions, spectroscopy) stopped at walls of principle, and the tools of the upper side (curvature, tubes, one-dimensional chains) all stopped where they had moved the difficulty into a single assumption. Where each stopped is as in the following table, tool by tool. The right column is the prospect of what changes when the same tool is brought to the model of the first step (the path measure of the double well in 0+1 dimensions, 10-04).

Some things do not change. That H↑ is an inequality over "every $f$", and that tools of the lower side cannot reach it in principle (GA1), are the same on the first step. So what the first step does is to build the assembly of the upper side first, in the form of propositions with their assumption rows named. Assumption H is a sentence independent of dimension (10-01 §05), so we build the same sentence in the smallest model where the ruler grows exponentially in the parameter, the slow motion is the motion of "changing phase", and the quantities compared are known exactly.

Even if the first step is closed, there is no guarantee that it lifts to the four-dimensional lattice. What has a prospect of lifting is the form of the sentence and the form of the tools; constants and ranges of parameters do not lift (10-03 §06). The table of walls in this chapter is kept as a table for reading which walls the tools of the first step might hit again on the second step.

Assumption H is one restatement chosen as a handle, and the roads and walls of this chapter too are within the scope of a study with H at its axis. Viewpoints other than H are dealt with in Part 11.

What this chapter can say

Sources and reproduction

Words used in this chapter — one step of bookkeeping, complete family, one-link integration, the shortfall $\delta(b)$, the form of the answer, the lattice directions G0–G15′, the lattice walls GA1–GE4, the renewal form, a lower bound on the overlap, $T(c,1)$, the projection assumption (P), topological charge, negative control. Their meanings are in A-02 Glossary.

To the next chapter — 10-03 The four steps of the staircase — why we start from 0+1 dimensions: divides Assumption H into four steps, and fixes the conditions for choosing the double well for the first step and the form of the statement of the first step (the power form of H and Proposition 4.5).

← 10-01 The line of construction and Assumption HContents10-03 The four steps of the staircase — why we start from 0+1 dimensions →