2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircasepatterns of insight, discarded hypotheses, small questions · Kiichi, Shiori and Rin

The whole book　Contents and reading orders

Insights and small questions — views that remain as procedures, and questions that can be settled

What this chapter shows — The views that repeatedly proved useful while studying the first step, gathered into twelve insights in the form "any problem needs this procedure". For each insight, one small question the reader can check by hand. A table of discarded hypotheses and the reasons. A list of questions that can be settled at the size of an undergraduate thesis, and a worked example that takes one question to the form of a settlement.

Prerequisite chapters — The material for the insights is in the chapters 10-06 to 10-11. At the end of each insight we write the chapter where that view worked. The prerequisites of each question are written in its entry.

Said first — The level is [Y2]–[Y3]. What is listed here are insights and questions before verification, neither theorems nor results. Whichever question is settled, it does not reach a proof of the mass gap. For each question we wrote what does not move even when it is settled.

The order of this chapter

How to read the insights — three states, and the criterion for keeping one (Table 1)
Twelve insights and small questions — ways of paying, addresses, thresholds, directions, order
Discarded hypotheses — the original form, the corrected form, the grounds (Table 2)
Questions at the size of an undergraduate thesis — a list of seventeen questions (Table 3)
Worked example: one question to the form of a settlement — $1-\lambda_k$ for the Hermite family (Table 4)

How to read the insights — three states, and the criterion for keeping one

In the course of a study, "shifts in how things look" get written down apart from results. They fall into three states: still an insight (not yet tested), checked form (checked once on a small model or a closed formula), and overlaps the known (the core was a restatement of a known theorem; not counted as an insight).

There is one criterion for keeping something in this chapter. Not who thought of it or when, but only what can be written in a form that lets us say another reader who meets a problem of the same type would also need to take this procedure. For example, "before trimming an upper bound, sort the factors of the needed value into structure and tools" is a procedure that keeps its meaning away from the double well. Conversely, the observation that some number happened to lie near a threshold is not kept.

In the previous edition, Part 10 read the insights on the side of the four-dimensional lattice in seven patterns (cannot be measured from below / drop to one dimension / drop the units / counterexamples and controls first / look at the power of the difference / H is a pair of measure and form / connectedness of the state space). Table 1 shows those seven and how they relate to the twelve insights of this chapter.

Table 1 The seven patterns of the previous edition, and the insights of this chapter (numbers are the insights of §02)
Pattern	One line	Insight continuing it in this chapter
Cannot be measured from below	Averages and test functions bound $C_P$ only from below	Insight 10 (one lower bound before trimming upper bounds)
Drop to one dimension	Tools for upper bounds on $C_P$ are available in one dimension	Insight 8 (cut shorter than the threshold of convexity)
Drop the units	Make dimensionless ratios the protagonists	Insight 3 (the length where $1/\hbar$ lives)
Counterexamples and controls first	First name the models where the proof fails	Insight 4 (first count the minima at the wall of conditioning)
Look at the power of the difference	What fixes the power is not the quantity but what is left after cancellation	Insight 9 (the stage of cancellation and the stage of direction)
H is a pair of measure and form	Name the normalisation of the Dirichlet form explicitly	Insight 12 (the truth of H changes with the coordinates)
Connectedness of the state space	Suspect topology before uniformity of constants	Insight 12, the note in 10-01 §05

Twelve insights and small questions

Insight 1 Are we paying for the same place twice?

In a long chain of inequalities, one bad place (near the top, for the double well) may be paid for by two stages under different names. In the sector framework, the Bochner stage and the Gibbs stage both paid for the same neighbourhood of the top, and the amplification factor of one of them multiplied the whole. Moving the negative curvature to the other side avoids passing through the amplification. In the same pattern, when using gradients in which two terms complement each other, sharing the denominator can sometimes fold $(1+\sqrt{C})^2$ into $(1+C)$ instead. As a procedure: before tightening a construction, write out place by place what each stage is paying for, and look for double payment (10-09 §02, type 3 in 10-10). State: checked form.

Small question 1. Compare the best right-hand side, when $\|u\|^2=\|v\|^2=\|w\|^2=1$ , of two constructions for the sum of three vectors $u+v+w$ : applying $\|u+v\|^2\le(1+t)\|u\|^2+(1+1/t)\|v\|^2$ ( $t\gt0$ ) twice in a row, and distributing all three terms at once as $\|u+v+w\|^2\le(1+s_1)\|u\|^2+\cdots$ . Where does the twice-in-a-row construction pay for the same quantity twice?

Insight 2 The currency of Poincaré and the currency of entropy

There are situations where one wants to pay for rare events with "rarity". But on the side of the Poincaré inequality, rarity is not a currency. Poincaré looks at the speed of exit, and putting mass on a rare state cancels the mass in both the variance and the Dirichlet form. Rarity can be a currency only on the side of entropy (the logarithmic Sobolev inequality), but there one pays instead $\ln(1/\nu)$ for the rarest state on which test functions can concentrate. As a procedure: decide between the entropy type and the variance type by setting two rarities side by side (the rarity of the defect one wants to pay for, and the rarity of the states on which test functions can concentrate) (type 4 in 10-10). State: checked form (the two-point model).

Small question 2. Consider the measure $\pi=(1-p,p)$ on two points $\{0,1\}$ and the Dirichlet form with flip rate 1, $\mathcal E(f)=\pi_0\pi_1(f_1-f_0)^2$ . Show that the Poincaré constant does not depend on $p$ . Check, by citing the known formula for the two-point space (Diaconis–Saloff-Coste), that the logarithmic Sobolev constant grows at the rate $\ln(1/p)$ as $p\to0$ .

Insight 3 What does $1/\hbar$ live in? — read the loss on the same length as the quantity it multiplies

On the plateau at the top, the test function that produces a constant of size $1/\hbar$ was not a function of the collective linear coordinate $\sum z_j$ but of the radius from the corner at the top, $\sum z_j^2$ . Choosing the linear collective coordinate gets the direction wrong. Also, $1/\hbar$ lives on the length of the plateau $\ell\approx0.34/\hbar$ , and when measuring a loss it must be measured on the same length as the quantity it multiplies. As a procedure: before splitting into two scales, check whether the function that produces the lower bound lives on the coarse coordinate or on the fibre (type 5 in 10-10, 10-09 §06). State: still an insight (in a form supported by numerical readings).

Small question 3. For the measure $\mu\propto e^{-V(|z|)/\hbar}$ on $\mathbb R^k$ with $V(r)=(r^2-1)^2/8$ , compare the ratios $\operatorname{Var}(\cdot)/\mathcal E(\cdot)$ ( $\mathcal E(f)=\hbar\int|\nabla f|^2d\mu$ ) of the linear function $f=z_1+\cdots+z_k$ and the radial function $g=|z|^2$ , for $k=1$ and $k=2$ , up to the leading term as $\hbar\to0$ . For $k=2$ , when $\mu$ concentrates on an annulus, which function captures the slow direction?

Insight 4 The wall of conditioning creates minima

When conditioning by fixing the ends of an interval exactly, all the minima at $\hbar=0$ may lie on the wall. The "slowness" or "bistability" one then sees is not a property of the model but a shadow created by the conditioning (the bistability at the negative end of the core was the shadow of confining, inside the interval of the core, a configuration in which the kink wants to shift by one site). As a procedure: each time the way of cutting sectors or conditioning changes, make the first check a count of the $\hbar=0$ minima lying on the wall (10-11 §06). State: checked form.

Small question 4. Where does the measure $\mu_\hbar\propto e^{-V/\hbar}\mathbf 1_{[0,b]}$ , $V(x)=(x^2-1)^2/8$ restricted to the interval $[0,b]$ ( $0\lt b\lt1$ ), concentrate as $\hbar\to0$ ? Near there, approximate $\mu_\hbar$ by an exponential distribution, and write the behaviour of the Poincaré constant of $\mu_\hbar$ (Dirichlet form $\hbar\int f'^2d\mu$ ) as $\hbar\to0$ in terms of $|V'(b)|$ . What changes when the wall is moved to $b=1$ ?

Insight 5 Before closing the inside uniformly, first write what multiplies it on the outside

In nested estimates, one can get stuck trying to bound an inner constant by $O(1)$ . But if, in the outer formula, that constant is multiplied by a small factor, the accuracy needed inside is much looser. In the proposition for the box gap, the local term is multiplied by $e^{-2S_0/\hbar}$ , so the constant of the core did not need to be $O(1)$ ; size $e^{\Delta S/\hbar}$ ( $\Delta S\lt2S_0$ ) was enough. As a procedure: before closing the inside uniformly, first write what multiplies that quantity in the outer formula, and fix the accuracy needed inside by its ratio to the outer scale (10-11 §06). State: checked form (the reading of the action barrier is the value at $\hbar=0$ ).

Small question 5. Write the condition on $B$ needed for $K(\hbar)=A+e^{-2S_0/\hbar}B(\hbar)$ to be bounded as $\hbar\to0$ . If it has the form $B(\hbar)\le c\,\hbar^{-p}e^{\Delta S/\hbar}$ , what must the relation among $\Delta S$ , $2S_0$ and $p$ be? What is needed in the exact case $\Delta S=2S_0$ ?

Insight 6 The sup over a parameter is attained just outside the threshold

When bounding the constant of a core of length $k$ uniformly in $k$ , the hard case was not a long core but just outside the threshold where convexity is lost ( $k=8$ , both ends at the top). Inside the threshold convexity makes it free; far outside it saturates by reduction to the box; and one point in between attains the sup. As a procedure: when stuck on a sup over a parameter, first find where the threshold is, then name the value just outside it and bound it separately (10-11 §05). State: checked form (the threshold is a closed formula).

Small question 6. Find the smallest $k$ for which $\kappa(k)=8\sin^2\!\bigl(\pi/(2(k+1))\bigr)-\tfrac14$ is negative (the same formula as in 10-09 §06). How does $\kappa(k)$ behave as $k\to\infty$ ? Are "the threshold is at $k=8$ " and "the sup is attained at $k=8$ " the same thing or different things?

Insight 7 The threshold of an excluded set is fixed by the mass the downstream inequality allows

When excluding a bad set and estimating on the rest, where to put the threshold is decided not by the convenience of the bad set but by what the downstream inequality, which receives the mass of that set, allows. In the sector framework, the threshold of the window mark and the threshold of the defect disagreed, so the only set determined by the mark that contained the defects was the whole space, and the closing condition, taken literally, gave nothing. As a procedure: check literally that each of the conditions being composed holds on the same definition of the threshold, and work the threshold back from the mass the downstream can pay (10-09 §04). State: checked form (one line of measurability).

Small question 7. At one site $z\in\mathbb R$ , consider the mark $\sigma=\mathbf 1\{z\le-\theta_\sigma\}$ and the defect set $N=\{-\theta_B\lt z\le-\theta_\sigma\}$ ( $\theta_\sigma\lt\theta_B$ ). List all sets determined by the value of $\sigma$ (sets measurable with respect to $\sigma$ ) that contain $N$ . What changes if $\theta_\sigma=\theta_B$ ?

Insight 8 The threshold of convexity is decided by whether the parameter enters the Hessian

When stuck on a sup over end values, the road divides according to whether the end value enters the Hessian. If the end value enters a nearest-neighbour quadratic coupling only linearly, the Hessian does not depend on it, and convexity is uniform in the end value. So cutting shorter than the threshold of convexity makes the sup over end values free, and the whole price moves to the side of the correlation across the cuts (the maximal correlation). As a procedure: when stuck at the end of a parameter, first check whether that parameter enters the Hessian; if it does not, cut shorter than the threshold of convexity, and account separately only for the correlation at the seams (10-11 §05). State: checked form.

Small question 8. For $S(x_1,\dots,x_k)=\sum_{j=0}^{k}\frac{(x_{j+1}-x_j)^2}{2\varepsilon}+\varepsilon\sum_{j=1}^kV(x_j)$ and the measure $e^{-S/\hbar}$ with $x_0=a$ and $x_{k+1}=b$ fixed, check that the Hessian does not depend on $a,b$ . When $V''\ge-\tfrac12$ (the value at the top of $V=(x^2-1)^2/8$ ), write the range of $k$ for which the smallest eigenvalue of the Hessian is positive, in terms of $\varepsilon$ (compare with $\kappa(k)$ of small question 6 at $\varepsilon=\tfrac12$ ).

Insight 9 Split the ratio of upper bound to reading into a stage of cancellation and a stage of direction

When an upper bound $U$ and a reading from below $L$ are far apart, looking at the ratio $U/L$ as a single number does not tell where one is losing. Inserting between them "the quantity along the same path before the cancellation is discarded", $R$ , and splitting into $R/L$ (the stage of cancellation) and $U/R$ (the stage of direction), for strand 3 the former was around 1.2 and the latter over 1000, so almost all the loss was at the stage where the scalar Grönwall discards direction (10-09 §05). As a procedure: split the gap between an upper bound and a reading into ratios per stage before choosing the target of improvement. State: checked form (ratios of readings).

Small question 9. For $\dot X=-CX$ , $C=\operatorname{diag}(\lambda,\Lambda)$ ( $0\lt\lambda\lt\Lambda$ ), with initial value $g=(0,1)$ (the stiff direction), compare the true value of $\int_0^T|X_t|\,dt$ with the upper bound coming from the scalar Grönwall $|X_t|\le e^{-\lambda t}|g|$ . How does the ratio grow with $\Lambda/\lambda$ ? What is recovered when estimating with the matrix kept as it is?

Insight 10 Before trimming an upper bound, measure one lower bound

When tightening the construction of an upper bound through many stages still does not reach, first measuring one lower bound of the same quantity with a test function tells whether the problem lies in the technique or in the target. If the lower bound exceeds the target, improving the upper bound is pointless. In the sector framework, it turned out that the lower bound on the propagation constant lay at several times the needed value, which changed the meaning of the work of trimming the floor of the upper bound. As a procedure: before trimming the floor of an upper bound, put a stage that sets the needed value beside the lower bound on the best constant (10-09 §03, 10-10 §02). State: checked form.

Small question 10. On a single road of $L$ cells (uniform distribution, $\mathcal E(f)=\frac1L\sum(f_{i+1}-f_i)^2$ ), show with the single test function $f_i=i$ that the target " $C_P\le L^2/12$ " is false ( $\operatorname{Var}(f)/\mathcal E(f)=L(L+1)/12$ ). Compare also with the true value $1/(2(1-\cos(\pi/L)))$ (10-00 §04).

Insight 11 Obstacles do not vanish but move — record, line by line, where they moved

Changing the formulation can make an obstacle seem to vanish while it appears elsewhere in the same form. In the first step, a single number — the curvature at one site at the top falls short of the coupling by $\tfrac14$ — came back three times in different forms: the Poincaré constant of the lump of the core, the contraction ratio on the slope approaching 1, and the ratio of the tail pinned at the top exceeding 1 (the common wall of 10-09 §06). As a procedure: each time the formulation changes, record line by line "where the obstacle moved", and check whether the formula where it moved can be written with the same number. If it can, that is not a defect of the tool but the structure of the object. State: checked form (the formula of the common wall).

Small question 11. Find in closed form the smallest eigenvalue of the tridiagonal matrix $M_k=\operatorname{tridiag}(-2,\,4-\tfrac14,\,-2)$ ( $k\times k$ , Dirichlet at both ends), and compare it with $\kappa(k)$ (small question 6). How is the failure of diagonal dominance, $\tfrac{15}{4}\lt4$ , related to the smallest eigenvalue becoming negative for $k\ge8$ ?

Insight 12 Whether H is true changes not only with the measure but with the coordinates (the Dirichlet form)

Assumption H is not a sentence about a measure but about a pair of a measure and a Dirichlet form. Even for the same equilibrium measure, depending on which variables the dynamics moves independently, the size of the gap can be $O(1)$ or inversely proportional to the square of the side of the volume. In a dynamics with a conserved quantity (a quantity only handed between neighbours), its unevenness is smoothed out only by diffusion. As a procedure: before asking about H, name the coordinates of the dynamics, and first check whether that dynamics has a conserved quantity and at what length the conservation breaks (10-01 §05; for how to compare with the length at which conservation breaks, see Part 11). State: still an insight (checked form in the free-field approximation).

Small question 12. For the product measure of $L$ independent standard normals, compare the gaps of (a) the dynamics that moves each coordinate independently by Ornstein–Uhlenbeck, and (b) the dynamics that hands quantity between neighbours and keeps the sum $\sum x_i$ (on the measure with the sum fixed). Check that the gap of (b) has the form $4\sin^2(\pi/(2L))$ , and that it is unrelated to the static correlation length (here 0).

Discarded hypotheses — the original form, the corrected form, the grounds

Discarded hypotheses and the reasons for discarding them are information for avoiding the same errors. The upper six rows of Table 2 were discarded in the study of the first step, and the lower four on the side of the four-dimensional lattice in the previous edition.

Table 2 Discarded hypotheses and corrected insights
Original form	Corrected form	Grounds
Local Poincaré comes from a convex core	The action of an excursion out of the core is much smaller than $S_0$ , and exponentially many excursions fit into an interval of length $\xi$	The reading of the action of excursions (towards the two scales of 10-06 §03)
Convexity improves away from the wall	The opposite. Pressing the end against the wall lowers the variance and raises the Hessian; moving it away removes only this help	Comparison of the Hessian for each end value (10-06 §04)
Making the lumps larger lets the chain be built from the convexity of each lump	The worst convexity of a lump decreases with the length of the lump, and the coupling between lumps does not decrease	The formula of the common wall (10-09 §06)
The contraction can be bounded by the smallest eigenvalue of the time-averaged operator	False. What can be used is only the time average of the smallest eigenvalue, and this discards direction	The algebra of the counterexample (Lean `mbar_bound_false`, 10-09 §05)
If the pointwise reading is singular at rare configurations, the true gradient is singular too	What fixes the upper limit of the true gradient is not the pointwise minimum but the decay rate of the flow of derivatives	The representation $\nabla\psi=\int\mathbb E[J_t^{\top}\nabla G]\,dt$ (10-09 §05)
The road of sharpening constants can lower $\hbar$ indefinitely	It stops at the convex radius (about 0.45); beyond that the error terms become genuinely large	The reading of the margin at the nodes (towards the $\hbar\lt0.005$ row of 10-06 §07)
$C_P\asymp\xi^2$ usually holds	H is a specific claim, the dynamical exponent $z=2$	$z\approx2.17$ for the heat bath of the two-dimensional Ising model (previous edition)
Abelian counterexamples show that "non-abelianness is essential"	"An input that forbids a second small parameter" is essential	Three-dimensional $\mathbb Z_2$ keeps a single scale even though it is abelian (7-12)
The holographic density is not log-concave	The sign is the opposite ( $W''=-2B''$ )	The judgement of the one-dimensional model (previous edition)
Gauge-fixed numbers are a property of confinement	Numbers for compact $U(1)$ in four dimensions; not measured for $SU(2)$	10-02 §04

There are two types of error, and both occur at the stage of raising something to a one-line summary. One drops the conditions (group, model, range); the other confuses "a pointwise quantity" with "a quantity over the measure". Of the upper six rows of Table 2, four are of the latter type.

Questions at the size of an undergraduate thesis — a list of seventeen questions

A "small question" is one that satisfies three things. The form of the answer (exists / does not exist, the value of a constant, whether an inequality holds) is fixed in advance. The tools are at hand (python3 with NumPy and SciPy, a SAT solver, interval arithmetic, Lean and Mathlib, a literature search). There is a prospect that a third- or fourth-year student of mathematics or physics can settle it within the period of an undergraduate thesis. They are numbered Q1–Q17 (question 10 of the previous edition, "the gap of the path measure of the double well", was removed because it became the central test bed of this part).

Table 3 Seventeen questions ("What does not move when settled" are the rows of this part that do not change even if the question is settled)
Q.	One line	Tools	Form of settlement	What does not move when settled
Q1	Does a complete family of $k=3$ cells exist in $d=9$ ?	SAT, Lean	A construction in the periodic box, or a record refuting satisfiability of the box	That one-link integration is a single step only (10-02 §02)
Q2	The largest dimension in which a complete family exists, $d_{\max}(k)$	Exhaustive search, paper	A table for $k=3,4,5$ , and the reason the upper bound $3k+1$ is not sharp	Same as above. Non-existence with period 2 (Lean `no_perfect_k4_d9`, `no_perfect_k5_d8`, assumption: period 2) already exists
Q3	The constants of $1-\lambda_k$ for the Hermite family (the worked example of §05)	Paper, python3	The two terms and the remainder of (5)	That test functions bound $C_P$ only from below
Q4	The $SU(N)$ version, and the Lean statement, of the ratio of functions on the forest saturating at $1/\lambda_1(G)$	Paper, Lean	The statement of Poincaré for products of Haar	H↑ (it only closes one family of test functions for H↓ on the four-dimensional lattice)
Q5	$C_P(\beta)$ in the link variables of two-dimensional lattice YM	Paper, python3	The power as $\beta\to\infty$	H in four dimensions (two dimensions is a control)
Q6	The best constant of the curvature after one-link integration, $c^*\in[21,27]$	Global optimisation, interval arithmetic	$c^*$ and the configuration, a computer-assisted lower bound	That one cannot get out from the strong-coupling side
Q7	The coefficient $C\approx320$ of $N^{-3}$ in the column model, analytically (finite-size expansion of the XX chain)	Paper, python3	A closed formula agreeing to 4 digits or more	The logic of the arrows of the tube estimate $T(c,1)$
Q8	In the column model, the sign of $1/P$ at finite $u$	python3	A table of signs for $u=0.1\sim0.3$	Same as above
Q9	How much does the shortfall $c_2$ change with the choice of gauge in the block transformation?	python3	A table of $c_2$ for each gauge, and whether $c_2\gt0$ is kept for positive kernels	The arrow of the sum of shortfalls (it does not move if $c_2\ne0$ )
Q10	$C_P$ and $\xi^2$ of the two-dimensional $O(3)$ sigma model	python3, literature	The slope, and whether a logarithm is present when the scale is changed	H in four dimensions (an asymptotically free control)
Q11	$C_P/\xi^2$ of the three-dimensional $\mathbb Z_2$ gauge theory	python3	A table, and the correspondence with the dynamical exponent on the dual Ising side	H in the non-abelian case
Q12	Monotonicity in volume of $C_P(V)$ , and boundary conditions (find all eigenvalues of the generator on small lattices)	python3	A counterexample to monotonicity, or a proof	The question itself of the order of volume in which H is stated (the note in 10-01 §05)
Q13	Collect the primary sources on the dynamical exponent of local algorithms for four-dimensional $SU(2)$	Literature	A table of about ten primary sources (exponent, observable, box, boundary)	The conditions of the reading for the periodic box (the part that is a conjecture)
Q14	Bound from above, by interval arithmetic, the two-point maximal correlation of stage 2 of the core ( $8\le k\le14$ )	Interval arithmetic, paper	Computer-assisted upper bounds at each point of $k$ and $\hbar$ , and a paper on the difference between lattice and continuum	[007] of road A (the side $k\ge15$ ) and [014] (10-15 Table 1)
Q15	Read the power of the prefactor $\hbar^{-p}$ of $C_P$ for the core and the window by the capacity of the degenerate point	python3, paper	The value of $p$ (a reading), and the form of the capacity formula	The side of the action barrier $\Delta S\lt2S_0$ (only one of the conditions of small question 5)
Q16	A comparison theorem showing the alternating maximal correlation between boxes $c_{\rm alt}\lt1$ , avoiding examples that do not reduce to two-point quantities	Paper	An upper bound on $c_{\rm alt}$ uniform for the reduced model	The rows of (H1)(H2) (only [006] moves)
Q17	In the dynamics with conservation of small question 12, when a small term that breaks the conservation is added, the length at which the gap departs from the square of the side	Paper, python3	The relation between the strength of the breaking term and the length	H in four dimensions (the smallest model for the questions of Part 11)

For every question, before starting, we put a stage of searching for whether an answer to the same question exists. Write the question in one sentence in the form of a theorem, search with combinations of words from both mathematics and physics, follow the references of reviews to the text of the primary sources, and judge it as one of: the same answer, a close but different result, or not found. If nothing is found, write "not found in the range searched", and leave the words and materials searched in the source table.

Worked example: one question to the form of a settlement — $1-\lambda_k$ for the Hermite family

To see the size of a question, we carry Q3 to the end. The Poincaré constant of the standard normal distribution is $C_P=1$ , with equality at $f=x$ (1-10). Let $\lambda_k$ be the maximum of the ratio $\operatorname{Var}(f)/\mathcal E(f)$ over the family $F_k=\operatorname{span}\{x^3,\dots,x^{2k+1}\}$ , which does not contain $x$ .

Normalising the Hermite polynomials to $\mathbb E[h_mh_n]=\delta_{mn}$ gives $h_n'=\sqrt n\,h_{n-1}$ , so the variance and the Dirichlet form of an odd function $f=\sum_{j=0}^ku_jh_{2j+1}$ are diagonal.

\operatorname{Var}(f)=\sum_{j=0}^k u_j^2,\qquad \mathcal E(f)=\sum_{j=0}^k(2j+1)\,u_j^2 \tag{1}

The condition for being in the family is the single linear constraint "the coefficient of the monomial $x$ is 0", $\sum_jw_ju_j=0$ , with $w_j=(-1)^j\sqrt{(2j+1)!!/(2j)!!}$ . Finding the minimum $\mu_k$ of $\mathcal E/\operatorname{Var}$ under the constraint by Lagrange multipliers leaves an equation in one variable Cited (the secular equation for stationary values of a quadratic form under a linear constraint, Golub 1973).

g_k(\mu):=\sum_{j=0}^k\frac{w_j^2}{2j+1-\mu}=0,\qquad 1\lt\mu_k\lt3,\qquad \lambda_k=\frac1{\mu_k} \tag{2}

Setting $\mu_k=1+\varepsilon$ , the $j=0$ term is $-1/\varepsilon$ and balances the rest.

\frac1\varepsilon=S_k+O(\varepsilon),\qquad S_k:=\sum_{j=1}^k\frac{(2j+1)!!}{(2j)!!\,2j} \tag{3}

$S_k$ closes by a generating function. Splitting each term in two by $(2j+1)!!/(2j)!!=(2j+1)\binom{2j}j4^{-j}$ , and using the partial-sum identity $\sum_{j=0}^k\binom{2j}j4^{-j}=(2k+1)\binom{2k}k4^{-k}$ (by induction on $k$ ) and the value $\ln2$ at $x=1$ of $\ln\frac2{1+\sqrt{1-x}}=\sum_{j\ge1}\binom{2j}j\frac{x^j}{2j\,4^j}$ , gives Proved here

S_k=2\sqrt{\frac k\pi}+(\ln2-1)+O(k^{-1/2}) \tag{4}

Substituting into (3) and expanding $1-\lambda_k=\varepsilon-\varepsilon^2+\cdots$ fixes the two terms Proved here.

1-\lambda_k=\frac{\sqrt\pi}{2\sqrt k}-\frac{\pi\ln2}{4k}+O(k^{-3/2}),\qquad \frac{\sqrt\pi}2=0.886227,\quad\frac{\pi\ln2}4=0.544397 \tag{5}

Table 4 $\lambda_k$ and a check of (5) Reading (values from solving for the roots of (2) numerically)
$k$	$\lambda_k$	$\sqrt k(1-\lambda_k)$	$k\bigl(1-\lambda_k-\frac{\sqrt\pi}{2\sqrt k}\bigr)$
1	0.555556	0.4444	−0.4418
10	0.774921	0.7118	−0.5517
1000	0.972524	0.8689	−0.5489
$10^4$	0.991192	0.8808	−0.5459
$10^6$	0.999114	0.8857	−0.5446
(5)	→ 1	0.8862	−0.5444

What does not move when it is settled is the fact that test functions bound $C_P$ only from below. Enlarging the family keeps $\lambda_k\lt1$ and only approaches 1 at the rate $k^{-1/2}$ ; that the limit is 1 can be said only from an input outside the family (Gaussian convexity, $V''=1$ ). The settlement of a small question often takes this form.

This part placed Assumption H at the axis of the study, and this chapter gathers the views that repeatedly proved useful on that basis. Questions that stand on views other than H are placed in Part 11.

What this chapter can say

Said	Not said
In which stage of which chapter of the body each of the twelve insights worked (the chapter at the end of each insight), and the state of each insight (still an insight, checked form)	That the insights work outside the first step (the four-dimensional lattice, other problems). Every insight is a view before verification
Proved here The formula for the threshold in small questions 6 and 11 ( $\kappa(k)\lt0\iff k\ge8$ ), and the skeleton of the derivation of (4)(5)	The value of the coefficient of the remainder $O(k^{-3/2})$ in (5)
Reading The numbers of Table 4	Table 4 consists of floating-point values of the roots of (2), not propositions of upper or lower bounds
The list of seventeen questions, and what does not move when they are settled	No question, even when settled, shows anything about the mass gap, the continuum limit or Assumption H. "The size of an undergraduate thesis" is a prospect

Sources and reproduction

Item	Kind	Source / tools
Material for the twelve insights	This book	The sections of the chapters given at the end of each insight (10-06 to 10-11)
The two-point logarithmic Sobolev constant of small question 2	Cited	P. Diaconis, L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains (bibliographic data not checked; the statement has not been checked in the original. It is an object to be checked on the side of the question)
The previous-edition rows of Table 2	This book	The table of discarded hypotheses in Part 10 of the previous edition (the chapter of the notebook of ideas). The $z$ of the two-dimensional Ising model is from M. P. Nightingale, H. W. J. Blöte, Phys. Rev. Lett. 76 (1996) 4548 (abstract only)
Q1–Q13 of Table 3	This book	The fourteen questions of the chapter of small questions in the previous edition, minus the one that became the test bed. Q14–Q17 come from the remaining rows of 10-09 to 10-11
The secular equation (2)	Cited	G. H. Golub, Some modified matrix eigenvalue problems, SIAM Review 15 (1973) 318–334 (bibliographic data only)
Table 4	Reading	Bisection for the roots of (2), cross-checked against the generalised eigenvalues of the Gram matrix of monomials for $k\le6$ (a copy of the table of the previous edition)

Terms used in this chapter — the three states of an insight, paying twice, the currency of Poincaré and the currency of entropy, the length where $1/\hbar$ lives, the wall of conditioning, the threshold of convexity, the stage of cancellation and the stage of direction, the moving of obstacles, dynamics with conservation, small questions. Their meanings are in the A-02 Glossary.

Next chapter to read: Part 11 Further considerations — views other than Assumption H, and a list of material for new views that take H as given. To go on to the appendices, A-01 Table of notation.

Revised 2026-10-09: first version (combines the former 10-09 "From the notebook of ideas" and the former 10-13 "Small questions").