2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircasesector framework, closing condition, the common wall · Kiichi, Shiori and Rin

The sector framework and the three strands of route 5

The core $Q_k=(-x_c,0)^k$ is an interval of $k$ sites where the path stays near the top, and here the Hessian of the action has negative directions. To control the gap of the measure of the core (row (T0′_P)) uniformly in $k$, one must pay using the fact that places with negative curvature are rare. The sector framework splits that payment into five steps.

1. Sum over sectors. Even when the space of configurations is split into a direct product of boxes, the Dirichlet form, the variance and the entropy split into sums over sectors, and the number of sectors does not enter the constants.
2. Gibbs in marginal form. The weight on a rare set $B$ (measure at most $e^{-\ell}$) can be paid with the entropy and $\ln2$: $\ell\int_Bg^2\le\mathrm{Ent}+\ln2\int g^2$.
3. Componentwise localisation. Applying 2. to each component of the gradient with the logarithmic Sobolev constant $A_W$ of the marginal of the marks on the 17 sites around that site (the site window), $k$ does not enter the sum.
4. Locality of volume. The rarity exponent $\ell_{\min}$ does not depend on $k$.
5. Composition. Combining the above with the two inequalities of Barta and Bochner gives, for the gap of the core $\lambda$, $\lambda(1+bA_W/\ell_{\min})\ge\kappa-b(A_Wc_H+D_W+\ln2)/\ell_{\min}$. The condition for the right-hand side to be positive is the closing condition $b(A_Wc_H+D_W+\ln2)\lt\kappa\ell_{\min}$.

The closing condition rewritten for $A_W$ is the needed value $A_{\max}=(\kappa\ell_{\min}/b-D_W-\ln2)/c_H$. At the three values $\hbar=3\cdot10^{-3}／10^{-3}／3\cdot10^{-4}$, the original form on the road that defines defects by three sites is negative at $3\cdot10^{-3}$ (it closes for no $A_W$), $0.20$ and $7.13$, and correcting two choices of tools raises it to $25.1／90.3／312.3$ Reading (arithmetic on the measured rarity exponents). On the other hand, in the current construction the upper bound on $A_W$ is of order $1857／2794／9106$ Reading (including estimates), 29–74 times short.

The one remaining proposition "$A_W\lt A_{\max}$" was attacked by three strands. The componentwise strand stopped at 1.4–5.5 times the needed value, the scalar strand became a proposition with a table of assumptions (Proposition P5) for $\hbar\in[3\cdot10^{-4},10^{-3}]$, and the Feynman–Kac strand lies within 0.54 times the threshold by a Monte Carlo reading, but its upper bound is $10^3$ times the threshold. All three stop at the same place: at the top, the diagonal at one site, $4-\tfrac14=15/4$, does not reach the sum of the couplings to the neighbours, $4$ (§06).

Setting. Take the core $Q_k=(-x_c,0)^k$ ($x_c=0.89$), fix the end values $z_0,z_{k+1}\le-x_c$, and let $\mu\propto e^{-S/\hbar}\mathbf 1_{Q_k}$ and $\mathcal E(f)=\hbar\int|\nabla f|^2d\mu$ (Neumann on the faces of the box). Let $\lambda$ be the first non-zero eigenvalue of $-L$, $f$ the eigenfunction, $u=\nabla f$, and put

. $N_j$ is the defect set of site $j$ (configurations where $j$ or one of its neighbours lies on the top side of the threshold $-\theta_B$, $\theta_B=0.57$). Attach a mark $\sigma_j=\mathbf 1[z_j\gt-\theta]$ to each site, and write $\sigma_{W_j}$ for the set of marks on the window $W_j$ around site $j$. A sector is a direct product of boxes for each type of marks, $\Omega_\sigma=\prod_jI_{\sigma_j}$.

(Σ.1) Sum over sectors

For any partition $\mathcal P$ into direct products of boxes, (a) $\sum_P\mu(P)\mathcal E_P(f)=\mathcal E(f)$ (faces have measure 0, and $|\nabla f|^2$ is the same function on each sector), (b) $\operatorname{Var}_\mu f=\sum_P\mu(P)\operatorname{Var}_{\mu_P}f+\operatorname{Var}_\nu(\mathbb E[f\mid\mathcal P])$ (the law of total variance; the entropy has a chain rule of the same form), (c) if $\operatorname{Var}_{\mu_P}f\le C_P\mathcal E_P(f)$ on each sector, the first term is at most $(\sup_PC_P)\,\mathcal E(f)$ Proved here. The number of sectors appears only in the weights $\mu(P)$, which sum to 1. Leanvar_split_cells (no assumptions), cell_sum_sup (assumptions: $w_i\ge0$, $E_i\ge0$, $C_i\le C_{\max}$).

(Σ.2) The Gibbs step in marginal form

The Gibbs variational inequality. For a probability measure $\nu$, $G\ge0$ and a function $F$,

Proved here Proof. Put $\tilde G=G/\int G$. Jensen's inequality gives $\ln\int e^Fd\nu\ge\ln\int_{\tilde G\gt0}e^{F-\ln\tilde G}\tilde G\,d\nu\ge\int(F-\ln\tilde G)\tilde G\,d\nu$. Multiplying both sides by $\int G$ gives (2). ∎

Taking $F=t\mathbf 1_B$ and $p=\nu(B)\le e^{-\ell}$ in (2) gives $\ln\int e^F=\ln(1-p+pe^t)$, and with $t=\ell$ this is $\le\ln2$ Leangibbs_ln2_cell (assumptions: $p\ge0$, $\ell\ge0$, $pe^\ell\le1$). If $B$ is a set determined by a coarse variable $\mathcal F$, taking $G=\mathbb E[g^2\mid\mathcal F]$ gives

. This is the form that pays for the weight on a rare set with the entropy of the marginal of the coarse variable.

(Σ.3) Componentwise localisation

As input we assume, for all $j$ and $g$, the logarithmic Sobolev inequality of the marginal on the window

Assumption, and let $N_j\subset B_j$ ($B_j$ a set determined by $\sigma_{W_j}$, $\ell_{\min}=\min_j\ln(1/\mu(B_j))$). Applying (3) to $g=u_j$ and summing over $j$, from $\sum_j\mathcal E(u_j)=Z$ and $\sum_j\int u_j^2=X$,

Proved here Leancomp_sum_le (assumptions: $0\lt\ell\le\ell_j$, the inequality for each component). $k$ does not enter. For contrast, if (3) is applied to a single union $\bigcup_jB_j$ with a scalar $g$, without splitting the gradient into components, just putting in the constant function $g\equiv1$ makes the defect term at least $\sum_j\mu(B_j)$, and it is not uniform in $k$ Proved here Leanscalar_not_uniform (assumption: $p\gt0$; the conclusion is "there is no $d$ with $kp\le d$ for all $k$"). Uniformity in $k$ and having $A_W$ enter the condition are a trade-off.

(Σ.4) Locality of volume

$\ell_{\min}$ is the rarity exponent of the defect set, and does not depend on $k$. On the road that defines defects by three sites, the measured values $\hbar\ln\mu(N_j)$ give $\ell_{\min}=38.3／121.9／406.8$ ($\hbar=3\cdot10^{-3}／10^{-3}／3\cdot10^{-4}$) Reading (floating-point transfer matrix; $3\cdot10^{-4}$ is an extrapolation, and the remeasured value is $420.0$). On the road that defines defects by lumps of length 8 or more, $\ell_{\min}\ge E_8/\hbar-\ln V_{\rm loc}$ ($E_8=0.402$, $\ln V_{\rm loc}=2.3$) Reading. To say the same on the bridge, one needs that the remainder of the transfer operator of a scrap is small uniformly in the end values (PF_seg) Assumption.

(Σ.5) Composition

We combine the following three inequalities.

• Barta: $\lambda X\ge\kappa X-bY$. It comes from the row inequality $(\mathcal M\psi)_j\ge(\kappa-b\mathbf 1_{N_j})\psi_j$ obtained by applying a weight $\psi$ to the componentwise operator $\mathcal M=-L+\operatorname{Hess}S$ ($u=\nabla f$ satisfies $\mathcal Mu=\lambda u$). $\kappa=3/1000$ is the lower bound on the rows outside the defects, and $b=493/1000$ is the depth of the defects, determined by the worst row over the whole box Assumption (the paper estimate of the rows is not placed in the book).
• (5): (Σ.3) itself.
• Bochner: $Z=\lambda X-\int u\cdot\operatorname{Hess}S\,u\,d\mu\le(\lambda+c_H)X$, $\operatorname{Hess}S\ge-c_H$, $c_H=\tfrac14$. The identity is the integrated form of Bochner's formula Cited (Proposition 1.3 of Ledoux's lectures, checked in the original text), and the vanishing of the boundary terms on the flat faces of the box and the smoothness of the eigenfunction up to the corners are Assumption.

Putting Bochner into (5), and that into Barta, gives

Proved here Leancomp_close (assumptions: $X\gt0$, $b,A\ge0$, $\ell\gt0$, and the three inequalities Barta, (5), Bochner). If the right-hand side is positive, that is, if

, the gap of the core $\lambda$ is bounded below by a positive number that does not depend on $k$. (7) is the closing condition. Of the five steps, the constructions of (Σ.1)–(Σ.3) and (Σ.5) have elementary proofs placed in the text, and what remains as assumptions is the inputs (LS_W), (PF_seg), the Barta rows, the regularity for Bochner, and whether $A_W$ is small enough to satisfy (7). What remains as an assumption is not structure but the size of a constant.

Solving (7) for $A_W$ gives the needed value

Proved here (rearranging (7)). That $A_W$ is smaller than this is the one remaining proposition (LS_W♭), route 5 of 10-07 §02.

Where the constants come from. We sort the six constants of (7) into those that come from the structure and those that come from the choice of tools.

The reason $A_W$ enters the condition multiplied by $c_H$ is this: when $A_WZ$ on the right of (5) is replaced by $(\lambda+c_H)X$ using Bochner, the $\lambda$ part can be moved to the left side, but $c_HX$ remains without a factor $\lambda$ and can only be paid by $\kappa$. The amplification $b/\kappa\approx164$ multiplies $A_Wc_H$ here.

Two directions — raising the needed side, lowering the insufficient side

Raising the needed side (correcting the choice of tools).

• Localising Bochner. The negative curvature $-w(z)=(1-3z^2)/4$ exists only for $z\gt-1/\sqrt3$, and that region is already inside the defect set $N_j$. From the one-line interval bound $-w(z)\le\delta+\gamma\,\mathbf 1[z\gt-\theta_R]$ ($\delta=\max(0,(1-3\theta_R^2)/4)$, $\gamma=c_H-\delta$) Proved here Leannegw_le_localized (assumptions: $\theta\ge0$, $z\le0$, $(1-3\theta^2)/4\le\delta\le\tfrac14$), and using that the kinetic term $2T_k$ is positive semi-definite, we get $$Z\ \le\ (\lambda+\delta)X+\gamma Y \tag{9}$$ Proved here Leanbochner_localized (assumptions: the identity $Z=\lambda X-H$, $-H\le\delta X+\gamma Y$). $Y$ is a term already paid by (5), so the coefficient of $A_W$ goes down from $bc_H=0.123$ to $b\delta+\kappa\gamma=0.0038$ ($\theta_R=0.57$; about 1/32).
• Optimising the Gibbs tilt. $t$ in (2) may also satisfy $t\lt\ell$, and at $t^*=\ell-\ln((b-\kappa)/\kappa)=\ell-5.10$, which maximises $\kappa t-b\ln(1+e^{t-\ell})$, the cost $b\ln2=0.342$ is replaced by $0.0183$ Proved here (maximum of a concave function of one variable) Leantilt_cost_le (assumptions: $b\ge0$, $L\le1/178$).

The closing condition with both combined is $b(A_W\delta+D_W+L(t^*))\lt\kappa(t^*-A_W\gamma)$, and the needed value is $A_{\max}=(\kappa\ell_{\min}-0.0183-bD_W)/(b\delta+\kappa\gamma)$. Raising the threshold $\theta$ further to near $1/\sqrt3$ gives $\delta\le0$, and the condition becomes in effect $A_W\lesssim4\ell_{\min}$.

Lowering the insufficient side (rebuilding the upper bound on $A_W$). The upper bound on $A_W$ for the three-site window $|W|=3$ is $1857／2794／9106$ in the current construction Reading (including estimates), which is 74 / 31 / 29 times outside the needed value with both combined, and 14.5 / 6.2 / 5.7 times outside the needed value with the threshold raised (the latter is an estimate that assumes the same upper-bound technique gives the same value when the threshold is raised). On the other hand, the lower bound on $A_W$ measured by test functions is $1.71／1.74／2.59$ and does not exclude the needed value. That the true $A_W$ lies within the needed value is not excluded; what falls short is the way the upper bound is built. This is the starting point of the three strands of route 5.

(Σ.3) uses $N_j\subset B_j$, with $B_j$ a set determined only by the marks of the window $\sigma_{W_j}$. In Table 1, $\kappa$, $b$ and $\ell_{\min}$ were values measured with the defect threshold $\theta_B=0.57$, while $A_W$ was measured with the mark threshold $\theta_\sigma=\tfrac12$. If $\theta_\sigma\lt\theta_B$, the atom of marks "all marks in the window on the valley side" contains points with $z_i\in(-\theta_B,-\theta_\sigma]$, that is, points of $N_j$. The only set determined by the marks that contains $N_j$ is the whole space, so $\mu(B_j)=1$ and $\ell=0$ Proved here (one line of measurability). With the two thresholds mismatched, (7), taken literally, gives nothing. The fix is not to lower $\theta_B$ to $\tfrac12$ (at $\theta=\tfrac12$ the lower bound $\kappa$ on the Barta rows becomes negative), but to measure $A_W$ with marks at $\theta_\sigma=\theta_B$, which is an order to the insufficient side. It is an example showing that, when numerical constructions are stacked, a step is needed that checks literally whether each of the conditions being composed holds on the same definitions.

The one remaining proposition was attacked in three forms that differ in the functional form of the condition. None of the strands has shown (Σ.5), (T0′) or (H2). All numbers are values on the 0+1-dimensional double well on the lattice $\varepsilon=\tfrac12$. The full list of detour plans and the types of breakage are sent to 10-10.

Strand 1 The componentwise form

On top of the localised Bochner and the Gibbs tilt, adding a form that pays the negative curvature at each point by the rarity of that point (paying the curvature with a function of height $h=(-w(z_j))_+$) raises the needed value to $537／1754／6052$ Reading. The upper bound is built in the product form of "a coarse step" and "a propagation step".

• The coarse step is exhausted. With a mark chain that is multi-valued only in the middle, the logarithmic Sobolev constant of the coarse step is $\le3.584／3.544／3.613$, a ratio of $1.016／1.005／1.003$ to the lower bound from the indicator of a single point Reading (a rational decomposition on double-precision data). Nothing more can be gained from this step.
• The propagation step has no upper bound. Moving the rest into a weighted conditional Poincaré constant $C_w$, the needed value is $C_w\le39.3／138.2／476.3$, and the lower bound from a family of test functions is $15.4〜15.6$ (almost independent of $\hbar$) Reading. With the Lyapunov-function criterion (of the Bakry–Barthe–Cattiaux–Guillin type Cited, checked in a restatement in a secondary source), the family of Lyapunov functions that do not depend on the far end was $16.6／14.7／14.7$ times outside the needed value on the top face $x\equiv0$. Measuring by duality the best of a split family extended to the far end goes beyond this leading term, but on the finest grid it stops at $215／355／654$ = $5.5／2.6／1.4$ times the needed value Reading (grid values; the lower $\hbar$ goes, the more sensitive to the grid). The floor that holds for all Lyapunov functions is $31$ (uniform in $\hbar$) Reading Leantilt_g1 (assumption: $\hat\theta\,\mathrm{Var}\le\hbar b$), and the form of the criterion itself can get below the needed value.

The remaining step: a Lyapunov function with a coupling term between neighbours that lowers the finite-$\hbar$ correction. Grades: Reading and Assumption.

Strand 2 The scalar form — Proposition P5

This form does not split the gradient into components: it applies Bakry–Émery to the convex bottom (the box on the valley side) and pays for the band at the top by the Gibbs step at each site. The condition is only $\ln k\lt\ell_{\rm 帯}$, and $A_W$, $\kappa$, $b$ and $c_H$ are not needed, but $k$ enters as $\ln k$, so it works only over a finite range of core lengths. The Poincaré constant of the core becomes the sum of $C_P^{(1)}\approx50〜64$ and a remainder (Rem₂), and, restricted to the first eigenfunction, the remainder only needs to be small, $R_2(f)/\int g^2\lt\tfrac14$. Proposition P5 turns this into a proposition with a table of assumptions.

The number of runs to move to interval arithmetic is 8 nodes × 14 runs = 112 runs plus the enclosures between the nodes, and not a single one has been run yet. Slightly beyond $10^{-3}$ the two conditions no longer hold together on the same window (they do not hold together at $1.35\cdot10^{-3}$ Reading), and it does not extend towards $\hbar\to0$ either, because the estimate in the direction of exit is not uniform, by $\hbar^{-1/2}$. Proposition P5 is a proposition with a table of assumptions on a finite range at the lower end of the window.

Strand 3 The Feynman–Kac form

In the form that keeps the Feynman–Kac weight on the diagonal of the comparison, what is needed is that the logarithmic Sobolev constant of the whole core is $A_{\rm core}\lesssim a/\hbar$ with $a\lt1.29／1.47／1.55$ (the question extends to the coefficient of $1/\hbar$). The lower bound from a family of test functions is $\hbar A_{\rm core}\approx0.78〜0.83$ Reading, a margin of 1.6–1.9 times. Splitting into radial chains, the chain constant is $0.54〜0.56$ times the needed value (on the model chain) Reading, and what remains is the quantity $K^2=\sup|\nabla\psi|^2$ in the Helffer–Sjöstrand representation Cited (bibliographic only; the statement is unchecked, so this book treats it as Assumption), with threshold $1+K^2\le W$, that is, $K^2\le W-1=0.83〜0.86$ ($W$ is the ratio of the needed value to the chain constant).

• Monte Carlo reading: values from below after searching for starting points, $L=0.4333\pm0.0067$ ($10^{-3}$) and $0.4163\pm0.0079$ ($3\cdot10^{-4}$); adding 3σ gives $0.453$ = 0.54 times the threshold $0.835$ Reading (the maximum over starting points, not an upper bound over the whole fibre = assumption (MC-sup)).
• Paper upper bound: a pathwise Grönwall-type upper bound counted at the same points along the same paths gives $U^2=798$ = about $10^3$ times the threshold Reading. Most of the width comes from the step where the scalar Grönwall discards the direction of the gradient and reduces every direction at the softest rate. The version that controls the contraction by the smallest eigenvalue of the time-averaged matrix is false Leanmbar_bound_false (a two-dimensional counterexample, no assumptions).

The remaining step: an upper bound that keeps the direction (a matrix Lyapunov function along the path, or an estimate split into soft and hard subspaces).

The three side by side

Table 4 is not a ranking of which strand is "close". Strand 2 has a form only at the lower end of the window, and only strands 1 and 3 have forms that cover the whole window. Read progress by the grade of each row.

The Hessian of the action of the core is, on the lattice $\varepsilon=\tfrac12$, the tridiagonal matrix

(the neighbour term $(z_{j+1}-z_j)^2/(2\varepsilon)$ gives $4$ on the diagonal and $-2$ next to it). Looking at one row, the diagonal is $4+w_j$, and the sum of the absolute values of the couplings to the neighbours is $2+2=4$. At the top $z_j=0$, $w_j=-\tfrac14$, so

, and diagonal dominance fails by $\tfrac14$ Proved here. Row-by-row estimates (Gershgorin discs) only go as far as "smallest eigenvalue $\ge\min_jw_j=-\tfrac14$", and in fact on the face $x\equiv0$ where every site is at the top, the smallest eigenvalue of a lump of $k$ sites with both ends fixed is $8\sin^2\frac{\pi}{2(k+1)}-\tfrac14$, negative for $k\ge8$ Proved here (the closed formula for the eigenvalues of $\mathrm{tridiag}(-2,4,-2)$). To check by hand: at $k=7$, $8\sin^2(\pi/16)\approx0.304\gt\tfrac14$; at $k=8$, $8\sin^2(\pi/18)\approx0.241\lt\tfrac14$.

The three strands came back to this one-line shortfall in changed forms.

• However the Barta weight is chosen, the floor of the row of the one-site weight at the top is $F(0,0)=w(0)=-\tfrac14$.
• The criterion of the uniform road of strand 1 was $A-2B=-\tfrac14$ with diagonal $A=4-\tfrac14$ and coupling $B=2$.
• The top face where the Lyapunov criterion of strand 1 stopped is exactly the face $x\equiv0$ where the gradient of the action vanishes.
• In strand 3, the coefficient of $1/\hbar$ lived in the radius from the corner at the top (the non-convex direction of the plateau).

Why the wall is universal (one line). Tools whose entrance is convexity (a positive lower bound on the Hessian) — Bakry–Émery, Brascamp–Lieb, the Barta rows, diagonal dominance — must, since the Hessian really has a negative direction on the plateau at the top, pay this $\tfrac14$ somewhere with the rarity (entropy) of the plateau, and $\hbar$ enters that payment. A detailed study of this view, and other viewpoints that do not rely on convexity, are treated in Part 11 (11-06).

The "Proved here" rows of Table 5 are all either algebra on assumption rows or general inequalities whose proofs are placed in the text. They do not mean that a lower bound on the gap of the core has been shown unconditionally.

The format is the same as 10-05 §07, and the numbers to the right of "⟸" are always placed smaller than the number on the left.

[01] ASM LSW    input (LS_W)                                      ħ: —
[02] ASM PF     (PF_seg)                                          ħ: —
[03] ASM BAR    Barta rows κ = 3/1000, b = 493/1000               independent of ħ
[04] ASM REG    Bochner boundary terms and regularity             ħ: —
[05] ASM PROX   Assumption rows of Proposition P5 (lifting assumption etc.)  ħ: —
[06] ASM LSWb   (LS_W♭) A_W < A_max                               ħ: —
[07] CIT BOC    integrated form of Bochner's formula              independent of ħ
[08] RDG ELL    ℓ_min = 38.3 / 121.9 / 406.8                      ħ: three points
[09] RDG AW     upper and lower bounds on A_W                     ħ: three points
[10] PRV S1     (Σ.1) sum over sectors                            independent of ħ
               Lean var_split_cells, cell_sum_sup
[11] PRV S2     (Σ.2) Gibbs variational inequality                independent of ħ
               Lean gibbs_ln2_cell (assumptions: p ≥ 0, ℓ ≥ 0, p·e^ℓ ≤ 1)
[12] PRV S3     (Σ.3) componentwise localisation (5) ⟸ [01][10][11]  independent of ħ
               Lean comp_sum_le
[13] PRV SK     the scalar form is not uniform in k               independent of ħ
               Lean scalar_not_uniform (assumption: p > 0)
[14] RDG S4     (Σ.4) ℓ_min does not depend on k ⟸ [02][08]       ħ: three points
[15] PRV R1     localised Bochner (9) ⟸ [04][07]                  independent of ħ
               Lean negw_le_localized, bochner_localized
[16] PRV R2     optimising the Gibbs tilt                         independent of ħ
               Lean tilt_cost_le
[17] PRV S5     composition (6), closing condition (7) ⟸ [03][04][07][12][14]  independent of ħ
               Lean comp_close (assumptions: Barta, (5), Bochner)
[18] PRV AMX    needed value (8) ⟸ [17]                           independent of ħ
[19] RDG AM2    needed value 25.1 / 90.3 / 312.3 ⟸ [08][15][16][18]  ħ: three points
               Lean judge_i_inside (assumption: lower bounds on the measured ℓ)
[20] PRV THR    with mismatched thresholds ℓ = 0                  independent of ħ
[21] ASM P5     Proposition P5 ⟸ [05]                             ħ: 8 nodes
               Lean path5_conditional, p5_assembly
[22] RDG ST1    how far strand 1 got ⟸ [19]                       ħ: three points
[23] RDG ST3    how far strand 3 got                              ħ: two points
[24] PRV WALL   diagonal dominance fails by ¼ at the top          independent of ħ
[25] PRV CORE   positive lower bound on the core gap uniform in k ⟸ [06][17]  ħ: —

How to check. The rows with ⟸ are [12] ([01][10][11]), [14] ([02][08]), [15] ([04][07]), [17] ([03][04][07][12][14]), [18] ([17]), [19] ([08][15][16][18]), [21] ([05]), [22] ([19]) and [25] ([06][17]). In each the numbers on the right are smaller than the one on the left, so there is no circle.

What to read. The conclusion [25] depends, through [17], on the structural steps ([10]–[14]) and the input assumptions ([01]–[04]), and further on the assumption row [06] (the one remaining proposition). The candidates to support [06] are the three strands [21][22][23], and none of them shows [06] ([21] is a proposition with a table of assumptions at the lower end of the window, [22][23] are readings). [20] is a checking row: unless [01] and [03] are measured at the same threshold, [12] is empty. [24] explains where the three strands stopped, and is a premise of no conclusion.

We also check that the statements of the assumption rows do not contain the quantity of the conclusion. The quantity of the conclusion is the gap $\lambda$ of the measure of the core. [01] and [06] are logarithmic Sobolev constants of the marginal of the marks on the window, [02] the remainder of the transfer operator of a scrap, [03] the lower bound on the weighted rows, [04] the boundary terms and regularity, [05] the remainder quantity and the proxy ratio. None of them contains $\lambda$.

The full list of detour plans tried for route 5 (twenty-two, plus two born along the way, and their derivatives) and the types of breakage that appeared repeatedly are treated in 10-10. The construction that controls the core constant in three steps from the side of the box is 10-11. This part places Assumption H at the axis of the study, and this chapter has said how far the core row of (H2) of its first step could be built. A viewpoint that handles the negative curvature at the top without relying on convexity is treated in Part 11 (11-06).

What this chapter can say

Sources and reproduction

Words used in this chapter — sector framework, sector, mark, site window, defect set, rarity exponent, Gibbs variational inequality, componentwise localisation, closing condition, needed value, choice of tools and structure, threshold mismatch, the three strands, Proposition P5, lifting assumption, the common wall, diagonal dominance. Their meanings are in A-02 Glossary.

Next chapter to read: 10-10 The list of detour plans — types of breakage — puts the detour plans into one table with "the line that killed it", and reads the three sides (the needed side, the form, the floor) and the types of breakage that appeared repeatedly.

← 10-08 What computer-assisted means — the conditions on the chain inside the windowContents10-10 The list of detour plans — types of breakage →