2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircasewhat remains, cost estimates, two roads · Kiichi, Shiori and Rin

The whole book　Contents and reading orders

What remains and its cost — the distance to a theorem

What this chapter shows — That the rows needed to reach the conclusion of the first step (the upper side of the power form of H) split two ways at the single fork of the integrated table. For each of the two roads (the core on the box side, the core on the sector side), the assumption and reading rows needed, and the kind of work needed to close each row together with cost estimates. What is needed to produce a $\hbar$ at which all rows are in place. Outside the first step, what the distance from the second step on looks like when measured by three levels of the goal on the four-dimensional lattice, and the meaning of the one line "what is missing is uniformity".

Prerequisite chapters — 10-12 Table 1 (the integrated table) and 10-12 §07 (the required sets of leaves). The content of each row is in the chapters 10-05 to 10-11. For the second step on, 10-01 and 10-03 §06.

Said first — The level is [Y3]. Every cost is an estimate, and nothing has been run. We do not add up numbers of rows or hours and read them as "so many tenths to go". The rows do not have equal weight, and only rows expected to close by computation have time estimates; rows whose method is unknown have no estimate at all. This part places Assumption H at the axis of the study, and what this chapter measures is what remains of the first step on that axis.

The order of this chapter

The conventions of this chapter — distance is a list, cost is an estimate
The fork and the two roads — the core on the box side, the core on the sector side (Table 1)
Comparing the two roads — counting by kind of work (Table 2, Figure 1)
Remaining work not on the road to the conclusion — work worth having (Table 3)
What is needed to produce a ħ that fits all — why computer assistance in the window cannot reach it
Conditions common to the four conditions — five premises (Table 4)
The distance from the second step on — three levels of the goal, and the number of steps in β (Tables 5 and 6) / Detour: the lesson of φ⁴₄
How to read the distance — a one-line conclusion

The conventions of this chapter — distance is a list, cost is an estimate

The "distance" to a theorem is a metaphor. In this chapter we call distance the list of those premises of the target proposition that have no proof.

\text{distance}\ =\ \bigl\{\text{the rows among the premises of the target proposition that have no proof}\bigr\}\qquad(\text{a set; we do not speak of near or far by its size}) \tag{1}

The goal of the first step is the upper side of the power form of H for the path measure of the double well, that is, the conclusion of Proposition 4.5, $C_P\le K\xi^2S_0/\hbar$ , and $\ln C_P/\ln\xi\to2$ , which follows from it (10-05 §04). The premises as a whole are the 111 rows of the integrated table (10-12 Table 1), and the rows without proof are the "Assumption" and "Reading" rows. The "Reading" rows are at hand as numbers, but they are not propositions of upper or lower bounds, so as premises of the conclusion they count as rows without proof.

To each row we attach the kind of work needed to close it. There are six: interval arithmetic (replace it by a computer-assisted inequality; 10-08), paper (write a proof), bibliography (check a known theorem in the original source and raise it to "Cited"), measurement (measure a quantity not yet measured), another method (it is known that the present tools cannot reach it, and another tool is needed), and method undecided (it is not yet decided with which tool to attack it). Time estimates are written only for rows with an estimate in the record, and rows with "no estimate" are left blank.

Example. Row [013], between the nodes of (H1), has the estimate "24 subintervals, 23–39 hours in total". Row [002] for the tube, on the other hand, has no estimate. It is wrong to read the former as "closer" than the latter. The former is work expected to close if computed; the latter is work for which it is not yet known what would close it. A sum of computing hours does not show the size of the distance.

The fork and the two roads — the core on the box side, the core on the sector side

Following the dependency graph of the integrated table from the conclusion [111] in the direction of the premises, there is only one fork with "or": row (T0′_P) (the Poincaré constant of the core is uniform in core length and end values, [103]). This row is supported in two ways (10-12 §07).

Road A (the core on the box side): supported by Proposition CORE3 (three stages for the constant of the core, [082]) and the vanishing of the local term as $\hbar\to0$ ([084]) (10-11 §05).
Road B (the core on the sector side): supported by a uniform lower bound for the gap of the core of the sector framework ([099]) and regularity at the seam (lim) (10-09 §02).

Whichever road is taken, the rows needed for the conclusion are fixed as one set. Table 1 lists all the "Assumption", "Reading" and "Computer-assisted" rows in the union of the two roads, and writes the work and cost needed to close each row. The table, the graphs and the counts were made from the same single set of data (rows, edges) as the integrated table, with a column of costs added.

Table 1 The assumption, reading and computer-assisted rows needed for the conclusion [111], and the work needed to close them (numbers and contents as in the integrated table. A = the road of the core on the box side, B = the road of the core on the sector side. Every cost is an estimate; nothing has been run)
No.	Content (integrated table)	A	B	Work needed to close it	Kind	Cost estimate	Text
Assumption Assumption rows (22 rows)
[001]	Uniqueness of the measure at T=∞, Markov property, extension from local functions, the limit step, regularity	○	○	Cite from the original source or show on paper the uniqueness of the measure (the principal eigenvalue of a positive integral kernel on ℝ), the Markov property, extension, the limit step and regularity	Bibliography, paper	Obtaining 1 original source (positive kernel) and paper. No time estimate	10-05
[002]	(H2tube) Poincaré for the tube, uniform in tube length ≤ Cξ	○	○	The Poincaré constant of the tube is uniform in tube length ≤ Cξ	Method undecided	No estimate (not started)	10-07
[003]	(KG) The box gap of the reduced kink gas is of the Glauber type	○	○	Show the box gap of the reduced kink gas without copying the values of the one-dimensional Ising model	Another method	No estimate (a comparison theorem is needed)	10-11
[005]	(X) Cross term ℰ_Y(E[f\|Y]) ≤ (1+χ)ℰ(f), χ uniform	○	○	χ of the cross term is uniform	Method undecided	No estimate (not started)	10-11
[006]	(MIX_alt) The alternating maximal correlation between boxes c_alt < 1 is uniform	○	○	The alternating maximal correlation between boxes c_alt < 1, uniformly	Another method	No estimate (a comparison theorem avoiding the example that does not reduce to two-point quantities is needed)	10-11
[007]	(MIX_core) Upper bound for the alternating maximal correlation in the core c̄ < 1 (k ≥ 15)	○	—	Upper bound for the alternating maximal correlation in the core c̄ < 1 (k ≥ 15)	Another method	No estimate (the saturation is only a floating-point reading)	10-11
[008]	(ΔS_core) C_P ≤ c ħ^(−p) e^(ΔS/ħ) for core and window	○	—	C_P ≤ c ħ^(−p) e^(ΔS/ħ) for core and window	Paper, measurement	The prefactor ħ^(−p) has not been measured. The next move is the capacity at the degenerate point (no estimate)	10-11
[009]	Continuous version of Holley's comparison (the tail lemma)	○	○	Continuous version of Holley's comparison	Bibliography	Confirming the version and the theorem number	10-06
[010]	The Poincaré–Miranda theorem (covering by boxes)	○	○	The Poincaré–Miranda theorem	Bibliography	Obtaining 1 original source	10-06
[011]	(T5) Regularity and density of the functions used	○	○	Regularity and density of the functions used	Bibliography, paper	No estimate	10-06
[012]	(T1b) Neighbour and end values > 2.5	○	○	Neighbour and end values > 2.5	Paper	No estimate (the road through monotonicity does not close, and the lemma of tangents to a convex tail is far from the margin)	10-06
[013]	(T1c) ħ between the nodes (in uniform form)	○	○	ħ between the six nodes (0.005–0.2)	Interval arithmetic	After one rerun of the frame (30–50 minutes), 24 subintervals, 23–39 hours in total (one core; cut into jobs of at most 25 minutes each, 60–170 jobs)	10-06
[014]	(T1c) ħ < 0.005 (in uniform form)	○	○	ħ < 0.005	Another method	Infinitely many subintervals would be needed; interval arithmetic cannot reach it. A paper on the asymptotics as ħ→0 is needed (not started)	10-06
[015]	(T3d) End values of the two sites > 4	○	○	End values of the two sites > 4	Paper	No estimate	10-06
[016]	Transfer from the ring to scraps (with end values)	○	○	From the statement on the ring to the statement on scraps (with end values)	Paper	No estimate	10-06
[018]	(lim), (END_rand), regularity at the seam of the ring	—	○	(lim), (END_rand), regularity at the seam	Paper	No estimate	10-07
[019]	The small assumptions of route 4 (inputs to the sectors)	—	○	The small assumptions of route 4 (the vertical term, (T1′), (a′), (Σ_low), (G_B), regularity)	Paper	A list only, not started (no estimate)	10-07
[021]	Input (LS_W) (the form of LSI for the marginals of site windows)	—	○	Input (LS_W)	Paper	No estimate	10-09
[022]	(PF_seg) (locality of volume on the bridge)	—	○	(PF_seg) locality of volume on the bridge	Paper	No estimate	10-09
[023]	Barta's row κ = 3/1000, b = 493/1000	—	○	Barta's row κ = 3/1000, b = 493/1000 (at threshold 0.57)	Paper, interval arithmetic	No estimate (the paper evaluating the row is not given in this book)	10-09
[024]	The boundary term of Bochner and regularity of eigenfunctions	—	○	The boundary term of Bochner and regularity of eigenfunctions (C³ up to the corners does not follow from the standard results)	Another method	No estimate	10-09
[026]	(LS_W♭) A_W < A_max(ħ): the remaining single proposition of route 5	—	○	The remaining single proposition of route 5, A_W < A_max(ħ)	Method undecided	No estimate. All three strands stop at the shortfall of ¼ in the curvature of the single site at the top	10-09
Reading Reading rows (need to be replaced by propositions of upper or lower bounds) (5 rows)
[044]	ln ξ ≥ s/ħ (ξ ∝ ħ^½ e^(S₀/ħ)), the size of D/(Cξ)²	○	○	ln ξ ≥ s/ħ as a lower-bound proposition	Bibliography, paper	Expected to rise to "Cited" if the original source of the semiclassical asymptotics of the tunnelling splitting is checked (no estimate)	10-04
[046]	Two-point maximal correlation of Stage 2 and B₂ (22.1 / 30.5 / 66.4)	○	—	The two-point maximal correlation of Stage 2 (8 ≤ k ≤ 14) as an upper bound	Interval arithmetic, paper	Interval arithmetic for each of the seven values of k and each point of ħ, and a paper on the difference between lattice and continuum (no time estimate)	10-11
[047]	c̄_(≥15) ≤ 0.535 (measurement of a lower bound)	○	—	c̄_(≥15) ≤ 0.535 as an upper bound	Another method	No estimate (at present only a measurement of a lower bound)	10-11
[055]	ℓ_min = 38.3 / 121.9 / 406.8	—	○	ℓ_min = 38.3 / 121.9 / 406.8 as a lower bound	Interval arithmetic	No estimate	10-09
[089]	(Σ.4) ℓ_min does not depend on k	—	○	(Σ.4) ℓ_min does not depend on k	Paper	No estimate	10-09
Computer-assisted Computer-assisted rows (done within the nodes and ranges) (2 rows)
[040]	(T1a) Margin ħρ − 2ħκ ≥ 0.067 (neighbour and end in [0,2.5]²)	○	○	(Done) computer-assisted at six nodes	—	Closed within the nodes. Outside the nodes, [013][014]	10-06
[041]	(T3) λ_c ≥ 1.3×10⁻⁴ (ends in [0,4]²)	○	○	(Done) computer-assisted row	—	The range of ħ as in the column of the integrated table	10-06

What to read from Table 1. (1) The rows common to the two roads are 13 assumption rows and 1 reading row: the common premise [001], the rows inside the box [002][003][005], [006] between boxes, [009]–[016] extending (H1), and the reading of how the correlation length grows [044]. These remain whichever road is taken. (2) Needed only for road A are four rows: the alternating maximal correlation of the core [007], the action barrier [008], and the readings of Stages 2 and 3 [046][047]. (3) Needed only for road B are the remaining single proposition of route 5 [026], 7 assumption rows including the inputs to the sectors [019][021][022][024], Barta's row [023] and regularity at the seam [018], and the two reading rows [055][089]. (4) The two computer-assisted rows [040][041] are done within the nodes and ranges, and common to both roads.

Comparing the two roads — counting by kind of work

We compare the costs of the two roads not by the number of rows but by the kind of work (Table 2).

Table 2 The assumption and reading rows of Table 1 counted by the kind of work needed to close them (a row with two kinds is counted in both; the two computer-assisted rows are done and excluded)
Kind of work	Common to both roads	A only	B only
Interval arithmetic	1 [013]	1 [046]	2 [023], [055]
Paper	6 [001], [011], [012], [015], [016], [044]	2 [008], [046]	6 [018], [019], [021], [022], [023], [089]
Bibliography	5 [001], [009], [010], [011], [044]	0	0
Measurement	0	1 [008]	0
Another method	3 [003], [006], [014]	2 [007], [047]	1 [024]
Method undecided	2 [002], [005]	0	1 [026]

There are three ways to read it.

First, the only row with a time estimate is [013], common to both roads (24 subintervals, 23–39 hours, not run). There are other rows expected to close by interval arithmetic ([046], [023], [055]), but there are as yet no estimates of the number of jobs or the time. And among the common rows is [014] ( $\hbar\lt0.005$ , infinitely many subintervals needed), which is known to be out of reach of interval arithmetic. So neither of the two is a road that "closes given enough computer time".

Second, the additions of road A lean towards "another method", and the additions of road B lean towards "paper". [007][047] of road A need a comparison theorem bounding the alternating maximal correlation of the core uniformly in $k$ , and it must take a form that avoids the example that does not reduce to two-point quantities (10-11 §05). Many of the additions of road B are papers on the inputs to the sectors, but behind them lies one "method undecided" row, [026]: the remaining single proposition of route 5, the place where all three strands stopped at the shortfall $\tfrac14$ in the curvature of the single site at the top (10-09 §06).

Third, road B, with more rows, is not necessarily heavier. The additional papers of road B are all rows for which what has to be shown is fixed as a statement. The two additional rows of road A, [007][047], and the one row of road B, [026], are all rows for which "what to show" is fixed but "how to show it" is not. The difference between the two roads is whether these rows of unknown method lie on the side of the maximal correlation of the core or on the side of the curvature at the top.

Figure 1 is the dependency graph for each of the two roads, keeping only the ancestors of the conclusion. From the graph of the integrated table (111 nodes), only the nodes reached by choosing one side of the "or" at [103] and following the direction of the premises are listed, with the same numbers.

Figure 1 Dependency graph for each road (number, grade, symbol, content; the numbers to the right of ⟸ are premises. The rule of numbering is the same as in the integrated table: the numbers of premises are smaller than that of the conclusion)

Road A: the core on the box side (55 nodes)

[001] ASM P0     uniqueness of the measure at T=∞, Markov property, extension from local functions, the limit step, regularity
[002] ASM TUBE   (H2tube) Poincaré for the tube, uniform in tube length ≤ Cξ
[003] ASM KG     (KG) box gap of the reduced kink gas is of the Glauber type
[005] ASM X      (X) cross term ℰ_Y(E[f|Y]) ≤ (1+χ)ℰ(f), χ uniform
[006] ASM MXA    (MIX_alt) alternating maximal correlation between boxes c_alt < 1 is uniform
[007] ASM MXC    (MIX_core) upper bound for the alternating maximal correlation in the core c̄ < 1 (k ≥ 15)
[008] ASM DSC    (ΔS_core) C_P ≤ c ħ^(−p) e^(ΔS/ħ) for core and window
[009] ASM A5     continuous version of Holley's comparison (the tail lemma)
[010] ASM A10    the Poincaré–Miranda theorem (covering by boxes)
[011] ASM A15    (T5) regularity and density of the functions used
[012] ASM B1     (T1b) neighbour and end values > 2.5
[013] ASM B2     (T1c) ħ between the nodes (in uniform form)
[014] ASM B3     (T1c) ħ < 0.005 (in uniform form)
[015] ASM B4     (T3d) end values of the two sites > 4
[016] ASM B5     transfer from the ring to scraps (with end values)
[030] CIT MOB    mobility of a single kink D = ħ/S₀ (at the level of the process)
[031] CIT BE     Bakry–Émery on convex domains (Kolesnikov–Milman)
[032] CIT A3     Muckenhoupt's bracket B ≤ C_P ≤ 4B
[033] CIT A4     correlation inequality for MTP₂ (monotonicity of e, ħκ ≥ 0)
[034] CIT A6     Poincaré version of Helffer's criterion (Ledoux Proposition 3.1)
[035] CIT A12    Brascamp–Lieb (the cross κ_f, (T4))
[040] CMP A1     (T1a) margin ħρ − 2ħκ ≥ 0.067 (neighbour and end in [0,2.5]²)
[041] CMP A9     (T3) λ_c ≥ 1.3×10⁻⁴ (ends in [0,4]²)
[044] RDG XI     ln ξ ≥ s/ħ (ξ ∝ ħ^½ e^(S₀/ħ)), the size of D/(Cξ)²
[046] RDG MC2    two-point maximal correlation of Stage 2 and B₂ (22.1 / 30.5 / 66.4)
[047] RDG CB     c̄_(≥15) ≤ 0.535 (measurement of a lower bound)
[058] PRV A2     Lemma K2 (second-order Taylor of the tilt and the Hölder remainder)
[059] PRV A8     (T2) the marginal of the kept sites is a nearest-neighbour chain
[060] PRV A14    (T5) decomposition of the variance ⟸ [011]
[061] PRV MARG   the continuum statement of the margin λ ⟸ [040][058][032][033][009]
[062] PRV A7     gap on the coarse side ≥ λ (Gershgorin on the ring) ⟸ [034][059][061]
[063] PRV A11    nesting λ_S ≥ 2.3×10⁻⁵ ⟸ [041][010][035]
[064] PRV A13    (T4) v/ħ ≤ 5888 ⟸ [041][035]
[065] PRV A16    algebra of rates C_P ≤ ε/t ⟸ [060][062][063][064]
[066] PRV A17    Proposition H1: C ≤ 1.6×10¹⁰ (ring version) ⟸ [065]
[068] PRV SEG    conditions (H1), (H1seg): assembly when the B rows hold in uniform form ⟸ [066][012][013][014][015][016]
[069] PRV TP     Lemma TP (two projections)
[070] PRV ESP    Efron–Stein for product measures (martingale differences and Jensen)
[071] PRV BOX    product inequality for boxes (conditional independence of even boxes) ⟸ [001][070]
[072] PRV H3     Proposition H3: C_P ≤ (1+1/n)C_box/(1−c_alt) ⟸ [006][069][071]
[073] PRV DEC    decomposition of the variance and splitting into a product on the fast side ⟸ [001]
[074] PRV KGC    c_kg = C²+π² (algebra of the limit of the closed formula)
[075] PRV H4     Proposition H4: C_box ≤ C_loc + (1+χ)C_kg ⟸ [068][002][003][005][030][070][073][074]
[076] PRV BV     Lemma BV (convexity for k ≤ 7) ⟸ [031]
[077] PRV MR     Lemma MR (reduction to the ends) ⟸ [001]
[078] PRV TB     Lemma TB (two blocks for 8 ≤ k ≤ 14) ⟸ [069][076][077]
[081] PRV BET    inequality of form (β) (k ≥ 15) ⟸ [001][069][070][076]
[082] PRV CORE3  Proposition CORE3: three stages for the constant of the core ⟸ [007][046][047][076][078][081]
[084] PRV CLOC   the local term vanishes as ħ→0 ⟸ [008][044]
[103] PRV T0P    row (T0′P): Poincaré for the core, uniform in core length and end values ⟸ [082][084]
[104] PRV K4     K₄ does not depend on ħ ⟸ [103][075]
[105] PRV P45    Proposition 4.5: C_P ≤ Kξ²S₀/ħ ⟸ [030][072][104]
[106] PRV BL     family of bridges, one-dimensional version: Proposition L (R ≥ 1) ⟸ [001]
[110] PRV RB     1 ≤ R ≤ K′/ħ ⟸ [105][106]
[111] PRV PH     the power form of H: ln C_P / ln ξ → 2 ⟸ [044][110]

Road B: the core on the sector side (59 nodes)

[001] ASM P0     uniqueness of the measure at T=∞, Markov property, extension from local functions, the limit step, regularity
[002] ASM TUBE   (H2tube) Poincaré for the tube, uniform in tube length ≤ Cξ
[003] ASM KG     (KG) box gap of the reduced kink gas is of the Glauber type
[005] ASM X      (X) cross term ℰ_Y(E[f|Y]) ≤ (1+χ)ℰ(f), χ uniform
[006] ASM MXA    (MIX_alt) alternating maximal correlation between boxes c_alt < 1 is uniform
[009] ASM A5     continuous version of Holley's comparison (the tail lemma)
[010] ASM A10    the Poincaré–Miranda theorem (covering by boxes)
[011] ASM A15    (T5) regularity and density of the functions used
[012] ASM B1     (T1b) neighbour and end values > 2.5
[013] ASM B2     (T1c) ħ between the nodes (in uniform form)
[014] ASM B3     (T1c) ħ < 0.005 (in uniform form)
[015] ASM B4     (T3d) end values of the two sites > 4
[016] ASM B5     transfer from the ring to scraps (with end values)
[018] ASM LIM    (lim), (END_rand), regularity at the seam of the ring
[019] ASM SM     the small assumptions of route 4 (inputs to the sectors)
[021] ASM LSWin  input (LS_W) (the form of LSI for the marginals of site windows)
[022] ASM PF     (PF_seg) (locality of volume on the bridge)
[023] ASM BAR    Barta's row κ = 3/1000, b = 493/1000
[024] ASM REG    the boundary term of Bochner and regularity of eigenfunctions
[026] ASM LSWb   (LS_W♭) A_W < A_max(ħ): the remaining single proposition of route 5
[030] CIT MOB    mobility of a single kink D = ħ/S₀ (at the level of the process)
[032] CIT A3     Muckenhoupt's bracket B ≤ C_P ≤ 4B
[033] CIT A4     correlation inequality for MTP₂ (monotonicity of e, ħκ ≥ 0)
[034] CIT A6     Poincaré version of Helffer's criterion (Ledoux Proposition 3.1)
[035] CIT A12    Brascamp–Lieb (the cross κ_f, (T4))
[036] CIT BOC    integral form of Bochner's formula (Ledoux Proposition 1.3)
[040] CMP A1     (T1a) margin ħρ − 2ħκ ≥ 0.067 (neighbour and end in [0,2.5]²)
[041] CMP A9     (T3) λ_c ≥ 1.3×10⁻⁴ (ends in [0,4]²)
[044] RDG XI     ln ξ ≥ s/ħ (ξ ∝ ħ^½ e^(S₀/ħ)), the size of D/(Cξ)²
[055] RDG ELL    ℓ_min = 38.3 / 121.9 / 406.8
[058] PRV A2     Lemma K2 (second-order Taylor of the tilt and the Hölder remainder)
[059] PRV A8     (T2) the marginal of the kept sites is a nearest-neighbour chain
[060] PRV A14    (T5) decomposition of the variance ⟸ [011]
[061] PRV MARG   the continuum statement of the margin λ ⟸ [040][058][032][033][009]
[062] PRV A7     gap on the coarse side ≥ λ (Gershgorin on the ring) ⟸ [034][059][061]
[063] PRV A11    nesting λ_S ≥ 2.3×10⁻⁵ ⟸ [041][010][035]
[064] PRV A13    (T4) v/ħ ≤ 5888 ⟸ [041][035]
[065] PRV A16    algebra of rates C_P ≤ ε/t ⟸ [060][062][063][064]
[066] PRV A17    Proposition H1: C ≤ 1.6×10¹⁰ (ring version) ⟸ [065]
[068] PRV SEG    conditions (H1), (H1seg): assembly when the B rows hold in uniform form ⟸ [066][012][013][014][015][016]
[069] PRV TP     Lemma TP (two projections)
[070] PRV ESP    Efron–Stein for product measures (martingale differences and Jensen)
[071] PRV BOX    product inequality for boxes (conditional independence of even boxes) ⟸ [001][070]
[072] PRV H3     Proposition H3: C_P ≤ (1+1/n)C_box/(1−c_alt) ⟸ [006][069][071]
[073] PRV DEC    decomposition of the variance and splitting into a product on the fast side ⟸ [001]
[074] PRV KGC    c_kg = C²+π² (algebra of the limit of the closed formula)
[075] PRV H4     Proposition H4: C_box ≤ C_loc + (1+χ)C_kg ⟸ [068][002][003][005][030][070][073][074]
[085] PRV S1     (Σ.1) sum over sectors
[086] PRV S2     (Σ.2) Gibbs variational inequality
[087] PRV S3     (Σ.3) localisation component by component ⟸ [021][085][086]
[089] RDG S4     (Σ.4) ℓ_min does not depend on k ⟸ [022][055]
[092] PRV S5     (Σ.5) composition and the closing condition ⟸ [023][024][036][087][089]
[099] PRV COREP  a positive lower bound for the gap of the core, uniform in k (sector side) ⟸ [026][092][019][018]
[103] PRV T0P    row (T0′P): Poincaré for the core, uniform in core length and end values ⟸ [099][018]
[104] PRV K4     K₄ does not depend on ħ ⟸ [103][075]
[105] PRV P45    Proposition 4.5: C_P ≤ Kξ²S₀/ħ ⟸ [030][072][104]
[106] PRV BL     family of bridges, one-dimensional version: Proposition L (R ≥ 1) ⟸ [001]
[110] PRV RB     1 ≤ R ≤ K′/ħ ⟸ [105][106]
[111] PRV PH     the power form of H: ln C_P / ln ξ → 2 ⟸ [044][110]

Example. In the graph of road A the premises of [103] are [082][084]; in the graph of road B they are [099][018]. The rest of the trunk (the gap of the box and the composition from [104] on) consists of the same nodes in both graphs, and the work before the fork acts in common on both roads.

Remaining work not on the road to the conclusion — work worth having

The integrated table contains work that is not on the road to the conclusion yet has been named as remaining. Closing any of it would not move the conclusion [111], but it is worth having for the certainty of the numbers or as material for other roads.

Table 3 Remaining work not on the road to the conclusion (costs copied from the estimates in the record; none has been run)
Remaining work	Row of the integrated table	Work and cost estimate	What could be said if it closed	Text
Putting the inner chain of the ring on interval arithmetic	[053], [054]	With 1–2 rounds of paper study and 1–2 rounds of scripting, build a machine separating the lattice of pins from the lattice of positions and a proposition for the layer of the outward-pushing wall; about 50 jobs, about 42 hours over the 25 subintervals of the window (about 85 hours without the paper on bands)	The coefficient of the inner chain $\bar q\approx1.3$ becomes an upper bound. Not a road that closes the closing condition of the sectors (Σ.5)	10-07 §03
Extending the pin of the boundary stage to $(0,\tfrac12)$	[054]	Part of the machine above (extending the domain of the outer machine)	The condition "pin $\le0$ " on the outer chain is removed	10-07 §03
The remaining numbers of route 2	[052]	About 42 jobs, about 21 hours, for the constants of the window, the minimal weight and the congestion. Since the floor of the product form is 4–5 orders of magnitude above the needed value, the decision is not to run them	The upper bound of the product form becomes Computer-assisted (it does not reach the conclusion)	10-07 §04
Aligning the thresholds and remeasuring $A_W$	[056]	Measure $A_W$ with the mark of the defect threshold $\theta_B=0.57$ (no estimate)	The closing condition (7) of 10-09 has literal meaning	10-09 §04
Making the rows of Proposition P5 computer-assisted	[025], [096]	12 jobs per node (3 for the bathtub column, 7 for rarity, 2 for non-convex fibres), estimated 60–250 for the nodes. The lower bound of the action difference of non-convex fibres is not yet in the form of interval arithmetic. The lifting assumption remains	The rows of Proposition P5 at the lower end of the window become Computer-assisted. There is no edge of implication to the remaining single proposition [026]	10-09 §05
Sharpening the constant of (H1)	[066]	Reworking (T3) (no estimate)	$C\le1.6\times10^{10}$ becomes smaller (the form of the conclusion does not change)	10-06 §07
Conversion between the two spacings	—	Connect the readings at lattice spacing $\varepsilon=0.25$ with the tables at $\varepsilon=\tfrac12$ (no estimate)	One fewer "not comparable" row in the reconciliation	10-14 §05
Obtaining bibliography	[010] and others	4 items to obtain (Poincaré–Miranda, elliptic regularity of the Neumann problem on a box, stochastic monotonicity, the principal eigenvalue of a positive integral kernel)	Rows that can be obtained rise from "Assumption" to "Cited"	10-14 §03

The first three rows of Table 3 are, in amount of computation, larger work than [013] of Table 1. Even so, they are not on the road to the conclusion because closing them does not remove a single "Assumption" row of the integrated table (the same reason as "the two Computer-assisted rows of (H2) are not on the road to the conclusion" in 10-12 §05). The size of the computation and the closeness to the conclusion are different rulers.

What is needed to produce a ħ that fits all — why computer assistance in the window cannot reach it

Even if all the rows of Table 1 closed, if the ranges of $\hbar$ in which each row holds do not intersect, the conclusion cannot be stated at even a single $\hbar$ . The present ranges are of four kinds (10-12 §06): the nodes of (H1) $\{0.2,0.1,0.05,0.02,0.01,0.005\}$ , the computer-assisted window $[3\cdot10^{-4},3\cdot10^{-3}]$ , the nodes in the range of Proposition P5 $[3\cdot10^{-4},10^{-3}]$ , and the three points of reading $\hbar=3\cdot10^{-3}/10^{-3}/3\cdot10^{-4}$ . The (H1) side has nothing below $0.005$ and the (H2) side has nothing above $3\cdot10^{-3}$ , so the intersection is empty.

There are only two roads to bring them into line: (a) extend (H1) to $\hbar\le10^{-3}$ (row [014]), or (b) extend the (H2) side to $\hbar\ge0.005$ . (b) was decided to be outside the scope of this part (route 7, 10-07 Table 1). (a) cannot be reached by interval arithmetic: the width of subintervals shrinks with $\hbar$ , and covering down to $\hbar\to0$ would need infinitely many subintervals. What is needed is another method, writing the asymptotics as $\hbar\to0$ on paper, and it has not been started.

Moreover, even if a $\hbar$ that fits all could be produced at a single point, the power form of H is a statement about the limit $\hbar\to0$ .

\lim_{\hbar\to0}\frac{\ln C_P(\hbar)}{\ln\xi(\hbar)}=2 \tag{2}

(2) does not follow from an inequality on any finite window. Computer assistance in the window only checks the inequality at each $\hbar$ in the window and says nothing about the outside of the window (10-08 §10). So to close the first step, every row of Table 1 has to be stated "in a form uniform as $\hbar\to0$ ", and the role of computer assistance is limited to taking on a finite part within such a uniform proposition (for example, the side $\hbar\ge\hbar_1$ ).

Example. Even if [013] is closed with 23–39 hours, what is closed is the continuum $0.005\le\hbar\le0.2$ . [014] remains untouched, and the limit of (2) is not a single step closer. What computing time can buy is "between the nodes", not " $\hbar\to0$ ".

Conditions common to the four conditions — five premises

Apart from the work row by row in Table 1, there are five premises that bear on all four conditions. None of them disappears when a single row is closed; they are conditions to be written into the statement when putting the first step in a form that can be called "closed".

Table 4 Premises common to the four conditions
Symbol	Premise	Present state	What is needed to close it
C1	Kind of number	The reading rows are floating-point values (the 19 Reading rows of the integrated table)	Replace the reading rows among the premises of the conclusion by upper and lower bounds from interval arithmetic or paper (the reading rows of Table 1)
C2	Bibliography of the cited theorems	Of the 26 main citations, 19 have their statements checked in an original or secondary source. 4 to obtain, 3 rows changed into assumptions (10-14 Table 2)	Obtain the 4 items and check their statements. If they cannot be obtained, they stay "Assumption"
C3	The range of machine checking	Lean covers only algebra and arithmetic. Analytic theorems are input as assumptions (10-13 §05)	Widening the range of machine checking does not reduce the conditions of the conclusion. What reduces the assumption rows is paper and bibliography
C4	A $\hbar$ that fits all	There is no $\hbar$ at which all rows hold (§05)	Another method to extend (H1) to $\hbar\to0$
C5	Limit and regularity	Existence of the limit $T\to\infty$ , regularity, and simplicity of the first eigenvalue are assumptions ([001] of the integrated table)	The same as [001] of Table 1

The distance from the second step on — three levels of the goal, and the number of steps in β

The distance of the first step was the list of rows of Table 1. From the second step (the four-dimensional lattice, non-abelian) on, the statements are not divided finely enough to be written as a list of rows. This section lays out what the goal on the four-dimensional lattice looks like when written at three levels, and the number of steps in $\beta$ , as a rough guide by physical extrapolation. What can be expected to carry over from the first step to the second step and beyond is only the form of the statements and the form of the tools; why the constants and the range of $\hbar$ do not carry over is in 10-03 §06.

"The mass of the lightest particle is positive" is the statement on the spectrum of the quantum Hamiltonian $H$ , $\operatorname{spec}(H)\subset\{0\}\cup[\Delta,\infty)$ , $\Delta\gt0$ (4-11). On the road entering from the lattice, the same goal splits into three levels.

\text{(T1)}\ \ \xi(\beta)\lt\infty\ (\forall\beta),\ \ \xi(\beta)\to\infty\ (\beta\to\infty);\qquad \text{(T2)}\ \ \frac{m_{\rm lat}(\beta)}{a(\beta)}\to\Delta_\infty\in(0,\infty);\qquad \text{(T3)}\ \ \text{gap positive in the continuum theory} \tag{3}

Table 5 Three levels of the goal, and their relation to Assumption H
Level	Content	What is added to the level above	Status in four dimensions	Relation to Assumption H
(T1) Lattice	Correlation length finite at each $\beta$ , diverging as $\beta\to\infty$	—	Unsolved (Chatterjee's Problem 5.1) Cited	H is a statement that presupposes the latter half of (T1)
(T2) Physical units of the lattice	$m_{\rm lat}/a$ tends to a positive finite value	How the lattice spacing $a(\beta)$ is determined	Unsolved	If H↑ and the one-sided bridge (10-01 §06) are both in place, the lower half follows
(T3) Continuum	A non-trivial theory on $\mathbb R^4$ with positive gap	Existence of the limit (S3), rotations (S6), the OS axioms (S7), non-triviality (S8)	Unsolved (the problem statement)	Does not follow even with H and the bridge in place (10-01 §08)

Next, we measure the gap between the $\beta$ where there are proofs and the $\beta$ on the side of the continuum limit by the number of steps that coarsen the lattice by a factor of 2. It is a physical extrapolation that extends two-loop asymptotic scaling, matched to the value $\xi=1/0.514$ of Lucini–Teper at $\beta=2.60$ ; it is not a distance to a theorem.

\xi(\beta)\ \propto\ \Bigl(\frac{6\pi^2\beta}{11}\Bigr)^{-51/121}e^{3\pi^2\beta/11},\qquad k=\log_2\frac{\xi_{\rm target}}{\xi_{\rm start}} \tag{4}

Table 6 Target correlation lengths, and the $\beta$ and number of steps needed Reading (values of the two-loop extrapolation. The window end is the end of the numerical window, $\xi=1.95$ ; the curvature end is $\xi\approx0.093$ , the end of $\beta\lt0.2722$ that comes from the lower bound of the curvature after one-link integration)
$\xi_{\rm target}$	$\beta$ (two loops)	From the window end	From the curvature end
10	3.243	2.36	6.75
100	4.136	5.68	10.07
1000	5.022	9.01	13.39
$10^4$	5.903	12.33	16.72
$10^6$	7.654	18.97	23.36

The numbers in Table 6 are all finite. Even so, the distance does not shrink, because what a theorem needs is not "up to some target" but "at every target, with the same constants". In metaphor, what is missing is not the number of steps on the staircase but the guarantee of not missing one's footing on any step. Improving the constants of the theorem on the strong-coupling side ( $\beta\lt1/12$ of Shen–Zhu–Zhu Cited) only widens the range of $\beta$ in hand and does not touch this uniformity. The curvature end $0.2722$ is a threshold that comes from the lower bound of the curvature (Lean), and the step towards the gap is a sketch Assumption.

Put back into the language of the first step, the same thing happens with $\hbar$ . As in §05, however wide the window of $\hbar$ is made, the power form of H (2) does not come out; what is needed is a proposition in a form uniform as $\hbar\to0$ . On the four-dimensional lattice, a form uniform as $\beta\to\infty$ ; in the first step, a form uniform as $\hbar\to0$ — on every step of the staircase, the shape of what is missing is the same.

Detour: the lesson of φ⁴₄ — even if the limit is reached, the place one arrives at may be trivial

Detour

A subsection you may skip. For the four-dimensional scalar field $\phi^4$ , cutting it off on a lattice and taking the limit naturally lands on a Gaussian (the free field) (Aizenman–Duminil-Copin 2021, 6-08) Cited. Even if the limit exists and the mass stays finite, if the interaction has vanished it is not a solution in the sense of the problem statement. This is why non-triviality (S8) is written separately in (T3) of Table 5, and it is the same caution as H↓ being a different statement from non-triviality (10-01 §06). In metaphor, one may think one has reached the summit and find it is flat ground.

How to read the distance — a one-line conclusion

We put the distance of this chapter into a single diagram of arrows.

\underbrace{\text{each row of the first step}}_{\text{Table 1}}\ \xrightarrow{\ \hbar\to0\ \text{in uniform form}\ }\ \text{power form of H (first step)}\ \cdots\ \underbrace{\text{theorems for each }\beta\text{ value}}_{\text{in hand}}\ \xrightarrow{\ \text{uniformity (H↑, bridge, band statement)}\ }\ \text{(T2)}\ \xrightarrow{\ \text{(S1)(S3)(S6)(S8)}\ }\ \text{(T3)} \tag{5}

The dotted line is not an implication; it shows that a statement of the same form is placed at two steps. None of the arrows of (5) has been proved. In the first step we have gone as far as listing the rows before the arrow, dividing them into two roads and attaching a kind of work to each row; the dividing is an arrangement, not a shortening of the arrow.

In one line, what is missing is not the number of steps but uniformity: holding with the same constants at every scale. In the first step it appears as (H1) and the row of the core uniform as $\hbar\to0$ ; on the four-dimensional lattice, as H↑ and the bridge uniform as $\beta\to\infty$ — the same thing in different forms.

This part placed Assumption H at the axis and divided that uniformity into the rows of the first step. Roads that ask about the same uniformity from viewpoints other than H are taken up in Part 11. A single table that rearranges, for each of the four steps of the staircase, the hurdle, the known reach, whether the tools of this book reach it, and the kind of cost is in 11-07 §04.

What this chapter can say

Can say	Cannot say
That the rows needed for the conclusion [111] split into two roads at the fork [103]. Road A has 15 assumption rows and 3 reading rows, road B 20 assumption rows and 3 reading rows (counted from the data of the integrated table)	Which road is easier. The number of rows does not show weight
The kind of work needed to close each row (interval arithmetic, paper, bibliography, measurement, another method, method undecided), and the cost estimates in the record	Whether the estimates are right. None of them has been run. The cost of rows without estimates
That there is no $\hbar$ at which all rows are in place, that bringing them into line needs another method to extend (H1) to $\hbar\to0$ , and that computer assistance in the window does not reach (2)	The asymptotics of (H1) as $\hbar\to0$ . (2) itself
Three levels of the goal from the second step on, and the number of steps by the two-loop extrapolation (reading) Reading	Table 6 is a rough guide by extrapolation, not a theorem on the correlation length. This part shows none of the three levels

Sources and reproduction

Item	Kind	Source, tool
Numbers, contents and grades of the rows of Table 1	This book	10-12 Table 1. The source of each row is in the source column of the integrated table
Work and cost estimates of Tables 1 and 3	This book	The sections on what remains in each chapter (10-06 §07, 10-07 §04, 10-09 §05, 10-11 §07). All are estimates; nothing has been run
Counts of Tables 1 and 2 and Figure 1	Text processing	A small script that adds a column of costs to the same row and edge data as the integrated table, chooses one side of [103], and collects nodes from the conclusion in the direction of the premises (not a mathematical computation)
(3), Table 5, Problem 5.1	Cited	A. Jaffe, E. Witten, Quantum Yang–Mills Theory, Clay Mathematics Institute (2000). S. Chatterjee, arXiv:1803.01950 §5
(4), Table 6	Reading	Two-loop coefficients (source table of 4-06) and the value at $\beta=2.60$ in Table 1 of B. Lucini, M. Teper, JHEP 0106 (2001) 050, hep-lat/0103027. The extrapolated numbers are copied from the table of the old version
The strong-coupling range $\beta\lt1/12$	Cited	H. Shen, R. Zhu, X. Zhu, Commun. Math. Phys. 400 (2023) 805–851, arXiv:2204.12737 (source table of 10-01)
Triviality of $\phi^4_4$	Cited	M. Aizenman, H. Duminil-Copin, Ann. of Math. 194 (2021) 163 (source table of 6-08)

Terms used in this chapter — distance (a list), fork, road A (the core on the box side), road B (the core on the sector side), kinds of work (interval arithmetic, paper, bibliography, measurement, another method, method undecided), a $\hbar$ that fits all, three levels of the goal, uniformity. Their meanings are in the A-02 glossary.

Next chapter: 10-16 What is not claimed — what this part does not show, what should not be read strongly, and what is not called false either, with a table of correspondence to sentences of the text.

Revision 2026-10-09: first edition (inheriting the three levels, the number of steps and the detour from the former 10-11 "The distance to a theorem").