Welcome to the last part: a sketch of the paper's Fig. 2(a) (four parallel lines of slope one half in log ell, one per n; small windows as open markers, large ones as filled squares); by the end of the hour you will read every mark on it, and you already own every block. From Part 2 the Bell (1/2 log(4 pi V)) and the Dial (offsets log n/(2(n-1))); from Part 3 the yellow line chi_zz, a variance per site; Baxter's machine, couplings in, exact inverse length M out; the two lightest particles, a pair of walls and the pink rung; the train of small tables the computer uses. One job: put the six blocks together and check against the computer three times. Route card: three ingredients, three tests, then the error bar, the bell for real, the knob, the finale; it docks top right.
Part 3 left a question: the Clock's bump has a width V, the charge Bell has a variance; for a pure window they agreed. Vote: for a mixed window, is the Clock's width V equal to the variance of the charge Bell? Yes, always | Only when the window is pure | Never for a mixed window. Expected split between the first two; the napkin answers the second, the chain the third softened.
Reveal: two gauges, V (the Clock's width) and Var(Q_A) (the Bell's variance): a pure pair of spins puts both needles at the same place (V = Var); a mixed pair puts them apart (V differs from Var). Eq. (V): V weighs each pair of eigenvectors of the table by ||^2, times (p_a - p_b)^2; for a pure window every difference is 1 or 0 and V is the variance; mix the window and the weights shrink: on a napkin the second option wins. On the chain both widths are straight lines in ell with the same slope, chi_zz/2, and a constant gap; zoomed out the gap is invisible: V = Var(Q_A)[1 + O(1/ell)]. So 'never' was right, by a constant that never grows. Why: chunks longer than 1/M are independent, logs add, both widths grow by the same amount per chunk, only the edges differ. Black box: transfer matrix ('logs add'), taken on faith.
Assemble: the Bell, Delta S^(2)_A = 1/2 log(4 pi V), V = ?; V is the variance up to the edges (4.2) and the variance is 1/2 chi_zz ell (3.11), so 4 pi V = 2 pi ell chi_zz. Split the two off: 1/2 log(pi ell chi_zz) + 1/2 log 2, and 1/2 log 2 is exactly the Dial's n = 2 offset; turning the Dial gives log n/(2(n-1)): 1/2 at n -> 1, 1/4 log 3 at 3, 1/8 log 5 at 5. The edges cost O(1/(M ell)): the kink's size 1/M laid against the window's two edges. The master formula in three colours, nothing fitted; on faith: 'logs add' and the n -> 1 continuation (Black-box card).
Test 1, the structure of the paper's Fig. 2(a) as a sketch (symbolic axes, unlabelled markers): open markers are small windows with their tables built from the charge drawers of Part 2 (shape = n: circle 2, diamond 3, triangle-down 5, triangle-up n -> 1); filled squares are large windows where the computer ran the Clock on the infinite chain. The master formula's lines (slope 1/2 on the log axis, intercept from the variance per site chi_zz and the Dial, nothing fitted) for n = 2, 3, 5 and n -> 1: four parallel lines, the points fall onto them. Headline: the gap between the n = 2 and n = 3 lines is 1/2 log 2 - 1/4 log 3, a pure number; measured on the same state the gap lands on it. The small windows sit a little above their lines; that excess delta_n against 1/(M ell) falls on one trend into the origin for small and large windows (inset: n = 2 and n -> 1, whose tail is smaller): the small-window gap is the white tail, no constant left over. Test 1 passed: slope one half, offsets from the Dial, nothing fitted.
Two routes to chi_zz at the home coupling: the correlator sum gives the variance per site directly; the n = 2 asymmetry minus half log ell settles to c_1, and chi_zz = e^(2 c_1)/(2 pi): two routes, one quantity, and the two agree; c_1 < 0 because chi_zz < 1/(2 pi). Repeats: the attractive side (the measured c_1 and the formula's 1/2 log(2 pi chi_zz) agree) and the free line (exact chi_zz = gamma/(1 + gamma), the asymmetry road lands on it); caution: at finite gamma the free line's chi_zz/M = (gamma/(1 + gamma))/atanh gamma is below 1; the star value 1 is the gamma -> 0 limit (4.10).
Part 3's closing sum returns: the variance is a sum over excited states, one term per state; call chi_zz the budget (the yellow outline). The factory's double flip: gamma flips two neighbours at once and the charge jumps by two (Q = q -> q + 2 on one z snapshot of Octo's stripes); the charge leaks, Q-dot = i[H, Q] != 0. One-line identity: for an eigenstate at energy E_k, = i E_k , so each term is a leak rate squared over an energy squared, and for the whole chain, per site, chi_zz = (2/L) sum ||^2 / E_k^2 (chi_zz = 2 Var(Q)/L, the chain version of 3.11's chi_zz = 2 lim Var(Q_A)/ell). Bars over energy, all positive, the lightest tallest (cartoon heights); stacked, they fill the line exactly. Key move: every bar is positive, so keeping only the ones you can compute gives a lower bound; the lightest two are next (4.7). Black box: the sum rule, its one-line step, and the list of the lightest states with a bar taken on faith from 3.10b (heavier states add more positive bars).
Energy axis in units of vM: the pink bound-pair bar at m_B1/M under the two-wall threshold at 2, the orange two-wall hump from the threshold up, the grey heavier tail from 2 + m_B1/M (unknown, positive). The budget divided by M is c; bar plus hump is a lower bound on c, not a prediction; the hump's shape is a function Lukyanov found, its size fixed by one more identity and the free point, nothing fitted.
Headline on the attractive side: the bar's share c_B1 plus the hump's share c_2k is the lower bound; the computer's c sits just above it, inside the thin corridor that 4.9 explains. Sweeping xi from the attractive end to the free point the bar slides up to the threshold and its weight shrinks to nothing, handing its slope to the hump; two lines, one family. Black box: a function Lukyanov found, size fixed by Hellmann-Feynman and the free point.
Baxter (1972) and Johnson, Krinsky and McCoy (1973), drawn as a white machine: two couplings in, an inverse length out; fed the home coupling it gives M exactly, a kink a few sites long, longer than Octo. Light 1: on the XY line (Delta_z = 0) it reduces to M = atanh gamma. Light 2: on log axes the exact M at three values of Delta_z straightens at small gamma with slope pi/(2 theta_0), theta_0 = arccos Delta_z (1 on the XY line, steeper as Delta_z grows; the gamma -> 0 exponents, not fits). Inside are elliptic functions (M = atanh k1', k1' an elliptic modulus); we never open the box and use only its output, exact to any number of digits, never a fitted mass.
Substitute chi_zz = c(beta^2) M (the budget, 4.7) and M's small-gamma law (4.8) into the master formula: Eq. (final), gamma and Delta_z in, the asymmetry out, beta^2 standing in for Delta_z (theta_0 = arccos Delta_z, beta^2 = 1 - theta_0/pi); valid for gamma -> 0 and ell M >> 1. The structure of Fig. 2(b) at fixed ell M: the n = 2 line uses the lower bound for c, n -> 1, 3, 5 are the same curve shifted by the Dial, the star is the free point, the squares (the computer's c, converted) are copied onto every branch by the same rigid Dial shift and ride on the lines or a hair above where the bound is not tight (most visibly at the repulsive end; the two marginal points of 4.9 a hair below). Finite-gamma corrections (a power series and a lattice-only fractional power) are named, used in the error bars (4.10), never shown.
Fig. 1 rebuilt as a sketch: beta^2 axis with the matching Delta_z from the Map on top (no numbers); the star at (1/2, 1), free fermions; the line is the bar plus the hump, a lower bound. Left of the star two more sum rules squeeze the unknown tail into a corridor thinner than a symbol; right of it only the dashed lower edge survives, a hair above the line. Black box: the moment corridor.
Eight squares, the computer's c at the paper's couplings, each a gamma -> 0 extrapolation, honour the line within the combined fit-and-extrapolation uncertainty; the four attractive ones sit inside the corridor. Two repulsive centres sit a hair under the line but the corridor's lower edge lies above the line there, so they are extrapolation systematics, not violations; 4.10 shows how an error bar is made.
Zoom at the most repulsive coupling: the computer's c against the bound, with the corridor's edge just above the line: a clear excess. That is the tail switching on as the square of the distance from the star: the grey parabola, the leading term, coefficient A, a Monte-Carlo estimate, with a rigorous floor from the corridor. Test 3: a lower bound honoured within the uncertainties at eight couplings; the corridor holds at all four attractive ones. (The grey curve is the line plus c_4k; its leading-order term overshoots at the two middle repulsive couplings, since the finite-xi dressing is omitted (paper, four-kink appendix), and lands on the far point. It is an estimate, not a fit.)
At the home Delta_z the computer's c = chi_zz/M at four values of gamma (the home coupling among them) falls as gamma grows (a sketch, unlabelled). Four roads to gamma = 0 (two quadratics, a cubic, the umklapp form with s fixed by Delta_z) arrive at slightly different answers. Adopted: the mean of cubic and umklapp; the error, half the spread of the four; the lower bound sits inside the bar. The computer's own convergence (chi_zz locks as the bond dimension D grows) is far smaller than the road's spread. Black box: the gamma -> 0 extrapolation forms.
The structure of Fig. 5 at the home coupling, as two sketches on symbolic axes, in front of the ghosted coins of Part 2 (bump in alpha, Bell in q). Panel (a): -log Z_2(alpha)/ell for windows of growing size falls onto the parabola (chi_zz/2) alpha^2, the Gaussian saddle of 2.10 that turned V into 1/2 log(4 pi V). Panel (b): P(Q_A) broadens with ell; the solid curves are bells of the measured variance, which tends to ell chi_zz/2. One quantity, chi_zz, governs both panels and fixes the asymmetry. No toy ring in this scene (P4-14).
The structure of the paper's Fig. 3 (a fixed window, the home Delta_z) as a sketch: at gamma = 0 the chain is symmetric and the meter reads zero; turning gamma up the meter rises, ordered by n (n -> 1 on top, then 2, 3, infinity). The ruler 1/M grows as gamma shrinks: shorter than the window at the starting gamma (white, the master formula applies), longer than the window as gamma falls (grey: the white tail takes over); the crossing is where 1/M equals the window. Long windows and small gamma do not commute; this plot is the lattice echo of that. The paper's curves are iMPS data for a window in the infinite chain, not an exact-diagonalization sweep.
Eq. (final) with every light on: 1/2 log(pi ell) the Bell (2.10, 4.3); c(beta^2) the budget, the bar and the hump (4.6-4.7); the bracket with gamma Baxter's box (4.8); the gold term the Dial (2.11); chi_zz = c(beta^2) M. Two legs: chi_zz at the home coupling by two routes, the correlations and the asymmetry (chi_zz = e^(2 c_1)/(2 pi)), and the two agree; c on the attractive side (gamma -> 0): the computer's c inside the corridor. The paper's Discussion sentence. Take-home: the meter reads half the log of the variance, nothing fitted; the variance is a budget two particles bound from below; Baxter's exact length closes the formula in lattice variables.
Triptych: general (the meter grows as half the log of the window; for the paper's states, half the log of the charge variance plus the Dial), specific (chi_zz = c(beta^2) M, M exact, c bounded below by two particles), open (the tail, Delta_z > 1, fields, quenches and the Mpemba baseline). The Caveats, read by title only, page 1 of 3: V = Var(Q_A) on faith; which of the two measured widths is V; Part 2's faith items (2023 definition, Mpemba, Delta S >= 0); Part 3's faith items (Jordan-Wigner, Hastings-Koma, the continuum dictionary, iDMRG); the Dial's replica integral and the n -> 1 continuation.
Caveats, page 2 of 3: the sum rule's step, the bars' states and Lukyanov's function, Hellmann-Feynman; the Fig. 1 line is a lower bound and the corridor is two-sided only for xi < 1; the four-kink A is a Monte-Carlo estimate with sizeable truncation systematics over a rigorous floor from the corridor; M from Baxter/JKM, an inverse length; Eq. (final)'s range of validity.
Caveats, page 3 of 3: error bars from four fit forms, the two marginal points stay out; on the free line at finite gamma chi_zz/M = (gamma/(1 + gamma))/atanh gamma is below 1 (chi_zz = gamma/(1 + gamma), M = atanh gamma); Fig. 3 and Fig. 5 are sketches; the free-line run's coupling is inferred; four helper symbols beyond the roadmap's four.
The open list: Delta_z > 1 beyond the Neel-x phase (staggered breaking, the same marginality); a field coupled to the broken charge; after a quench, Eq. (final) is the equilibrium baseline for Mpemba-type behaviour; the tail (resum the four-kink term, a repulsive-side f-sum rule would close the corridor). Cliffhanger: what does the meter read beyond the Neel-x phase, in a field, or after a quench? That is the research frontier, where this series ends (all four dots lit).