We define and study infinite families of all-loop planar, dual conformal invariant (DCI) integrals, which contribute to four-point Coulomb-branch amplitudes and correlators in ${\cal N}=4$ supersymmetric Yang-Mills theory, by solving ``boxing'' differential equations via \texttt{HyperlogProcedures}~\cite{hyperlogprocedures}; The resulting single-valued harmonic polylogarithmic functions (SVHPL) are nicely labeled by ``binary'' strings of $0$ and $1$ without consecutive $1$'s. These functions are special cases of the so-called generalized ladders studied in~\cite{Drummond:2012bg}, where extended Steinmann relations (no consecutive $1$'s) are imposed due to planarity. Our results can be viewed as ``two-dimensional'' extensions of the well-known ladder integrals to many more infinite families of DCI integrals: the ladders have strings with a single $1$ followed by all $0$'s, and the other extreme, which nicely evaluate to the ``zigzag'' SVHPL functions with alternating $1$'s and $0$'s, are nothing but the four-point DCI integrals from the very special family of anti-prism $f$-graphs (while all other binary DCI integrals lie in between these two extreme cases). We also study periods of these integrals: while their periods are in general complicated single-valued multiple zeta values (SVMZV), the ``zigzag'' DCI integrals from anti-prism gives exactly the famous ``zigzag'' periods proportional to $\zeta_{2L{+}1}$, and empirically it provides a numerical lower-bound for $L$-loop periods of any binary string, with the upper-bound given by that of the ladder (also proportional to $\zeta_{2L{+}1}$). Based on $f$-graphs as a tool for studying these periods, we discuss several interesting facts and observations about these (motivic) SVMZV and relations among them to all loops, and enumerate a basis for them up to $L=10$.
An infinite family of all-loop planar DCI four-point integrals can be solved via boxing differential equations to yield SVHPLs labeled by binary strings (no consecutive 1s); the zigzag end of this family coincides with anti-prism f-graph integrals and has periods proportional to ζ_{2L+1}, while ladders provide a numerical upper bound and zigzag a numerical lower bound on periods of any binary string at L loops.
The lower/upper period bounds (zigzag ≤ binary-string period ≤ ladder) are stated as empirical observations, not proven; extension to all loops relies on extrapolation from finite-L data. Also, identification of the "no consecutive 1s" labeling with extended Steinmann relations is asserted via planarity rather than derived in the abstract.
1/ In planar N=4 super Yang-Mills, four-point dual-conformal integrals on the Coulomb branch satisfy "boxing" differential equations. He & Jiang solve infinite families of these to all loops using HyperlogProcedures, getting single-valued harmonic polylogs. 2/ The integrals are indexed by binary strings of 0s and 1s with no two adjacent 1s (a Steinmann/planarity condition). One extreme — a single 1 — gives the classical ladders. The other — alternating 10101... — reproduces anti-prism f-graph integrals and zigzag SVHPLs. 3/ Their periods (numbers you get integrating over a point) are generally complicated single-valued MZVs, but the zigzag case is exactly ζ_{2L+1}. Empirically the zigzag/ladder periods bracket all other binary-string periods, and they enumerate a motivic SVMZV basis up to 10 loops.
Physicists found a family of hard particle-physics integrals labeled by 0/1 strings, and computed their answers up to ten loops.
Abstract-only review. The paper sits within a well-developed program (DCI integrals, ladder/zigzag periods, f-graphs, SVHPLs, SVMZVs in planar N=4 SYM) and uses an established computational tool (HyperlogProcedures by Schnetz) to solve boxing differential equations. The claims are concrete and falsifiable: (i) an infinite family of all-loop DCI integrals labeled by binary strings without consecutive 1s, (ii) zigzag = anti-prism f-graph integrals evaluate to ζ_{2L+1}, (iii) empirical lower/upper period bounds between zigzag and ladder, (iv) basis enumeration of relevant motivic SVMZVs up to L=10. Items (i),(ii),(iv) are computational/structural claims standard for this community; the empirical bounds (iii) are explicitly empirical and merit the "empirical conjecture" label. I can't verify computations from the abstract; the framework and methods are mainstream hep-th and have been validated in related work (Drummond et al., Brown-Schnetz zigzag conjecture proven by Brown-Schnetz). Confidence LOW because text is unavailable. No obvious red flags from abstract; "empirically" hedge is appropriate. Verdict UNVERDICTED given abstract-only access.