d4Max: Multi-Domain Methods in Science and Engineering

• d4Max is a multifaceted term describing high-dimensional maximal structures, from maximum weighted model counting in quantum circuits to maximal supergravity in four dimensions.
• It also denotes a fully fourth-order, energy-stable finite difference method for Maxwell–Drude systems and optimal combinatorial constructions in finite-group theory and Hamming cubes.
• d4Max further appears in computational chemistry and algorithm design, linking degree-4 bottlenecks in MaxSAT problems and dispersion corrections with high precision.

Searching arXiv for papers associated with “d4Max” and its major technical usages. In the cited literature, the label d4Max is used in several distinct senses rather than as a single standardized object. The most concrete usage is as a maximum weighted model counter employed as a backend for quantum circuit synthesis, but the same label also appears as shorthand for maximal supergravity in four space-time dimensions, for a fully fourth-order accurate energy-stable finite difference method for Maxwell–Drude systems, and for several notions of “$d=4$ maximality” in algebra and combinatorics (Zak et al., 1 Aug 2025, 0705.2101, Sakkaplangkul et al., 2019, Lucchini et al., 2023). Across these domains, the common motif is not a shared formal definition but a field-local abbreviation attached to high-dimensional, degree-4, or four-dimensional maximal structures.

1. d4Max as a maximum weighted model counter

In quantum circuit synthesis, d4Max is the backend solver used for maximum weighted model counting (MWMC). The underlying weighted model counting problem is defined on a CNF $F(A)$ with literal-weight function $W$, with

$$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$

MWMC extends this by splitting variables into disjoint sets $A$ and $B$, and asking for an assignment to $A$ that maximizes the weighted model count over the remaining variables. In "Reducing Quantum Circuit Synthesis to #SAT" (Zak et al., 1 Aug 2025), d4Max is used to optimize over gate-selection variables in Clifford$+T$ synthesis, so that exact and approximate depth-optimal synthesis reduce to a single MWMC instance.

The solver was extended in that work to support negative weights for Pauli-basis encodings and complex weights for computational-basis encodings. The implementation uses arbitrary precision arithmetic from the GMP library, as in the original version of d4Max. Because upper bounds obtained by treating subformulas as tautologies are no longer straightforward in the presence of negative weights, the original pruning optimization was disabled; the ability to compute intermediate approximations before processing all connected components was also removed, making the modified solver slower at providing intermediate solutions (Zak et al., 1 Aug 2025). Within the synthesis reduction, exact equivalence is encoded by cyclic or linear-cyclic constraints, while approximate synthesis is expressed through the Jamiołkowski fidelity

$Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$

with the computational-basis formulation maximizing $|c|^2$ and the Pauli-basis formulation maximizing a real weighted count (Zak et al., 1 Aug 2025).

2. d4Max as maximal supergravity in $F(A)$0

In supergravity, d4Max is used as shorthand for maximal $F(A)$1 supergravity, i.e. $F(A)$2 supergravity in four space-time dimensions. The ungauged theory contains the graviton, eight gravitini, 28 electric vectors together with their magnetic duals in duality-covariant form, 56 spin-$F(A)$3 fermions, and 70 real scalars parametrizing the coset

$F(A)$4

The general gauging is encoded by an embedding tensor $F(A)$5, subject to a linear constraint selecting the $F(A)$6 of $F(A)$7 and quadratic constraints enforcing gauge closure and mutual locality. In the universal electric/magnetic-covariant formulation, magnetic charges require dual gauge fields and two-form tensor fields in the adjoint of $F(A)$8, yielding a frame-independent bosonic Lagrangian and scalar potential (0705.2101).

A separate line of work studies whether $F(A)$9 is exceptional from the viewpoint of ultraviolet counterterms. "Is $W$0 Maximal Supergravity Special?" argues that candidate counterterms below a critical loop order $W$1 are only linearized or harmonic-superspace invariants and therefore break nonlinear local supersymmetry and local $W$2 symmetry; adding them would be BRST-inconsistent (Kallosh, 2023). The critical loop order is

$W$3

when $W$4 is not an integer, and for $W$5 one has $W$6. Thus $W$7, so the first eligible geometric counterterm in $W$8, $W$9 begins at eight loops. The paper emphasizes that divergences with $$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$0 occur in maximal supergravities for all integer $$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$1, but not for $$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$2 in $$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$3 so far, which renders the four-dimensional case special (Kallosh, 2023).

3. d4Max as a fully fourth-order Maxwell–Drude discretization

In numerical electromagnetics, d4Max denotes a fully fourth-order accurate, energy-stable finite difference method for time-domain Maxwell’s equations in Drude metamaterials. The continuous model is the source-free, collisionless Maxwell–Drude system for electric and magnetic fields $$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$4 together with polarization and magnetization current densities $$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$5, with periodic boundary conditions and $$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$6 (Sakkaplangkul et al., 2019). The formulation is reorganized into two decoupled second-order pairs, $$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$7 and $$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$8, and the corresponding continuous energies are conserved under periodic boundary conditions.

The discretization combines a fourth-order staggered spatial layout, generalizing Yee staggering, with a modified equation approach to achieve fourth-order temporal accuracy. The resulting scheme is explicit, two-step, and uses fourth-order curl operators in the principal terms together with second-order operators in the $$\#SAT_W(F) = \sum_{\tau\in \mathrm{SAT}(F(A))} W(\tau),\qquad W(\tau)=\prod_{a\in A} W(a,\tau(a)).$$9 corrections. The fully discrete method preserves a discrete analogue of the continuous energy; in one dimension the positivity argument requires

$A$0

The paper reports fourth-order convergence in one- and two-dimensional periodic tests and relative energy errors $A$1 in the one-dimensional experiments, with long-time stability maintained over extended integrations (Sakkaplangkul et al., 2019). In this usage, d4Max is effectively a label for a high-order, energy-conserving Maxwell–Drude FDTD scheme.

4. d4Max in finite-group theory and $A$2-group structure

In finite-group theory, d4Max refers to finite 4-maximal groups, where a group $A$3 satisfies $A$4 and every proper subgroup $A$5 has $A$6. The paper "On finite $A$7-maximal groups" proves that every finite $A$8-maximal group is supersolvable, and for the non-nilpotent case establishes that $A$9 has the form

$B$0

where $B$1 is a maximal $B$2-pair of rank $B$3 (Lucchini et al., 2023). For 4-maximal groups this yields a detailed rank-3 analysis: if $B$4, then $B$5 has nilpotency class at most $B$6; if $B$7 and $B$8, then necessarily $B$9 and $A$0 is unique up to isomorphism. The nilpotent case reduces to 4-maximal $A$1-groups; $A$2 is a basic example, and for odd $A$3 such groups satisfy class $A$4, whereas for $A$5 class $A$6 examples exist (Lucchini et al., 2023).

A related but more general theory concerns $A$7-maximal $A$8-groups with operator groups. "A note on $A$9-maximal $+T$0-groups I" defines $+T$1 to be $+T$2-maximal for $+T$3-subgroups if $+T$4 for every proper $+T$5-invariant subgroup $+T$6 (Aiech et al., 2022). For odd $+T$7, if $+T$8 is a $+T$9-group acting on $Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$0 and $Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$1 is $Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$2-maximal for $Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$3-subgroups, then $Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$4 has nilpotency class at most $Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$5. For $Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$6, the paper proves

$Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$7

and, more strongly, $Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$8 for $Fid_J(U,V)=|\langle \varphi_U|\varphi_V\rangle|^2,$9 (Aiech et al., 2022). These results feed directly into the structure theory of 4-maximal groups, especially in the 2-group case.

5. d4Max in combinatorics: Hamming cubes and $|c|^2$0-permutations

In extremal combinatorics, d4Max denotes 4-maximal sets in Hamming cubes. A set $|c|^2$1 is $|c|^2$2-maximal if $|c|^2$3 and adding any point increases the diameter. Specializing to $|c|^2$4, the paper "Maximal sets of a given diameter in Hamming cubes" proves that if $|c|^2$5 is 4-maximal and contains no 1-ball, then

$|c|^2$6

It also shows that the $|c|^2$7-dimensional cube

$|c|^2$8

is 4-maximal of size $|c|^2$9, establishing asymptotic tightness of the $F(A)$00 growth rate (Bukh et al., 14 Jul 2025). In the binary case, the even-$F(A)$01 construction gives a 4-maximal set of size $F(A)$02, matching the cube size $F(A)$03 for $F(A)$04 (Bukh et al., 14 Jul 2025).

A different combinatorial usage appears in higher-dimensional permutations. "Max-tree for $F(A)$05-permutations and pattern avoidance" defines a generalized max-tree with $F(A)$06 children, one for each direction in $F(A)$07 having last sign negative (Muller, 18 May 2026). For $F(A)$08, the unrestricted construction therefore has eight child directions. When restricted to 4-permutations avoiding $F(A)$09 and $F(A)$10, only the four prefix-$F(A)$11, suffix-$F(A)$12 directions survive, so the max-tree collapses to a 4-ary tree. The corresponding class is counted by the Fuss–Catalan number

$F(A)$13

with initial terms $F(A)$14 (Muller, 18 May 2026). In this setting, d4Max identifies the $F(A)$15 specialization of a higher-dimensional max-tree bijection.

6. Secondary shorthand usages in chemistry and algorithms

In computational chemistry, one compact technical briefing uses d4Max as a label for the D4 dispersion correction as employed in r2SCAN-D4 (Ehlert et al., 2020). In that formulation, D4 is a semi-classical, atom-pairwise London dispersion correction with charge-dependent $F(A)$16 coefficients and an added three-body Axilrod–Teller–Muto term, coupled non-self-consistently to the non-empirical meta-GGA r2SCAN. The specific D4 flavor is D4(EEQ)-ATM, with optimized parameters $F(A)$17, $F(A)$18, and $F(A)$19 Bohr, while $F(A)$20 are fixed (Ehlert et al., 2020). Reported performance includes $F(A)$21 kcal/mol on GMTKN55, $F(A)$22 kcal/mol on MOR41, $F(A)$23 kcal/mol on L7, and $F(A)$24 kcal/mol on DMC8, with the abstract summarizing the method as having “the speed of generalized gradient approximations while approaching the accuracy of hybrid functionals” (Ehlert et al., 2020).

In discrete algorithms, the label is also used in summaries centered on degree-4 or dimension-$F(A)$25 bottlenecks. "Dealing With 4-Variables by Resolution: An Improved MaxSAT Algorithm" focuses on degree-4 variables in parameterized MaxSAT and obtains an $F(A)$26 algorithm by combining resolution, kernelization, and branching rules tailored to $F(A)$27-literals (Chen et al., 2015). "Multivariate Analysis for Computing Maxima in High Dimensions" studies Maxima for $F(A)$28 and gives the deterministic DPC-Maxima algorithm with entropy-sensitive running time

$F(A)$29

and worst-case bound $F(A)$30 (Barbay et al., 2017). These usages do not define a common object named d4Max; rather, they attach the label to technically distinct problems where “4” or “$F(A)$31” marks the difficult regime.