The face-magic hypercube module
Abstract
All-ring coordinates classify unrestricted face-magic hypercubes.
Definition 1.1 (Common face sums).
Formalization. D5/S3/Combinatorics/Graph/Hypercube/FaceMagicModule.FaceMagic (✓ std3).
Citation. Manjul Bhargava (2026). The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes. URL: https://arxiv.org/html/2609.26762v1.
Commentary.
FaceMagic n f means that there is one C in the ring such that every two-coordinate face S, S∪{i}, S∪{j}, S∪{i,j}, with distinct i and j outside S, has sum C. The carrier is the actual function space of arrays admitting such a C; C is not stored in the carrier.
Definition 1.2 (The face-magic submodule).
Formalization. D5/S3/Combinatorics/Graph/Hypercube/FaceMagicModule.faceMagicModule (✓ std3).
Source. Repository-derived.
Acknowledgement. Manjul Bhargava (2026). The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes. URL: https://arxiv.org/html/2609.26762v1.
Commentary.
The arrays satisfying the common-face condition form an R-submodule by adding witnesses and scaling a witness.
Theorem 1.3 (All-ring coordinate classification).
Proof. Machine-checked in Lean as D5/S3/Combinatorics/Graph/Hypercube/FaceMagicModule.face_magic_module_structure (✓ std3). ∎
Resolves. Problems/bhargava-2026-facemagic-module (proved) by D5/S3/Combinatorics/Graph/Hypercube/FaceMagicModule.face_magic_module_structure.
Source. Repository-derived.
Acknowledgement. Manjul Bhargava (2026). The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes. URL: https://arxiv.org/html/2609.26762v1.
Commentary.
The source question is: What is the structure of the R-module U_n(R) of face magic hypercubes of dimension n over R when the characteristic is even and positive? The theorem answers it over every unital ring. In dimension zero evaluation at the empty subset is an R-linear equivalence with R. In dimension one the empty and singleton entries give an equivalence with R × R. For n ≥ 2, evaluation at the empty subset, at each singleton, and on the fixed {0,1}-face gives an equivalence with R × (Fin n → R) × R. The inverse is the explicit integer-coordinate formula: for k = |S| it is (-1)^k ((1-k)a − Σ_{i∈S} b_i + floor(k/2) C), interpreted by additive integer coefficients. No division by two is used, so zero divisors and every characteristic are included. The face identity follows by the parity second difference of floor(k/2), and the inverse law follows by strong induction on the subset cardinality. Bhargava Section 2, equations (4), (8), (9) and Theorem 13 supply the face recurrence and induction method. The empty-entry, singleton-entry and common-face-sum coordinates instead give an integral reconstruction, avoiding the invertibility of two required by the source’s determinant-two coordinates.
For the ordinary Boolean cube with literal complement edges, the exact symmetry model used alongside this module is the following. For every even n with 2 ≤ n, write Bin n = Fin n → ZMod 2 and identify Boolean vertices with Bin n by the coordinatewise equivalence boolVector. The folded graph is the ordinary hypercube joined with the complement relation, and its generators are direction none (the all-one vector) together with direction (some i) (the i-th unit vector), indexed by Option (Fin n). Every actual graph automorphism has one and only one pair (t, π), with t : Bin n and π : Perm (Option (Fin n)), and acts by x ↦ t + (j ↦ E_x(π⁻¹(some j)) + E_x(π⁻¹(none))), where E_x none = 0 and E_x (some j) = x j; every such affine action is an automorphism. The proof uses parity to separate the all-one-plus-unit sums from two-unit sums, the uniform extended-coordinate cancellation formula for composition, and generation by the directions. The ordinary-cube subgroup is exactly π none = none, equivalently π = σ.optionCongr for a unique σ : Perm (Fin n), with coordinate action x j ↦ t j + x (σ⁻¹ j). This symmetry statement is an application boundary for the face-magic module: an arbitrary folded automorphism that moves none may exchange cube edges with complement edges and therefore does not automatically preserve ordinary coordinate-face incidence; only the stabilizer conclusion is used when transporting FaceMagic arrays. The Boolean-to-ZMod2 identification, the all-even family, and the folded action are exact graph facts; no claim about the separate real-array reconstruction, offsets, distinct labels, free orbits, or counting problem is included here.