A Discrete-Substrate Perspective on the Yang–Mills Mass Gap Problem

The Clay Yang–Mills mass gap problem asks for a constructive quantum field the-

ory on R4 for any compact simple gauge group G, satisfying the Streater–Wightman or

Osterwalder–Schrader axioms, and exhibiting a strict mass gap ∆ > 0. We argue that

the Holographic Circlette (TCH) discrete-substrate framework based on the bipartite ten-

sor network Z3 ⊗Q3 does not solve this problem — and cannot, because the framework

is discrete from the start, satisfies Lorentz invariance only emergently, and is built around

SU(2)χ ×SU(3)c rather than arbitrary G. However, we identify three structural insights

that complement the constructive–QFT programme. First, in the TCH framework a finite

mass gap is the structural default and gaplessness requires a symmetry reason: the T1u

photon multiplet is protected by a representation-parity theorem, and only the E = +1

transmission resonance of the macroscopic gauge web admits a flat-band crossing. Second,

confinement arises mechanistically from a 3D string-tension argument on the parity-check

graph of the [[8,4,4]] code, giving a substrate-level lower bound for closed-loop excitations

(the structural analogue of a glueball mass floor). Third, the framework’s finite-dimensional

local Hilbert space places it in the decidable regime of the Cubitt–P´erez-Garc´ıa–Wolf unde-

cidability theorem, providing worldview-level evidence that the Yang–Mills mass gap is in

principle answerable rather than blocked by Turing undecidability. We conclude that the

framework offers a structural complement to constructive QFT, not a substitute, and that

the open targets it identifies (§15 items 71 and 73 in the framework’s anchor document)

constitute concrete next steps for a substrate-level approach to the Clay question.