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. 2026-06-20 legacy canon revision: This is a canon-reconciled legacy version. SMG/TCH continuum frontier changed; likely stale The paper retains its historical derivation trail but carries a 2026-06-20 canon revision note identifying current status and superseded claims. 2026-06-21 canon refresh: This version incorporates the 2026-06-21 ANCHOR/DRIFT/PTMS canon refresh and rebuilt local PDF.