We propose a novel computational interpretation of the Yang–Mills mass gap problem using a bounded information framework informed by Resolution Memory Theory (RMT) and the Bounded Simulation Framework (BSF).
Our contribution is not a conventional proof, but a demonstration that the apparent mass gap in non-abelian gauge fields arises naturally within any simulation constrained by finite symbolic memory and resolution limit. Within such systems, confined gluon-like excitations acquire effective mass through information-curvature compression and symmetry-bound scroll depth.
This model preserves gauge invariance while introducing bounded curvature oscillations as a substitute for infinite-dimensional Hilbert spaces. Though we make no claims of absolute mathematical proof in the conventional sense, we show that within the BSF and RMT environments, massless Yang–Mills fields constrained by resolution bounds cannot remain massless without violating the constraints of confinement and energy quantization. The paper provides structured simulation arguments, citations from both quantum field theory and formal bounded logic, and a testable research pathway forward.
1. Introduction
The Yang–Mills Mass Gap problem remains one of the Clay Millennium Prize challenges. The question, originally posed by Jaffe and Witten [1], asks whether a quantum Yang–Mills theory with a compact non-abelian gauge group in four dimensions admits a mass gap—i.e., whether all excitations have strictly positive mass despite the theory being classically massless.
While mathematical physics continues to seek rigorous constructions within Wightman or Osterwalder–Schrader frameworks [2], we instead explore the question through the lens of bounded symbolic computation, inspired by Resolution Memory Theory (RMT) [3] and our previously formalized Bounded Simulation Framework (BSF) [4]. These systems enforce hard computational boundaries on symbolic scrolls (state evolution traces), simulating how physical law may appear to evolve in finite-resolution computational universes.
2. Framework and Approach
2.1 The Bounded Simulation Environment
Our core assumption, following prior work [4], is that all physically realized computation—including the structure of a gauge field—must operate within symbolic resolution limits. Let R be the highest scroll depth permitted for any symbolic excitation, corresponding to the number of traceable computational steps (akin to Planck-level granularity for discrete systems).
The Bounded Simulation Framework defines a 3-layer architecture: Layer 1: A formally verified bounded arithmetic interpreter [5]
Layer 2: Type-safe representation of tensorial field structures
Layer 3: Scroll simulation with enforced guard constraints (max steps, stack, nat-depth)
2.2 Yang–Mills Representation Under RMT
We simulate SU(3) gauge fields as scroll-constrained symbolic bundles over discrete lattice-like structures. These symbolic bundles are evolved using finite, rule-based propagators analogous to lattice gauge theory [6], but limited by: Scroll length
Symbolic bandwidth
Recursive gate depth This results in an effective curvature bound, expressed not as a field strength, but as an information-theoretic contraction: beyond a certain point, recursive oscillations collapse, unable to sustain infinite symmetry group excitations.
3. Simulation Result: Emergence of the Mass Gap
In a scroll-bounded environment, massless gauge bosons (such as gluons) cannot propagate indefinitely due to: Resolution-bound curvature limits (analogous to UV cutoff)
Recursive collapse at finite stack depth (no infinite ladder of excitation)
Information-dissipation at non-abelian loop junctions (topological bounds) Using symbolic implementations of Wilson loops under RMT-style constraints, we observe: Non-trivial minimum energy configurations
Quantized oscillatory return paths
Lack of long-range correlation beyond a resolution boundary In effect, gluon-like symbolic excitations acquire minimum resolvable energy, which in conventional field theory would manifest as a mass gap.
4. Philosophical Framing and Limitations
We do not claim that this framework proves the existence of a mass gap in SU(3) Yang–Mills theory in the rigorous sense of axiomatic quantum field theory. Instead, we argue:
Any computation of Yang–Mills behavior under bounded symbolic simulation (as implemented in BSF) exhibits a mass gap
The massless assumption breaks under finite recursion and symbolic collapse
This suggests a deep tie between energy quantization and information-bound symmetry This is in line with the modern view of computational irreducibility in physics (Wolfram [7]) and bounded arithmetic in complexity theory (Buss [8]).
5. Conclusion and Future Work
We believe this approach offers a new perspective: the mass gap is not derived from symmetry breaking or renormalization, but from bounded scroll recursion under symbolic simulation. By analyzing the collapse points and resolution curvature of Yang–Mills bundles under bounded simulation, we uncover a framework that necessitates mass gaps.
Future directions include: Formal verification of SU(3) bundle behavior under bounded tensor rewriting
Integration with interactive theorem provers to extend RMT simulation proofs
Educational tools for visualizing gauge symmetry in finite resolution systems