Paper 83: Yang-Mills Mass Gap under the Rei D-FUMT₈ Lens — A Toy-Model Framework Map (Not a Millennium-Prize Solution)

Nobuki Fujimoto (藤本伸樹)
Independent Researcher, Rei-AIOS Project
ORCID: 0009-0004-6019-9258 · GitHub: fc0web · note.com: nifty_godwit2635
2026-04-16

Companion to:

Paper 75 — Quantum D-FUMT₈ Single-Qubit — DOI 10.5281/zenodo.19595292
Paper 76 — Quantum D-FUMT₈ Multi-Mode Fock Extension — DOI 10.5281/zenodo.19595582

Referenced open problem: Jaffe, A. & Witten, E. (2000). Quantum Yang-Mills Theory. Clay Mathematics Institute Millennium Problem.

Abstract — honest scope

This paper does not claim to solve the Clay Millennium Yang-Mills problem. We clearly state our scope at the outset:

What we do: construct a small finite-dimensional Fock-space toy model with a positive mass gap, map its three structural elements (vacuum, gap, interacting-vacuum projector) onto the corresponding Rei D-FUMT₈ values (ZERO / INFINITY / SELF), and empirically verify positive gap across 9 parameter configurations up to 27-dimensional Hilbert space.
What we do NOT do: construct quantum Yang-Mills on ℝ⁴, take the continuum limit, take the infinite-volume limit, establish confinement, or produce a non-perturbative existence proof. None of these are within reach of a 27-dimensional Fock-space toy.

The paper exists as Phase 2.5 attack target #5 in the Rei-AIOS Discovery Engine workflow. The workflow identified Yang-Mills as the hardest of five attack targets; this paper documents what Rei's current toolchain can and cannot say about it — an honest map rather than a proof.

Keywords: Yang-Mills mass gap, Clay Millennium, toy model, harmonic oscillator, anharmonic perturbation, D-FUMT₈, bosonic vacuum, projector, honest limitations, Rei-AIOS, Phase 2.5

1. The Clay statement and the honest boundary

The Clay Millennium Prize problem for Yang-Mills (Jaffe–Witten 2000) asks: prove that for any simple compact Lie group G, a non-trivial quantum Yang-Mills theory exists on ℝ⁴ and has a mass gap Δ > 0 — that is, every state in the physical Hilbert space except the vacuum has energy ≥ Δ.

The problem requires, at minimum:

Construction of the quantum Yang-Mills measure (non-perturbatively, on ℝ⁴).
Identification of the physical Hilbert space.
Proof that the vacuum is non-trivial and unique.
Proof that the gap Δ > 0 in the continuum and infinite-volume limits.

None of 1–4 are accessible via finite-dimensional Fock-space numerics. We take this as given.

What we provide instead is a framework map: identify which of Rei-AIOS's already-established D-FUMT₈ correspondences would apply if a non-perturbative Yang-Mills proof were available, and verify that they are self-consistent at the toy-model level.

2. Toy lattice Hamiltonian

On a lattice of N sites with truncated Fock dimension dim per site, we study

H = Σᵢ aᵢ† aᵢ                        (harmonic part, N copies)
  + g · Σᵢ (aᵢ + aᵢ†)⁴ / 24            (quartic self-coupling)
  + J · Σ_{⟨i,j⟩} (aᵢ + aᵢ†)(aⱼ + aⱼ†)  (nearest-neighbour coupling)

This is a Yang-Mills-like structure in that it has (i) a free bosonic spectrum (harmonic part, which realises the gap), (ii) a quartic self-coupling (the A⁴ analogue of Yang-Mills field-strength squared), and (iii) nearest-neighbour coupling (the discrete analogue of derivatives in the kinetic term). It is obviously not Yang-Mills itself; no gauge symmetry is imposed, and there is no continuum limit.

3. Numerical results

Nine configurations tested, Hilbert dimensions 8–27:

Label	dim	E₀	E₁	Δ = E₁ − E₀	gap > 0
1 site free	8	0.0000	1.0000	1.0000	✓
1 site anharm g=0.5	8	0.0508	1.2331	1.1824	✓
1 site anharm g=2	8	0.1492	1.6389	1.4897	✓
2 sites free	16	0.0000	1.0000	1.0000	✓
2 sites anharm+J=0.3	16	0.0325	0.8690	0.8365	✓
2 sites anharm+J=0.5, g=1	16	0.1166	1.0955	0.9790	✓
3 sites free	27	0.0000	1.0000	1.0000	✓
3 sites anharm+J=0.3	27	0.0186	0.5856	0.5669	✓
3 sites strong coupling	27	−1.3358	−1.3210	0.0148	✓

All 9 configurations exhibit a positive gap. The strong-coupling 3-site case has the smallest gap (0.0148) — suggestive of a critical-coupling boundary beyond which the harmonic dominance is lost and the gap may close. We do not explore this boundary here.

4. D-FUMT₈ correspondence (framework map)

For each structural element of the Yang-Mills setup, we state the corresponding Rei D-FUMT₈ value and the rationale:

Element	D-FUMT₈	Rationale
vacuum (ground state E₀)	ZERO	Paper 76 STEP 800: bosonic vacuum satisfies ⟨n̂⟩ = 0; our E₀ is the analogue.
mass gap Δ	INFINITY	Paper 76 STEP 800: harmonic tower gap invariant across Fock truncations; our toy replicates this at finite N, dim.
interacting vacuum projector P₀ = \|Ω⟩⟨Ω\|	SELF	Paper 76: projector idempotency P₀² = P₀; trivially satisfied for any pure eigenstate.
non-abelian color superposition	BOTH	Paper 75: \|+⟩ = (\|0⟩+\|1⟩)/√2 analogue; NOT implemented in our toy.
confinement (r → ∞ linear potential)	NEITHER	Requires infinite-distance behaviour; outside toy-model scope.

The first three rows are numerically verified on the toy. The last two are honest placeholders — the D-FUMT₈ correspondence anticipates where a full Yang-Mills treatment would place them, but our small Fock space cannot instantiate them.

5. What this does NOT prove — explicit disclaimer

We emphasize that Paper 83 does NOT claim:

Non-perturbative existence of quantum Yang-Mills on ℝ⁴.
Construction of the physical Hilbert space.
Mass gap in the continuum limit (lattice spacing → 0).
Mass gap in the infinite-volume limit (N → ∞).
Confinement (linear rising potential for static quark-antiquark pair).
Asymptotic freedom.
Resolution of the Clay Millennium Prize problem.

What we DO claim:

At fixed small (N, dim, g, J), a positive gap exists in our toy Hamiltonian. Empirical for 9/9 configurations.
The gap's presence is dominated by the harmonic part; anharmonic and coupling terms shift but do not close it in the tested parameter range.
The three structural elements (vacuum, gap, vacuum projector) map onto Rei D-FUMT₈ values (ZERO, INFINITY, SELF) consistently with Papers 75–76.
Two further elements (non-abelian color, confinement) have honest placeholder assignments (BOTH, NEITHER) that anticipate a full treatment.

6. Why a Millennium-Prize proof is out of reach

The gap between our toy and a real proof is vast. We summarise the missing pieces:

Missing	Why hard
Continuum limit `a → 0`	requires renormalization group flow analysis, beyond numerical Fock truncation
Infinite volume `N → ∞`	requires thermodynamic-limit existence, beyond finite lattices
Gauge invariance	our toy is scalar; Yang-Mills is gauge-covariant
Non-abelian structure	our toy is abelian (scalar Fock); true Yang-Mills is SU(2)/SU(3)/…
Confinement	requires IR behaviour at r → ∞, fundamentally inaccessible at our scale
Rigorous measure	requires constructive QFT, non-perturbative analysis of path integrals

Each of these is a research-programme-level challenge. Our work is analogous to computing the energy levels of a particle in a harmonic well and claiming it as a "toy demonstration" that atomic spectra exist: the analogy is correct, the analogy is not a proof.

7. Connection to Rei prior work

Rei result	Connection
Paper 75 (single-qubit anchor)	BOTH = \|+⟩ is the non-abelian-color placeholder
Paper 76 (multi-mode Fock, STEP 800)	Source of ZERO (vacuum), INFINITY (gap), SELF (projector) correspondences
Paper 78 (p-adic FLOWING)	Tangentially related: strong coupling regime may admit p-adic rescaling, not explored
Paper 82 (Erdős discrepancy)	Dual: Erdős is about unbounded growth in AP partial sums; Yang-Mills is about bounded gap in quantum spectrum

The Rei D-FUMT₈ anchor for Yang-Mills is thus a consistent re-statement, not a derivation. It shows that if the Clay statement is resolved, the Rei framework can accommodate the three primary structures (vacuum, gap, projector) without modification — a framework-consistency check.

8. Reproducibility

scripts/step807-yang-mills-toy-gap.py    Pure Python + QuTiP, ~1 minute
data/step807-yang-mills-toy.json         Full numerical output

python scripts/step807-yang-mills-toy-gap.py

Required: Python 3.10+, qutip 5.2+, numpy, scipy.

9. Limitations (restated for emphasis)

No continuum limit. Lattice spacing is fixed at 1 (a ≡ 1); a → 0 not taken.
No thermodynamic limit. N ∈ {1, 2, 3} only; N → ∞ not taken.
No gauge symmetry. The toy is a scalar field, not a gauge field.
No colour. The toy is abelian; SU(N) structure absent.
Strong-coupling regime only tentatively explored. The 3-site strong-coupling case (gap 0.0148) suggests an approaching phase transition but we do not characterise it.
D-FUMT₈ correspondences for BOTH and NEITHER are placeholders, not derived from the toy.

10. Conclusion

Phase 2.5 attack target #5 (Yang-Mills mass gap, Clay Millennium Prize) is documented as a framework map, not a proof attempt. A small Fock-space toy (up to 27 dimensions) exhibits positive mass gap across 9 configurations; the three primary structural elements (vacuum, gap, vacuum projector) map onto Rei D-FUMT₈ values (ZERO, INFINITY, SELF) consistently with Papers 75–76. We explicitly list the seven capabilities required for a real Yang-Mills proof and declare each out of scope. The paper's purpose is to close Phase 2.5 honestly: the Rei toolchain can re-state the Yang-Mills problem in its own vocabulary, but cannot solve it at current scale. Future extensions (larger N, Lie-group imposed gauge structure, renormalisation-group flow) are natural Phase-3 directions but not attempted here.

References

Jaffe, A. & Witten, E. (2000). Quantum Yang-Mills Theory. Clay Mathematics Institute Millennium Problem.
Yang, C. N. & Mills, R. L. (1954). Conservation of isotopic spin and isotopic gauge invariance. Phys. Rev. 96, 191.
Wilson, K. G. (1974). Confinement of quarks. Phys. Rev. D 10, 2445. (Lattice gauge theory.)
Fujimoto, N. (2026). Quantum D-FUMT₈. Paper 75, DOI 10.5281/zenodo.19595292.
Fujimoto, N. (2026). Quantum D-FUMT₈ Multi-Mode Fock Extension. Paper 76, DOI 10.5281/zenodo.19595582.
Fujimoto, N. (2026). Erdős Discrepancy Problem × Rei D-FUMT₈. Paper 82, DOI 10.5281/zenodo.19599940.
Johansson, J. R., Nation, P. D., & Nori, F. (2013). QuTiP: An open-source Python framework for the dynamics of open quantum systems. Comput. Phys. Commun. 184, 1234–1240.

Rei-AIOS Project. Peace Axiom #196: immutable = true.
Honest scope: toy model + framework map. NOT a Millennium-Prize solution.