>[!warning] >This content has not been peer reviewed. # Yang–Mills existence and mass gap — RST application The **Yang–Mills existence and mass gap** problem (Clay Millennium Prize) asks for a proof that (for a suitable gauge group, e.g. SU(3)) **quantum Yang–Mills theory** exists as a well-defined quantum field theory and exhibits a **mass gap** — i.e. the spectrum has a gap above the vacuum (no massless gluons in the physical spectrum). In RST, the **gauge sector** is **relational maintenance**: gauge degrees of freedom are the substrate’s way of keeping relations consistent across scales. The **mass gap** is the **resolution floor** — the substrate cannot sustain arbitrarily low-energy (long-wavelength) excitations in the gauge channel without paying a minimum workload; that minimum is the mass gap (e.g. glueball scale, or \(\Lambda_\text{QCD}\) in the strong sector). --- ## I. Mapping Yang–Mills to the relational axioms | YM / QFT | Relational identity | RST axiom | |:---|:---|:---| | **Gauge field / connection** | Relational “glue” — how the format keeps relations consistent between neighbouring nodes. | A1 (Format), A3 (Translation) | | **Gauge invariance** | Redundancy of description; the substrate’s accounting is invariant under re-parameterisation of the same relational state. | A2 (Resolution) | | **Mass gap** | **Minimum workload** to excite the gauge sector — no “free” signal at zero cost; resolution floor (like \(a_0\) for gravity). | A2, Noise Floor | | **\(\Lambda_\text{QCD}\)** | Strong-sector scale where fidelity drops (confinement) — already in RST as a substrate eigenvalue. | RST Core Reduction, Depth Identity | --- ## II. RST reading - **Existence:** A well-defined YM theory = the substrate’s gauge “bookkeeping” is consistent and finite (no infinite renormalisation in the relational picture — the cutoff is the substrate’s resolution). - **Mass gap:** The substrate cannot render **zero-cost** gauge excitations; there is a minimum **noise floor** in that channel, so the spectrum is gapped. Analogous to \(a_0\) in gravity: a minimum scale below which the substrate does not “resolve” the field. - **Link to RST eigenvalues:** \(G\), \(\Lambda_\text{QCD}\), \(M_W\) etc. are derived as **substrate eigenvalues**; the mass gap in the strong sector is then the natural scale of that eigenvalue. RST does **not** supply a constructive proof of YM existence or mass gap in the sense of the Clay problem (which requires a formulation in standard QFT/axiomatic framework). It gives a **conceptual** picture: gauge = relational maintenance; mass gap = resolution floor. --- ## III. Scope and limitations - **No proof.** Conceptual reframe only. The Clay problem demands a mathematically rigorous construction (e.g. on a lattice or in continuum with Osterwalder–Schrader axioms); RST does not provide that. - **No numerical verification.** Lattice QCD computes the mass gap; RST does not replace that. Optional future: compare RST-predicted scale (from substrate eigenvalues) to \(\Lambda_\text{QCD}\) or glueball mass — **qualitative** only. --- ## Links - **Applications Roadmap:** [[../../Applications Roadmap]] - **Core theory:** [[expanded theory/Relational Substrate Theory (RST)]] - **RST Core Reduction:** [[expanded theory/RST Core Reduction]]