A discrete Yang–Mills proposal: which gap is being proved?

The claim and the model must be separated

The public catalogue lists Discrete Inner–Time Yang–Mills Models and a Rigorous Mass Gap Via Variational Analysis. The full older argument was not retrieved for this expansion. This page therefore expands the reading method in A208, using explicit elementary spectra rather than certifying the claim in the title.

A gap means a positive separation above the vacuum in a stated physical spectrum. Its value depends on the Hamiltonian, constraints, volume, regulator, and units. Defining those objects is the first substantive step, not a preliminary formality.

Calculate the compact-mode threshold

On a circle of circumference 2πR, Fourier modes e^inθ have eigenvalues n²/R² under the positive Laplacian −R⁻²∂²_θ. The first nonzero eigenvalue is 1/R², but the constant mode n=0 remains.

Choosing an energy operator proportional to the square root would make nonzero energies proportional to |n|/R after units are fixed. That is a separate dynamical choice. A source must justify both the physical operator and why allowed zero or collective modes do not destroy the claimed threshold.

Check what Gauss law actually removes

A gauge constraint restricts physical states to an invariant sector. To infer a gap, write the constraint action and determine that sector explicitly. A charged mode can have a positive threshold while neutral composites or collective fluctuations remain arbitrarily soft.

Even a single compact example illustrates the issue: selecting nonzero Fourier modes creates a gap by restriction, but it does not explain why the physical vacuum and allowed observables support precisely that restricted space. The newer finite-cycle work distinguishes charged thresholds from complete physical-sector conclusions for this reason.

A family of positive gaps can converge to zero

Consider the positive Laplacian on a periodic spatial interval of length L. Its first nonzero eigenvalue is (2π/L)². Every finite L has a positive spacing, yet that spacing tends to zero as L grows. Positivity at each regulator value is therefore weaker than a uniform positive lower bound.

For a continuum gauge theory, a bound must also survive removal of the short-distance regulator in fixed physical units. One must retain a nontrivial theory rather than a limit in which all fluctuations vanish. These requirements explain why finite diagonalization and continuum existence are genuinely different achievements.

What a trial state can tell us

A variational trial state gives an upper bound on the lowest energy in the allowed sector. A mass-gap proof needs to control all states orthogonal to the vacuum from below. Such a result may use a coercive quadratic-form estimate, an operator inequality, or another uniform argument.

SMM/DTT can contribute a physically motivated recurrence scale and a constructive constrained state space. The next task is to show how that same source fixes spatial propagation, interaction, and scale conversion. A compact temporal radius is a useful ingredient only when its role in the full physical spectrum is established.

Preserve exact model results while extending them

Read A208 with A195–A200. Separate the finite center-selection calculation, local stability, charged thresholds, effective monopole matching, and the remaining full-gap obligation. Each result answers a different question and can remain useful even if a stronger claim is not established.

The next rigorous target should name one finite physical sector, prove a lower bound there, and track its constants across an explicitly defined family. Does the bound deteriorate with volume? Are neutral states controlled? Which physical scale is held fixed? Answers to these questions would make the proposal substantially more informative without importing a continuum theorem from its title.

Continue with the articles

• Reading a discrete Yang–Mills mass-gap proposal

  Prepare a careful audit of the listed mass-gap proposal. Distinguish a regulated model result from the continuum four-dimensional Yang–Mills problem, and compare the claim with newer gap-obligation drafts.

• Center selection on a finite SU(2) cycle

  Introduce the exact minimization problem for the supplied one-loop potential with deformation. Explain how finite cycle size changes the center-selection threshold.

• Local stability versus the global minimum

  Explain why local stability boundaries and global phase selection answer different questions. Use the finite-cycle model to organize the possible competing extrema.

• A charged threshold is not a full mass gap

  Explain the importance of neutral modes and centralizer kernels when interpreting a cycle spectrum. A positive threshold in one sector need not gap every physical excitation.

• Monopole matching and the dual-photon model

  Introduce the regulated monopole-gas matching prescription as a conditional infrared application. Explain why the effective model still requires supplied couplings and weights.

• Kink tension, mass, and higher harmonics

  Explain the classical dual-photon relation between kink tension, stiffness, and mass in the specified model. Discuss why additional harmonics change the relation and require an error analysis.

• The remaining gap-proof obligation

  Summarize the draft’s distinction between finite-volume constructions, screening results, and a volume-uniform quantum gap. Explain the connected-expansion step that remains unproved and the separate gauge-matching problem.

Sources and reading

• Author’s papers catalogue: Discrete Inner–Time Yang–Mills Models and a Rigorous Mass Gap Via Variational Analysis

Read the original source for its exact formulation and edition. The examples on this page are explanatory reconstructions.