The continuum limit: convergence, gap transfer and identification

Fix the scales before claiming convergence

The focused source chooses one diagonal regulator sequence. In units set by an inverse length Λ_*, its physical and outer-time lattice spacings are a_n=a_t,n=2⁻ⁿ/Λ_*. The spatial side contains L_n=2²ⁿ sites. The auxiliary spacing is a_τ,n=2⁻²ⁿ/Λ_*, with N_τ,n=2ⁿ sites.

The physical box grows while its lattice spacing shrinks, and the auxiliary circle shrinks while its resolution increases. These are simultaneous limits along a declared family. Separate fixed-volume or fixed-circle results need uniform estimates before they can be combined along it.

The bare coupling is also fixed by a specified two-loop-inspired reference relation, yielding g_n²∼1/(2b₀n log 2). The source warns that this numerical prescription does not prove nonperturbative convergence or identify Λ_* with a dynamically generated mass.

Source: Vacuum obstructions and form Feshbach reduction, §2.2–2.4, pp.4–6.

Remove induced interactions in a local norm

Suppose a nonlinear completion band has been isolated and eliminated. The desired effective action has an ordinary gauge part plus connected auxiliary-induced interactions Φ_n,X supported on finite physical regions X.

A bare closure coefficient tending to zero is inadequate because the number of sites tends to infinity. Even small terms at every site can produce an extensive effect, and connected cumulants can modify relevant or marginal physical coefficients.

This is the volume-independent type of estimate requested by the source. It controls locality and the summed effect near every site rather than only the coefficient of one term.

The same frozen sequence must produce a diverging exact fast gap, vanishing cross-form error and vanishing induced auxiliary interactions. Berry, Born–Huang, holonomy and overlapping-band corrections all belong in that accounting.

Gauge invariance constrains which effective operators may occur. It does not establish their locality, coefficient size or irrelevance. Those are dynamical estimates that have to hold together with the desired four-dimensional renormalization trajectory.

Source: Auxiliary-time lattice gauge framework, §19, printed pp.47–49.

Declare the observables that will converge

Convergence of bare link measures is not enough because continuum observables require their own scaling and renormalization. The source proposes a countable algebra built from rational polygonal Wilson loops, a separating family of finite-dimensional representations, bounded curvature transforms and their finite products, adjoints and rational outer-time translates.

At finite lattice spacing, each continuum loop has a declared polygonal approximation and a renormalization factor. Perimeter and cusp contributions must be handled when required. Curvature insertions use a declared small-loop reconstruction and compatible renormalization conditions.

The bounded transform is useful for a self-adjoint curvature observable because its norm is at most one. Boundedness does not eliminate the need to define the underlying smeared field or prove convergence of its approximation.

A countable core makes the convergence problem concrete, but countability alone does not imply density. The completion topology must make the relevant products and functionals continuous, and the core must be cyclic in the reconstructed vacuum representation. Otherwise it can fail to see part of the theory.

Source: Auxiliary-time lattice gauge framework, §27, printed pp.56–57.

Recover curvature with the correct kind of loop data

For a smooth connection and a small square of side ε, the untraced holonomy in a chosen local frame has the expansion Hol(γ)=I+ε²F_μν+O(ε³). This is a matrix-valued statement. Its leading coefficient transforms by conjugation, just as curvature does.

A traced Wilson loop is a scalar. For a simple Lie algebra, Tr dρ(F)=0, so the linear area term disappears after tracing. It cannot be divided by ε² and identified directly with the matrix F.

The second expression shows the leading invariant quadratic response in the smooth small-loop regime, with anti-Hermitian curvature convention. A rigorous quantum construction needs renormalized smeared versions and appropriate invariant combinations; the formal expansion alone does not define a distribution-valued field.

This distinction corrects an overly compressed scalar-to-curvature formula in the umbrella source. The projected synthesis’s bounded-curvature approximation programme should be read as requiring explicit admissible precursor observables and controlled functional calculus, not as recovering a matrix field from one scalar trace.

The order of lattice removal, loop shrinking and smearing must also be specified. Exchanging these limits can change both ultraviolet divergences and the operator topology in which a statement is meaningful.

Source: Auxiliary-time lattice gauge framework, §31.1, printed p.61; see also projected synthesis §9, pp.46–49.

Reflection positivity is one closed property

Let S_n be the finite-regulator vacuum functionals on a common algebra. For a finite collection of positive-time observables A_j and coefficients c_j, reflection positivity gives Σ over j,k of overline{c_j}c_kS_n(ΘA_jA_k)≥0.

If every term converges, the finite sum has a nonnegative limit. Thus reflection positivity passes to the limiting functional on that algebra. This useful closure fact is exact.

The remaining continuum properties require more work. The source asks for normalization, Euclidean translation and rotation covariance, regularity and temperedness, locality after reconstruction, clustering, a unique vacuum and nontrivial connected correlations. It also requires independence from admissible auxiliary regulator choices.

Full four-dimensional Euclidean covariance is not automatic from spatial cubic symmetry and outer-time evolution. The temporal and spatial normalizations must approach a common finite physical scale, and every allowed anisotropic relevant or marginal term must be controlled by independently fixed renormalization conditions.

A tight sequence of probability measures or convergence of a handful of loops does not establish this package. A limit containing only scalar observables can satisfy positivity while failing the required nontrivial field theory.

Source: Auxiliary-time lattice gauge framework, §28, printed pp.57–58.

Use recovery sequences to transfer a uniform gap

The Hilbert spaces vary with n. Introduce isometries I_n:ℋ_n→ℋ̂ and I:ℋ→ℋ̂ into a common comparison space. Identified Mosco convergence combines a weak lower-semicontinuity condition with strong recovery sequences whose energies converge.

For each u in the limiting form domain, the recovery condition gives u_n with I_nu_n→Iu and q_n[u_n]→q[u]. Assume also I_nΩ_n→IΩ strongly and a uniform finite-regulator inequality

Strong convergence carries norms and vacuum overlaps to the limit. Passing along the recovery sequence therefore gives q[u]≥Δ(∥u∥²−|⟨Ω,u⟩|²). If q[Ω]=0, the associated Hamiltonian has the same gap Δ above Ω.

For an explicitly excited recovery sequence, subtract ⟨Ω_n,u_n⟩Ω_n. Its energy is unchanged because the vacuum has zero form energy, and strong vacuum convergence preserves the desired limit.

This is a proved abstract transfer mechanism. Constructing the comparison maps, recovery sequences, observable convergence and uniform Δ for the auxiliary lattice family remains a separate programme requirement.

Source: Auxiliary-time lattice gauge framework, §29, printed p.59.

Remove neutral and geometric auxiliary remnants

A gapped limit of the enlarged regulator need not be pure four-dimensional Yang–Mills. Diverging nonneutral thresholds remove charged auxiliary states below fixed energy, but leave the neutral sector.

The source therefore also asks for holonomy concentration and scalarization of every bounded local neutral auxiliary observable between physical-core states. It requires disappearance or admissible absorption of completion-band Kato holonomy, control of Born–Huang corrections and compatibility with the selected complex structure.

These requirements address possible observable remnants of the regulator. A neutral holonomy operator can remain gauge invariant; a degenerate band can carry an extra unitary connection; a real positive band can leave a scalar kinetic correction. None is removed merely by averaging the cyclic symmetry.

Route independence then asks that the limiting physical correlations be unaffected by admissible changes of auxiliary resolution, circumference, closure potential, gauge fixing and band trivialization. The continuum local algebra should be generated by the declared four-dimensional loop and curvature observables.

If an auxiliary feature survives with a finite observable effect, it must be identified as part of a different physical model or derived as intended physical content. It cannot simultaneously be treated as an arbitrary removable regulator.

Source: Auxiliary-time lattice gauge framework, §30, printed pp.59–61.

Identify Yang–Mills without a circular target

There are two legitimate forms of identification. If an independent target vacuum theory is available, equality of continuous reflection-positive functionals on a dense OS-cyclic core defines an isometry [A]₁↦[A]₂. Equality of core inner products makes the null spaces agree, and cyclicity extends the isometry to a unitary.

If the same time translations are represented, that unitary intertwines the transfer semigroups and Hamiltonians. Spectra and gaps then agree. The theorem is powerful because it converts observable equality into full vacuum-theory equivalence.

It is circular to define the target functional to equal the proposed limit and then use their equality as proof of physical identification. The umbrella source therefore prefers an intrinsic route: prove the prescribed gauge group and global form, the correct local observables, pure Yang–Mills Ward and Schwinger–Dyson relations, ultraviolet normalization, locality, nontriviality and universality.

The projected synthesis supplies conditional finite-band and bounded-curvature mechanisms intended to establish full-core matching. Its localization, approximation and continuity assumptions must still be verified on the actual continuum family.

Source: Auxiliary-time lattice gauge framework, §31, printed pp.61–62.

A useful order for completing the programme

The first stage is the exact nonlinear fast-sector problem: derive the conditional auxiliary Hamiltonian, prove its uniform band and control projection derivatives. The next stage is global patching and form-level elimination with errors that vanish in a local interaction norm.

Then the surviving four-dimensional gauge theory needs its independent nonabelian renormalization and infrared estimates. Uniform clustering or a global physical Poincaré inequality must be obtained with constants tracked through all scales.

Finally, construct nontrivial renormalized Schwinger functionals, restore continuum covariance, build the comparison-space convergence and prove auxiliary invisibility and intrinsic Yang–Mills identification. The order prevents a desired continuum conclusion from being used to justify an earlier regulator estimate.

The existing papers establish the exact finite framework and several rigorous interfaces between these stages. They formulate the remaining analytic packages as programme requirements. The resulting dependency map is a research agenda with concrete falsifiers and milestones, rather than a completed continuum construction.

Source: Auxiliary-time lattice gauge framework, §32, printed pp.62–64.

Continue reading

• The common exact finite-regulator generator
• The first nonlinear fast-sector milestone
• What a physical gap requires
• The complete claim-status ledger
• Resolvents, moving Hilbert spaces and stable spectral bands
• A concrete roadmap for the lattice proof obligations