A concrete roadmap for the lattice proof obligations

Freeze the mathematical question before optimizing it

Start with one compact group, one finite physical block, a declared auxiliary cycle and the source’s specified regulator sequence. Record the kinetic coefficients, spatial and mixed plaquette weights, closure potential, transfer step and normalization. State exactly which boundary, magnetic and center-flux sectors the first result covers. A successful estimate for one sector can then be extended deliberately rather than silently treated as universal.

The source proposes G = SU(2), trivial magnetic and center sector, and one fixed physical block as the first concrete milestone. This restriction is useful because it isolates a nonlinear completion-band question without pretending to settle the full infinite-volume theory. Uniformity in the physical base field remains part of even this first theorem.

The temporal interpretation does not determine missing coefficients or estimates. The six inner levels are nested stages with a seventh completion, and the auxiliary circle is a declared model coordinate. The transfer direction and its time unit are specified separately. A theorem should therefore begin from the exact mathematical family it studies, while keeping its proposed SMM/DTT interpretation visible as a distinct claim.

Source: Auxiliary-time lattice framework, Requirement 16.8, printed p.43 / PDF p.50; Warning 2.3, printed p.7 / PDF p.14.

Produce an exact finite generator and vacuum ledger

The first deliverable is an exact closed, semibounded form on the gauge-invariant Hilbert space, with its Friedrichs generator and normalized vacuum semigroup. The source already constructs this for the declared finite regulator. An application should record its common core, gauge projection, form-domain invariance and the precise normalization that places the vacuum energy at zero.

Keep the exact generator separate from a finite time-discretization formula. A single symmetric product of kinetic and potential transfers has a logarithm, but that logarithm is not automatically the original kinetic-plus-potential Hamiltonian. If an approximation is used, its error must be assessed in the topology needed later.

The vacuum ledger should state whether each proposed projection contains Ω, splits Ω, or fails to preserve the form domain. The three-state puncture is a compact checkpoint for the split-vacuum case: reconstruct QΩ from PΩ through the high block, verify the zero Schur line, and test strict positivity only after puncturing. Those equalities should be established before numerical evidence is interpreted as coercivity.

Source: Vacuum obstructions and form Feshbach reduction, §7.1, printed/PDF pp.19–21, and §10, pp.23–25.

Build the actual conditional fast operator

A local fast Hamiltonian must come from disintegrating the exact gauge form. Write the physical base variable as u and the remaining auxiliary fiber as Y_B,n(u;σ), where σ labels the boundary and flux data. Supply the conditional measure, gauge-slice charts and their overlap maps. The unitary or form identification must include the density and derivative terms created by those choices.

The deliverable is a conditional operator C_B,n(u;σ) with a specified domain and spectrum, together with its complete lowest-band projection Π_B,n(u). Compact resolvent at fixed parameters gives discrete eigenvalues, but does not make their separation uniform in u or n. Gauge singularities, holonomy strata and remote wells are part of the parameter space, not exceptions to be dropped after a favorable local calculation.

The first source milestone asks for rank one and E₁−E₀ ≥ Γ_τ,n → ∞, uniformly in u, with gauge-covariant derivative bounds strong enough to make the later geometric and cross-form corrections small. Listing these outputs makes clear what a nonlinear band proof must deliver beyond a quadratic tangent spectrum.

Source: Auxiliary-time lattice framework, Requirement 14.6, printed p.38 / PDF p.45; Requirements 16.4–16.8, printed pp.42–43 / PDF pp.49–50.

Turn local stiffness into a global fast inequality

The tangent estimate controls a neighborhood of the coherent stratum. To turn it into a nonlinear spectral statement, prove a localization scheme with explicit errors. For a partition χ₀²+χ₁² = 1, an IMS identity has the form

If the two localized forms have lower bounds m₀ and m₁, and E_IMS[ψ] ≤ e_IMS‖ψ‖², the global lower bound is min(m₀,m₁)−e_IMS. The required inputs are the nonlinear near-region estimate, a complement estimate excluding other low wells, and a localization cost smaller than both margins. The abstract identity proves none of those model-specific inequalities by itself.

The band derivatives need a separate estimate. A normalized positive real eigenvector has zero Abelian Berry one-form, but its orthogonal derivative can remain nonzero and produce a Born–Huang contribution. Thus uniqueness and positivity can remove a phase ambiguity while leaving an energetic correction that must be controlled relative to the physical form.

A useful report supplies the localization regions, Jacobian bounds, error exponents and worst allowed base configurations. A plot of the Hessian at one minimum is a diagnostic input, not the requested theorem.

Source: Auxiliary-time lattice framework, Lemma 16.3, printed p.42 / PDF p.49; Proposition 16.5 and Proof Audit 16.9, printed p.43 / PDF p.50.

Patch local bands without losing the vacuum or volume control

The next deliverable is a global completion projection Π_n^cb that contains the exact vacuum, preserves the form domain and respects gauge transformations, spatial symmetries and auxiliary translations. Local bands overlap through shared links and Gauss-law constraints. Their tensor product is not automatically the physical global low space.

The source asks for control of boundary modes, overlap maps, correlations and topological labels through a genuine patching argument. It also asks for a high complement bound Γ_τ,n−C* with a constant C* independent of the growing volume. A local statement with an error repeated once per site may not satisfy that requirement.

For induced interactions, the relevant small quantity is a local norm such as sup_xΣ_X∋xexp(μ diam(X)/a_n)‖Φ_n,X‖_Cʳ. A coefficient ε_n tending to zero at each site does not control a total error |Λ|ε_n uniformly over volumes. The local norm must be bounded after connected auxiliary integration, including closure, geometric and overlap contributions.

This stage converts a family of local spectral observations into an operator and interaction estimate that can be consumed by the four-dimensional theory.

Source: Auxiliary-time lattice framework, Requirement 17.3 and Proof Audit 17.4, printed pp.44–45 / PDF pp.51–52; Requirement 19.3, printed p.48 / PDF p.55.

Prove the normalized cross-form estimate

With an admissible global split, write the shifted form as q_z[p+r] = a_z[p] + d_z[r] + 2 Re b[p,r]. The high form must be closed and strictly positive. The cross functional must be continuous in its form norm, giving a Riesz representative K_zp through d_z[K_zp,r] = b[p,r].

The exact identity is f_z[p] = a_z[p]−d_z[K_zp]. A useful theorem bounds the subtracted term by a controlled fraction of the low form, or by the source’s vanishing normalized error plus a stated scalar remainder. The constants must cover the same vacuum, boundary and flux sectors as the preceding construction.

If a hard window splits the vacuum, transport and remove its low image before claiming a strict margin. If the global band contains the vacuum exactly, preserve that property through every comparison. The source’s corrected transfer theorem then gives an explicit order-a_t,n punctured margin when its contraction and mixing hypotheses hold. Calling those hypotheses “expected” does not provide the inequality.

Source: Auxiliary-time lattice framework, Theorem 18.3 and Requirement 18.5, printed pp.46–47 / PDF pp.53–54; vacuum-obstructions Proposition 6.8, printed/PDF p.18.

The nonabelian physical estimate remains a new task

Eliminating the fast completion sector leaves a physical gauge theory. It does not create a proof of its infrared gap. The source requires a compatible scaling window in which the exact nonlinear fast gap diverges, normalized adiabatic errors vanish, and auxiliary-induced local interactions vanish or are absorbed into declared Yang–Mills parameters.

The next independent deliverables are gauge-covariant block estimates, an RG trajectory reaching a controlled massive domain, and a multiscale Poincaré inequality for physical neutral observables. Their constants must be tracked in physical units and remain uniform in volume. A reference two-loop bare sequence supplies a chosen input trajectory; it does not establish the nonperturbative RG theorem.

The charged complement and the neutral physical sector need separate entries in the proof ledger. The moving-character and neutral-soft-mode examples show why charged divergence alone cannot supply the physical Poincaré bound. Likewise, making the closure coefficient larger may strengthen a fast band while leaving unacceptable finite corrections in the physical action. One fixed sequence must satisfy both requirements without retrospective retuning.

Source: Auxiliary-time lattice framework, Requirements 19.4–19.5, printed pp.48–49 / PDF pp.55–56; Requirements 24.2–26.1, printed pp.54–56 / PDF pp.61–63.

Specify the endpoint evidence and the next achievable result

The continuum deliverable consists of renormalized correlations on a separating Wilson–curvature core, a nontrivial reflection-positive limit with the remaining OS properties, and comparison maps that transport forms, normalized vacua and local observables. Recovery energies carry an already uniform gap to the limiting form. Norm-resolvent estimates with an isolated contour stabilize the low spectral clusters needed downstream.

Identification then requires auxiliary scalarization, disappearance or classification of completion-band geometry, route independence and intrinsic pure-Yang–Mills properties. Equality with a target defined to be the same limit is not an independent identification. The source’s endpoint theorem is conditional on all of these packages; it does not assert they have been proved.

Three near-term outputs are especially reviewable: the uniform one-block SU(2) band with derivative bounds; a common-space resolvent estimate for a concrete dangerous-cluster model; and a vacuum-compatible cross-form bound on an explicit gauge-invariant core. Each can be published with its exact assumptions, parameter scope and constants. Together they would advance the programme while leaving its remaining nonabelian and continuum obligations stated accurately.

Source: Auxiliary-time lattice framework, §29–§32, printed pp.59–63 / PDF pp.66–70; projected-lattice Remark 4.13, printed/PDF p.28.

Continue reading

• The nontrivial finite puncture
• Common-space spectral transport
• Charged and neutral estimates
• The complete research-status ledger