Protected completion bands and their geometric corrections

Decide what the band is for

A carrier band is retained as part of the physical theory. Its internal unitary transport can represent genuine gauge data. A completion band is introduced to isolate an auxiliary regulator and remove its finite-energy excitations. The same mathematical bundle construction can occur in both cases, but its physical role differs.

For a retained rank-m Hermitian band, holonomy lies in U(m). Further parallel structures can reduce that group. A rank-two band with a parallel quaternionic structure admits Sp(1)≅SU(2), and a rank-three band with a parallel complex volume form admits SU(3). These are conditional reductions after ranks and structures have been supplied.

For a removable completion band, an additional finite-energy U(m) connection would be unwanted unless it is identified with already declared physical gauge data. The source therefore requires its geometric effect to vanish, become an admissible scalar flat contribution, or belong to an infinitely costly sector.

Neither a three-dimensional spatial displacement space nor the number of nested inner levels determines the rank of a complex internal band. Those objects require separate definitions and separate derivations.

Source: Auxiliary-time lattice gauge framework, §5, printed pp.11–13.

The proved tangent estimate

Near a coherent trivial-holonomy configuration, decompose an auxiliary tangent sequence into its mean ξ̄ and relative part. The normalized quadratic model is

The discrete Poincaré estimate controls the zero-mean modes; the closure potential controls the coherent holonomy mean. The physical τ-constant gauge field is a base variable and must not be counted as an auxiliary direction that this estimate removes.

For the frozen sequence, the source obtains a relative stiffness 8Λ_*/g_n² and mean stiffness (κ_*Λ_*/2)N_τ,n^σ_H, with 0<σ_H<1. Both diverge along the chosen sequence.

This calculation is exact for the quadratic model. It demonstrates that the chosen scaling is algebraically compatible with a fast tangent sector. It is not a proof about the difference between the two lowest eigenvalues of the full nonlinear quantum Hamiltonian.

A quantum band estimate also involves the kinetic metric, the measure and possible competing wells. These data cannot be read off from the Hessian at a single coherent configuration.

Source: Vacuum obstructions and form Feshbach reduction, §8, pp.21–22.

What a nonlinear band theorem must add

The required slow–fast decomposition identifies the exact conditional auxiliary form at each physical base field u. It must remove gauge-orbit directions, retain the physical τ-constant directions and reconstruct the original global form with controlled cross and overlap terms.

Near the coherent set, a uniform nonlinear normal form should compare the full potential with a coercive transverse quadratic form and bound the remainder relatively by a factor θ<1. Far from that set, it must exclude additional low wells. Gauge-chart metrics, densities, Jacobians and transition maps must remain controlled after physical rescaling.

A large Hessian alone does not settle this. A double-well model can have steep individual wells while its first two quantum levels are split only by a small tunnelling amplitude. A local expansion around one well misses that mechanism entirely. Likewise, an omitted holonomy stratum can invalidate a uniform claim even when the trivial stratum is well behaved.

The source proposes near/far localization through an IMS identity. If two localized estimates have floors m₀ and m₁ and the localization error is at most e_IMS, the global floor is min{m₀,m₁}−e_IMS. The identity is proved; obtaining the required nonlinear estimates and controlling that error uniformly remain open tasks.

Source: Auxiliary-time lattice gauge framework, §§14.6–16.4, printed pp.38–42.

Specify an actual isolated quantum band

For fixed regulator, a compact-resolvent local Hamiltonian C(u) has discrete eigenvalues. A band is a chosen finite cluster separated from all other eigenvalues. If a contour Γ stays in the resolvent set and encloses the cluster, its projection is

Norm-continuity of the resolvents gives norm-continuity of Π. Projections whose norm distance is less than one have the same rank, so a connected parameter region with a common isolated contour has constant finite rank. This is a precise sense in which the band is protected.

A local eigenvalue gap at one u does not guarantee a contour that works for all u. The band can collide with another level as the physical field changes. The programme needs uniform separation and quantitative first and second derivatives of the projection over every admitted sector.

If a compact symmetry commutes with C(u), it commutes with its resolvents and Π(u). The band therefore decomposes into symmetry types, and equivariant effective operators act on their multiplicity spaces. This supplies a useful simplification after the band itself has been constructed.

An external band gap also differs from internal splitting. The matrix diag(0,ε,Γ) has an isolated rank-two low band when Γ is large, regardless of how small ε is. The projection onto that band can remain perfectly constant while its internal excitation gap closes as ε→0. A retained degenerate band may therefore contain physical low-energy dynamics. For a removable auxiliary ground band, every such internal degeneracy must be classified and its physical role specified. Bounding the rank and the separation from the complement does not itself prove a positive gap above the unique full vacuum.

Source: Auxiliary-time lattice gauge framework, §§16.1–16.2, printed pp.41–42.

The Kato connection acts on sections

The ranges E_u=Ran Π(u) form a Hermitian vector bundle. A section s(u) satisfies Πs=s. The natural connection differentiates the section in the ambient space and projects the result back:

The local frame matrix A is skew-Hermitian because the frame is orthonormal. Parallel transport is therefore unitary. Under a frame change its components obey the connection transformation law; the underlying projected derivative remains the same geometric object.

A tempting expression is ΠdΠ restricted to Ran Π. It is identically zero: differentiating Π²=Π and multiplying on both sides by Π gives Π(dΠ)Π=0. The connection is Πd acting on a varying section, not a nonzero endomorphism obtained by sandwiching dΠ.

If a fixed orthogonal complex structure J commutes with all C(u), the resolvent and band projection commute with J. Because J does not depend on u, the Kato connection is J-linear. If J varies with the base, extra derivative terms must be analyzed; the fixed-J conclusion cannot simply be reused.

Source: Vacuum obstructions and form Feshbach reduction, §9, pp.22–23.

A positive band can leave a kinetic correction

If a rank-one band has a globally normalized positive real eigenvector φ(u), then ⟨φ,dφ⟩=0. Normalization makes its real part zero, and reality makes the inner product real. The Abelian Berry one-form therefore vanishes in this positive gauge.

The orthogonal derivative need not vanish. Its squared norm enters the Born–Huang tensor G_BH(X,Y)=⟨(1−Π)∇_Xφ,(1−Π)∇_Yφ⟩. An exact two-component example makes the difference visible:

Take C(u)=Γ[I−|φ(u)⟩⟨φ(u)|]. It has a unique positive ground vector and an arbitrarily large gap Γ. Nevertheless, for a retained wavefunction ψ(u), the ambient kinetic form satisfies ∥∂_u(ψφ)∥²=|ψ′|²+|ψ|². The extra scalar term does not shrink merely because Γ grows.

This example does not model the full lattice regulator. It exactly demonstrates the remaining estimate: the dependence of the fast ground state on physical fields must be controlled relative to the physical kinetic form. Zero Berry phase alone cannot establish auxiliary decoupling.

Source: Auxiliary-time lattice gauge framework, Proposition 16.5 and Program Requirement 16.6, p.43.

Local bands need a global compatibility theorem

If local projections commute, their product is a projection onto the simultaneous low space. In a simple commuting positive Hamiltonian with each local excitation costing at least γ, any state outside that simultaneous low space costs at least γ. This benchmark is easy because all local tests can be diagonalized together.

Overlapping gauge blocks need not commute or factorize. Gauss constraints connect boundary data, and local bands can carry incompatible topological labels. A product of their projections need not be self-adjoint or idempotent, so it cannot be called a global band without further work.

One useful abstract repair begins with a self-adjoint contraction 0≤A≤I satisfying ∥A²−A∥≤ε<1/4. Its spectrum lies near zero or one. Define α_±=(1±√(1−4ε))/2 and Π=1_[α₊,1](A). Then ∥A−Π∥≤α₋. Any symmetry commuting with A also commutes with Π.

The source asks for a gauge-compatible global projection with a diverging high completion gap, exact vacuum containment ΠΩ=Ω, and an isometric identification of its range with a four-dimensional gauge space. Almost-projection estimates are useful tools, but constructing A and controlling all overlaps are genuine model-specific tasks.

Source: Auxiliary-time lattice gauge framework, §17, printed pp.44–45.

The first concrete test of the proposal

The source isolates a manageable milestone: G=SU(2), one fixed physical block, trivial magnetic and center sector, and the already frozen scaling. Construct the conditional auxiliary Hamiltonian from the exact global form. Prove a unique positive fast ground state and a gap E₁,n(u)−E₀,n(u)≥Γ_τ,n→∞ uniformly in u.

Then prove gauge-covariant derivative bounds strong enough to suppress the Born–Huang and cross-form corrections. The positive eigenvector must agree consistently across gauge-chart overlaps. The connected interactions induced by holonomy closure must vanish in a volume-independent local norm.

Success would show that the auxiliary construction supplies a real removable fast sector. It would still leave the nonabelian four-dimensional infrared estimate, nontrivial continuum construction and intrinsic Yang–Mills identification to be proved.

This milestone is useful because it tests the specifically auxiliary proposal before committing to the full continuum problem. It can fail through an extra low well, uncontrolled projection derivatives, a surviving geometric field or a closure-induced interaction. Each failure would identify a concrete modification or limitation of the proposed regulator.

Source: Vacuum obstructions and form Feshbach reduction, Open Problem 11.1, pp.25–26.

Continue reading

• Spectral bundles and retained gauge carriers
• Geometric phase and holonomy
• Elimination through quadratic forms
• Removing the auxiliary regulator