The auxiliary cycle: exact spectra and their limits

What the auxiliary coordinate represents

At a fixed regulator, one outer-time slice contains a spatial lattice Σ and a finite cyclic label τ. The configuration therefore has links on Σ×C_N. Ordinary Euclidean time labels the succession of these slices and supplies the reflection used in transfer reconstruction. The cyclic label sits inside each slice.

This construction is mathematically meaningful without an ontological interpretation. Within SMM/DTT, it is proposed as a regulator for recurrent completion. The six nested inner levels describe an ordered internal organization, and the seventh stage completes the record. The parameter τ does not turn those levels into independent time axes, and its N lattice sites do not count ontological levels.

Both N and a_τ matter. Their product β_τ=Na_τ is the circumference in the declared units. Refining N at fixed β_τ, shrinking β_τ, and changing the kinetic coefficient are different operations. A claimed fast scale should specify all three rather than invoking compactness alone.

Source: Vacuum obstructions and form Feshbach reduction, §§1–2, pp.2–6.

Use one endpoint convention throughout

We write U_yx:V_x→V_y and compose paths from right to left. If the local frame changes by g_x, then U′_yx=g_yU_yxg_x⁻¹. The lattice source often writes a link from the opposite endpoint order, with transformation g_xU^src_xyg_y⁻¹. In this guide U_yx=(U^src_xy)⁻¹. This conversion reverses products while leaving traced loop defects unchanged.

Let U_τ transport along a spatial link x→y at auxiliary step τ. Let V_x,τ transport from (x,τ) to (x,τ+1), and similarly at y. There are two paths from (x,τ) to (y,τ+1): first go spatially and then around the auxiliary step, or perform those moves in the opposite order.

Both products have the same endpoints, so their difference transforms covariantly. Its Hilbert–Schmidt norm is gauge invariant. A nonzero difference detects failure of the square to close covariantly; it does not represent a breaking of gauge symmetry.

Source: Vacuum obstructions and form Feshbach reduction, §§3.1 and 5.1, pp.6,11–12.

The plaquette identity is exact

For a faithful representation ρ, normalize the defect by ℓ_ρ(g)=c_ρ⁻¹Re Tr[I−ρ(g)], where −Tr[dρ(X)dρ(Y)]=c_ρ(X,Y). Unitarity gives a simple matrix identity:

Here W is the closed product comparing the two routes. Its orientation can be reversed because Re Tr ρ(W⁻¹)=Re Tr ρ(W). No small-field expansion is used. The mixed potential is therefore exactly a sum of squared covariant interruption defects at every configuration.

The identity compares numerical functions of link configurations. If K multiplies a wavefunction by the total mixed energy, its Hilbert-space quadratic form is ∫K(X)|ψ(X)|²dμ(X). This is different from the quadratic form of a difference matrix acting on a sequence of matrices, and different again from the kinetic-plus-potential form of the quantum Hamiltonian.

The distinction is consequential. An exact identity in classical link variables does not establish a spectral decomposition of the quantum Hamiltonian. A common-space realization and comparison theorem are needed before a fiber estimate can constrain quantum excitation energies.

Source: Vacuum obstructions and form Feshbach reduction, §5.1, pp.11–12.

Axial gauge leaves one genuine holonomy

A gauge change along each auxiliary chain can set the first N−1 auxiliary links to the identity. The closing link then contains the ordered product around the circle. In our convention, H_x=V_x,N−1⋯V_x,0. Residual gauge changes are independent of τ and conjugate H_x at its base point.

For a mixed-flat configuration, the nonclosing plaquettes force the spatial links to be independent of τ. The closing plaquette imposes

Thus the auxiliary holonomies must be related by parallel transport. They need not all be the identity. The zero-mixed set contains different conjugacy strata, including central or commuting holonomies compatible with the spatial configuration.

On the trivial-holonomy stratum, H_x=I at every site. The remaining variables are ordinary spatial gauge links on a single outer-time slice. Their spatial plaquettes can still have curvature. Consequently, making every mixed plaquette flat does not eliminate the ordinary gauge dynamics or assign it a positive mass.

This observation identifies the role of a separate holonomy-closure potential. It penalizes the unwanted continuous auxiliary holonomy moduli while leaving the physical spatial links as base variables. Whether its quantum effects disappear after elimination is a further universality question.

Source: Vacuum obstructions and form Feshbach reduction, §§5.2 and 7.1, pp.12–13,19.

Derive the twisted Fourier spectrum

After axial gauge, choose an eigenmatrix of the closing twist with eigenvalue e^iθ. With the orientation of the difference chosen consistently, the scalar sequence satisfies a twisted closing condition f_N=e^iθf₀. Reversing the cycle changes θ to −θ and permutes the same spectrum. The normalized Fourier sequences have components

Indeed, applying the forward difference multiplies f by [exp(i(2πk+θ)/N)−1]/a_τ. Taking the squared modulus gives the displayed eigenvalue. Each twist eigenmatrix contributes one such tower, including its multiplicity.

The kernel consists of constant sequences in the invariant eigenspace θ=0. For the untwisted scalar N=4 cycle with a_τ=1, the spectrum is 0,2,4,2. The first nonzero eigenvalue is 2. This exact finite example is useful for checking normalizations before considering a changing regulator.

At fixed circumference and small mode number, the eigenvalues approach (2πk+θ)²/β_τ² as N grows. This is a continuum approximation to the auxiliary difference operator. It says nothing by itself about a continuum four-dimensional gauge theory.

For a conjugation twist B↦HBH⁻¹, diagonalize H with eigenangles φ_j. The matrix unit |j⟩⟨k| then has twist angle φ_j−φ_k. Every diagonal matrix unit has angle zero, and degeneracies of H add further invariant directions. For H=diag(e^iφ,e^−iφ) in the two-dimensional representation, the off-diagonal angles are ±2φ while the diagonal directions remain untwisted. Reversing the cycle changes their signs and permutes the same eigenvalues. This elementary example makes clear that one auxiliary holonomy can contain both untwisted directions and nonzero twist directions; there is no single universal shifted frequency for the whole fiber.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 5.5, p.13.

Why removing the exact kernel is insufficient

Take k=0 and a small nonzero twist θ. Then

This is a positive eigenvalue, so removing the exact kernel does not remove it. Allowing holonomy phases to approach zero produces arbitrarily soft positive modes. A compact auxiliary cycle therefore does not supply a twist-uniform lower bound without a separate condition on the allowed holonomies.

The smallest positive eigenvalue can even behave discontinuously as a diagnostic: when θ reaches zero, the soft branch becomes an exact zero mode and is omitted from a list labelled “positive modes.” The next listed eigenvalue then jumps upward. A spectrum explorer should show the zero branch continuously rather than concealing it through that selection rule.

The source also proves that plaquette multiplication potentials can have spectral infimum zero but no nonzero L² vector at zero energy. Their classical flat configurations occupy a Haar-null set. Thus a geometric zero set is not automatically a quantum ground space on which a frustration-free projector method can operate.

There are two separate obstructions: near-zero twisted fiber modes obstruct a uniform classical spectral floor, while measure-zero minima obstruct the naive construction of local quantum ground projections. Adding a genuine kinetic operator addresses the second issue; controlling all twists and nonlinear wells remains necessary for the first.

Source: Vacuum obstructions and form Feshbach reduction, §§5.3–5.4, pp.13–14.

What the linear closure estimate proves

For a vector sequence ξ, decompose ξ_r=ξ̄+(ξ_r−ξ̄). The mean and zero-mean parts are orthogonal. Fourier analysis and sin(π/N)≥2/N give the cyclic Poincaré estimate

The difference term sees only relative modes. Adding a mean penalty gives q(ξ)=αΣ∥Δξ∥²/a_τ²+κN∥ξ̄∥², and hence q(ξ)≥min{16α/β_τ²,κ}Σ∥ξ_r∥². Both terms are needed to control every auxiliary tangent direction.

For a holonomy closure coefficient κ_H, the quadratic expansion of the product around N sites contains (κ_H/2)N²∥ξ̄∥². Thus κ=(κ_H/2)N in the normalized expression. The N² factor is essential: a product around a longer cycle amplifies a coherent mean displacement.

The frozen scaling in the source makes both relative and mean tangent stiffness diverge while the one-step bare closure coefficient vanishes. This is an algebraic scaling window. The estimate is local and quadratic; it does not prove a nonlinear quantum spectral band, control distant wells, or bound the induced physical interactions.

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

From a cycle calculation to a physical statement

To turn the auxiliary calculation into a result about physical dynamics, construct the exact conditional fast Hamiltonian from the same global form that defines the theory. Show that its low band is isolated uniformly in the physical base field, boundary sector and regulator. Control how that band changes with the base field, since its derivatives enter effective kinetic and geometric terms.

Next show that local bands patch into a gauge-compatible global projection containing the physical vacuum exactly. Eliminate the high band with a form-level estimate and prove that all induced auxiliary interactions become small in a volume-independent local norm.

The ordinary four-dimensional gauge sector remains after this elimination. Its neutral excitation spectrum requires its own nonabelian coercivity and infrared analysis. A successful fast auxiliary band can simplify that problem without solving it automatically.

The practical reading rule is therefore precise: the cycle spectrum is an exact finite-matrix result; the closure stiffness is a tangent-space result; a completion band is a nonlinear quantum claim; and a physical mass gap is a statement about the vacuum-orthogonal Hamiltonian after all limits and identifications are controlled.

Source: Auxiliary-time lattice gauge framework, §§14.6–19.5 and 24.3, printed pp.38–49,55.

Continue reading

• Holonomy and closed comparison
• The exact finite-regulator Hamiltonians
• From tangent stiffness to a protected band
• Charged thresholds and neutral modes