Character sectors: exact reduction and uniform estimates

Derive the character projections before estimating them

Let R be a unitary auxiliary shift with R^N = I. For k = 0,…,N−1, define

The finite geometric sum proves P_kP_ℓ = δ_kℓP_k, P_k* = P_k, and ΣP_k = I. On Ran P_k, the shift is multiplication by exp(2πik/N). These are quantum Hilbert-space character sectors, obtained from the action on wavefunctions. They are not the eigenangle fibers of a configuration-dependent twisted difference matrix.

If R strongly commutes with H, each projection reduces the Hamiltonian and its spectral calculus. For an unbounded H, this includes the appropriate domain invariance; a formal commutator on an unspecified domain is not enough. Once reduction holds, an inequality can be proved and applied separately on each complete sector, including its multiplicities and all other quantum degrees of freedom.

The neutral projection P₀ is the vector Reynolds average. Its formula supplies a subspace, but neither its dimension nor a positive energy on its vacuum-orthogonal part follows from the averaging identity.

Source: Vacuum obstructions and form Feshbach reduction, §4.2, equations (39)–(40) and Proposition 4.4, printed/PDF p.11.

Why the source vacuum is neutral

Commutation and vacuum uniqueness imply RΩ = χΩ for some phase χ. Those two facts alone do not force χ = 1: a one-dimensional vacuum representation could carry a nontrivial character. The source has an additional input. Its auxiliary shift acts by relabeling configuration variables, so it preserves the positive cone. The unique normalized vacuum wavefunction is strictly positive.

Thus RΩ is another positive normalized vacuum. Positivity excludes a nontrivial complex or negative phase, and uniqueness gives RΩ = Ω. This establishes P₀Ω = Ω for the actual finite regulator. It is a statement about the specified symmetry action and positive ground state, rather than an arbitrary convention for assigning charge zero.

Neutrality also does not mean independence of auxiliary variables. A function of a closed auxiliary holonomy can be invariant under the cyclic relabeling while remaining a nonconstant function of that holonomy. Likewise, correlations among relative auxiliary configurations can be invariant. The neutral space is therefore a physical candidate space, whose unwanted invariant observables still need to be eliminated by a separate dynamical and observable argument.

Source: Vacuum obstructions and form Feshbach reduction, Proposition 4.4 and its scope paragraph, printed/PDF p.11.

The bound needed for exact low-energy exclusion

Suppose the restricted Hamiltonians obey H|_{Ran Pk} ≥ γ_n,k for k ≠ 0 at regulator n. The bound needed on the whole complement is Γ_n = inf_k≠0γ_n,k, including every admitted character and multiplicity. If Γ_n → ∞, reduction gives

This is exact exclusion from a spectral window, not merely a small population estimate. For a normalized vector known only to satisfy 〈H_n〉 ≤ E, positivity instead gives ‖(I − P₀)ψ‖² ≤ E/Γ_n. A state with small mean energy can contain a small high-energy tail even when a sharp spectral window cannot.

If the projection does not reduce H, these conclusions change. With a high block D ≥ Γ and a bounded cross block B, an eigenvector of energy λ < Γ satisfies Qψ = −(D−λ)⁻¹B*Pψ. Its high component is bounded by ‖B‖/(Γ−λ), but it need not vanish. Exact character reduction and adiabatic suppression are distinct mechanisms.

Source: Auxiliary-time lattice framework, Lemma 30.1, printed p.60 / PDF p.67; Theorem 18.1, printed p.45 / PDF p.52.

A soft character can move with the cutoff

Let N_n = 2n+1 and take one state for each cyclic character. Give the neutral state energy zero. Give characters k = n and k = n+1 energy 1/n, and every other nonneutral character energy n. The two soft characters are conjugate partners, so the construction also respects the pairing expected in a real description.

For every fixed nonzero signed integer label k, its representative eventually lies away from the two middle characters and has energy n. Every such fixed label therefore diverges. Yet the minimum energy over the nonneutral complement is always 1/n. A statement checked one fixed label at a time misses the moving soft pair.

This example identifies the quantifier needed when the auxiliary resolution changes. One common Γ_n must bound the full complement, or an argument must prove that the omitted moving labels are uniformly harmless. Bounds over a fixed finite collection cannot establish exclusion of an expanding collection.

The source’s auxiliary-invisibility requirement displays a common diverging scale for nontrivial characters. Reading that scale uniformly is essential for applying its spectral-decoupling lemma to the complete low-energy space.

Source: Auxiliary-time lattice framework, Requirement 30.2, item AI1, printed p.60 / PDF p.67.

Charged divergence leaves a neutral problem

A separate four-state example keeps the character set fixed. Let H_n = diag(0,1/n,n,n), and let R act with eigenvalues 1,1,exp(2πi/N),exp(−2πi/N), for N ≥ 3. Every charged excitation becomes expensive. The vacuum is unique at each n. The neutral excitation still closes the full gap.

Define a neutral observable B that exchanges the first two basis vectors and vanishes on the charged states. It commutes with R, has zero vacuum mean and unit vacuum variance. Its connected Euclidean two-point function is exp(−s/n), which approaches one at every fixed s. Charged decoupling has done nothing to scalarize this invariant observable.

This is why the source asks separately for a neutral physical Poincaré inequality and for scalarization of bounded neutral auxiliary insertions. The latter requires 〈A₁(B−〈B〉I)A₂〉 → 0 against physical-core observables, not merely 〈B〉 = 0. A vanishing one-point function can coexist with a surviving fluctuation, response or additional low-energy state.

Source: Auxiliary-time lattice framework, Requirement 26.1, printed p.56 / PDF p.63; Requirement 30.2, AI3, and Proof Audit 30.3, printed pp.60–61 / PDF pp.67–68.

What a charged correlator estimate actually proves

Suppose a bounded operator O_k transforms by R O_k R⁻¹ = exp(2πik/N)O_k. Since the vacuum is neutral, O_kΩ belongs to character k. For k ≠ 0 its vacuum expectation vanishes. If that sector has lower energy γ_k, the spectral theorem gives, for s ≥ 0,

This is a meaningful decay bound for the specified charged insertion, with its own normalization and sector threshold. It does not control the decay of an arbitrary neutral observable. Even a charge-balanced product can create a neutral excitation whose scale is unrelated to γ_k.

Nor does suppression of charged intermediate population require every induced neutral interaction to vanish. Virtual excursions can leave a return term B(D−z)⁻¹B*. If B grows with the charged scale, that term can approach a finite effective interaction. Uniform estimates must therefore track both the complement energy and how physical insertions or dynamics couple to it.

Source: Auxiliary-time lattice framework, §18.1, printed pp.45–46 / PDF pp.52–53, and §30.1, printed p.60 / PDF p.67.

Averaging operators retains an invariant algebra

The vector projection P₀ should be distinguished from the operator average E(O) = N⁻¹ΣR^jOR^−j. The latter is a unital completely positive idempotent map onto operators commuting with R. It fixes every invariant operator, including the nontrivial neutral B in the preceding example.

Because Ω is neutral, E(O)Ω = P₀OΩ. This identity relates the two averages on vacuum-created vectors without identifying the entire invariant operator algebra with a one-dimensional scalar algebra. For invariant A and B, the averaging also has the bimodule property E(AOB) = A E(O)B. Thus it preserves precisely the invariant structure that auxiliary scalarization must later address.

A symmetry filter therefore removes charged components from a declared description while preserving neutral correlations. It does not by itself implement a physical measurement, select a symmetry-broken phase, or prove disappearance of auxiliary degrees of freedom. Those interpretations require additional state and dynamical assumptions beyond the finite Reynolds identity.

Source: Auxiliary-time lattice framework, Proposition 2.2 and Warning 2.3, printed pp.6–7 / PDF pp.13–14.

A usable sector estimate has three separate entries

A proof ledger should record a complement-wide charged threshold, a vacuum-orthogonal neutral physical bound, and control of invariant auxiliary observables. Each entry answers a different question: whether charged states remain at fixed energy, whether the retained physical theory is gapped, and whether the retained algebra contains unwanted auxiliary information.

At transfer step a_t,n, a sector energy bound γ becomes T|_sector ≤ exp(−a_t,nγ). The source’s neutral Poincaré target uses the same physical-time normalization. A positive number in a dimensionless cycle spectrum cannot be substituted without a common-space comparison and an energy calibration.

Requirement 26.1 is explicitly restricted to physical gauge-invariant observables in the auxiliary-neutral sector. Its subsequent gap implication should preserve that restriction unless the complement is also controlled. Together with uniform charged divergence and the later scalarization requirements, it contributes to the intended four-dimensional theory. None of these ingredients is supplied solely by the existence of cyclic characters or by the six nested inner levels and seventh completion used in the temporal interpretation.

Source: Auxiliary-time lattice framework, Requirement 26.1 and Proposition 26.2, printed p.56 / PDF p.63; Requirement 30.2, printed pp.60–61 / PDF pp.67–68.

Continue reading

• The introductory sector distinction
• Counterexamples to nonuniform thresholds
• Vacuum neutrality and reconstruction
• The required lattice estimates