Counterexamples that locate a gap proof’s missing premise

A sequence of positive gaps can close

Consider H_n = diag(0,1/n,2) on a fixed three-dimensional space. Every H_n is nonnegative, has a unique vacuum and has a strictly positive gap. The sequence converges in operator norm, yet its limit diag(0,0,2) has an additional zero vector. The statements “for every n there exists Δ_n > 0” and “there exists Δ > 0 for every n” are different statements, even with the Hilbert space and vacuum held fixed.

The finite-regulator compactness argument in the source gives the first statement. A continuum or thermodynamic application requires a lower bound expressed in physical units with the second quantifier order. Taking logarithms of transfer eigenvalues does not repair a missing uniform bound.

For T_n = exp(−a_nH_n), the leading excited eigenvalue is exp(−a_n/n). A test that only records whether it is below one misses both the shrinking excitation energy and the changing step length. The appropriate threshold is exp(−a_nΔ) for one declared positive Δ; the actual eigenvalue eventually violates that comparison.

Source: Vacuum obstructions and form Feshbach reduction, Definition 3.4 and Proposition 3.5, printed/PDF p.9.

Deleting an exact zero does not control nearby modes

For a cycle with N sites, circumference β and spacing β/N, the twisted difference spectrum is

The constant branch vanishes at θ = 0. At any small nonzero θ it is positive, but λ₀(θ) approaches θ²/β² as the lattice is refined. Choosing θ_N = 1/N produces a positive eigenvalue of order 1/(N²β²). Removing only the exact kernel at every parameter value therefore leaves a sequence of arbitrarily soft modes.

The distinction is spectral: a projection onto the complement of the exact kernel can change discontinuously when a branch reaches zero, while a uniform positive threshold must control that branch before it reaches zero. A protected band normally needs an isolated contour or another quantitative separation condition, not merely a declaration that its eigenvalues are nonzero.

This is a finite difference-fiber example. It does not identify the auxiliary-translation characters of the quantum Hamiltonian. Any argument that transfers the fiber estimate to a wavefunction operator must supply the common-space realization and its comparison inequality separately.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 5.5 and Remark 5.6, printed/PDF p.13.

A sharp window imposes a vacuum-tail constraint

On L²([0,1]), let K multiply by x² and let Ω be the normalized constant function. The exact kernel of K is zero: an L² function supported at the single point x = 0 vanishes almost everywhere. Nevertheless the spectrum starts at zero. For 0 < ε < 1, the spectral window P = 1_[0,ε](K) retains the interval [0,√ε], so p = ‖PΩ‖² = √ε. Both vacuum pieces are nonzero, and the low space has infinite rank.

A useful strengthened necessary bound follows directly from the abstract transfer assumptions. If T ≥ 0 and TΩ = Ω, self-adjointness makes Ω⊥ reducing and gives T ≥ |Ω〉〈Ω|. Therefore QTQ ≥ |QΩ〉〈QΩ|. If QTQ ≤ ρQ with ρ < 1, its norm must satisfy 1 − p ≤ ρ:

This sharpens the weaker necessary bound displayed in Proposition 6.6 using its stated operator-positivity premise. It remains necessary, not sufficient. For T = |Ω〉〈Ω| + t(I − |Ω〉〈Ω|), direct calculation gives ‖QTQ‖ = 1 − (1 − t)p, hence p ≥ (1 − ρ)/(1 − t). This bounded operator is an abstract diagnostic, not the source’s compact heat-kernel regulator.

Source: Vacuum obstructions and form Feshbach reduction, Proposition 5.7, p.14; Theorem 5.10, p.15; Proposition 6.6, p.17, printed/PDF pagination.

Compression and exponentiation do not commute

Let H = diag(0,100) and retain the one-dimensional direction e = (1,1)/√2 as the high projection Q. Its compressed energy is 〈e,He〉 = 50. But its compressed unit-step transfer value is

The scalar exp(−50) is entirely different. The vector e includes half of the vacuum weight, which remains unchanged under the semigroup. A large expected energy contributed by the other half does not remove that vacuum component.

The spectral measure of e explains the direction of the error: averaging the convex function exp(−E) gives a value at least exp(−〈E〉), rather than the upper bound a contraction argument would need. An operator lower bound on QHQ can be exponentiated inside the restricted high theory only when Q reduces H. Without that reduction, Q exp(−aH)Q includes excursions outside Q.

The source accordingly has two legitimate interfaces: transfer-level contraction hypotheses and Hamiltonian form-level coercivity hypotheses. Each has its own exact Schur identity. Replacing one interface by the other requires a theorem about the projection and functional calculus.

Source: Vacuum obstructions and form Feshbach reduction, §6.3, p.18, and §10, pp.23–25, printed/PDF pagination.

A diverging high block can leave a finite correction

Take the symmetric matrix H_Γ = [[1,c√Γ],[c√Γ,Γ]] at threshold zero. Its high diagonal tends to infinity, while the exact Schur value remains 1 − c². For 0 < c < 1, the lower eigenvalue tends to 1 − c² rather than one. At c = 1 there is an exact zero eigenvalue for every Γ. For c > 1 the matrix has a negative eigenvalue and therefore fails the assumed nonnegativity of a physical Hamiltonian.

The failure mechanism is the ratio B²/D, not the size of D alone. The off-diagonal matrix element grows as √Γ, exactly fast enough to leave an order-one return contribution. Saying that the high direction is expensive therefore does not determine whether it can be neglected, retained as a finite correction, or used in a threshold argument.

At a general real threshold z below the high block, the relevant expression is B²/(D − z). A uniform estimate needs a lower bound on that denominator and an upper bound on the normalized cross term. The form version replaces a bounded B by a continuous cross functional measured in the high-form norm.

Source: Auxiliary-time lattice framework, §18.1, equations (132)–(140), printed pp.45–46 / PDF pp.52–53.

A successful puncture can violate a sufficient premise

The three-state nonempty punctured window supplies a complementary diagnostic. Set p = 0.01, t₁ = 0.2, t₂ = 0.5, r = 0.8 and α = π/4. Its vacuum-corrected operator is nonnegative, its high block is D = 0.0045, and its exact punctured coefficient is c = 0.4. Yet its uncorrected high transfer compression is 0.9935, larger than r.

There can therefore be no direct estimate QTQ ≤ exp(−a m_Q)Q with m_Q exceeding the chosen threshold energy. The rank-one vacuum correction supplies the positive high block instead. This does not contradict the source’s conditional theorem: that theorem lists a sufficient package, and a successful exact model need not satisfy every stronger hypothesis in the package.

For research, this distinction identifies two legitimate strategies. One can prove the stronger direct contractions and gain a convenient uniform implication. Alternatively, one can estimate the corrected high form directly, tracking the transported vacuum and its effect on the inverse. The second strategy needs its own quantitative constants; the example does not provide them for a lattice.

Source: Vacuum obstructions and form Feshbach reduction, Proposition 6.8, printed/PDF p.18.

A threshold needs an error budget

Even a correct reduced formula needs a margin larger than its unresolved errors. For a scalar Schur form F = A − B²/D with D > 0, suppose the entries have errors bounded by ε_A, ε_B and ε_D, with ε_D < D. A direct resolvent subtraction gives the conservative bound

The denominator shows why a small high-block error can matter near a singular split. A computed positive coefficient is a certificate only after the analytic approximation error and numerical uncertainty are smaller than that coefficient. Conversely, a tiny negative floating-point residual at an exact zero is not evidence of a physical instability.

The explorer keeps raw residuals and exact toy formulas separate. In a theorem, the corresponding separation is between an established identity, a proven bound on the omitted terms, and the final strict inequality. A useful audit records all three rather than reporting only a sign from a numerical diagonalization.

Source: Projected lattice gauge synthesis, §3.4–§3.6, band comparison and dangerous-cluster margin, printed/PDF pp.20–25.

Turn each failure into a precise next test

These examples suggest a disciplined order for a proposed estimate. First fix the Hilbert space and identify the vacuum. Then state whether the projection is a multiplication window, a character projection, or a spectral band. Specify the time and length units before comparing thresholds. Finally identify the quantifiers: which parameters are fixed, which limits are taken, and which constants must survive all of them.

Ask the estimate to pass the exact three-state vacuum transport, the small-twist branch, and the √Γ mixing family. Each tests a distinct issue. Failure on one of them identifies a missing premise; success on all of them remains an algebraic and numerical checkpoint, rather than a proof of the nonlinear source hypotheses.

The next substantive task is to establish a comparison on the actual gauge-invariant form domain with a uniform high-block bound and a controlled punctured cross form. Continuum convergence, neutral-sector coercivity and auxiliary invisibility enter later as additional requirements. Keeping those steps separate makes a failed calculation useful: it tells the programme which estimate must change and prevents a local success from carrying a larger claim than it supports.

Source: Vacuum obstructions and form Feshbach reduction, §11, printed/PDF pp.25–26.

Continue reading

• The exact three-state puncture
• Character sectors and uniform bounds
• Resolvent and form convergence
• What a mass-gap statement must establish