Module 9 — Relative entropy: the conjecture and its bracket

The fidelity coefficient is a theorem. The relative-entropy coefficient is not. Two independent measurements land on c/12 to about one per cent, and the best bracket anyone here can derive around them is a factor of twenty-seven wide — and even its two ends are conditional, on a hypothesis that is open and on a source that was never obtained.

conjectural s₂ = c/12 for a general net

Along the zero-collar recovery curve ω_s = ω ∘ Ad U_s, the conjecture is s2 = lim D(omega_s || omega)/s² =? c/12. Believed, consistent with both measurements, and not proved for any net. This is the one headline number of the programme that is not a theorem.

Where this comes from: claim page · rigor/kubo_mori_gauge_lemma.tex

conditional The bracket, and what each end depends on

Neither end of 0.0338c ≤ liminf ≤ limsup ≤ 0.924c is unconditional.

Upper end, 0.924c: proved under hypothesis (V), uniform integrability along the orbit. (V) is open — Gap RK1 — and is known only at σ = 0. Two hypotheses of the earlier route, (U_ε) and (A_ε), are refuted for the big interval algebra: locality in K forbids them.

Lower end, 0.0338c: proved modulo Araki's perturbation expansion, which is hypothesis (ii) of the lower-bound theorem. The locators are Araki's 1973 paper and Bratteli–Robinson II §5.4.1; neither was opened here (NOT OBTAINED), so it is carried as a hypothesis and not as a checked citation. That is Gap RK2, and it is recorded rather than glossed.

So the honest reading of the bracket is: conditional at both ends, with named hypotheses. It is still worth having — it is what makes the second variation finite at all — but it is not the same kind of statement as Theorem A.

Where this comes from: rigor/kubo_mori_gauge_lemma.tex §“The lower bound” (Gap RK2) and §“Summary, and what is open”

refuted The Kubo–Mori gauge lemma by restriction to A(K)

What was claimed. The Phase-2b route to s₂ = c/12 had three steps: (a) choose an interval K ⊇ I that also carries the implementer, so that Ad U_s is an automorphism of A(K) and monotonicity applies; (b) evaluate the result to second order; (c) the gauge lemma: the infimum of the Kubo–Mori form over admissible gauges is c/6.

What is true. Steps (a) and (b) are correct. Step (c) is false. The infimum exists, is attained in the limit by an explicit one-parameter family, and equals (11+5√5)c/12 at R* = (3+√5)/2 — a factor 11.09 above c/6.

Why. In the modular frame of any interval K ⊋ I the Kubo–Mori form is the local Sobolev functional g_KM(T(f)Omega) = (c/12) integral [f''² + f'²] dxi, and the fixed part of the field — g_tot = −x² on the compressed interval, which no gauge may alter — already costs 11 + 5√5 units, where c/6 corresponds to 2. Note 7's c/6 was the analytic continuation of a divergent integral. Kosaki and Petz–Donald give suprema, so they bound from below only.

This card is kept because the record is part of the result: the route was tried, the constant it needed is false by a factor of 11.09, and what survives is the finite bracket above, not the conjecture.

Where this comes from: the card in the claim map (a refuted card gets no page under results/, so the map is where it lives) · rigor/kubo_mori_gauge_lemma.tex §4–5

proved The exact identity on unitary orbits — and why it does not apply here

For ω_s = ω(e^−isA · e^isA) with A affiliated with the algebra, D(omega || omega_s) = || h(Delta)^(1/2) (e^{isA} - 1) Omega ||², h(lambda) = lambda - 1 - log lambda. It gives the lower bound unconditionally and the full second-order expansion under modular analyticity. It is not applicable to the recovery curve, whose generator is not affiliated with the interval algebra — which is exactly why the gauge route was tried at all.

Where this comes from: claim page · rigor/kubo_mori_second_variation.tex

The bracket is a factor of 27 wide; the conjecture and both measurements sit close to its lower end. With JavaScript enabled the same picture carries the file and line each number was read from.

The bracket against the conjecture

Both ends of the bracket and the conjecture are exactly linear in c, which is proved; the two measurements were made at c = 1 and are drawn there, not scaled along the line.

Central charge c the marker only; nothing is rescaled

What the fermion measurement rests on

The nine rigorous window values and the extrapolation drawn through them. The measured s₂ is a two-point 1/κ_c extrapolation, and it lies well above the last window value: that gap is the reason its error bar is what it is.

Window row nine rows, three window depths

The refuted lemma, in numbers

What the route needed, what it got, and what that leaves.

Claims this module carries

claim	status
s₂ = c/12 for general nets	conjectural
the upper end 0.924c	conditional on (V), open
the lower end 0.0338c	conditional on Araki's expansion, not obtained
the exact identity on unitary orbits	proved
the gauge lemma by restriction to A(K)	refuted

Compare module 2: the fidelity coefficient f₂ = c/(12π²) is a refereed theorem. The relative-entropy coefficient is the open one.