Module 2 — The quadratic law

The recovery error depends on the geometry through one cross ratio. As that cross ratio goes to zero the error is exactly quadratic, with a coefficient fixed by the central charge; over three decades of measurement you can watch the ratio settle on it, and then watch it leave again as the cross ratio grows.

refereed Theorem A

lim_ζ↓0 [−log F(ω, ω̃_ζ)] / ζ² = c / (12π²), c = r. A data-processing lower bound in the Legendre form of the SLD metric, and a matching Uhlmann upper bound built from a Bogoliubov extension of the compression whose vacuum overlap is a Fredholm determinant. No interchange of limits and no tail bound are needed for the theorem.

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

certified numerical What the curve below actually is

Two different things are drawn, and they are not interchangeable. A bracket is an error bracket, not a rigorous enclosure: its upper end (for small ζ the window value plus the tail estimate ζ²/(15κ_c²), for large ζ the extrapolation Φ_∞) is not a proved bound; its lower end is the recovery error restricted to the spectral window ε < w < 1−ε, which is a proved lower bound for the full value. A best estimate is an extrapolation with an error bar — a measurement, not a theorem. The plot draws brackets as bars with a shaded interior and best estimates as filled dots.

Where this comes from: certified enclosures · the ultraviolet tail bound · numerics/certified/cert_tail_hp.py

The recovery error against the cross ratio on logarithmic axes, with the quadratic asymptote and the measured remainder. — Φ(ζ) against ζ, the ζ² asymptote, and the measured remainder. With JavaScript enabled the same data become a sweep, and every point names the file and line it was read from.

Sweep the measured cross ratio ζ twelve measured values, 0.0083 to 17.07

Show brackets and best estimates are drawn differently and never merged

The ratio Φ/ζ²

Theorem A is the statement that this ratio has a limit as ζ → 0, and that the limit is c/(12π²). The measured ratio is not constant — it is not supposed to be; only its limit is claimed.

Where the quadratic law stops

The local exponent d log Φ / d log ζ measured at large cross ratio. It drifts away from 2 as soon as ζ is of order one: the quadratic law is a statement about ζ → 0 and nothing else.

Claims this module carries

claim	status
Theorem A: the quadratic law	refereed
f₂ = c/(12π²)	proved
the one-particle metrics g_B, g_KM	proved
the remainder: integer powers, c₃ = −0.99	proved
the ultraviolet tail bound	proved
exact root fidelity from the symbols	proved
the type-III second-variation lemma	proved
certified enclosures of the windowed error	numerical
Φ at large ζ and the drifting exponent	numerical