Discretization: continuous relations → discrete grids (BZ mesh, ω/E grid, finite-N)

[sotashimozono]

The concern (user-flagged)

Many relations here are stated in the continuum: Brillouin-zone integrals (ChernFromBerryCurvature C = (1/2π)∫Ω), frequency integrals (SpectralSumRule ∫A dω = 1), energy integrals (density of states), the finite-size-scaling forms, the Kubo/multi-time transforms. A many-body calculation (ED, MPS, TPQ) that uses this library to verify its measured quantities works on discrete data: a finite k-mesh, an ω or E grid, a finite system size N.

So each continuum relation needs a discretization contract when consumed:

• BZ integrals → discrete sums with the right measure; the Fukui–Hatsugai–Suzuki chern_number is already the discrete-exact answer, but ChernFromBerryCurvature takes a supplied continuum flux — document the mesh convention / provide a discrete companion.
• Frequency / energy integrals (sum rules, DOS, KK) → quadrature weight + broadening conventions; finite-η Lorentzian vs delta.
• Derivatives (∂S/∂E, ∂M/∂h, ∂F/∂T) → finite-difference stencils and their O(h²) error, so check(...; atol=...) is set from the stencil, not guessed.
• Finite-N corrections — ensemble equivalence, TPQ self-averaging (CanonicalTPQ), and the FSS forms all carry O(1/N) or O(e^{−N}) corrections; the library should expose the expected correction scale so a consumer's tolerance is principled.

What to build (later)

A thin discretization layer: for each continuum relation, a documented discrete evaluation contract (quadrature/measure/stencil) and, where useful, a discrete companion relation whose residual accounts for the leading grid/finite-size correction. Keep the continuum relation as the source of truth; the discrete layer is how a real MPS/ED/TPQ measurement plugs in.

Not blocking the accumulation of continuum relations — this tracks the bridge to the actual measured-data use case.

[sotashimozono]

Status: parked (deferred by design, not blocked). The continuous→discrete-grid layer (quadrature / measure / stencil / finite-N correction per relation) is a real enhancement, set aside for now.

The interface half of its boundary already shipped: src/evaluation.jl — principal_value_hilbert / spectral_moment over an AbstractResponse — is the seam the discretized numerics plug into (the scope line #14/#50: definitional pointwise/supplied-value relations owned here; the functional transforms/quadratures evaluated by a ParaLA-based sibling). So the supplied-integral relations (sum rules, KK, f-sum) are already turnkey from a supplied scalar; #19 is the remaining piece that computes those scalars from a raw grid. Prioritize when that functional package is built.

[sotashimozono]

Evidence for the "expose the expected correction scale" bullet, from one relation where
the answer is that no correction scale exists at reachable sizes.

ActivatedFiniteSizeScaling's docstring names its 1D instances as Iglói–Monthus
Eq. (4.12) for the gap and Eq. (4.6) for the surface magnetisation, both for free
boundary conditions. Taking the first at face value: exact free-fermion gaps of the
critical random transverse-field Ising chain (J_i, h_i iid uniform on (0,1]),
3000 realisations per size, L = 16…384, fitting
log(-<log Δ>) = log a + ψ log L + b·x(L):

correction x(L)	ψ	b	χ²/dof
none	+0.5309 ± 0.0029	—	16.25
L^{-1/2}	+0.3984 ± 0.0135	−2.128	4.07
1/log L	+0.4015 ± 0.0135	−2.177	4.91
1/L	+0.4667 ± 0.0074	−3.638	5.88

No form fits, and which one you assume decides the exponent (0.40 to 0.53 against
ψ = 1/2). Sliding a fit window instead gives the same picture from the other side:
ψ_eff drifts monotonically 0.533 → 0.470 as the smallest sizes are dropped, by more
than the statistical error at every step.

For contrast, the same exponent from a quantity with no boundary in it — the typical
correlation between a pair at the centre of an L = 512 chain, 600 realisations — is
fitted by the same forms with residual RMS ≈ 0.02 and gives 0.47–0.55.

So the practical reading is that the gap instance on an open chain is not a usable probe
of ψ at sizes a numerical calculation reaches, while a bulk correlation is. That is a
statement about which observable the relation should be pointed at, and it is the kind
of thing a consumer currently has to discover by fitting both. Worth carrying next to
the correction scale this issue asks for.