Give the response genealogy more than one way to be evaluated

[sotashimozono]

Stacked on #161 (base feat/conventions), which it needs only for the version line.

structure/response.jl declares WHICH derivative a response function is. There was exactly one way to evaluate it, nested forward-mode AD in a package extension, so a project holding a measured potential rather than a differentiable one could not traverse the declared genealogy at all.

Finite differences were already here three times, written inline and unnamed:

relations/region_entropy.jl:598	(S2 - S1) / (log(ℓ2) - log(ℓ1))
relations/finite_size.jl:44	(log(O2) - log(O1)) / (log(L2) - log(L1))
relations/finite_size.jl:51	(log(-log(O2)) - log(-log(O1))) / (log(L2) - log(L1))

Those difference two MEASURED points and stay where they are, but they are why a route is a declared thing now.

No duplicated genealogy

The AD path had already factored the genealogy out, so the route is the only thing that differs:

return -_nth(F, x, derivative_order(q, e.field()))

_genealogy_derivative now holds that once and both paths call it. The extension supplies its nth and nothing else; its 44 tests are unchanged and green.

Routes: AutoDiff (the extension), CentralDifference(h), Richardson(h; levels). A new one is a subtype plus an nth_derivative method.

F(h) = -log(2cosh(h))
thermal_derivative(Magnetization(:z), F, 0.3, Richardson(1e-2))   # no backend needed

The trust number is on the normal return path

Measured on F(h) = -log(2cosh h) at h = 0.3, where M = -F'(h) = tanh(0.3):

route	error	observed_order
AutoDiff()	0	n/a, no step
Richardson(1e-2)	6.3e-14	n/a, at machine precision
CentralDifference(1e-2)	8.9e-6	2.00
CentralDifference(1e-9)	2.7e-8	0.32

The last row is the point. Its error is smaller than the row above it and its step is on the roundoff side, and only the order says so. A threshold would have passed it.

The control the diagnostic needs is a function it must fail on: on a kinked potential a central difference is first-order exactly, and observed_order reports 1.

derivative_report runs several routes and reports each value with its order, so a route that cannot run (missing backend) is a NaN row rather than an aborted sweep.

Mutation

Against test/core/test_derivative_routes.jl (26 assertions, runs with no backend):

mutation	assertions failed
drop the genealogy's minus sign	5
_route_order always 1	2
central difference to forward difference	8

test/ext/test_derivative_routes_ad.jl (21 assertions) pins AD against Richardson across seven genealogy edges, comparing the sign before the magnitude.

Additive (git diff -- src/ | grep -E "^-[^-]" is empty), so 0.7.15 -> 0.7.16.

🤖 Generated with Claude Code

[github-actions]

📚 Docs preview: https://qatlashub.github.io/AbstractQAtlas.jl/previews/PR162/

(updates on each push to this PR)

[codecov]

Codecov Report

❌ Patch coverage is 94.80519% with 4 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/derivative_routes.jl	94.73%	4 Missing ⚠️

📢 Thoughts on this report? Let us know!