Second-Order SLT: fluctuation theory of singular learning on the exact normal form

Four research notes and the verification suite that regenerates every table in them.

What this is

Working at the Gaussian-observation normal form of singular (Watanabe) learning theory --- where the rescaled parameter sees an exactly solvable Gaussian channel --- this stack develops the fluctuation theory of the Bayes generalization error: its equations of state, its second cumulant, the trajectory law sigma^2 = 2*lambda, a separation theorem for Bayes-factor paths, and instruments (a shifted bootstrap ladder; drift/quadratic-variation reads) that measure (lambda, nu, M, h) and pairwise model distances from data. Every quantitative claim carries a verification record; corrections and retractions are kept in-paper with root causes. Nothing here claims scale: hypotheses (realizability, multiplicity one, homogeneity) are stated where they bind, and the two open prior-art passes are flagged in the notes.

The stack (dependency order; builds frozen, see papers/FREEZE.md)

1. Companion --- diffusion_tensor.pdf (fb2e03b7): chart-free convergence of the second cumulant, normal-crossing residue formula, off-class closure with the -T/3 vertex.
2. Fluctuation equations of state --- fluctuation_eos.pdf (9e454431): joint limit law, bootstrap dichotomy, calibration covariance.
3. Tilted equations of state --- tilted_eos.pdf (cf5cf35a): resonance and temperature spectroscopy, the shifted-ladder protocol, the statistical-dimension identity.
4. Trajectory second cumulant --- trajectory_cumulant.pdf (dc7f1237): sigma^2_traj = 2*lambda, free energy as Poisson potential, the separation theorem, reciprocity and the nu-identity, Gram spectroscopy, charge theory.

Reading paths

Theorists. Read the four notes in dependency order (companion -> fluctuation -> tilted -> trajectory). Everything lives on the Gaussian-observation normal form, where n*G_n => (1/2)||m(G)||^2 is exactly solvable; each note carries its epistemic tags (PROVEN / DERIVED / VERIFIED / CONJECTURE / RETRACTED) and keeps corrections in-paper with root causes. Every quantitative claim regenerates from verify/ --- run python verify/run_all.py <group> next to the note you are reading (groups: trajectory, separation, second, charge_panel, temperature, companion, uv, alpha).

Practitioners. The notes double as instruments for LLC-adjacent estimation on the normal form:

• notebooks/calibration_check.ipynb --- what resampling does to the temperature. An m-out-of-n resample runs a beta-tempered posterior at effective temperature (1+m/n)beta, so the naive bootstrap (m=n) measures fluctuation functionals at 2beta; the protocol is to freeze (beta,gamma) and resample at the operating point, with m ~ sqrt(n).
• the shifted-ladder (M,h) readout (tilted note, S7.2) and the drift / quadratic-variation reads of (lambda,nu) from a single data ordering (trajectory note, SS5-8) --- with kappa2 irreducibly between-orderings, so error bars require replication.

Results are toy-exact on the normal form; off-class corrections are O(n^{-1/2}) per cumulant with computed constants.

Reproduce

pip install -r requirements.txt   # numpy>=2, scipy>=1.11
make verify     # runs verify/run_all.py: regenerates the section-6 spot-check table
                # (deterministic checks at 5e-4; seeded Monte-Carlo checks within 3 sigma)
make hashes     # confirms the four frozen PDFs (and .tex) against papers/FREEZE.md

make verify runs the whole battery; the alpha-stable occupation check alone takes ~2 min. To run one group: python verify/run_all.py trajectory (also separation, second, charge_panel, temperature, companion, uv, alpha).

Repository layout

• papers/ --- the four frozen PDFs, their .tex sources, and FREEZE.md (md5-8 manifest; byte-immutable).
• verify/ --- evaluators.py (the shared channel/denoiser block) and the run_*.py modules that regenerate the appendix tables; run_all.py asserts them; check_freeze.py backs make hashes.
• raw/ --- the unmodified session scripts that produced the tables, kept as the provenance layer (MANIFEST.md maps each script to its paper table).
• notebooks/calibration_check.ipynb --- a calibration check: an m-out-of-n resample runs the tempered channel at (1+m/n)*beta, not beta.
• docs/ --- BUILD_REPORT.md (integrity, acceptance results, and reproducibility notes).

Provenance and AI assistance

This research is directed and owned by Justin Sheek. Derivations, verification code, and drafting were developed in working sessions with Claude (Anthropic), as acknowledged in each note. The basis for trust here is the reproducibility stack, not the authorship mode: every number regenerates from verify/, the builds are hash-frozen, and errors --- four were found and corrected during the program, each recorded with root cause --- are the owner's.

License and citation

• Code (verify/, raw/, notebooks/) --- MIT, see LICENSE.
• Research notes (papers/) --- Creative Commons Attribution 4.0 International, see papers/LICENSE.

Cite via CITATION.cff.