Full proofs

Each statement marked Proved links to its complete written proof; this page lists them all, grouped by argument in the order of the manuscript. Next to Proved, each statement says who checked the proof, and when:

• agent review (model, date): an AI agent, run without the conversation that produced the proof, checked it line by line against the statement; the link opens its report.

• reviewed by a name: a person checked it the same way and wrote the report.

• accepted by a name: a person read the proof and vouches for it, without a report.

Proved (from a preprint) marks a preprint’s result whose proof has been written out and checked here. Preprint, not yet checked here marks one that has not, and that no proof here relies on; Established in the literature marks a published result, cited and not reproved. An agent review is not journal refereeing, and no person has yet reviewed the proofs of KLS below.

Song–Zhang, first version¶

The spectral criterion and the iterated-logarithm bound of Chapter Song–Zhang, first version: polynomial estimates and curvature:

1. Analytic foundations and Appell variance estimates.

2. Polynomial coefficients control the spectral gap.

3. The iteration of curvature profiles.

4. Gaussian transfer, the all-depth bound and its affine form.

5. KLS and exponential growth of Appell coefficients, the end point that the next proofs reach.

Bizeul–Klartag–Lehec¶

The proof of Chapter Bizeul–Klartag–Lehec: cumulants and suspension, which explains how its two branches, the tilt criterion and the cumulant bound, meet in the suspension:

1. Analytic foundations and tensor symmetrization.

2. The tilt criterion and Appell duality.

3. Inverse-covariance cumulant dynamics.

4. Cumulant energy estimates.

5. The all-order cumulant bound.

6. Suspension, uniform coefficients and KLS.

7. A consequence: uniform conditional initialization.

Song–Zhang, second version¶

The proof of Chapters Song–Zhang, second version: repeated refinement with summable losses and Song–Zhang, second version: technical estimates; it uses no BKL conclusion. The first-version proofs above stay attached to the first version.

1. The common coefficient radius and static transfer, joint frames and skew credit.

2. The improved inner iteration and its dimension bound.

3. Block construction and propagation, joint loss estimates, and fixed-cost repeated height reduction.

4. Finite chains with retained bounds and near-unit refinement.

5. Summable budgets and starting depths, the uniform Poincaré bound, and KLS.

Balasubramanian–Kasiviswanathan¶

The proof of Chapter Balasubramanian–Kasiviswanathan: compatible integration; it uses Letwin’s quadratic inequality and the Appell normalization, but no BKL or SZ v2 conclusion, and its explicit constant $1+2\cdot10^{16}$ comes from its own integration estimate.

1. Approximation with covariance and curvature control.

2. Compatible-tensor Hodge estimates and domains, then integration and Appell observations.

3. Uniform operator powers and the quadratic seed.

4. Covariance-normalized localization and moving Appell variance, then reverse coefficient transfer.

5. The explicit degree induction, the Poincaré and Cheeger constants, and KLS.

Results of the literature¶

Preprint results that the chapters use, with their proofs written out here; each statement’s status says whether its proof has been checked:

• Letwin’s matrix and quadratic estimates, his third-moment bound and covariance control, and his bound $\CP\lesssim\sqrt{\log n}$ (Chapters Family 4: moment maps, Monge–Ampère, and Stein kernels and Family 2: stochastic localization).

• The Chen–Klartag moment-Hessian, thin-shell and third-tensor bounds (Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels).

• Klartag–Lehec: stopped rank tails and integrated covariance (Chapter Small-time operator-norm control of the covariance).

The moment map¶

The results of Chapters The moment map: the deterministic inequality–The moment map: appendix:

• The moment-Hessian inequality, its operator data, and $\CPaff\le\CMH$.

• Exact constants on the line and on products, and the bound 4 on Dirichlet laws.

• The linear test and the third-moment tensor, and its spectral resolution.

• The exponential cones and cones over products of simplices.

• Approximation closure, the recovery calculus, and lower semicontinuity of the affine Poincaré constant.

The fixed eigenfunction¶

The results of Chapter The fixed eigenfunction: following one eigenfunction through localization:

• Spectral sufficiency: the occupation estimate implies KLS.

• The posterior eigenfunction defect, restart deweighting, and a small-gap fourth-moment bound.

• The time-weighted source budget, the stopped source bound, and the occupation implication.

Conditional fibers¶

The results of Chapter Conditional fibers: inverse-variance frames of line resamplings:

• The resampling form and the simplex root obstruction.

• A uniform floor at every fixed polynomial degree.

• The degree-two value for the root frame.

Foundations and the fixed-cut archive¶

The localization identities of Chapters Analytic conventions and the two-color localization setup–Model geometries: the Gaussian and product brackets, and the results of the fixed-cut archive, which opens with Chapter The fixed cut: approach and lessons:

• Foundations: localization and Riccati identities, full matrix dissipation, quadratic chaos, Stein contrast and boundary flux, consequences of Letwin’s quadratic estimate, and profile curvature and the model geometries.

• From survival of one cut to KLS: balanced survival, Carleson control implies centroid control, centroid control implies KLS, and the weighted near-Cheeger implication.

• The bootstrap: the near-worst bootstrap, its stopped covariance quantity, and its residual dichotomy.

• Budgets and products: the scale-weighted source budget, Lyapunov–Stein duality, product coordinate budgets, split-class screened supply, and the excess and the perimeter martingale.

• Counterexamples: the exponential-spectator obstruction and its superlinear-remainder form.