| # STATE — kcnuX4xEpL (QOT Localization Bounds & Affine Case) |
|
|
| ## Claim → experiment map |
|
|
| Both extracted claims are **proven theorems** validated by ONE synthetic experiment. |
| The manager tag `[ALGORITHMIC]` is really `[THEOREM]`-with-empirical-validation; I treat |
| each as a theorem (derivation + numerical lemma checks) PLUS a shared compute discriminator. |
|
|
| | Claim | Paper result | Decided by | |
| |-------|--------------|-----------| |
| | C1: sptπ_ε cannot concentrate around grT faster than ε^(1/(d+2)) (directed Hausdorff LOWER bound) | Theorem 3.3 (+Lem 3.1, 3.2, Eckstein–Nutz rate 3.2) | DERIVATIONS D1–D3 (lemma/rate checks) + exp01 (measured slope β̂ ≤ ~1/(d+2)) | |
| | C2: affine Brenier regime, sharp pointwise tube bound ε^(1/(d+2)) for Gaussian→Gaussian (UPPER bound) | Theorem 3.7 (+Lem A.2, Prop 3.8, Wiesel–Xu) | DERIVATIONS D4–D6 (reduction/rate checks) + exp01 (measured slope β̂ ≈ 1/(d+2)) | |
| |
| Both claims are about the SAME rate exponent 1/(d+2). In the affine regime lower and upper |
| bounds coincide (r ≍ ε^(1/(d+2))), so the single measured dbias-vs-ε slope across d probes |
| both. No second compute experiment is justified: any support/tube statistic Claim 1 would |
| need is computable from exp01's saved per-ε dbias trajectories. |
| |
| ## In-regime note (critical, honest) |
| Strict theoretical regime is ε ≤ ε_0 = λ_{μA} κ_A ω_d r_A^(d+2). ω_d and r_A^(d+2) collapse: |
| for d=100, ω_100 ≈ 2.3e-40, r_A≈0.08, r_A^102 ≈ 1e-112 ⇒ ε_0 ≈ 1e-152 — far below any |
| representable grid ε (~1e-11). NO accessible ε is in the strict asymptotic regime for ANY d. |
| exp01 therefore tests the PRE-ASYMPTOTIC manifestation exactly as the paper does; the |
| theorems' correctness is decided by the derivation checks (proofs are constant-explicit). |
| The exp01 acceptance gate reproduces the paper's REPORTED empirical pattern, NOT a strict |
| β̂=1/(d+2) equality (see spec). |
|
|
| ## Ledger |
|
|
| | exp | spec | build_status | run_status | gate | |
| |-----|------|--------------|------------|------| |
| | exp01 | specs/exp01_affine_scaling.md | done | todo | toy-pass | |
|
|
| Derivations (no compute job; checks live in DERIVATIONS.md, runnable by writer): D1–D6. |
|
|
| exp01 projected full-scale: 4.6 h at 8 cores (toy-measured) |
|
|
