Record evidence ledger and wrap-up corrections

Note 36 consolidates the full evidence chain (numbers, figure sources,
negative results, claim discipline) as the paper-writing base; note 37
records the two audit fixes; superseded banners on notes 00/03; README
index updated. Superseded alignment_recovery_dynamics probe removed.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

5 files changed, 282 insertions, 0 deletions

diff --git a/notes/00_project_overview.md b/notes/00_project_overview.md
index 3f8696e..b6a19c8 100644
--- a/notes/00_project_overview.md
+++ b/notes/00_project_overview.md
@@ -1,5 +1,9 @@
 # Project Overview
 
+> **Superseded.** This note's framing (redundancy-exhaustion phase transition,
+> no estimator, no closed form) is stale. Current contributions and numbers:
+> notes 35 + 36 (evidence ledger). Kept for history.
+
 ## Working Title
 
 Distributional Capacity Bounds for Feedback Alignment in Multilayer Perceptrons
diff --git a/notes/03_paper_outline.md b/notes/03_paper_outline.md
index 40ab86f..4a7deb8 100644
--- a/notes/03_paper_outline.md
+++ b/notes/03_paper_outline.md
@@ -1,5 +1,9 @@
 # Paper Outline
 
+> **Superseded.** This outline predates the soft-ramp revision, the e0
+> theorem, the closed form, and the estimator. Write from notes 36 (evidence
+> ledger) + 35. Kept for history.
+
 ## Title
 
 Distributional Capacity Bounds for Feedback Alignment in Multilayer Perceptrons
diff --git a/notes/36_evidence_ledger.md b/notes/36_evidence_ledger.md
new file mode 100644
index 0000000..63e797d
--- /dev/null
+++ b/notes/36_evidence_ledger.md
@@ -0,0 +1,184 @@
+# Evidence Ledger (Writing Base for the AAAI Paper)
+
+Authoritative consolidation of every result, number, and figure source in this
+project, compiled directly from notes 00-35, the scripts, and the raw outputs.
+**Write the paper from this file**, not from any earlier draft: the earlier
+LaTeX draft was written without a careful pass over the experimental record and
+omits much of the evidence chain below. Status of every claim is marked.
+
+Numbers updated 2026-06-09 after the wrap-up corrections (note 37): the
+closed-form comparison is now data/init-matched, and the e0-distribution
+sampler runs the real FA backward pass.
+
+---
+
+## 0. Scope and claim discipline
+
+- Setting: MLPs, squared loss `L = ||r||^2/(2N)`, full-batch training,
+  feedback alignment = fixed random backward matrices `B_l` replacing
+  `W_{l+1}^T`. Synthetic random-label / teacher tasks + MNIST-subset
+  validation (note 38).
+- The paper quantifies the **optimization (train-loss) gap** FA pays relative
+  to BP. It does NOT claim FA fails to converge (compatible with Song-Xu-
+  Lafferty), does not predict test accuracy, and the estimator is
+  **conditional on two early kernel snapshots** — not architecture-only.
+- "No fit" everywhere below means: no post-hoc scalar was tuned to match the
+  reported data. Predictions are computed from initialization (and, for the
+  estimator, one early snapshot).
+
+## 1. Theorems (proof status)
+
+| # | statement | status | source note |
+|---|---|---|---|
+| T1 | `Q = cos^2(a,b) ~ Beta(1/2,(D-1)/2)`, `E[Q]=1/D` for independent isotropic directions | exact | 00/01 |
+| T2 | log-volume costs add: `C_all = sum_l C_l = Theta(L n^2)`; surprisal `S_l ~ Exp(1)`, `sum ~ Gamma(L,1)` | exact | 01 |
+| T3 | prior-free minimax: `sup_mu inf_a E[(a^T b)^2] = 1/D`, attained by isotropic | exact | 01 |
+| T4 | null model: Haar-random k-dim deletion gives `e(r) ~ Beta(k/2,(P-k)/2)`, `E[e]=k/P`, `P(e=0)=0` | exact (within null model) | 26 |
+| T5 | **initial-erosion theorem**: `E_B[e_0 | W,r] = 1 - ||g_out^BP||^2 / sum_l ||g_l^BP||^2` (hidden speed share), any depth, any independent zero-mean feedback | exact | 29 |
+| T6 | one-hidden-layer Gaussian FA: `e_0 | W,r` exactly Gaussian with explicit mean/std (linear in B) | exact | 32 |
+| T7 | closed-form gap under A1 (lazy) + A2 (scalar erosion `S_FA ~= rho K_BP`): `gap_T = (1/2N) sum_i c_i^2 [(1-eta rho lam_i/N)^{2T} - (1-eta lam_i/N)^{2T}]`; analytic => soft ramp, no threshold unless `rho -> 0`; converged regime `log gap_T ~= const - (2 eta T/N) rho lam_min` | exact under A1+A2 | 34 |
+
+A2 is exact in the residual direction by T5 (`rho = 1 - E_B[e_0]` = output
+share); extending it to all eigenmodes is the one modeling step. A1 is the
+standard lazy approximation whose breakdown is *measured* (item 3c below).
+
+## 2. Static geometry: validation record
+
+| claim | evidence | numbers | output dir |
+|---|---|---|---|
+| Beta law | 700k samples, D = 64..4096 | KS stat 0.002-0.004, p 0.17-0.99; mean/q99 match to 4 digits | `capacity_empirical_validation` |
+| capacity tail cost | same run | max smoothed cost error 0.20 nats, mean 0.019; multilayer product max err 0.46 nats | same |
+| `Gamma(L,1)` surprisal | 100k per (D,L), D in {64..4096}, L in {1..16} | max KS **0.0042**, mean abs mean-error 0.0042 | `multilayer_capacity_distribution` |
+| minimax | 5 schemes, D=32/128 | isotropic lam_min 0.0291 vs 1/32=0.0313; rademacher 0.0289; anisotropic 0.0061; subspace/axis 0 | `minimax_initialization`, `initialization_distribution_matching` |
+| functional overlap | P=96,d=24, trials=100 | hard loss exact (0 until k>P-d, then k-(P-d)); soft overlap kd/P to 3 digits | `functional_capacity_overlap` |
+
+## 3. The gap pillar: four-level evidence chain
+
+### (a) Null model + its honest failure (negative result, keep in paper)
+
+- T4 proves nonzero soft cost in the cleanest model; `P(e=0)=0`.
+- **Hard-k overpredicts real FA**: predicted gap 0.040 at margin +130 vs
+  measured 0.00016 (250x); ~2x on local erosion (measured local erosion 0.23-
+  0.39 vs Beta-mean 0.63-0.73) (note 25, `soft_erosion_theory_vs_empirical`).
+  Role in paper: motivates the *operator* object; hard constraint count is
+  not the effective burden.
+
+### (b) Exact at t=0 (T5 + T6)
+
+- Mean theorem, 2-hidden-layer, real FA backward, 6 widths x 6 inits x 512
+  draws: `E_B[e_0] = 0.4378 +/- 0.008` vs theory `0.4403`; per-width errors
+  0.0003-0.008 (note 29, `actual_fa_initial_operator_moments`).
+- Independent t=0 anchor from the (superseded) recovery probe: 0.4026+/-0.0066
+  vs 0.4071 (run log, note 37).
+- Full distribution (T6), **sampler = real FA backward after note-37 fix**:
+  4 widths x 3 inits x 4096 draws, KS p in [0.13, 0.999], all means/stds match
+  to 3-4 digits; linear identity `<C,B>` verified against the real backward to
+  max err 8.9e-16 (`actual_fa_initial_erosion_distribution`).
+- **MNIST (note 38)**: 16 configs (width {64,256} x depth {1,2} x 4 inits,
+  512 real-backward draws each): hidden share spans **0.39-0.82** and
+  `E_B[e_0]` lands on it with |diff| 0.0001-0.0089 (all within ~2 stderr) —
+  the theorem holds on real data with no fit
+  (`real_data_validation_mnist/e0_mnist_rows.csv`).
+
+### (c) Finite time: the lazy chain (previous draft omitted ALL of this)
+
+1. **T=5 local exactness** (note 10): frozen K(0), 256 traj, 8 N values:
+   corr **0.99947**, MAE **0.00083**, mean ratio 0.993. Horizon table: ratio
+   1.028/0.987/0.942/0.884/0.814 at T=1/5/10/20/50. => the operator object is
+   right; freezing is a *local* theorem.
+2. **Drift is the whole error** (note 13): at T=50, oracle time-varying K(t)
+   product cuts gap MAE 0.0277 -> **0.000273** (100x); BP loss MAE 0.0254 ->
+   0.00024. => no higher-order Taylor mystery; model K_t and you are done.
+3. **Estimator** `K_hat_t = K_0 + (t/s)(K_s - K_0)` (notes 15-17): 256 traj,
+   T=50, s=20: MAE **0.00189**, bias -0.0015, corr **0.99885**; fixed-K(0)
+   MAE 0.0185 (10x worse); re-tangent MAE 0.0049 biased high => **no-fit
+   bracket** `linear <= empirical <= retangent`.
+   Stress grid (7 settings): depth 1/2/3, width 32/64/96, horizon 25/50/100:
+   linear MAE 0.00023-0.0057, corr 0.994-0.99989
+   (`operator_stress_grid_summary`).
+4. **Bias structure** (note 17): residual -0.0021 gap bias = BP-loss
+   overprediction (+0.0026), FA near-unbiased (+0.0005); cause: drift velocity
+   decays after s (alpha_t < t/s; e.g. true alpha_50 = 1.46 (BP) / 1.17 (FA)
+   vs linear 2.5). Honest error story, no fitted correction.
+5. **MNIST replication (note 38)**: same protocol, width 256, depth 2,
+   N in {128,256,512}, 48 traj: see `real_data_validation_mnist/`.
+
+### (d) The soft ramp + closed form (T7)
+
+- Dense sweep (note 22): T=30000, 11 widths (20..40), 352 traj: gap
+  0.1468 -> 0.000162 across margin -190 -> +130, smooth, log-linear
+  (`log gap ~ -0.0209*margin`, R^2 **0.993**), **no kink at margin 0,
+  nonzero at positive margin** => refutes the redundancy-exhaustion phase
+  transition (and notes 19-21 document how the kink hypothesis died as T grew:
+  near-margin-0 gap 0.297 at T=3000, 0.051 at T=10^4, 0.0037 at T=3x10^4).
+- Closed form vs measured (**matched data + init after note-37 fix**):
+  - matched single init: corr(log,log) = **0.933**;
+  - 5-init geometric mean (same data): corr **0.982**; ensemble-mean
+    lam_min(w) rises 0.026 -> 0.763 and explains the ramp at corr **0.987**;
+    rho stable 0.68-0.79 across widths.
+  - the previously reported corr 0.977 came from a comparison computed on the
+    wrong data seed and init seed; retire that number (note 37).
+  - frozen-init compresses the range (under-predicts large-gap end, over-
+    predicts small-gap tail) — exactly the drift the estimator corrects;
+    state this as mechanism-vs-magnitude.
+  (`closed_form_soft_ramp/closed_form_vs_measured.csv`, `closed_form_per_seed.csv`)
+
+## 4. Negative results worth reporting (reviewer-proofing)
+
+| failed approach | number | note |
+|---|---|---|
+| hard-k as gap predictor | overpredicts 2x-250x | 25 |
+| infinitesimal derivative extrapolation `K_0 + t dK_0` | MAE **270** vs 0.024 baseline (diverges) | 31 |
+| scalar directional-gain model across N | corr 0.124 (vs operator 0.999) | 14 |
+| trajectory bridge on long Adam horizons | shape corr 0.97 but 2-3x scale error | 02/08 |
+| phase-transition (kink) hypothesis | dissolves as T grows; dense T=30000 smooth | 19-22 |
+
+These define why the final objects (operator erosion + early velocity) are
+what they are; (31) and (25) deserve explicit mention in the paper.
+
+## 5. Real-data and generalization coverage (new, note 38)
+
+- MNIST e0 + estimator: `outputs/real_data_validation_mnist/` (script
+  `scripts/real_data_validation.py`, self-test verifies kernels vs autograd
+  at 1e-15).
+- Teacher-task test gap (SGD, widths 8..96): `outputs/teacher_test_gap_sgd_T8000/`
+  — addresses "what about generalization": train-gap story plus a descriptive
+  test-MSE companion. Random-label sweeps measure optimization gap by design.
+
+## 6. Figure plan (sources verified on disk)
+
+| fig | content | source |
+|---|---|---|
+| F1 | static law: Beta hist + QQ (+ Gamma(L,1) inset or appendix) | `capacity_empirical_validation` |
+| F2 | dense soft ramp + closed form (matched + 5-init mean + band) + lam_min ramp law | `closed_form_soft_ramp/closed_form_vs_measured_ramp.png` |
+| F3 | e0: moment calibration + Gaussian distribution panels (real backward) | `actual_fa_initial_operator_moments`, `actual_fa_initial_erosion_distribution` |
+| F4 | estimator: predicted-vs-empirical scatter, 256 traj + stress grid summary | `compressed_operator_s20_256traj_T50_width64_plots`, `operator_stress_grid_summary` |
+| F5 | MNIST: e0 calibration + estimator scatter (real data) | `real_data_validation_mnist` |
+| F6 (optional/appendix) | T=5 exactness + oracle-K(t) diagnostic (the lazy chain) | notes 10/13 outputs |
+
+## 7. Main text vs appendix split
+
+- Main: T1-T3 (brief), T4 + failure (a), T5 (centerpiece theorem), T7 +
+  Corollaries, estimator + stress grid, dense ramp, MNIST.
+- Appendix: full proofs (T1-T7 from note 33 + 34), T6 derivation + KS table,
+  the finite-time evidence chain details (T=5 table, oracle diagnostic, bias
+  decomposition, alpha_t curvature), negative results (31, 25), experiment
+  configs/seeds, teacher test-gap figure.
+
+## 8. Open problems (state, do not solve)
+
+Architecture-only finite-time theory: predict `rho` and the drift `K_s - K_0`
+from initialization statistics (infinite-width recursions / deep-linear order
+parameters); per-mode spectral erosion `rho_i` refining A2. (note 35)
+
+## 9. Known booby traps for the writing pass
+
+- Do not cite corr 0.977 for the closed form (mismatched; see note 37).
+- Do not present the estimator as architecture-only; it conditions on (K_0,K_s).
+- Do not present hard-k as a gap predictor; it is the null model only.
+- `E[e]=k/P` and `rho` are different objects (null model vs measured share).
+- The e0-distribution figure must be the post-fix (real-backward) rerun.
+- Dense-run measured gaps live in `width_summary.csv` (train_gap_mean), 1 init
+  x 32 feedback seeds — say so in the caption.
+- All speeds/losses use `L = ||r||^2/(2N)`; `e_0` is defined from speeds, not
+  losses.
diff --git a/notes/37_wrapup_corrections.md b/notes/37_wrapup_corrections.md
new file mode 100644
index 0000000..671c237
--- /dev/null
+++ b/notes/37_wrapup_corrections.md
@@ -0,0 +1,79 @@
+# Wrap-Up Corrections (pre-writing audit fixes)
+
+Two integrity issues found in the pre-writing audit (2026-06-09) and fixed.
+Both were flagged as cleanup items in note 35; this note records the fixes and
+the corrected numbers. The evidence ledger (note 36) already uses them.
+
+## 1. e0-distribution sampler was partly circular (fixed)
+
+**Problem.** `scripts/actual_fa_initial_erosion_distribution.py` sampled the
+erosion via the analytic linear form `speed_FA = output_speed + <C, B>`. Since
+a linear functional of Gaussian `B` is Gaussian by construction, the KS test
+against the Gaussian theory was close to tautological — it validated the
+implementation of `C`, not the claim about real FA.
+
+**Fix.** `sample_erosions` now runs the **real FA backward pass** for every
+feedback draw (`delta_hidden = (delta_out @ B^T) * gate`, then
+`<g_hidden^BP, g_hidden^FA>`). The analytic form is demoted to an internal
+consistency check: the per-draw deviation between the real backward and
+`<C, B>` is tracked and reported as `linear_identity_max_err`.
+
+**Result (rerun, same seeds).** Identical numbers to the published table in
+note 32 — as they must be, since the linear identity holds exactly:
+`linear_identity_max_err <= 8.9e-16` (float64 roundoff) across all panels.
+KS p-values 0.13-0.999 over widths {16,32,64,128} x 3 inits x 4096 draws.
+The figure is now non-circular: histogram from real backward, density from the
+theorem. Old outputs preserved in
+`outputs/actual_fa_initial_erosion_distribution_pre_fix_backup/`.
+
+For one hidden layer the linearity in `B` is exact algebra, so the empirical
+content of this experiment is (i) the identity holds for the real backward
+implementation and (ii) mean/std match the explicit formulas. Both now tested.
+
+## 2. Closed-form comparison was computed on mismatched data and init (fixed)
+
+**Problem.** `scripts/closed_form_soft_ramp.py` hard-coded the dense-sweep
+gaps (copied from note 22) and evaluated the closed form with its own defaults
+`data_seed=123`, `init_seed=0`. The dense T=30000 sweep actually used
+`data_seed=0` and init seed `10000` (= `10_000 + init_seed_offset`,
+`downstream_capacity_sweep.main`). So prediction and measurement lived on
+different data and different initializations, and the reported
+`corr(log pred, log measured) = 0.977` (notes 34/35) was partly luck.
+
+**Fix.** The script now loads the dense run's `summary.json` (config) and
+`width_summary.csv` (measured gaps + stds) from `--dense-dir`, reproduces the
+exact training data, evaluates the closed form at the **matched init seed**,
+and additionally at 4 extra init seeds on the same data to expose the
+init-to-init variability of the no-fit prediction.
+
+**Corrected numbers (rerun).**
+
+| quantity | value |
+|---|---|
+| corr(log pred, log measured), matched single init | **0.933** |
+| corr(log pred, log measured), 5-init geometric mean | **0.982** |
+| corr(lam_min, -log measured), matched init | 0.948 |
+| corr(lam_min mean over inits, -log measured) | **0.987** |
+| measured log-linearity (unchanged, data-only) | R^2 = 0.993 |
+| ensemble-mean lam_min(w) | 0.026 -> 0.763 (monotone) |
+| rho across widths (5-init mean) | 0.68-0.79 |
+
+Reading: a single init's frozen `lam_min(w)` is noisy across widths, while the
+T=30000 measured ramp is smooth (drift + feedback-averaging smooth it), so the
+matched-single-init correlation is lower (0.933). The init-ensemble closed
+form — still zero fitted parameters — tracks the ramp at 0.982 and isolates
+`lam_min` as the driver at 0.987.
+
+**For the paper:** report 0.982 (ensemble) and 0.933 (matched) and retire
+0.977; the figure now shows measured (error bars), matched-init curve,
+ensemble geometric mean, and the init-seed band.
+
+## 3. Related cleanup executed with this audit
+
+- `scripts/alignment_recovery_dynamics.py`: superseded probe (note 35);
+  its divide-by-zero guard was fixed earlier but the rerun was never completed
+  and its outputs are empty. Removed; its one useful datum (independent t=0
+  anchor `E_B[e_0] = 0.4026 +/- 0.0066` vs `hidden_share = 0.4071`) is
+  recorded here and in the ledger.
+- Notes 00/03 carry a superseded banner pointing at notes 35/36.
+- Headline-number corrections appended to notes 34/35.
diff --git a/notes/README.md b/notes/README.md
index db6bcfa..158a8c5 100644
--- a/notes/README.md
+++ b/notes/README.md
@@ -6,6 +6,17 @@ This folder collects working notes for the project tentatively titled:
 
 Use this directory as the project memory. Keep rough ideas here before promoting them into a paper draft or experiments.
 
+## Start here (current state)
+
+- [36_evidence_ledger.md](36_evidence_ledger.md): **the writing base** — full
+  evidence chain, corrected headline numbers, figure sources, claim discipline.
+- [37_wrapup_corrections.md](37_wrapup_corrections.md): pre-writing audit
+  fixes (e0 sampler now real FA backward; closed-form comparison now
+  data/init-matched — corr 0.977 retired, 0.933 matched / 0.982 ensemble).
+- [35_current_results_consolidated.md](35_current_results_consolidated.md): contribution snapshot (numbers corrected per note 37).
+- [34_closed_form_soft_ramp_theorem.md](34_closed_form_soft_ramp_theorem.md): closed-form gap law that quantifies the burden and explains the soft ramp.
+- [18_contribution_roadmap.md](18_contribution_roadmap.md): contribution roadmap (includes the tangent-operator estimator).
+
 ## Files
 
 - [00_project_overview.md](00_project_overview.md): high-level thesis, contribution map, and scope.