# 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.