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# 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.
## 4. e0 moment headline "0.4378/0.4403" not reproducible (retired)
Found 2026-06-13 while building the master technical reference (note 42).
The figure `E_B[e_0] = 0.4378 +/- 0.008` vs hidden share `0.4403`, cited in
notes 29/35/36/40/41 and the old `paper/body.tex`, does NOT appear anywhere in
the current `actual_fa_initial_operator_moments/initial_operator_moment_rows.csv`
(6 widths {16,24,32,48,64,96} x 6 init seeds; no row or aggregate is ~0.44;
closest is width 16 init 0: predicted 0.4519 / empirical 0.4443). It is
presumed to come from a superseded run that was overwritten, or from the
AI-written draft.
Retired. The reproducible Thm 3 evidence is the full 6x6 table (note 42 §6):
the no-fit prediction sits within Monte-Carlo error of the measurement, **max
|error| <= 0.008**, with predicted erosion spanning 0.15-0.45 (a wide-range
calibration, stronger than a single point). Living docs (36, 40, 41, README)
updated to this statement; historical notes 29/35 left as-is with this pointer.
## 5. 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.
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