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+# Closed-Form Soft-Ramp Gap Law
+
+> **Correction (2026-06-09, note 37):** the verification table below was
+> computed on mismatched data/init seeds. Matched rerun: corr(log,log) =
+> **0.933** (matched single init) / **0.982** (5-init geometric mean);
+> `corr(lam_min_mean, -log gap) = 0.987`. Use these numbers, not 0.977.
+
+This note gives a closed-form burden that quantifies the finite-time FA/BP train
+gap and explains the empirical soft ramp (notes 22, 25) as a smooth spectral
+phenomenon rather than a phase transition. It connects the exact initial-erosion
+theorem (note 29) to the finite-time gap and sits one level below the early
+operator-velocity estimator (notes 15-17): the closed form gives the *mechanism
+and shape*; the estimator supplies the drift-corrected *magnitude*.
+
+## Assumptions
+
+**A1 (lazy / frozen tangent operators).** Over the horizon the tangent operators
+are approximately frozen at initialization, so for squared loss
+`L = ||r||^2/(2N)` the BP residual evolves as
+
+```text
+r_t^BP = (I - eta K_BP / N)^t r_0,    K_BP = J J^T.
+```
+
+This is the standard lazy/NTK approximation. Operators actually drift; that
+residual is exactly the `K_s - K_0` the estimator corrects (notes 13, 15).
+
+**A2 (scalar erosion model).** The FA tangent operator `K_FA = J J_tilde^T` is
+non-symmetric; only its symmetric part `S_FA = (K_FA + K_FA^T)/2` drives the
+first-order loss change. We model
+
+```text
+S_FA ~= rho K_BP,    rho = output_share = r^T K_out r / r^T K_BP r in (0,1).
+```
+
+This is *exact in the residual direction* by the initial-moment theorem
+(note 29): `r^T E_B[S_FA] r = ||g_out^BP||^2 = r^T K_out r = rho * r^T K_BP r`,
+i.e. `rho = 1 - E_B[e_0]`. Treating `S_FA` as a global scalar multiple of
+`K_BP` extends this identity to all eigenmodes and is the one approximation
+beyond A1.
+
+## Theorem (closed-form FA/BP gap)
+
+Under A1-A2, in the BP eigenbasis `K_BP v_i = lam_i v_i` with `c_i = v_i^T r_0`,
+
+```text
+gap_T = L_FA(T) - L_BP(T)
+      = (1/2N) sum_i c_i^2 [ (1 - eta rho lam_i / N)^{2T}
+                           - (1 - eta lam_i / N)^{2T} ].
+```
+
+**Proof.** `L_rule(T) = ||r_T^rule||^2/(2N)`. Expanding `r_0 = sum_i c_i v_i`,
+
+```text
+r_T^BP = sum_i v_i (1 - eta lam_i/N)^T c_i
+=> ||r_T^BP||^2 = sum_i c_i^2 (1 - eta lam_i/N)^{2T}.
+```
+
+Under A2, `S_FA = rho K_BP` shares eigenvectors `v_i` with eigenvalues
+`rho lam_i`, so
+
+```text
+r_T^FA = sum_i v_i (1 - eta rho lam_i/N)^T c_i
+=> ||r_T^FA||^2 = sum_i c_i^2 (1 - eta rho lam_i/N)^{2T}.
+```
+
+Subtract and divide by `2N`. ∎
+
+## Corollary 1 (soft, not hard)
+
+Because `rho < 1`, every mode decays no faster under FA than under BP, so each
+bracket is `>= 0` and `gap_T >= 0`. `gap_T` is real-analytic in `(rho, {lam_i},
+T)`. Therefore under any smooth deformation of the spectrum (e.g. width or
+capacity margin) the gap moves smoothly: it is a **soft ramp**. A discontinuous
+transition would require `rho -> 0` (FA fully stalls), which never happens since
+`rho >= output_share > 0`. The hard-margin threshold
+`Delta d_hard = max(0, k - (P - d))` is recovered only in that degenerate limit.
+
+This is the operator-level version of the "no free capacity" statement
+(note 26): nonzero `rho < 1` gives a positive contribution from every mode at
+every margin.
+
+## Corollary 2 (the ramp law: where it comes from)
+
+Let `tau_i = N / (eta lam_i)`.
+
+- **Underparameterized side** (many `tau_i >~ T`): each bracket
+  `~= 2 eta T (1 - rho) lam_i / N`, so
+
+  ```text
+  gap_T ~= (1 - rho) (eta T / N^2) sum_{slow i} c_i^2 lam_i,
+  ```
+
+  residual-mass limited; large at negative margin (ill-conditioned spectrum,
+  residual stuck in slow modes).
+
+- **Converged side** (BP fit, `L_BP(T) ~= 0`): the gap is the slowest surviving
+  FA mode,
+
+  ```text
+  gap_T ~= (c_min^2 / 2N) exp(-2 eta rho lam_min T / N),
+  i.e.  log gap_T ~= const - (2 eta T / N) rho lam_min.
+  ```
+
+The soft ramp is therefore driven by `lam_min(w)` (smallest BP-NTK eigenvalue)
+growing smoothly with capacity; the two regimes cross over smoothly.
+
+## Verification (no fit)
+
+Script: `scripts/closed_form_soft_ramp.py`.
+Figure: `outputs/closed_form_soft_ramp/closed_form_vs_measured_ramp.png`.
+Rows: `outputs/closed_form_soft_ramp/closed_form_vs_measured.csv`.
+
+Frozen-initialization closed form (`rho`, `lam_i`, `c_i` all measured at init,
+no fitted parameters) versus the dense `T=30000` ramp (note 22):
+
+| quantity | value |
+|---|---|
+| `corr(log pred, log measured)` across `w=20..40` | **0.977** |
+| measured `log(gap) ~ -0.021*margin`, fit `R^2` | **0.993** (log-linear ramp) |
+| `lam_min(w)` over the sweep | 0.07 -> 0.99 |
+| `corr(lam_min, -log measured)` | **0.963** (ramp driven by `lam_min`) |
+
+## Scope / honesty
+
+- Frozen-init captures the ramp **shape** (corr 0.977) but **compresses the
+  range**: it under-predicts the large-gap (negative-margin) end and
+  over-predicts the small-gap (positive-margin) tail. That residual is operator
+  drift `K_s - K_0`; the early-velocity estimator (notes 15-17) corrects it and
+  supplies the quantitative magnitude. Closed form = mechanism + shape;
+  estimator = number.
+- A2 is exact only in the residual direction; a per-mode `rho_i` (spectral
+  erosion) refinement is the natural next step.
+- Lazy regime, full-batch SGD, synthetic random-label MLP scope, as elsewhere.
+
+## Role in the paper
+
+This is the quantitative core of contribution 2 (soft erosion). It converts the
+exact scalar burden `rho = 1 - E_B[e_0]` (note 29) into a finite-time gap and
+proves the soft ramp is a spectral consequence of `rho < 1` plus a smoothly
+growing `lam_min`, not a redundancy-exhaustion threshold. Pairs with: note 29
+(supplies `rho`), notes 15-17 (drift-corrected magnitude), notes 22/25 (the
+empirical ramp it explains).