# Minimal Transition Theory

The cleanest correction is not to add a separate FA burden coefficient first.
Start by calibrating the task dimension from BP alone.

## Problem

The hard theory used:

\[
d_{\mathrm{task}}=Nn_L=512.
\]

With this value, the smooth theory peak is near:

\[
M_{\mathrm{FA}}\approx -273,
\]

while 256 empirical FA trajectories peak at:

\[
M_{\mathrm{FA}}=-318.
\]

## BP-Only Calibration

Fit the BP train curve with:

\[
L_{\mathrm{BP}}(P)
\approx
s\cdot \operatorname{softplus}_{\tau}(d_{\mathrm{task}}^{\mathrm{eff}}-P).
\]

The BP-only fit gives:

\[
d_{\mathrm{task}}^{\mathrm{eff}}\approx 437,
\qquad
\tau\approx 50.
\]

So BP is stronger than the naive parameter count suggests:

\[
d_{\mathrm{task}}^{\mathrm{eff}}
\approx
0.85\,Nn_L.
\]

## Minimal FA Prediction

Use the same hard FA burden:

\[
K_{\mathrm{FA}}=\sum_l(D_l-1),
\]

but replace the task dimension:

\[
M_{\mathrm{FA}}^{\mathrm{min}}
=
P-K_{\mathrm{FA}}-d_{\mathrm{task}}^{\mathrm{eff}}.
\]

Equivalently, when plotted on the original hard-margin axis

\[
P-K_{\mathrm{FA}}-Nn_L,
\]

the theory shifts left by:

\[
Nn_L-d_{\mathrm{task}}^{\mathrm{eff}}
\approx
75.
\]

This explains the observed left shift without introducing a fitted FA burden
coefficient.

Generated figure:

`outputs/downstream_capacity_random_main_fast/capacity_transition_bp_calibrated_p2_256.png`

## Recommendation

Use this as the main downstream theory:

1. Matrix/capacity theory remains exact and uses the hard/log-volume definitions.
2. Downstream performance uses a BP-calibrated effective task dimension.
3. FA capacity loss is then tested with no extra FA-specific fitting parameter.

Only add an effective FA burden coefficient

\[
K_{\mathrm{FA}}^{\mathrm{eff}}=\alpha K_{\mathrm{FA}}
\]

as an appendix ablation if needed. It improves fit but makes the theory look
more ad hoc.