# ADR-018 graph BL/solve certificate controller draft

Status: private design only. Shared test source is frozen while the configured
Debug focused run is active.

## Scope

Stored `ConstraintRowOperators::{gradient,hessian,time_gradient,third_directional_gradient}`
are exact binary64 inputs.  The draft certifies only fixed-basis contraction and
linear graph-solve arithmetic.  It does not certify AD/FE operator formation,
root branch sensitivity, or nonlinear history.

## Test-local data

`GraphArithmeticCertificate` should retain dense-small matrices/vectors:
`Ghat`, `P_hat`, `BG`, `BP`, `bhat`, `Bb`, `R`, `Hbar`, `Q`, `rbar`, `Bx`.
It is private to `hht_shell_physics_test.cpp`.

For every graph row and basis column, replay source order:

1. Iterate `row.gradient`, then `column.support`; match full DOF.
2. Product radius is `ProductRadius(C, 0, L, BL)`.
3. Reduce matching products in `RateSumRadius` source order.

This yields stored `Ghat=C*Lz`, `P_hat=C*Lp`, and entrywise `BG/BP`.
The same replay builds RHS radii for `P*vp`, `P*ap`, Hessian/time contractions.
No zero-BL fallback is allowed.

## Inverse action

For each canonical unit RHS, use the existing graph solver and N2-L check to
obtain one stored column of `R`.  Replay `I-Ghat*R` with product/sum radii to
make nonnegative `Hbar`.  Require finite values and `thetaH=||Hbar||inf<1`.
Construct

`Q[i,j] = abs(R[i,j]) + (abs(R)*Hbar*1)[i] / lower(1-thetaH)`.

All bound additions/divisions use existing outward helpers.  Failure of the
positive lower denominator or either theta is INCONCLUSIVE, not a radius reset.

## Uncertain solve

Replay `rhat=bhat-Ghat*xhat`; combine absolute residual and rounding into
`rbar`.  With actual RHS bound `Bb`:

`y = Q*(rbar + Bb + BG*abs(xhat))`, `A=Q*BG`, `thetaG=||A||inf`.

Require `thetaG<1`; set

`Bx[i] = y[i] + rowSum(A,i)*normInf(y)/lower(1-thetaG)`.

Undo any actual rate scaling before using `Bx` in physical lifted radii.

## Tests

Add only after the current focused run closes:
- small chart with plane coefficient value zero / positive BL, proving BG/BP
  remain nonzero after an omitted-BL mutation;
- 1x1 and 4x4 source-order graph replay, N2-L checked;
- one thetaH and one thetaG failure must return incomplete;
- use a known 4x4 perturbation from the independent 32-case probe as a
  behavior oracle, without copying implementation equations as its oracle.

No production API, tolerance, grid, or sparse global inverse is added.