docs: consolidate solver knowledge and preserve formulation originals

2026-10-17 22:34:12 +09:00
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+# 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.