Objective
Compute block-relative deterministic curvature at the incumbent ep06c_polished in 54-D phi-space on the Hamilton surrogate, producing eigendirections and feasible radii for potential restart preconditioning.
Setup
- Model: 54 free parameters, Hamilton surrogate eval_nll_hamilton (no cap rates), fully deterministic
- Evaluator: forward-FD diagonal Hessian H[i,i] = 2(f+ - f0)/h^2 with block-relative steps h_i = h_rel * S_b
- Comparators: central-FD Hessian (auditor), 08a curvature at theta_true
- Acceptance: nonzero_feasible_radius_top10 >= 8 for USE_FOR_RESTART_SCALING; >= 3 for DIAGNOSTIC_ONLY
Procedure
- Block-relative step scaling across 9 blocks (Risk premia to Regime persistence)
- 54-coordinate diagonal Hessian + 1431 cross-term pairs, h_rel=1e-4
- Eigendecomposition, top-5 stiff/sloppy selection, binary-search feasible radii
- Verifier: independent driver reproduces bit-identically (deterministic surrogate)
- Auditor: central-FD pipeline (5833 evals), algebraic contamination diagnostic, h_rel sweep
Results
- Forward-FD: 30 pos / 24 neg / 25 null eigenvalues; condition 1.57e26; 0 shrinks, 0 blocked coords
- Central-FD (audit): 37 pos / 17 neg / 29 null; condition 1.12e24
- Ranks 1-2: stencil-robust (Monetary means, |lambda| ~ 5.6e9 / 5.1e8, same sign)
- Ranks 3-5: sign-flipped under central FD; magnitude displaced 2.5x / 27.5x / 143x
- h_rel sweep confirms: forward-FD ranks 3-5 scale as 1/h (ratios 12.38/12.26/10.05 = gradient contamination)
- Gradient contamination: ||grad f_Hamilton||_inf ~ 1.9e4 (Monetary means); 2*grad/h ~ 3.8e8 comparable to rank-2
- Algebraic identity H_B[i,i] - H_A[i,i] = 2*grad_i/h_i verified to 2.98e-8 on all 54 coords
- Nonzero feasible radii: 6/10 (fwd) / 7/10 (central) — both < 8 threshold
- Null space: exactly Income (24) + Risk price (1) = 25 coords, all zero under Hamilton surrogate
- Slice validation: abandoned (Path A) after spec cost model found off by ~50x (118s/eval, not 3s)
Analysis
The spec-level FD stencil bug is the key finding: the forward-FD formula 2(f+ - f0)/h^2 is unbiased only at a critical point of the same objective, but ep06c_polished was polished on eval_nll_sim (MC + Hamilton), not eval_nll_hamilton alone. The Hamilton-surrogate gradient at the incumbent is O(1e4) in Monetary means, producing O(1/h) contamination that dominates ranks 3-5. The feasible-radius gate (6/10 < 8) is a geometry fact independent of the stencil bug — block-relative scaling fixes step sizes but not the knife-edge feasibility geometry along stiff directions. The DIAGNOSTIC_ONLY outcome is robust under both stencils.
Claim updates
- forward-solution-unique-fundamental-ree-with-nbc: Curvature analysis at ep06c_polished reveals a 25-dimensional null space (Income + Risk price blocks) on the Hamilton surrogate, consistent with the forward-solution/NBC structure where these blocks are invisible to the macro+yield likelihood. The knife-edge feasibility geometry (6-7/10 nonzero radii in top-10 stiff directions) confirms that the NBC boundary is locally close to the incumbent.
Follow-up
- Exp 11b: central-FD rerun if curvature directions are needed for operational restart scaling
- Downstream consumers of rotation_scaling.json must restrict to ranks 1-2 + top-4 block labels
- Forward-FD stencil must not be used at non-critical points of the evaluation objective