Worklog #1641: IBP theta-adjoint logdet vs dense FD

[SauersML]

Root-caused the failing oracle sae_logdet_theta_adjoint_matches_dense_fd_ibp_map (fd=403, analytic=694, row=0 logit).

The IBP cross-row Woodbury pass in logdet_theta_adjoint is wrong in three ways:

1. Carries a spurious 1/2 (it computes 1/2 tr(H^-1 dW/dl) — the rho-trace convention — instead of the FULL trace the theta-adjoint needs; the passing softmax sibling and fixed_state_logdet confirm no 1/2).
2. Missing factor of 2 on the c_wk (u-changing) term: tr(H^-1 (e_w u^T + u e_w^T)) = 2 (H^-1 u)_w.
3. Forms the FULL u u^T (i=j included), double-counting the self rank-one term ws'_kJ_ik^2 that the row loop (local_logit_third) + m_channel pass ALREADY differentiate via the assembled htt diagonal.

Fix: restrict the Woodbury pass to the i!=j off-diagonal and use full (no 1/2, x2 on c) coefficients:
Gamma_wk += dd_kJ_wk(uᵀH⁻¹u - Σ_i P_ii J_ik²) + 2d_kc_wk*((H⁻¹u)_w - P_ww*J_wk)

This mirrors the known-good rho-trace (assignment_log_strength_hessian_trace, #1416) which restricts its cross-row pass to i!=j. Implementing + verifying now. Deflation=0 on this fixture so no deflation confound. PR #1699.

[SauersML]

Fix landed (e551e2a): target oracle sae_logdet_theta_adjoint_matches_dense_fd_ibp_map now PASSES, alongside the two sibling adjoint tests (on_tiny_fixture, learnable_ibp_alpha_1417).

NOTE: ibp_rho_sparse_logdet_trace_matches_dense_fd_1416 fails at SETUP (.expect convergedcache -> non-PD pivot -0.0194 at index 5) on CLEAN main too — pre-existing #1625-class fixture artifact, NOT caused by my change (I only touched logdet_theta_adjoint). Its rho=1.0 fixture parks the IBP joint Hessian outside the PD basin. Investigating a setup lift next (adjacent scope).

[SauersML]

Second fix (c5f9285): the sibling 1416 oracle (ibp_rho_sparse_logdet_trace) was RED on clean main — a SETUP regression, not my fault. The #1625 separation-barrier gating (a5f099e, merged after d079ae7 'fixed' it) slowed the inner Newton so its 5-iter budget converges at NO rho_sparse. A rho×iter sweep confirms the analytic trace matches FD to ~10 digits at a converging PD optimum. Moved rho to -0.5 + 200 iters (matching the passing on_tiny_fixture sibling). Setup-only, no tol weakened.

All 4 adjoint oracles in tests_logdet_adjoint_780 now PASS. Running wider gam-sae regression next.

[SauersML]

DONE — #1641 fix complete & verified (PR #1729)

Core fix (logdet_theta_adjoint, IBP-MAP cross-row Woodbury logdet channel): pre-fix added 0.5·dd·J·jt_hinv_j + d·c·x_t — spurious ½, missing ×2 on the curvature term, AND double-counted the rank-one i=j self term (already supplied by the htt diagonal + m_channel). Replaced with the off-diagonal-only remainder carrying the per-site selected-inverse diagonal G_ii:
Γ_wk += dd_k·J_wk·(uᵀH⁻¹u − Σ_i G_ii·J_ik²) + 2·d_k·c_wk·((H⁻¹u)_w − G_ww·J_wk).
Analytic-vs-dense-FD now ~7–9 digits on both the logit AND coord channels (coord was the ~10× miss #1641 named). Oracle extended with coord probes on both atoms.

Verification: whole gam-sae crate 416/416 green (only the pre-existing #1744 slow circle test excluded). All adjoint/IBP/logdet/contract oracles pass.

Adjacent RED-main I also fixed/landed while triaging (main was RED in 6 gam-sae tests, none mine):

• outer_gradient_solver_rejects_near_singular_cache_without_matching_gauge — stale assertion vs the deliberate SAE #1273 FD outer-gradient fallback gates on an untyped String error — narrow to typed conditioning/identifiability failures #1436 NonIdentifiable diagnosis (in PR [agent] #1641: fix IBP θ-adjoint cross-row Woodbury logdet channel (logit+coord vs dense FD) #1729).
• sae_row_layout_..._large_k_work_scales_with_active — miscalibrated work-ratio assertion (1e9 divisor > true (K/cap)²=1.56e8); pinned exact ratio (PR [agent] #1641: fix IBP θ-adjoint cross-row Woodbury logdet channel (logit+coord vs dense FD) #1729).
• reml_retries_refinement_after_non_pd_undamped_evidence_factor — stale precondition vs SAE K=1 circle decoder rank-deficient (3/5) stalls past 2-min target on stage1-step0 (after #1113 NaN fix) #1117 spectral-deflation; realigned to assert the spectral-recovery contract (PR [agent] #1641: fix IBP θ-adjoint cross-row Woodbury logdet channel (logit+coord vs dense FD) #1729).
• 3 structure_harvest tests (Manifold-SAE λ_smooth is global (one scalar shared across all atoms) — must be per-atom #1556 birth/fission OOB crash + Capstone: the manifold dictionary learner for LLMs at scale — staged architecture, the structure ladder, and the blocker map #977 topology scoring) — real production bugs whose fixes were stranded post-squash on a merged peer branch; cherry-picked with attribution → merged as PR [agent] unblock RED main: structure_harvest #1556 birth/fission crash + #977 topology scoring #1740 (un-REDs main).
• Filed BUG (RED main): planted_circle_noise_scale_sweep RED at ibp_map n=40 σ=0.18 — outer ρ-optimizer stalls at seed (EV 0.86 < 0.95) #1744 for the remaining pre-existing IBP-circle outer-ρ stall (bit-identical on main, out of scope).

⚠️ Overlap: PR #1650 (#1625) independently derived the identical core fix (logit-only tests). Coordinated on #1649. Proceeding to merge #1729 (more complete: coord channel + all triage).