feat(cam-lower): seam-jump term from endpoint kappa'' (droppable: negligible yield) #132
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SE(3) extension S-i, design-logged tightening lever — HONEST RESULT: negligible (1.0004x gentle / 1.0058x stiff). The seam term was never the dominant slack; the kappa-third product in Omega_sup is, and the next real lever is the design S5 alternative (bound TV(zeta) directly). Ships because the machinery is conceptually right (the endpoint value IS the jump — no sup to take) and the interior-sup cap is load-bearing: measured endpoint-estimate overshoot 7.93845876e7 vs sup 7.93829933e7, and the cap is what makes new <= old a theorem (min of same-rigor bounds) rather than a fixture observation. Reviewer recommendation: land last, DROPPABLE at zero cost if the queue is tight — the subtlety is captured here either way. Opus review PASS.
Certificate tightening lever 2, the design-logged one (docs/superpowers/se3-lowering-design.md section 3.3a: "reading actual seam-endpoint values from the same 1024-sample sweep is the logged tightening"). The seam jump at a host/blend boundary is |kappa''_blend(seam) - kappa''_host| = |kappa''_blend(seam)|, because a Line has kappa == 0 and an Arc has kappa constant, so both have kappa'' == 0 exactly. The endpoint value IS the jump; there is no sup to take. The previous form, 2 * sum over blends of sup|kappa''|, replaced each endpoint with the blend's interior maximum, which for an eta^3 blend (kappa' runs 0 -> peak -> 0, so kappa'' is largest somewhere inside) over-counts. NUMERICAL CARE, since Eta3Spline exposes no kappa'' accessor. A one-sided stencil exactly at the endpoint would be first-order accurate and evaluated where the parameterization is least well conditioned. Instead the outermost CENTRED difference from S-c's existing 1024-sample sweep is used -- an estimate of kappa'' half a sample step inside the seam -- plus an explicit margin of 0.5 * ds * sup|kappa'''| to step it back to the seam. kappa'' is C1 inside the blend, so over a half-step it moves by at most the local kappa''' times that half-step, and blend_kappa_third_sup already bounds kappa'''. The correction is therefore bounded by machinery S-c certifies, not by assumption. The result is then CAPPED at the blend's own interior sup. That cap is trivially sound (the seam is inside the blend) and is what makes "never above the superseded form" a theorem rather than a fixture observation: without it the margin can overshoot on a blend whose kappa'' peaks AT the seam, where the estimate is already the maximum and the margin pushes it past the interior sup's own smaller Lipschitz margin. That overshoot was real on the 60x40 mm fixture -- 7.93845876e7 against a sup of 7.93829933e7 -- so the cap is load-bearing, not defensive decoration. ACHIEVED IMPROVEMENT -- NEGLIGIBLE, and reported as such because the numbers go to Patrick: gentle (200x150 mm, 1 mm corridor, 50 mm/s, dt 0.5 ms) orientation 3.758043e-2 -> 3.7565905e-2 rad (1.0004x, +0.04%) omega 2.942506e-3 -> 2.9413682e-3 rad/s (1.0004x, +0.04%) stiff (60x40 mm, 0.2 mm corridor, 0.35 m/s, dt 1 ms) omega 9.859359e0 -> 9.8024359e0 rad/s (1.0058x, +0.58%) Well under the 2x bar -- this is rounding, not a lever. The reason is structural and worth recording: the seam term enters as (dt^2/8) * J_ang while the bound is dominated by the windowed (dt^2/12) * sum(Omega_p * D_p) term, whose leading factor is |kappa'''| v^4. Tightening the seam term cannot move a total it is a small fraction of. Combined with S-h the two levers give 1.74x (gentle) and 2.14x (stiff); essentially all of that is S-h. WHERE THE REMAINING HEADROOM ACTUALLY IS, for whoever picks this up: the |kappa'''| v^4 term itself. Either a tighter kappa''' sup than S-c's dense-sampled Lipschitz heuristic, or the design's logged S5 alternative (bound TV(zeta) directly, needing only kappa'' and no fourth-order finite difference), which would replace the term rather than shave it. Revisable: the half-step margin and the cap are both choices; the cap is required for the "never above" claim, the margin size is not. Co-authored-by: patman-assist <patrick-ai@kgroo.co>