Time rescaling integration: derived timesteps via a warp (phased plan — NT data layer → DTO evaluator → Piccolo templates → Altissimo)

[aarontrowbridge]

Important

Problem — Free-time in the stack has three formulations of the same physics with wildly different numerics: per-knot free Δt decision variables (the N-variable block — measured stalls, exploitation, det drift), the all-equal constraint (uniform mesh, free duration — correct semantics, but parameterized as N equality-chained variables), and pinned bounds. The measured evidence (Piccolo.jl#317): pinned-but-variable stalls at F=0.65 where a derived-timestep engine (Δt never a decision variable; duration as ONE coordinate on a fixed rational weight vector) solves the identical problem to 1e-12; the hub's bordered Jacobian adds a measured 1.05–1.9× wall (K-stable); a complete validated prototype exists (GriddedDiscretization, 451-test suite independently reproduced).
Approach — Integrate time rescaling (TR) as a first-class time mode in three phases, ordered to be collision-free with the open-core split (slices 3b–3e own the Piccolo-template + Piccolissimo-integrator refactor queue; this track owns the NT data layer + DTO evaluator/integrator plumbing):

Phase 1a — NamedTrajectories: derived timestep rows. A trajectory may carry a monotone warp t = w(s) with a small parameter vector (v1: GlobalScale, w(s)=T·s on uniform rational weights). The timestep component stays present (get_times, plotting, rollouts keep working) but becomes DERIVED: excluded from the decision vector, recomputed on unpack (sync_timesteps!), never optimizer data. This is the prototype's §5.4 "derived pinned rows" pattern — it kills the N-variable block AND the error-tuning channel by construction, and makes the AL KKT-plateau stall on free-Δt structurally unreachable.

Phase 1b — DTO plumbing. Integrators read derived Δt; the defect Jacobian/Hessian gain the warp-parameter column (exact entries w_n·∂Φ/∂Δt_n — a dense border column, O(Nb) measured-cheap); regularizer weights use derived Δt with the single ∂J/∂T chain term (kills the explicit per-knot ∂J/∂Δt scatter); TimeConsistencyConstraint, TimeStepsAllEqualConstraint, and the auto-Δt≥0 bound become obsolete under derived mode; MinimumTimeObjective → D·T exactly.

Phase 2 — Piccolo template surface. timesteps_all_equal=true REMAPS to Phase-1 derived-TR: identical documented semantics (uniform mesh, free duration), better structure (one variable, no degenerate block). Fixed-T = derived with T as data. Bounded per-knot stays as the free-time default per #317. The three-mode statement (#317) becomes the kwarg surface.

Phase 3 — Altissimo. The warp column as a global reduction (supported pattern, KnotLayout halo unchanged for uniform mesh); matrix-free JVP/VJP/HVP gain the T-slot via the exact chain rule w_n·∂/∂Δt_n.

Amicode side (parallel, cheap): skills carry the three modes + TR terminology (landed: armonissima#75/#79); the produced template switches to pinned-or-TR (amicode#613); formulation vocabulary gains time_mode: rescaled.

Acceptance Criteria

• Phase 1a: trajectory with a warp exposes derived Δt rows; vec(traj) excludes them; unpack! + sync_timesteps! restore consistency; all existing NT consumers (get_times, plotting, rollouts) unchanged on the derived rows
• Phase 1b: a 1q unitary problem solves end-to-end through the DTO evaluator with the warp as the only free time quantity; JVP/VJP parity vs FD including the T-column; the drift-dominated benchmark converges free-T where per-knot free-Δt stalls (the 0.65-vs-1.0 evidence reproduced in-stack)
• Phase 2: all-equal problems produce derived-TR construction (no Δt decision variables; one duration variable); bit-parity of objective/Jacobian values at equal T
• F_pre convergence-suite parity maintained (fidelity_regression suite green)

Testing Decisions

• NT: roundtrip/sync/derived-row testitems (prototype §8 patterns, exact == assertions)
• DTO: integrator FD-parity incl. the T-column; #172-pending constraint suite untouched; fidelity_regression as the parity gate
• Prototype as oracle: GriddedDiscretization.jl (branch dev/time-dependent) is the reference implementation — port semantics, not code, where licenses/structure demand

Key Decisions

• Phase order 1a → 1b → 2 → 3 chosen because NT is outside the open-core split's refactor queue (collision-free), and each phase is independently mergeable behind defaults
• GlobalScale only in v1; PiecewiseLinearWarp interface lands with it (prototype-tested) but is not wired into the NLP
• The all-equal constraint is NOT deleted in Phase 2 — it remaps; its documented semantics (#317) are preserved

Prior Art

• GriddedDiscretization.jl PLAN §5.1–5.4 (grid/warp/trajectory core), §7 (objectives/NLP), §10 (ten verified migration call sites); NOTE §2–§6 (two-error-channel theory, geometric-vs-Hamiltonian scaling, Magnus separability)
• Piccolo.jl#317 (the three-mode semantics), DirectTrajOpt.jl#145 (default Δt window — pre-derived guard), #464 (the AL stall this cures structurally)
• ADR-0001/0008/0009/0010/0011 (Piccolissimo-side conventions for Phase 3)

Source

• Campaign: piccolissimo-matrixfree-autodev (session-20260827); user directive 2026-08-28: "figure out how best to integrate the new time rescaling and then do it"