Map: allocator translation cost is linear (core r2 probes, tolerance, chosen constants)
Imagination pass, aetheria-release, session self-2026-10-09-morning. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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@ -7073,6 +7073,66 @@ That file's Cut 4 (AI heading planner) and Cut 5 (combat facing) are carried as
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allocator's input and rewrites throttle-lock's two mixer-shaped tests in allocator terms. One owner of
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`Ship.Update`'s thruster block at every step.
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### Revision: translation error costs linearly (core r2)
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Pass: Imagination, session `self-2026-10-09-morning`, after Hands stopped on `cut-thrust-allocator-core.r1`
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(question `allocator-translation-cost-l2-has-no-onset`, ruling `allocator-l1-translation-cost`, operator
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ruling `turn-authority-full`). Probes ran the real `CultMath.BoundedLeastSquares` (CultLib `614fd445`, the
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pinned 0.3.0) in a scratch console on Yggdrasil, logs `l1probe1.log` to `l1probe3.log` in the session
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scratchpad (`imag-allocator-r2`). Column sets: TA3 (Duel Longinus); TA3 with the clockwise Talaria zeroed;
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a symmetric Djinni-like hull (mains (0,50,+-20), fore laterals (+-20,0,+-80), aft laterals (+-20,0,-+60),
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bow (0,-30,0)); drives 0.3% apart for the tiny-imbalance hold.
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- **TA10. A squared translation cost has no onset** (Hands' probe, confirmed): r1's TA-R2 mixes the
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drive into a turn from Turn 0 for any weights, so it cannot meet `turn-authority-full`.
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- **TA11. The split form works only when the solver resolves it.** Variables: throttles and four error
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variables e+x, e-x, e+y, e-y in [0,2]; rows H*(B_r u / s_r - e+ + e-) = H d_r / s_r, the yaw row
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Wy * B_z u / s_z = Wy d_z / s_z, a cost row w*e = -w*kappa per error variable (linear slope
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L = 2 w^2 kappa, quadratic part at most e/(2 kappa) of it), a ridge row rho*u per throttle.
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At the solver's fixed KKT tolerance (1e-6 of the gradient at the origin) every one of 84 settings in two
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grids failed somewhere: Hands' (H 300, Wy 1, w .01, kappa 2, rho .01) and the first choice (H 100, Wy 10,
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w .01, kappa 10, rho 1e-3) gave all-zero throttles for move (0,1) and for a strafe, because the cost
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signal L is below the tolerance against the H^2 and Wy^2 terms; larger L settings served translation
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but gave throttles that depended on the warm start by up to 0.62 (ties the ridge cannot resolve below
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the tolerance). The chain the form needs, tolerance << rho^2 << L << H^2, Wy^2, with L/H^2 about 1e-4
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for a sharp onset and L/Wy^2 about 1e-6 for the hold, spans more than the 1e6 that tolerance leaves.
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- **TA12. With the tolerance at 1e-12 everything holds** (`l1probe3.log`, same grid, scratch change of
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the constant): warm-start difference 0 in every setting, IterationLimit never reached, at most 10
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iterations. Chosen H 100, Wy 1000, w .3, kappa 10, rho .1:
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| Check | Result |
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|---|---|
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| Duel onset, clockwise / counter-clockwise | .772 / .691 (columns: 166.26/215.50 = .7715, 124.63/180.49 = .6905) |
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| Duel, clockwise Talaria lost | drive from Turn > 0, Turn .1 gives drive .0999; Turn 1 gives 49.23 deg/s |
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| Symmetric hull | no translation up to Turn .88 (pure-torque pairs, then mains cancelled by the bow); strafe .95 of the extreme |
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| Hold, forward, yaw rate (deg/s) | intact -.0003, strong drive off .0003, weak drive off -.0002, drives .3% apart .0003 |
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| Forward served, move (0,1) | 1.000 of the forward extreme |
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| Largest throttle step per .001 of Turn over [-1,1] | .0133 (the steepest column ratio; Duel's drive slope is .0044) |
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| Largest yaw residual in the sweeps | .0006 deg/s |
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| cond(A), estimated from A^T A | 1.0e4; the solver documents exact results through 3e5 |
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Neighbouring settings fail one check each: H 30 leaks the drive at Turn .001; w 1 or 3 leaks it at
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.0014 throttle before the onset (L/H^2 too large); rho .03 makes throttles jump up to .12 per .001 of
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Turn (near-ties resolved too steeply); Wy 300 lets the hold drift .003 deg/s.
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- **TA13. The Duel Longinus does not strafe.** Under the linear cost a starboard strafe would need the
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counter-turning drive at full to hold heading, which costs more forward error than the strafe gains,
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so move (1,0) gives no throttle. Today's mixer gives none either (TA4). The symmetric hull strafes.
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Rationale changes:
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- **The solve keeps one call; the tolerance is the solver owner's to expose.** CultMath's
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`BoundedLeastSquares.Solve` gets an optional relative KKT tolerance (default unchanged, 1e-6), filled in
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CultLib, never worked around in Aetheria; the allocator passes 1e-12 and relies on the solver's
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progress rule to stop at its noise floor. Follow-up `cultmath-bls-kkt-tolerance` carries it; Aetheria
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takes it by a pin bump in the core cut's first commit.
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- **The onset is an ordering, not a label.** Under a linear cost the actuator with the least normalised
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translation per unit of turn serves first and the next joins only when it saturates; on the Duel that
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is the Talaria, so the onset is the Talaria's share of the turn extreme and moves with the columns
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(damage, heat, brownout). Nothing classifies a thruster as attitude or drive.
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- The bullets "One solve with a large yaw weight" and "The ridge" above describe r1 and are superseded
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by this revision: the yaw weight is 1000 against a ridge of .1, and uniqueness of the answer depends on
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the tolerance as well as the ridge.
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## Spec refresh 2026-10-09: scarce consumables, the fog bank, multiply, tiered boosters, the shipped AetherDb
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Pass: Imagination `imagination-spec-refresh`, session `self-2026-10-09-morning`, follow-up
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