“Speed derate” is an impedance derate — why softening gains holds the singularity floor and throttling the command collapses it

Date: Jun 18, 2026 (EDT). Question: the shipped derate γ(σ_G) is named a “speed derate” (current_sota.md §6.3 title; symbol table line 95) but is applied to the control gains, K_e, D_e → γK_e, γD_e. Are these the same thing? If not, which one is right, and why does the code’s choice hold the singularity floor while the literal “speed derate” collapses it?

regen: /Users/antoniahoffman/miniforge3/envs/new-pin-env/bin/python validation/test_derate_mechanism.py (every algebraic claim below is a passing pin in that file; the empirical numbers are the committed cascade A/B, analysis.speed_derate_study logs/logs_Jun18_26 speed_derate_cascade, HEAD 8bef81a+).

Verdict up front. They are not the same operation, and the difference is decisive:

• Impedance derate — what the code does (γ on the gains). Provably holds the floor: it relocates the cruise-lag equilibrium to a larger lag under the same restoring force (eq 4.14), so the arm goes slack rather than driving itself into the singular direction. min s_min_J stays at 0.0268.
• Speed derate — what the name implies (γ on the commanded twist ν_e). Provably cannot beat the amplification bound (eq 6.1) and in practice collapses the floor to 1.0e-5, because the throttle’s own pose-error growth crashes σ_min far harder than γ throttles the speed (a ~644× net loss).

So §6.3’s title and the symbol table are a misnomer: γ is an impedance derate that produces an emergent slowdown, not a commanded one. The fix is a rename (§4 below), and the cascade A/B is the receipt.

Vocabulary (derate, DLS, σ engagement bands) is in wiki/terminology.md.

1. The two operations are different equations

The singularity proximity scalar is σ_G ≡ σ_min(Γ) (current_sota.md §6, Spearman ≥ 0.997 with σ_min(J⁺)). The ramp γ(σ_G) ∈ [0.25, 1] (eq 6.4) is the same in both cases — scale_by_svd (utils/sampling.py:70). What differs is what γ multiplies:

The impedance derate touches the force the arm applies; the speed derate touches the target it chases. That distinction is the whole story.

2. Impedance derate holds the floor (eq 4.14)

The quasi-steady cruise-lag error (current_sota.md eq 4.14, computed by the controller’s own cruise_lag_floor, GNC/guidance/analytic_feedforward.py) is

\[ \tilde{\boldsymbol{x}}_{ss}=-\big(\boldsymbol{J}_{\tilde{x}}^{\top}\breve{\boldsymbol{K}}\big)^{-1}\,\boldsymbol{b}, \qquad \boldsymbol{b}=\breve{\boldsymbol{C}}\breve{\boldsymbol{v}}_d+\big(\boldsymbol{C}_c+\breve{\boldsymbol{D}}\breve{\boldsymbol{G}}_{v_c}\big)\dot{\tilde{\boldsymbol{x}}}_c . \]

The forcing b carries no K. So scaling the stiffness K̆ → γK̆ gives, exactly,

\[ \tilde{\boldsymbol{x}}_{ss}(\gamma)=-\big(\boldsymbol{J}_{\tilde{x}}^{\top}\gamma\breve{\boldsymbol{K}}\big)^{-1}\boldsymbol{b} =\tfrac{1}{\gamma}\,\tilde{\boldsymbol{x}}_{ss}(1). \tag{lag scales 1/γ} \]

At the derate floor γ = 0.25 the lag grows 4×. But the restoring wrench is γ-invariant:

\[ \boldsymbol{w}(\gamma)=\boldsymbol{J}_{\tilde{x}}^{\top}(\gamma\breve{\boldsymbol{K}})\,\tilde{\boldsymbol{x}}_{ss}(\gamma) =\boldsymbol{J}_{\tilde{x}}^{\top}(\gamma\breve{\boldsymbol{K}})\cdot\tfrac{1}{\gamma}\tilde{\boldsymbol{x}}_{ss}(1) =\boldsymbol{J}_{\tilde{x}}^{\top}\breve{\boldsymbol{K}}\,\tilde{\boldsymbol{x}}_{ss}(1)=\boldsymbol{w}(1). \tag{wrench γ-invariant} \]

Both identities are pinned (test_gain_derate_scales_lag_by_inverse_gamma, test_gain_derate_leaves_the_restoring_wrench_invariant). The reading: reduced authority, same force. The arm settles at a slacker equilibrium (larger trailing lag) but is pushed there by the same gentle restoring wrench that just balances the cruise’s Coriolis forcing C̆v̆_d. It has no extra drive toward the singular (full-extension) direction — it relaxes rather than fights. That is the “natural slowdown near singularities” the sheet describes (eq 6.4 commentary): an emergent slowdown from the larger lag, never a commanded one. Empirically the gain variant holds min s_min_J at 0.0268 with γ engaging to a 0.58 median.

3. Speed derate cannot beat the amplification bound (eq 6.1), and collapses the floor

The velocity reconstruction from the SVD Γ = Σ sᵢ uᵢ vᵢᵀ amplifies by 1/sᵢ (current_sota.md eq 6.1):

\[ \boldsymbol{x}=\sum_i\frac{\boldsymbol{u}_i^{\top}\boldsymbol{z}}{s_i}\boldsymbol{v}_i \quad\Longrightarrow\quad \lVert\boldsymbol{x}\rVert\le\frac{\lVert\boldsymbol{z}\rVert}{\sigma_{\min}(\boldsymbol{\Gamma})}=\frac{v}{\sigma_{\min}}, \]

with equality when z aligns with u_min, the dying left-singular direction (both pinned: test_velocity_reconstruction_obeys_the_sigma_min_bound, test_bound_is_tight_along_the_dying_direction). The bound is tight, not conservative — exactly the directions a near-singular trajectory excites.

Now throttle the command, v → γv. The bound becomes γv/σ_min — naively safer. But the throttle does not act on σ_min, and here is the trap: a throttled EE cannot hold the target against the full-speed orbit, so its pose error grows; the retained full stiffness K closes that error by extending the arm; extension crashes σ_min. The denominator moves far more than the numerator. Quantify it from the committed A/B (speed_derate_cascade_{gain,speed}): the speed variant’s net velocity bound relative to the gain variant is

\[ \frac{\gamma\,v/\sigma_{\text{speed}}}{v/\sigma_{\text{gain}}} =\gamma\,\frac{\sigma_{\text{gain}}}{\sigma_{\text{speed}}} =0.25\times\frac{0.0268}{1.04\times10^{-5}}\approx 644 . \]

So the 4× throttle leaves the speed variant’s worst-case joint-rate bound ≈644× worse, not safer (test_speed_derate_throttle_is_overwhelmed_by_the_sigma_collapse). The realized trajectory confirms the mechanism (analysis.lockin_fingerprint, CLAIMS ~16:00): during the near-singular episode the arm is not frozen — q̇ runs 2.27× the healthy rate and reach is 1.14× higher. The throttle causes the over-extension it was meant to prevent. Speed-derate is falsified: min s_min_J 0.0268 → 1.0e-5, nsf 0.77%.

4. The fix: rename, don’t re-engineer

The math says the operation is correct (impedance derate holds the floor); only the label is wrong. Proposed edits to GNC/equations/current_sota.md:

• §6.3 title “Layer 2 — speed derate through impedance” → “Layer 2 — impedance derate (emergent slowdown)”, with one line: γ scales the gains, not the command; the slowdown is the larger cruise-lag this produces (eq 4.14), not a throttled reference. A literal speed derate (γ·ν_e) is a different operation and collapses the floor — see derate_impedance_vs_speed.md.
• Symbol table line 95 “speed-derate ramp (scales gains/wrench)” → “impedance-derate ramp (scales gains; emergent slowdown)”.

This is the inconsistency the project flagged as indefensible: a name asserting one mechanism (commanded slowdown) over code that runs another (impedance softening). Renaming makes the sheet defensible without touching a single control gain.

5. Consequences for the code

• The singularity floor is arm-geometric. Both the impedance- and speed-derate analyses route through the same σ_min(Γ), set by the arm’s reach/posture, not by either block’s stiffness. The gainbase A/B confirms it from the other side: also derating the base gains (the live γ·τ_b^⊕ via base_breve_gains, GNC/breve_controller.py:425) moves the arm floor by +0.1% — i.e. nothing — while degrading base pointing z_b by +23%. The base gain-derate is internal-consistency-only (γ now genuinely scales the base K/D block in the RHS, so the previously logged-but-inert wrench is no longer a phantom); recommend controller.base.derate.gain stay default-off.
• The lever for the floor is geometric, not a throttle. Over-extension — not lock-in — is the failure mode (the q̇ 2.27× / reach 1.14× fingerprint), so the actionable knob is reach margin / q0 posture (bounding how far the arm extends), not tuning γ or the onset thresholds. Throttling makes it worse.
• The derived onsets stand: σ_crit,1 = v/x_max = 0.143 (eq 6.2), σ_crit,2 = σ_crit,1² ≈ 0.02 (test_singularity_floors.py). They size when the impedance derate engages; this document explains why engaging it that way (gains, not command) is the floor-holding choice.

Provenance / cross-references

• Algebra: validation/test_derate_mechanism.py (5 pins, all PASS).
• Empirics: analysis/speed_derate_study.py (cascade A/B table + gain-vs-gainbase line); analysis/lockin_fingerprint.py (amplification fingerprint); analysis/crossbox_check.py (Δ=0.0000).
• Equations: GNC/equations/current_sota.md §4.3–4.5 (closed loop, eq 4.14), §6.1 (bound), §6.3 (the ramp).
• Investigation context: generated_reports/GNC/Jun18_26/derating_Jun18.md (one γ, three application sites).
• Threshold sizing (companion): generated_reports/math/threshold_derivation.md.