Michael Quoc Founder & CEO · Axioms.ai
Vector 1 of 4: Tire force envelope

Axiomatic Intelligence · An investigation

The Physics of Driving at the Limit

What happens at the limit of grip: a tire's footprint is already sliding at its rear from the first instant of slip, and asking for more moves that boundary forward until force flattens and then fades as the sliding rubber heats; braking, turning and accelerating share one load-dependent, direction-sensitive capacity because they bend the same rubber, and the friction circle is only its teaching picture; the limit sits at a different slip, force and temperature for every load, heat state and surface, and rain or ice changes its law rather than shrinking it; the tire answers after a rolling distance and its steering torque falls before its force peaks, the one output that leads.

The question this page asks: what is actually happening at the edge of grip, and why does a track day feel the way it does?

“Track days are a full sensory assault. Adrenaline, focus, and the physics of grip at the edge of adhesion.”

“But a Ferrari is not a scalpel. It is an instrument.”

This page rests on four research vectors and 32 axioms, forged on October 4, 2026 under the current version of the method. Every axiom lists the mechanism, the underlying physics, and the primary sources it rests on. Nothing on this page is taken on faith; it can be read one piece at a time.

What governs the force a tire can make, how braking, turning and accelerating share it, and how that envelope changes with load, temperature and surface at and past the limit?

Tire force envelope · eight axioms

The mechanism under test is tire grip: the combined-slip force envelope, load sensitivity and temperature. The easy answer this vector was told not to settle for: “Grip is a fixed friction budget and crossing the circle means a sudden universal collapse”
1.1 A tire's contact patch is never fully stuckVERIFIEDForce is a traction field integrated over a patch that slides at its trailing edge from the first instant of slip, and saturation is a boundary migrating forward, not a state the tire enters.Open this axiomClose this axiom
The axiom in full

In brush contact mechanics (S11, the openai leg's own derivation) the patch has an adhesion region at the leading edge where tread elements stay pinned and deflect linearly with distance travelled, and a sliding region behind it where the deflection-vector traction has reached mu times the local pressure. With a parabolic pressure distribution the pressure is zero at the trailing edge, so a sliding region exists for any non-zero slip; raising slip moves the stick-to-slide boundary forward and the force-versus-slip curve flattens as the sliding fraction grows. Measurements agree on the shape: Holmes and Stone's 1969 wet-surface loci show a progressive reduction of resultant force with braking slip (S1); the 2006 NHTSA flat-track data show longitudinal force saturating while lateral force decays smoothly as longitudinal slip rises (S2); the 2018 instrumented-tire rig shows lateral force cresting at about 12 degrees of slip angle at 1,000 N and about 11 degrees at 750 N (S19), one passenger tire, within-tire design. Population: single passenger tires on test rigs (INTERVENTION) read against brush and Magic Formula models (MODEL). Observation: the measured curves rise, flatten and crest smoothly. Inference: the smoothness is the forward migration of the sliding boundary. Boundary: hard surfaces; on a deformable surface the boundary picture is replaced by terrain shear (A6).

The physics underneath

Shear traction per unit area is capped locally by friction times contact pressure. Pressure tapers to zero at the patch exit, so the cap is reached there first; total force is the integral of a field whose capped fraction grows continuously with slip, which cannot produce a step.

The sources
This is the first axiom on the page.Next axiom: 1.2, Braking and cornering couple through one shared deflection vector in the patch, not through a scalar force budget; the friction circle is a theorem only under isotropic stiffness and friction, and opened measurements show the real envelope is not an ellipse and depends on both slip angle and slip ratio →Link to this axiom: #ax-1-1
1.2 Braking and cornering couple through one shared deflection vector in the patch, not through a scalar force budget; the friction circle is a theorem only under isotropic stiffness and friction, and opened measurements show the real envelope is not an ellipse and depends on both slip angle and slip ratioVERIFIEDStick or slide is decided by the magnitude of one shared deflection vector, so braking and turning couple through deflection, not a budget; anisotropy makes the envelope non-circular.Open this axiomClose this axiom
The axiom in full

The local stick-or-slide test is on the magnitude of the deflection vector driven by the slip vector (sigma_x, sigma_y): sliding begins where sqrt(q_x^2 + q_y^2) exceeds mu times local pressure (S11). If bristle stiffness and friction are isotropic the force depends only on the slip magnitude, and the outer envelope is the circle of radius mu F_z by the triangle inequality on the integral (openai leg derivation). Real tires break each assumption: Brach and Brach's 2011 comparison with measured passenger-tire data finds the F_y-F_x relationship is not a true ellipse and the limit depends on slip angle and traction slip, so the idealized circle 'does not represent actual tire behavior, even approximately' (S7, MODEL versus data); Holmes and Stone's Figure 12 loci do not satisfy the elliptical family (S1, 1969 wet measurement); an Auburn test wheel on clay yields forces the idealized circle underestimates (S9); in the NHTSA 2006 combined tests lateral force at a fixed 2 to 6 degree slip angle falls as longitudinal slip rises while longitudinal force saturates, the two components not vanishing together (S2). The industry's own fitted model does not impose a budget: MF 5.2 combined slip is weighting functions Gxa and Gyk plus an additive term Svyk, a lateral force generated by longitudinal slip alone at zero slip angle (S10). Population: single passenger and test tires on rigs; outcome: tire force against combined slip; evidence: INTERVENTION measurements read against MODEL fits.

The physics underneath

Force is the integral of a vector traction field; the field's direction follows the local deflection, and its cap follows local pressure and friction. Only when stiffness and friction are direction-blind does the integral collapse to a single radius; anisotropic tread, belt and camber effects rotate and reshape the locus.

The sources
← Previous axiom: 1.1, A tire's contact patch is never fully stuckNext axiom: 1.3, Post-peak decline is imported from the friction law, not produced by saturation: a constant-friction patch plateaus, the decline needs friction falling with sliding speed and flash temperature, the sliding-to-peak ratio is a fitted per-tire parameter, and the sudden 'collapse' lives in the wheel's feedback loop →Link to this axiom: #ax-1-2
1.3 Post-peak decline is imported from the friction law, not produced by saturation: a constant-friction patch plateaus, the decline needs friction falling with sliding speed and flash temperature, the sliding-to-peak ratio is a fitted per-tire parameter, and the sudden 'collapse' lives in the wheel's feedback loopVERIFIEDConstant friction saturates to a plateau; the fade past the peak comes from flash-heated rubber losing friction with sliding speed; the cliff is the wheel's feedback, not the rubber's.Open this axiomClose this axiom
The axiom in full

With constant local friction the brush model's force is constant above the full-slide threshold (S11; openai leg derivation): saturation alone yields a plateau. A declining branch requires the sliding friction itself to fall. Persson's flash-temperature theory (S5, MODEL) gives the mechanism: friction rises with sliding speed to a maximum then falls as asperity-scale heating lowers the viscoelastic modulus; Selig et al. 2014 measured tread temperature rising from about 38 to about 75 C during cornering on an indoor sandpaper rig with the friction coefficient falling at high cornering slip, and found friction depends on sliding history (S8, INTERVENTION, one tire); Miyashita and Persson 2026 separate background from flash temperature and show branch transitions that depend on history, not an instantaneous mu(v,T) (S6, MODEL). In the Magic Formula the decline is one fitted parameter: F = D sin(C atan(...)) with the sliding asymptote D sin(C pi/2), so C = 1 gives no decline and C = 1.3 retains about 89 percent of peak (S10; arithmetic on the opened formula, one tire's fitted C). The cliff drivers describe is the wheel's: open-loop braking is stable on the rising branch and unstable on the falling branch of force versus slip, where more slip gives less force gives more slip until lock at slip 1.0 (S16, MODEL review); a slip-controlled actuator on the same rubber shows no cliff. Population: single tires on rigs and wheel-dynamics models; outcome: force versus slip past the peak; evidence: MODEL plus INTERVENTION.

The physics underneath

Dissipation at asperity contacts heats the rubber surface in milliseconds; the modulus at the excitation frequency drops with temperature by time-temperature equivalence, so the local friction cap falls as sliding speed rises. A falling force-slip slope under near-constant torque is positive feedback on wheel spin; the rubber supplies the slope, the wheel supplies the runaway.

The sources
← Previous axiom: 1.2, Braking and cornering couple through one shared deflection vector in the patch, not through a scalar force budget; the friction circle is a theorem only under isotropic stiffness and friction, and opened measurements show the real envelope is not an ellipse and depends on both slip angle and slip ratioNext axiom: 1.4, Vertical load deforms the envelope non-affinely →Link to this axiom: #ax-1-3
1.4 Vertical load deforms the envelope non-affinelyVERIFIEDPeak force rises with load at a diminishing rate on dry surfaces, the sign of the normalized change is condition-dependent (wet reversed it in one opened test), and cornering stiffness itself peaks and then falls with load.Open this axiomClose this axiom
The axiom in full

On the NHTSA 2006 flat-track tests across five loads the dry lateral peak coefficient (peak F_y divided by F_z) decreased with load while the wet coefficient increased over the same load range at 30 mph (S2, INTERVENTION, one passenger tire, within-tire). Holmes and Stone's wet loci likewise vary with load (S1). The received statement that friction coefficient falls with load (S21, ARGUMENT, the exponent unsourced) therefore holds as a dry-surface tendency, not a law. Mechanism (observation versus inference): the patch cannot widen in proportion to load because of carcass and belt, so it lengthens and contact pressure rises; friction-induced heating adds a second load effect (S5, S6: in the rough-contact regime microscopic contact area grows about proportionally to load, so footprint length and heating, not area, carry the tire-level effect). Stiffness is non-monotonic by the fitted model's own form: MF 5.2 cornering stiffness K_y = p_ky1 F_z0 sin(2 atan(F_z / (p_ky2 F_z0))) attains its maximum at F_z = p_ky2 F_z0 and declines beyond it (S10). The 2018 rig shows the peak slip angle rising from about 11 degrees at 750 N to about 12 degrees at 1,000 N (S19). The axle consequence, that a concave peak-force law makes lateral load transfer lower the summed capacity of an axle, is an allocation result the car imposes and belongs to the load transfer and balance vector; the tire merely supplies the concave function.

The physics underneath

A capped traction field integrated over a footprint whose area grows slower than load gives force per unit load that falls; where water carries part of the load, more load restores rubber contact and reverses the sign. The fitted stiffness form is a sine of an arctangent, which must peak.

The sources
← Previous axiom: 1.3, Post-peak decline is imported from the friction law, not produced by saturation: a constant-friction patch plateaus, the decline needs friction falling with sliding speed and flash temperature, the sliding-to-peak ratio is a fitted per-tire parameter, and the sudden 'collapse' lives in the wheel's feedback loopNext axiom: 1.5, Temperature acts through two channels with different clocks →Link to this axiom: #ax-1-4
1.5 Temperature acts through two channels with different clocksVERIFIEDBulk tread temperature lowers cornering stiffness while surface and flash temperature set peak friction, the two can move in opposite directions in one warming run, and the optimum temperature is a compound property spanning roughly 20 to 120 C with no universal window.Open this axiomClose this axiom
The axiom in full

Goodyear's cornering tests on a P265/40R19 at 90 percent rated load (S12, INTERVENTION, one tire, cold-to-hot sweeps) state the split: a high bulk temperature lowers tread rigidity and cornering stiffness; a high surface temperature lowers the peak friction coefficient. Singh and Sivaramakrishnan's Flat-Trac warming sweeps measured cornering stiffness about 28 to 29 percent lower and peak grip about 26 percent higher after repeated sweeps than cold, within the same tire (S3, 2023, INTERVENTION): the tire got softer in response and stronger at the limit at once. Alcazar Vargas et al. measured the lateral coefficient at -8 degrees slip, 60 km/h and 4,000 N rising from about 0.7 to about 1.05 with warming and then declining, their temperature fit peaking at 88 C; tread regions reached about 120 C in a prolonged slip test (S4, 2022, INTERVENTION, one tire). Goodyear's quadratic fit of peak grip against temperature has p1 = -0.0003365 with a 95 percent confidence interval from -0.0006418 to -3.125e-05, a factor of about 20, so the data establish downward curvature but do not locate the optimum closely (S12). Michelin states the compound optimum ranges from about 20 C to about 120 C by tread material (S13, ARGUMENT). Mechanism: the viscoelastic modulus E(omega, T) at the excitation frequency sets both stiffness and hysteretic friction, with time-temperature equivalence making speed and temperature one axis, and flash heating acting on the surface in milliseconds while bulk heating takes far longer (S5, S6, MODEL; S8 measured 38 to 75 C in one cornering run).

The physics underneath

Rubber's storage modulus falls with bulk temperature, softening the carcass-tread spring that sets cornering stiffness; the surface's instantaneous modulus at asperity frequencies sets hysteretic friction, which is maximal in a compound-specific temperature-frequency window and falls on either side.

The sources
← Previous axiom: 1.4, Vertical load deforms the envelope non-affinelyNext axiom: 1.6, The surface changes the tire's law, not a scalar in front of it: wet friction falls continuously with water depth, speed and texture because water carries part of the load, ice removes most skid resistance at 1 mm, and a deformable surface replaces the peak with terrain shear →Link to this axiom: #ax-1-5
1.6 The surface changes the tire's law, not a scalar in front of it: wet friction falls continuously with water depth, speed and texture because water carries part of the load, ice removes most skid resistance at 1 mm, and a deformable surface replaces the peak with terrain shearVERIFIEDWater or ice carries part of the wheel load and shrinks real rubber contact with speed and depth; soft terrain shears in its own law; surface rewrites the curve.Open this axiomClose this axiom
The axiom in full

On wet pavement part of the vertical load is carried by fluid pressure, so the rubber-contact area that generates traction shrinks and the same dry-contact friction applied to total wheel load misrepresents the force (openai leg's vertical-equilibrium argument). Measurements: Holmes and Stone's 1969 wet-road loci depend on texture and speed (S1, INTERVENTION); the HVE wet procedure, built on wet test data, states the reduction 'is not a step function, but rather a continuous decrease' with water depth, tread depth, speed, tread design, patch dimensions and texture, with full hydroplaning the limit (S17, MODEL); the NHTSA 2006 wet tests at 30 mph gave a load dependence of opposite sign to dry (S2). Ice: British Pendulum Number on four laboratory asphalt specimens fell from about 70 to 80 wet to about 10 under a 1 mm ice layer, recovering as the ice melted from -5 to 10 C (S18, INTERVENTION, lab pavement specimens, a pendulum tester not a rolling tire). Deformable ground: on a single test wheel on remolded clay the idealized force circle underestimates the measured forces, and the maximum force and the slip at which it occurs follow the terrain's shear strength (S9, INTERVENTION); the force then grows with sinkage and slip rather than cresting at low slip, so the hard-surface peak ceases to exist. Population: test tires and pavement specimens; outcome: friction against water depth, ice, terrain; evidence: INTERVENTION and MODEL. Inference: the surface changes which mechanism dominates (lubricated contact, bulk terrain shear), so curves cannot be scaled from dry.

The physics underneath

Traction needs rubber-to-asperity contact; water and ice interpose a load-bearing film whose thickness relative to texture sets the real contact area, and that area falls with speed as drainage time shrinks. Soft terrain fails in its own shear, so the force law becomes the soil's, not the rubber's.

The sources
← Previous axiom: 1.5, Temperature acts through two channels with different clocksNext axiom: 1.7, The tire is a first-order filter whose lag is a rolling distance: lateral force reaches 63 percent of its new steady value after one relaxation length, about 0.3 to 0.8 m across 2,000 to 7,000 N on passenger tires, so the time lag shrinks with speed and grows with load, and shortens again as the tire saturates →Link to this axiom: #ax-1-6
1.7 The tire is a first-order filter whose lag is a rolling distance: lateral force reaches 63 percent of its new steady value after one relaxation length, about 0.3 to 0.8 m across 2,000 to 7,000 N on passenger tires, so the time lag shrinks with speed and grows with load, and shortens again as the tire saturatesVERIFIEDForce builds only as new footprint is laid down, so lateral force lags by a rolling distance set by cornering over carcass stiffness, longer under load, briefer at speed.Open this axiomClose this axiom
The axiom in full

Lateral tire dynamics are described to a reasonable approximation by a first-order equation whose time constant is relaxation length divided by forward speed; relaxation length is the rolling distance to reach 1 - 1/e of the steady lateral force after a step in slip angle, and in the point-contact idealization equals cornering stiffness divided by lateral carcass stiffness (S20, S4). Measured: Alcazar Vargas et al. fit relaxation length as a function of load and speed on a test bench spanning 2,000 to 6,000 N, with about 0.71 m at 4,000 N and 60 km/h, a time constant of about 43 ms, in the linear regime (S4, 2022, INTERVENTION, one tire); passenger-tire values of the order of 0.3 m at 2,000 N rising to 0.8 m at 7,000 N, increasing with load and speed whether read as distance or time (S20, INTERVENTION). Consequences by arithmetic on the opened numbers: at 0.5 m the lag is about 10 ms at 50 m/s and about 50 ms at 10 m/s, so slow corners filter inputs harder than fast ones; the loaded outside tire in a corner lags more than the unloaded inside tire. The leg-reported statement that Pacejka's stretched-string model shortens relaxation length at high slip (anthropic leg, UNVERIFIED primary) is consistent with the mechanism (cornering stiffness falls as the sliding region grows while carcass stiffness does not), so near the limit response quickens as force gradient falls. Boundary: the first-order law is validated in the linear regime (S4 says so); it is not validated through saturated combined slip. Population: passenger tires on benches; outcome: force build-up against distance; evidence: INTERVENTION plus MODEL.

The physics underneath

A steered wheel lays down a new footprint only as it rolls, so the already-pinned tread cannot jump to the new slip angle; the carcass spring in series with the patch makes force build over a distance equal to the ratio of the two stiffnesses.

The sources
← Previous axiom: 1.6, The surface changes the tire's law, not a scalar in front of it: wet friction falls continuously with water depth, speed and texture because water carries part of the load, ice removes most skid resistance at 1 mm, and a deformable surface replaces the peak with terrain shearNext axiom: 1.8, The aligning moment peaks before the lateral force →Link to this axiom: #ax-1-7
1.8 The aligning moment peaks before the lateral forceVERIFIEDPneumatic trail shrinks as the sliding region grows from the patch exit, so steering torque falls before peak friction, the one tire output that leads the limit rather than flattening at it.Open this axiomClose this axiom
The axiom in full

In the brush picture the traction field is largest toward the patch exit in adhesion and capped in the sliding zone, so the lateral force resultant acts behind patch center by the pneumatic trail; as slip grows and the sliding zone eats forward the resultant moves toward center and the trail collapses before the force itself crests (S11, S10 where trail t multiplies F_y in the aligning moment). Stated as an estimator by Hsu and Gerdes: pneumatic trail decreases even in the linear handling region, so steering torque encodes early friction information that identifies slip angle and peak lateral force before the limit is reached (S15, MODEL plus vehicle test, Stanford). Stated in control-engineering language by Sentient: 'the steering wheel torque will decrease before the position where the peak friction is located' as a result of the pneumatic trail and side-force saturation (S14, ARGUMENT, patent). Whether a driver perceives that torque drop, and how mechanical trail and power steering dilute it, is the sensing the limit vector's question; this axiom holds for the tire's output. Observation: trail and aligning moment peak at smaller slip angles than lateral force. Inference: the geometry of a sliding region growing from the exit. Population: passenger tires and test vehicles; outcome: aligning moment against slip angle; evidence: MODEL plus field estimation.

The physics underneath

The moment arm of the lateral force is the offset of the traction field's centroid behind patch center; a sliding zone that caps traction at the rear pulls the centroid forward as it grows, so the moment falls while the force is still rising.

The sources
← Previous axiom: 1.7, The tire is a first-order filter whose lag is a rolling distance: lateral force reaches 63 percent of its new steady value after one relaxation length, about 0.3 to 0.8 m across 2,000 to 7,000 N on passenger tires, so the time lag shrinks with speed and grows with load, and shortens again as the tire saturatesNext axiom: 2.1, Longitudinal load transfer is a moment balance set by acceleration times centre-of-gravity height over wheelbase; brake bias and suspension motion do not change it →Link to this axiom: #ax-1-8

What governs load transfer under braking, turning and acceleration, and how do mass distribution and the car's own force allocation set a mid-engine hybrid's understeer, oversteer and yaw response near the limit?

Load transfer and balance · eight axioms

The mechanism under test is load transfer and balance: mass distribution, trail braking and the car's own force allocation set understeer, oversteer and yaw response. The easy answer this vector was told not to settle for: “Mid-engine placement alone makes a car snap into oversteer”
2.1 Longitudinal load transfer is a moment balance set by acceleration times centre-of-gravity height over wheelbase; brake bias and suspension motion do not change itVERIFIEDLongitudinal acceleration acting through centre-of-gravity height creates a moment the axles must react, so normal load shifts between axles by m·a·h/L regardless of suspension or brake bias.Open this axiomClose this axiom
The axiom in full

On a level road with aerodynamic forces and pitch acceleration neglected, the normal load moved between axles under longitudinal acceleration a_x is ΔF_z = m·a_x·h/L; it exists in a rigid car with no suspension travel, so dive and squat are symptoms of the transfer, not its cause. At the same acceleration and geometry, brake bias changes which tires must produce the braking force, not how the axle loads are shared. Mass cancels from the load fractions: for the 296 GTS's published 40.5/59.5 static split and 2.600 m wheelbase (S1) and an ASSUMED h = 0.50 m (Ferrari publishes no centre-of-gravity height; an illustration, not a measurement), a 1.0 g stop moves the front axle's share of total weight mg from 40.5 percent to about 59.7 percent and 0.8 g of acceleration moves it to about 25.1 percent, by calculation under those stated assumptions. Population: any four-wheeled rigid body; evidence class: MODEL (first principles) with one INTERVENTION-class confirmation (S8: the autonomous racing controller's corner-entry tracking failed until longitudinal weight transfer was modelled explicitly).

The physics underneath

Linear momentum balance in x and moment balance about the contact line: the inertial force m·a_x acting at height h must be reacted by a couple of normal-load changes ±ΔF_z spaced L apart, so ΔF_z·L = m·a_x·h. Springs and dampers decide how fast the body reaches the new attitude and through which load paths, never whether the couple exists.

The sources
← Previous axiom: 1.8, The aligning moment peaks before the lateral forceNext axiom: 2.2, Lateral transfer's total is fixed by geometry, its front/rear split by roll stiffness, and tire load sensitivity makes the axle that takes more of it lose capacity →Link to this axiom: #ax-2-1
2.2 Lateral transfer's total is fixed by geometry, its front/rear split by roll stiffness, and tire load sensitivity makes the axle that takes more of it lose capacityVERIFIEDRoll stiffness decides which axle absorbs the geometrically fixed lateral load transfer, and tire force's concave dependence on load makes that axle lose cornering capacity.Open this axiomClose this axiom
The axiom in full

In a corner the total load moved from inside to outside tires is fixed by m·a_y·h divided by track (S5, S6), but its division between front and rear axles is set by roll-centre heights and roll-stiffness distribution (springs, anti-roll bars), not by static weight distribution (S5, S7). A tire's peak force rises with normal load at a diminishing rate: Milliken's tabulated racing-tire data carried by S2 show peak F_y/F_z falling from 1.10 at 900 lbf to 1.08 at 1350 lbf to 0.97 at 1800 lbf (one tire, three loads, a controlled measurement); the quoted exponent of 0.7 to 0.9 is a rule of thumb S2 marks 'citation needed', so it enters as a range of fitted exponents, not a measured constant. Because peak force is concave in load, two tires at unequal loads sum to less capacity than at equal loads, so the axle receiving a larger share of lateral transfer loses capacity relative to the other; stiffening the front in roll moves steady-state balance toward understeer and stiffening the rear toward oversteer (S3 illustration; S4, S7 model). Population: pneumatic tires in their typical operating range; outcome: steady-state axle lateral capacity; evidence class: MODEL with one tabulated measurement.

The physics underneath

Rubber grips by adhesion and by deforming around surface asperities; as load rises the contact patch saturates those asperities and the friction coefficient falls. Concavity of P(F_z) gives P(W/2+Δ)+P(W/2−Δ) < 2P(W/2) for any Δ ≠ 0.

The sources
← Previous axiom: 2.1, Longitudinal load transfer is a moment balance set by acceleration times centre-of-gravity height over wheelbase; brake bias and suspension motion do not change itNext axiom: 2.3, Longitudinal force at a tire consumes its lateral capacity, so a rear-drive car's exit penalty lands on the rear axle and a braking car's penalty lands on both axles at once →Link to this axiom: #ax-2-2
2.3 Longitudinal force at a tire consumes its lateral capacity, so a rear-drive car's exit penalty lands on the rear axle and a braking car's penalty lands on both axles at onceVERIFIEDOne load-dependent force limit per tire is shared between braking or drive and cornering, so pedal demand at an axle subtracts from that axle's cornering capacity.Open this axiomClose this axiom
The axiom in full

A tire's braking or drive force and its cornering force draw on one load-dependent capacity, approximated in teaching by (f_x/P_x)² + (f_y/P_y)² ≤ 1 (S4, S9); the true boundary needs measured combined-slip curves, so the ellipse is a bound on reasoning, not a constitutive law. Under power on a rear-wheel-drive car the whole drive demand lands on the rear axle, which simultaneously gains load; the SIMEA 2026 low-order model (S10, linear tires with a friction ellipse, midsize-car parameters, acceleration cases only) finds the lateral-capacity penalty concentrated at the rear axle for RWD, the largest total reduction for FWD (front axle unloaded and traction-limited together) and the most neutral response for AWD. Under braking the front gains load but spends capacity on braking while the rear loses load and may also spend capacity on friction braking or regeneration (S1 documents rear regeneration during normal braking and during ABS on the 296 GTS). Which axle's combined demand crosses its capacity first decides whether the instantaneous response is understeer or oversteer; neither drive layout nor engine position decides it alone. Peng and Hu (S9) show the physical upper bound on combined motion comes from allocating force across all four tires inside their circles. Population: four-wheel vehicles; outcome: axle lateral capacity under combined demand; evidence class: MODEL (S4, S9, S10) with S8's limit experiments confirming that allocation, not label, governs.

The physics underneath

The contact patch's shear force has one magnitude limit set by load and friction; its direction is shared between slip ratio and slip angle, so a longitudinal component shortens the lateral component available at the same load.

The sources
← Previous axiom: 2.2, Lateral transfer's total is fixed by geometry, its front/rear split by roll stiffness, and tire load sensitivity makes the axle that takes more of it lose capacityNext axiom: 2.4, Braking into a corner moves normal load forward while the lateral-force requirement stays split by centre-of-gravity position, so the unloaded rear's utilization rises and corner-entry balance depends on which axle saturates first →Link to this axiom: #ax-2-3
2.4 Braking into a corner moves normal load forward while the lateral-force requirement stays split by centre-of-gravity position, so the unloaded rear's utilization rises and corner-entry balance depends on which axle saturates firstVERIFIEDBraking moves load off the rear while the rear still owes its centre-of-gravity share of cornering force, so rear utilization rises and corner-entry balance turns on which axle saturates first.Open this axiomClose this axiom
The axiom in full

With zero yaw acceleration and no control-generated yaw moment, the lateral forces the front and rear axles must supply are in the ratio b:a (b the centre of gravity's distance to the rear axle, a to the front), which for the 296 GTS's 40.5/59.5 static split means the rear axle owes 59.5 percent of total lateral force at every instant of a steady turn (S1 for the split; S4, S6 for the moment balance). Braking does not change that requirement; it moves normal load forward (A1). In A1's illustration (assumed h = 0.50 m, 1.0 g) the rear carries about 40.3 percent of normal load while still owing 59.5 percent of lateral force, and it may also be braking or regenerating (S1), so its combined utilization climbs. Three outcomes are possible on entry, decided by which axle's utilization reaches one first: front saturation gives entry understeer; rear unloading plus rear braking demand gives entry oversteer; releasing the brake restores lateral margin while also reducing front load. In the only opened controlled experiment at the limit (S8, an autonomous research car on oval tracks), corner-entry trail braking required a correct entry speed and an accurate model of longitudinal weight transfer, and the controller added weight transfer to its feedforward to minimize oversteer caused by reduction in rear normal load. Ferrari's description of ABS Evo (S1) names the same compromise at the rear axle between braking performance and lateral stability and redistributes brake force so the driver can trail brake toward the apex. Population: four-wheel vehicles on corner entry, S8 the only measured case (not a human driver); outcome: yaw response on entry; evidence class: MODEL plus one INTERVENTION.

The physics underneath

The moment balance about the centre of gravity (a·F_yf = b·F_yr when yaw acceleration and external moments are zero) is independent of the normal-load balance; the two diverge under braking, and the axle whose required force rises relative to its capacity is the one that lets go.

The sources
← Previous axiom: 2.3, Longitudinal force at a tire consumes its lateral capacity, so a rear-drive car's exit penalty lands on the rear axle and a braking car's penalty lands on both axles at onceNext axiom: 2.5, Mass centralization lowers yaw inertia, raising yaw acceleration per unit moment in both directions; whether a car spins is set by the rear axle's cornering-stiffness slope at saturation and the correction's latency, not by where the engine sits →Link to this axiom: #ax-2-4
2.5 Mass centralization lowers yaw inertia, raising yaw acceleration per unit moment in both directions; whether a car spins is set by the rear axle's cornering-stiffness slope at saturation and the correction's latency, not by where the engine sitsVERIFIEDCentralized mass lowers yaw inertia so any yaw moment acts faster; the car spins only if the rear's cornering-stiffness slope collapses before a correcting moment arrives.Open this axiomClose this axiom
The axiom in full

Yaw inertia is I_z = Σ m_i r_i², so packaging engine, gearbox and battery near the centre of gravity lowers I_z relative to the same masses placed farther out (S6). Yaw acceleration is ṙ = M_z/I_z, so for the same tire-generated moment a low-inertia car yaws faster whether the moment is destabilizing (a rear that has let go) or corrective (counter-steer, a brake-generated moment) (S6, S4). In the locally linear bicycle model the tire contribution to yaw-rate damping is −(a²C_f + b²C_r)/(I_z·u) (S4): as the rear axle approaches its force peak its incremental cornering stiffness C_r falls toward zero and past the peak turns negative, so damping collapses at the rear's saturation; low I_z then shortens the time available to apply a correcting moment. The sign of the response, understeer or oversteer, is set by the understeer-gradient term (b/C_f − a/C_r) and its nonlinear extension, that is by weight distribution relative to axle stiffness, never by weight distribution alone (S4). Population: rigid-body vehicle models; outcome: yaw response; evidence class: MODEL, with S16's research-vehicle envelope-control experiments as the measured statement of the stable region. No opened source measured the 296 GTS's yaw inertia, yaw-rate gain or sideslip trace, so this axiom carries no 296-specific number.

The physics underneath

Angular momentum balance: the same moment produces more angular acceleration on a smaller inertia; stability of the yaw and sideslip states depends on the slopes of the tire curves at the operating point, which vanish at the peak.

The sources
← Previous axiom: 2.4, Braking into a corner moves normal load forward while the lateral-force requirement stays split by centre-of-gravity position, so the unloaded rear's utilization rises and corner-entry balance depends on which axle saturates firstNext axiom: 2.6, The 296 GTS's documented controls reallocate tire utilization and add yaw moments within existing grip, so the driver feels car plus controller; the controller's contribution to the GTS's balance is unmeasured beyond manufacturer and simulation figures →Link to this axiom: #ax-2-5
2.6 The 296 GTS's documented controls reallocate tire utilization and add yaw moments within existing grip, so the driver feels car plus controller; the controller's contribution to the GTS's balance is unmeasured beyond manufacturer and simulation figuresVERIFIEDChassis controllers reallocate wheel torque and brake pressure from a sideslip and grip estimate, moving utilization and adding yaw moments inside existing grip; the driver meets car plus controller.Open this axiomClose this axiom
The axiom in full

Ferrari's press kit (S1, 2022-04-19) documents for the 296 GTS a single electric motor driving the rear wheels only; regenerative braking at the rear in normal braking and when ABS intervenes, plus rear over-braking on lift-off; a brake-by-wire unit blending hydraulic and electric braking in all modes including ABS; an ABS controller (ABS Evo, Race position and above) fed by the six-axis 6w-CDS (acceleration and rotation rate on three axes) that optimises braking distribution and 'allows the grip limits of the rear tyres to be pushed still further ... as it permits the driver to trail brake towards the apex'; grip estimation integrated with the electric power steering, stated as 35 percent faster on track than previous applications; and the eSSC suite (eTC, eDiff, SCM, FDE 2.0, EPS, ABS Evo, 6w-CDS). The tire does not distinguish electric from hydraulic braking torque; both change wheel slip and spend the rear's combined capacity. A controller cannot create grip; it reduces demand, moves utilization between wheels and generates yaw moments from left-right force differences within existing capacity (S9, S16). Measured effect sizes exist only as manufacturer claims (S1: 200-0 km/h stopping distance 8.8 percent shorter and repeat-braking efficiency 24 percent better than the F8 Spider, denominator the F8 Spider's own figures, design a manufacturer comparison) and as simulation (S11, CarSim co-simulation of a brake-by-wire passenger vehicle under cornering braking: load-based distribution plus a yaw-moment allocator cut yaw-rate RMS error 76.2 to 98.6 percent and peak sideslip magnitude 20.9 to 98.6 percent relative to proportional distribution across eight speed, demand and adhesion conditions, with braking duration moderately longer in some cases). Two road testers report the lived effect without instruments: ABS Evo compensating for late braking and turn-in without understeer (S15) and a GTS confidence-inspiring up to and just beyond the limits of lateral grip (S14). Population: the production 296 GTS (architecture) and a simulated passenger car (effect); outcome: braking distribution, yaw-rate and sideslip error; evidence class: ARGUMENT (manufacturer, reviewers) plus MODEL (simulation); no independent instrumented measurement of the GTS controller contribution was opened.

The physics underneath

Wheel torque, friction or electric, sets longitudinal slip and so the tire's longitudinal force; a left-right difference in longitudinal force at track width t makes a yaw moment (t/2)(F_x,outer − F_x,inner); an estimator of sideslip and grip lets the controller choose these allocations before the driver can.

The sources
← Previous axiom: 2.5, Mass centralization lowers yaw inertia, raising yaw acceleration per unit moment in both directions; whether a car spins is set by the rear axle's cornering-stiffness slope at saturation and the correction's latency, not by where the engine sitsNext axiom: 2.7, A skidpad g or a stopping distance cannot identify a car's yaw dynamics; the 296's published instrumented figures bound its grip, and its balance at the limit is known only from reviewers' observations with the controls active →Link to this axiom: #ax-2-6
2.7 A skidpad g or a stopping distance cannot identify a car's yaw dynamics; the 296's published instrumented figures bound its grip, and its balance at the limit is known only from reviewers' observations with the controls activeVERIFIEDPeak lateral g and stopping distance are scalars on the force envelope; yaw behaviour at the limit lives in the yaw-rate and sideslip history, which no opened 296 source recorded.Open this axiomClose this axiom
The axiom in full

A steady-state lateral acceleration or a straight-line stopping distance fixes one point on the force envelope and leaves yaw inertia, yaw damping, rear slip progression and controller contribution unidentified; identifying those needs synchronized histories of steering, speed and sideslip, yaw rate, wheel load and slip, brake pressure and delivered torque (S4 on what a static model can predict; S16 on the yaw-rate and sideslip state as the measured object). The opened 296 measurements are bounds on grip, not statements of balance: Car and Driver's instrumented 296 GTB on optional Michelin Pilot Sport Cup 2 R tires stopped from 70 mph in 130 feet and held 1.12 g on the skidpad (S12, one car, one test); MotorTrend's 296 GTB Assetto Fiorano averaged 1.15 g on the skidpad and ran its figure eight in 21.7 seconds at 1.04 g average (S13, one car, one test); both coupes, neither a GTS. The balance observations are reviewers' reports with controls active: hot tire pressures eroding grip and introducing mild understeer and, in CT Off, slight power oversteer never threatening or twitchy (S13, GTB); the GTS showing a faint inertia on turn-in and a slightly greater willingness to push into understeer than the stiffer coupe (S14); ABS Evo removing late-braking understeer (S15). No opened source supplies a 296 GTS understeer gradient, yaw-rate gain, sideslip trace or trail-braking telemetry; the Gemini leg's account of a 296 that rotates too quickly to catch rests on an anonymous forum thread (HTTP 403 to this writer), class ARGUMENT, with the controls' state unrecorded. Population: one 296 GTB and one GTB Assetto Fiorano (instrumented), one GTS (two road testers); outcomes: skidpad g, stopping distance, figure-eight time, qualitative balance; evidence class: CORRELATIONAL (n=1 tests) and ARGUMENT (reviewers).

The physics underneath

The envelope's radius (peak g) and the dynamics inside it (how yaw rate and sideslip evolve toward and past the peak) are different quantities; the first is a scalar, the second needs the state history.

The sources
← Previous axiom: 2.6, The 296 GTS's documented controls reallocate tire utilization and add yaw moments within existing grip, so the driver feels car plus controller; the controller's contribution to the GTS's balance is unmeasured beyond manufacturer and simulation figuresNext axiom: 2.8, Hybrid mass changes the absolute forces and, through where it is installed, centre-of-gravity height and yaw inertia; it cancels from the load-transfer fractions, so its handling effect is a placement question, not a weight question →Link to this axiom: #ax-2-7
2.8 Hybrid mass changes the absolute forces and, through where it is installed, centre-of-gravity height and yaw inertia; it cancels from the load-transfer fractions, so its handling effect is a placement question, not a weight questionVERIFIEDAdded hybrid mass scales every tire force but cancels from load-transfer fractions; only its installed height and radius change balance, through centre-of-gravity height and yaw inertia.Open this axiomClose this axiom
The axiom in full

The 296 GTS press kit (S1) documents a 7.45 kWh high-voltage battery located on the floor behind the seats, a the arousal attention and workload vector that weighs 30 kg less than the the track day exposure and harm vector of previous berlinettas, a wheelbase 50 mm shorter than previous mid-rear-engined berlinettas, and a dry weight of 1,540 kg at 40.5/59.5 (the 1,540 kg marked as achieved with optional lightweight content). From A1 and A2, mass cancels from the fraction of weight transferred at a given acceleration and h/L, so the hybrid system's mass raises every absolute tire force without changing the load split; its effect on balance enters only through the height and position at which it is installed (h, the a:b split, and I_z = Σ m r²). A floor-mounted battery between the axles adds mass at low height and small radius, so it raises h and I_z less than the same mass in the nose or tail would (S6 for I_z; S1 for placement). Component masses quoted by the Gemini leg (185 kg engine, 73 kg battery, 22 kg motor) and a '10 mm lower centre of gravity' come from secondary sites and are not in the press kit this writer opened; they stay UNVERIFIED. Population: the production 296 GTS; outcome: static loads and inertia; evidence class: ARGUMENT (manufacturer specification) plus MODEL.

The physics underneath

Load-transfer fractions depend on a·h/L and a·h/t, mass-free; inertia depends on mass times radius squared; so the same kilograms placed low and central change the car's dynamics least.

The sources
← Previous axiom: 2.7, A skidpad g or a stopping distance cannot identify a car's yaw dynamics; the 296's published instrumented figures bound its grip, and its balance at the limit is known only from reviewers' observations with the controls activeNext axiom: 3.1, Utilization is a ratio and every inertial cue reports only its numerator, so no force-proportional sensation can separate demand from capacity →Link to this axiom: #ax-2-8

Which cues let a driver estimate remaining grip, how accurately does a perceived limit track the measured one, and when do those estimates become delayed, biased or ambiguous?

Sensing the limit · eight axioms

The mechanism under test is sensing the limit: the driver's estimate of grip in use from steering, inertial, visual and acoustic cues. The easy answer this vector was told not to settle for: “Seat-of-the-pants feel reliably reveals remaining grip”
3.1 Utilization is a ratio and every inertial cue reports only its numerator, so no force-proportional sensation can separate demand from capacityVERIFIEDInertial cues scale with the force a tire produces, never the force it could produce, so remaining grip is unobservable from motion and must be inferred.Open this axiomClose this axiom
The axiom in full

Per tire, utilization is force produced over force available (mu times vertical load). Lateral acceleration, yaw rate, seat pressure and body loading are functions of the force produced and carry no information about the divisor: a measured 0.9 g is 90 percent of a dry limit and over 100 percent of a damp one with identical cue histories. Two surfaces with 0.5 g and 1.0 g limits are indistinguishable at a 0.3 g demand (utilization 60 versus 30 percent) from any cue that scales with force. Holds for every population (professional, club, track-day amateur, simulator subject) because it is a property of the signal, not the sensor; outcome governed: tire force and slip against perceived margin; evidence class MODEL (the estimator literature) grounded in the planar vehicle equations.

The physics underneath

m(dv_y/dt + v_x r) = sum F_y and I_z dr/dt = a F_yf - b F_yr: the body's motion is set by the forces acting now. The maximum force is a parameter of the tire's force-slip curve whose distinguishing feature, saturation, appears only near saturation; in the linear region the slope (cornering stiffness) is nearly insensitive to mu. The denominator is unobservable from motion alone, and an electronic estimator reading the same channels faces the same degeneracy, so the human constraint is the plant's constraint.

The sources
← Previous axiom: 2.8, Hybrid mass changes the absolute forces and, through where it is installed, centre-of-gravity height and yaw inertia; it cancels from the load-transfer fractions, so its handling effect is a placement question, not a weight questionNext axiom: 3.2, Tire capacity becomes identifiable only as the tire is excited into its nonlinear range, so every sub-limit confidence is a prior and incident-free laps cannot calibrate it →Link to this axiom: #ax-3-1
3.2 Tire capacity becomes identifiable only as the tire is excited into its nonlinear range, so every sub-limit confidence is a prior and incident-free laps cannot calibrate itVERIFIEDBelow the limit the force-slip curve is linear and mu-blind, so only excitation toward saturation identifies capacity; incident-free laps raise confidence without calibrating margin.Open this axiomClose this axiom
The axiom in full

The friction limit is a parameter of a curve that is linear below the limit; the data that distinguish a high-mu curve from a low-mu one exist only where it bends. For any driver on any surface (MODEL evidence, estimator literature) the margin estimate at low utilization is a prior drawn from surface appearance, temperature and earlier laps, not a posterior from the current cues. In the one randomized training study opened (INTERVENTION, S15: 45 driving-simulator novices, 15 per group, four 10-minute practice sessions in a Formula 3 model on an 800 m oval at 76, 100 or 150 percent of normal grip, then two retention sessions) confidence and performance moved apart: the low-grip group reported higher confidence and lower frustration after both retention sessions yet was significantly slower than the high-grip group in the first. Practice that never reached the limit produced confidence without lap time.

The physics underneath

Parameter identification needs persistence of excitation: the input must drive the system through states where the parameter changes the output. Below the limit the output is a stiffness times slip with little mu dependence, so laps at 60 percent utilization carry almost no information about the remaining 40 percent. Calibration is supervised learning that needs a labelled error, and below the limit every lap's label is 'no incident' whether the margin was 40 percent or 4 percent. The limit is learned by approaching it, which also shrinks the time and force reserve for correction: observability and controllability are coupled.

The sources
← Previous axiom: 3.1, Utilization is a ratio and every inertial cue reports only its numerator, so no force-proportional sensation can separate demand from capacityNext axiom: 3.3, The collapse of pneumatic trail in steering aligning torque is the one cue that leads the force peak; it reaches the driver only from the front axle, through the steering path, and is diluted by mechanical trail, combined slip, load transfer and assist →Link to this axiom: #ax-3-2
3.3 The collapse of pneumatic trail in steering aligning torque is the one cue that leads the force peak; it reaches the driver only from the front axle, through the steering path, and is diluted by mechanical trail, combined slip, load transfer and assistVERIFIEDSliding starts at the trailing edge, so aligning moment falls before lateral force peaks; that drop reaches the hands only from steered tires, diluted by assist and caster.Open this axiomClose this axiom
The axiom in full

Lateral force is the zeroth moment of the contact-patch shear stress and aligning moment its first moment; sliding starts at the trailing edge and moves forward, so the stress centroid (pneumatic trail) migrates and aligning torque falls while lateral force still rises. This is the only cue in the car that changes before the force peak (MODEL, S1, S2, S4; validated on an instrumented test vehicle in S2). Bounds: (1) it exists only for the steered axle; the rear axle, whose saturation destabilizes the car, has no haptic path and is inferred through yaw and sideslip residuals; (2) hand torque equals (mechanical trail + pneumatic trail) times F_y less assist plus column friction, so caster and assist shrink the mu-sensitive fraction; (3) braking force lowers and driving force raises aligning moment, and load transfer scales M_z with F_z, so trail braking and an unloaded inside wheel both mimic lost grip; (4) in a dynamic simulator (INTERVENTION, S24, small sample, sub-limit curve following) drivers controlled the car under linear or nonlinear torque laws alike while zero or inverted torque made driving nearly impossible, so the hand needs torque for stabilization and is indifferent to its calibration; in S17 (INTERVENTION, 42 participants, single-blind within-subject) non-road vibrations raised ratings of steering feedback, so a rating tracks texture, not grip information.

The physics underneath

Moment hierarchy: higher moments of a distribution being eaten from one edge change before its integral does. The migration begins when trailing-edge rubber crosses from adhesion to sliding, a function of local slip and local mu; so the trail cue is the one place mu enters a signal before the force saturates. Everything between contact patch and palm is a filter: caster adds a term proportional to F_y with no mu information; electric assist subtracts a programmed torque; column friction adds hysteresis.

The sources
← Previous axiom: 3.2, Tire capacity becomes identifiable only as the tire is excited into its nonlinear range, so every sub-limit confidence is a prior and incident-free laps cannot calibrate itNext axiom: 3.4, The sensory channels differ in delay and bandwidth →Link to this axiom: #ax-3-3
3.4 The sensory channels differ in delay and bandwidthVERIFIEDVestibular and haptic cues arrive in tens of milliseconds, vision in about 120 ms, and the vestibular sense is high-pass, so slow grip loss converts a detection threshold into a latency and sustained rotation fades from perception.Open this axiomClose this axiom
The axiom in full

Three dated laboratory measurements bound the driver's sensing. (1) Visual feedback in a human steering task is modelled with about 120 ms delay against about 10 ms vestibular (S6, 2025, healthy adults, within-person). (2) Yaw-rotation direction thresholds in the dark rise from 0.4 deg/s at 2 Hz to 2.8 deg/s at 0.05 Hz, consistent with a high-pass filter with a 0.71 deg/s plateau, 0.23 Hz cut-off and about 0.70 s time constant (S7, 2008, seven healthy subjects, adaptive 2AFC on a motion platform). (3) Sustained yaw rotation without vision fades with an average time constant of 17.2 s, SD 6.8 s (measurement cited in S19, 2023). The driving consequence is measured only indirectly: in evasive manoeuvres on snow and ice with real-world data from skilled test drivers, waiting for visual cues is too slow and drivers act on steering torque and vestibular motion (S10, 2016), and in a motion simulator drivers adapt to simple scaled-motion conflicts more readily than to filtered motion (S16). Arithmetic, not a measurement: at a 0.1 Hz-class onset the threshold is about 2.5 deg/s, so a yaw error building at 1 deg/s per second passes about 2.5 s before detection is possible. Bounded to the laboratory populations named; no opened study measured these latencies in a race car.

The physics underneath

The semicircular canals are mechanical differentiators whose perceptual output is further high-pass filtered, so a steady state reads as no state. Vision reports position and flow after roughly a tenth of a second of processing. A control loop with a 120 ms sensor inside a yaw dynamic of a few hundred milliseconds is near its phase margin, which is why visually guided limit corrections overshoot. The grip-loss shape that matters most on a long corner, a slow quasi-steady drift, lives below 0.2 Hz where the vestibular sense is least sensitive and where only the trail cue or late vision can carry it.

The sources
← Previous axiom: 3.3, The collapse of pneumatic trail in steering aligning torque is the one cue that leads the force peak; it reaches the driver only from the front axle, through the steering path, and is diluted by mechanical trail, combined slip, load transfer and assistNext axiom: 3.5, Detection, recognition and magnitude estimation are different acts with thresholds an order of magnitude apart, and a reported limit carries a response criterion, so a vivid sensation is not a calibrated estimate →Link to this axiom: #ax-3-4
3.5 Detection, recognition and magnitude estimation are different acts with thresholds an order of magnitude apart, and a reported limit carries a response criterion, so a vivid sensation is not a calibrated estimateVERIFIEDDetection needs only signal above noise, recognition needs its direction and estimation needs a trained scale, so sensing a change is cheap while estimating the margin is uncalibrated.Open this axiomClose this axiom
The axiom in full

For yaw rotation on a motion platform, two-interval motion-detection thresholds were on average 56 times smaller than direction-recognition thresholds in four healthy subjects (S8, 2013). For the steering wheel, difference thresholds for steady-state force and angle were about 15 percent in a laboratory task (S23, 2007), with force sensation growing faster than linearly. Perceived steering torque depends on arm posture: with a 2.0 Nm reference straight ahead, test torques of 1.1 to 2.9 Nm were misjudged at 30, 60, -30 and -60 degree postures (S18, 2019, EMG and staircase). In vestibular psychophysics responses are biased against the preceding response, not the stimulus, and ignoring this overestimates thresholds by 5.5 percent for yaw and 7.1 percent for interaural translation (S20, 2023). So 'something is changing' (detection) can be true and early while 'the rear is going' (recognition) is 56-fold harder and 'I have 15 percent left' (estimation) rests on a scale no driver was trained on. Every number estimates a laboratory threshold in healthy adults, not a racing driver's grip accuracy; none converts to a grip-margin error.

The physics underneath

Signal detection theory separates sensitivity from criterion; a report is both. Detection needs only energy above noise in any channel; recognition needs the sign of a direction-specific channel, degraded by vibration; magnitude estimation needs a learned mapping from sensation to a physical scale, and A2 shows the labels for that mapping are withheld below the limit. The 15 percent wheel-force JND cannot become a 15 percent margin accuracy without the sensitivity of torque to utilization and its nuisance variability, and posture bias shifts the mapping between corners of opposite hand.

The sources
← Previous axiom: 3.4, The sensory channels differ in delay and bandwidthNext axiom: 3.6, In simulators with a known limit, motion cues shift exploitation of that limit by about 2 percent with opposite signs by corner phase, drivers prefer settings that do not perform best, and down-scaled motion yields yaw underestimation with a sub-unity optimum; simulators measure cue dependence, never calibration →Link to this axiom: #ax-3-5
3.6 In simulators with a known limit, motion cues shift exploitation of that limit by about 2 percent with opposite signs by corner phase, drivers prefer settings that do not perform best, and down-scaled motion yields yaw underestimation with a sub-unity optimum; simulators measure cue dependence, never calibrationVERIFIEDScaled, filtered simulator motion shifts limit exploitation about 2 percent with phase-dependent sign and biases perceived yaw, so simulators reveal cue dependence while the real limit stays unmeasured.Open this axiomClose this axiom
The axiom in full

Where the limit is known exactly (a tire model), cue manipulation measures how performance depends on cues. (1) Professional race drivers with and without motion changed a model-based Tire Potential Exploitation Rating by +1.86 percent at corner exits and -2.52 percent at corner entries (S11, 2020, within-person; percentages of the TPER metric, not perceptual error). (2) Three motion-cueing settings, 26 amateurs and 2 professionals: both professionals preferred A, 46.2 percent of amateurs preferred C, performance descriptively favoured A, most differences not significant (S12, online 2026-02-27; two professionals permit no expertise claim). (3) A neurobiologically grounded model fitted to prior slalom data reproduces yaw-rate underestimation under motion down-scaling and a local optimum of down-scaled motion beating one-to-one motion (S9, 2019). (4) Twenty-four participants braking from 120 km/h: higher motion levels gave braking closer to a real-car trace, with exceptions (S22, 2023). (5) Drivers adapt to simple scaled conflicts more readily than complex filters (S16). The inversion: a simulator knows the limit but washes out the cues; a real car has full cues and an unknown drifting limit; neither measures how a perceived limit tracks a measured one.

The physics underneath

Motion platforms cannot sustain 1 g, so cueing algorithms scale and high-pass the motion, introducing gain and phase errors; a driver with an internal model tuned to one-to-one motion reads scaled motion as slower rotation and advances steering, which compensates the simulator's own latency, hence the sub-unity optimum. Preference tracks comfort and familiarity, and reports carry criteria (A5), so a preferred setting is not a validated one.

The sources
← Previous axiom: 3.5, Detection, recognition and magnitude estimation are different acts with thresholds an order of magnitude apart, and a reported limit carries a response criterion, so a vivid sensation is not a calibrated estimateNext axiom: 3.7, Cross-sectional differences between racing and non-racing drivers are not learning, and lap time is not perceptual accuracy: experts reach similar lap times by different paths, amateurs lose 10 to 20 percent mainly in handling, and the one randomized longitudinal study shows confidence and speed moving apart →Link to this axiom: #ax-3-6
3.7 Cross-sectional differences between racing and non-racing drivers are not learning, and lap time is not perceptual accuracy: experts reach similar lap times by different paths, amateurs lose 10 to 20 percent mainly in handling, and the one randomized longitudinal study shows confidence and speed moving apartVERIFIEDLap time maps many-to-one onto perception and cross-sectional contrasts mix selection with learning, so expert-amateur gaps measure what experts do, not what they sense or learned.Open this axiomClose this axiom
The axiom in full

Over 1,200 simulator laps from seven professional race drivers and two amateurs, the amateurs were 10 to 20 percent slower, principally through a lower handling score (closeness to physically optimal tire-force use) rather than line choice; the 2 to 5 percent RMSE belongs to a neural-network surrogate for the physics optimiser, not to any human estimate (S13, 2022, CORRELATIONAL). Seven racing and ten non-racing drivers differed in simulator control and gaze (S14, 2017, cross-sectional). In open race data from a real circuit, professionals' path dispersion is lowest where curvature is highest and two professionals reached similar lap times with distinct styles, so the limit admits a family of solutions (S26, Stanford DDL, CORRELATIONAL, real car). None of these designs shows that learning particular cues produced the difference; the only randomized longitudinal evidence is S15 (45 novices, three grip conditions, retention tests) where confidence rose and lap time did not under low-grip practice. Holds for professional and amateur simulator and race populations on lap-time and handling outcomes; says nothing about how fast a track-day amateur learns, which no opened source measured.

The physics underneath

The mapping from perceptual quality to lap time is many-to-one: a driver can be slow with good perception and poor control, or fast on a conservative, well-practiced line; where many paths give one time, time cannot be inverted to perception. Selection, prior training and risk appetite all enter a cross-sectional contrast and none separates without a within-person series. A2 explains why the one longitudinal study found confidence detached from speed: sub-limit practice cannot label the margin.

The sources
← Previous axiom: 3.6, In simulators with a known limit, motion cues shift exploitation of that limit by about 2 percent with opposite signs by corner phase, drivers prefer settings that do not perform best, and down-scaled motion yields yaw underestimation with a sub-unity optimum; simulators measure cue dependence, never calibrationNext axiom: 3.8, No opened source maps tire squeal onset to a measured utilization; the measured effect of in-car sound on driving is on speed perception, where attenuation lowers perceived speed and raises chosen speed, which enters lateral utilization as speed squared →Link to this axiom: #ax-3-7
3.8 No opened source maps tire squeal onset to a measured utilization; the measured effect of in-car sound on driving is on speed perception, where attenuation lowers perceived speed and raises chosen speed, which enters lateral utilization as speed squaredVERIFIEDTire squeal is tire-specific stick-slip with no measured link to utilization, while engine and tire sound set perceived speed, so attenuated sound lowers perceived speed and raises chosen speed.Open this axiomClose this axiom
The axiom in full

Squeal is stick-slip vibration at the sliding part of the contact patch, of the same physical family as the trail cue, but its onset, loudness and pitch depend on tread stiffness, compound, surface, manoeuvre and transmission to the cabin; eleven professional drivers, analysts and instructors interviewed in 2024 named squeal as a limit cue 'on certain tires' alongside pitch rises and steering-torque loss (S25, ARGUMENT, qualitative). No study opened by any leg measured squeal onset against tire utilization, so no universal margin attaches to any sound. What is measured is sound as a speed cue: across the studies reviewed in S21 (2024; laboratory and simulator; between- and within-subject; mostly students and road drivers), removing or attenuating engine sound led drivers to underestimate speed and drive faster, and a 5 dB reduction in in-car noise made participants judge video speeds significantly slower. Since lateral acceleration is speed squared over radius, a speed estimate biased low by 5 percent biases inferred lateral utilization low by about 10 percent with the lateral channel untouched (arithmetic). Holds for the populations and outcomes in S21; untested with racing drivers or on track.

The physics underneath

Tire noise arises where tread elements slide and re-stick; its frequency content follows element stiffness and sliding speed, both tire-specific, so one tire howls at 80 percent utilization and another is quiet to the peak. Speed perception is multisensory and engine and tire noise are among its inputs; attenuate them and the estimate drops. Modern cabins, active noise control and hybrid powertrains all attenuate.

The sources
← Previous axiom: 3.7, Cross-sectional differences between racing and non-racing drivers are not learning, and lap time is not perceptual accuracy: experts reach similar lap times by different paths, amateurs lose 10 to 20 percent mainly in handling, and the one randomized longitudinal study shows confidence and speed moving apartNext axiom: 4.1, Curvature taxes the tire before any tire model is written →Link to this axiom: #ax-3-8

What makes one path through a corner faster than another, and how do tire, power, braking and track constraints couple path curvature to speed in a minimum-time line?

Minimum time line · eight axioms

The mechanism under test is the line: minimum-time motion through a corner under tire, power, braking and track constraints. The easy answer this vector was told not to settle for: “Outside-inside-outside is always the fastest line”
4.1 Curvature taxes the tire before any tire model is writtenVERIFIEDA_n = v^2 kappa, so speed can change only where the force envelope has tangential room left.Open this axiomClose this axiom
The axiom in full

For a vehicle whose center of mass follows a planar path with curvature kappa at speed v, the acceleration is v-dot along the path plus v^2 kappa toward the center of curvature; this is a kinematic identity (MODEL, derivation) independent of tire, car or driver. Under an aggregate combined-force envelope of radius A (the friction-circle approximation) the constant-speed ceiling is sqrt(A/|kappa|) and any speed change must fit inside |v-dot| <= sqrt(A^2 - v^4 kappa^2): at 95 percent lateral use 31 percent of A remains for braking or acceleration, at 98 percent 20 percent (arithmetic on the circle). At full lateral saturation nothing remains, so a car cannot ride a lateral ceiling that is itself changing and must sit below it to change speed. The circle is the model-dependent part: per-tire ellipses, load sensitivity, aero load and yaw-moment balance move the envelope but not the identity that curvature at speed is a force demand. Population: any wheeled vehicle; outcome: tire force demand and feasible acceleration; evidence class: MODEL, used as the constraint in every fixed-path minimum-time formulation opened (S1, S2, S14).

The physics underneath

Differentiating r(t) with ds/dt = v gives r-ddot = v-dot t + v^2 kappa n; Newton's law converts the geometry into a force the tires must supply, and the combined-force bound couples the two components so lateral demand caps tangential authority.

The sources
← Previous axiom: 3.8, No opened source maps tire squeal onset to a measured utilization; the measured effect of in-car sound on driving is on speed perception, where attenuation lowers perceived speed and raises chosen speed, which enters lateral utilization as speed squaredNext axiom: 4.2, Elapsed time is the integral of ds/v over the whole segment; a larger radius or a higher apex speed is not a minimum-time objective →Link to this axiom: #ax-4-1
4.2 Elapsed time is the integral of ds/v over the whole segment; a larger radius or a higher apex speed is not a minimum-time objectiveVERIFIEDLap time integrates 1/v over distance, so a wider, faster arc loses when its extra meters cost more time than its extra speed saves.Open this axiomClose this axiom
The axiom in full

The quantity a line minimizes is T = integral of ds/v(s) between common timing gates, so a path trades distance against feasible speed and the time-optimal path is neither the widest nor the one with the highest apex speed. Velenis & Tsiotras (ISIE 2005, bicycle model with suspension dynamics, MODEL) report that a larger-radius trajectory could carry more speed but would travel farther, and that the minimum-time trajectory tends to a single circular arc whose radius compromises between the highest possible travel speed and the shortest distance travelled. The worked annulus case makes it explicit: at a lateral limit A = 10 m/s^2 a steady 25 m radius lap takes 2 pi sqrt(R/A) = 9.93 s at 15.8 m/s while a 50 m radius lap takes 14.05 s at 22.4 m/s, faster in speed and slower in time (a comparison of two legal periodic candidates, not the optimum of every annulus). Field correlate (CORRELATIONAL, autonomous Audi TTS at Thunderhill, planned mu 0.90): Kapania et al.'s two-step line had slightly lower cornering speeds but slightly higher top speeds and drove 0.6 s faster (138.6 s vs 139.2 s) than the gradient-descent line.

The physics underneath

Time weights every meter by 1/v, so a small speed deficit over a long stretch outweighs an apex gain over a short one; with path length L proportional to R^alpha and lateral-limited speed proportional to R^(1/2), T is proportional to R^(alpha - 1/2), so only the length exponent decides whether more radius helps.

The sources
← Previous axiom: 4.1, Curvature taxes the tire before any tire model is writtenNext axiom: 4.3, On a fixed path the speed profile is the lower envelope of three propagations, so the braking point is set downstream; fixing the path makes the problem convex, freeing it makes every solver's line model-optimal only →Link to this axiom: #ax-4-2
4.3 On a fixed path the speed profile is the lower envelope of three propagations, so the braking point is set downstream; fixing the path makes the problem convex, freeing it makes every solver's line model-optimal onlyVERIFIEDThe feasible speed trace is the minimum of a curvature ceiling, a forward traction reach and a backward braking reach, so where braking starts is dictated by what lies downstream.Open this axiomClose this axiom
The axiom in full

For a given path the feasible speed is the pointwise minimum of (i) the curvature ceiling sqrt(A/|kappa|), (ii) a forward pass propagating the highest speed reachable under traction and power from upstream, and (iii) a backward pass propagating the highest speed from which the next lower requirement can still be met under braking. Because (iii) runs backward, where braking begins is a property of the downstream curvature profile and brake capacity, not of the corner's entry. Velenis & Tsiotras (JOTA) derive the necessary optimality conditions and construct the profile semi-analytically from maximum-acceleration, maximum-deceleration and active-bound arcs; Kapania et al. and Heilmeier et al. run the forward-backward solver on real autonomous cars. Lipp & Boyd show that in squared speed the fixed-path problem is convex, so a fixed-path profile can be globally optimal for that path and model; once the path is free the joint problem is nonconvex (van den Eshof et al. 2025 introduce a nonconvex centrifugal term and solve by sequential convex programming), so a converged free-path solution is model-optimal within its demonstrated scope and never certified as the uniquely fastest line. Evidence class: MODEL with algorithms validated on autonomous vehicles (Audi TTS, Roborace DevBot) and an F1 model.

The physics underneath

The three constraints act on the same variable from different directions; the minimum of their envelopes is the only speed trace that is simultaneously reachable, holdable and stoppable. Convexity in q = v^2 with dq/ds = 2 a_t gives the fixed-path problem a single global optimum; path freedom reintroduces products of curvature and speed and the guarantee is lost.

The sources
← Previous axiom: 4.2, Elapsed time is the integral of ds/v over the whole segment; a larger radius or a higher apex speed is not a minimum-time objectiveNext axiom: 4.4, A minimum-curvature line is slower than a minimum-time line because its curvature peaks fix corner minimum speeds that acceleration limits cannot recover: about 4 percent of lap time in the one F1 model opened, and a tenth of a second against a human in the one field run opened →Link to this axiom: #ax-4-3
4.4 A minimum-curvature line is slower than a minimum-time line because its curvature peaks fix corner minimum speeds that acceleration limits cannot recover: about 4 percent of lap time in the one F1 model opened, and a tenth of a second against a human in the one field run openedVERIFIEDMinimizing curvature alone accepts local curvature peaks that fix corner minimum speed, and bounded acceleration cannot buy that speed back, so the lap loses time.Open this axiomClose this axiom
The axiom in full

Separating geometry (minimize curvature) from speed (fit a profile afterward) loses time because the geometric step cannot see the longitudinal envelope. van den Eshof, van Kampen and Salazar (arXiv 2511.13522, Nov 2025; MODEL: quasi-steady-state single-track F1 model, Spa-Francorchamps geometry from rFactor2) report that the minimum-curvature line prefers short peaks in curvature whose minimum speeds the car cannot recover from under acceleration and braking limits, producing a lap-time deficit of around 4 percent against the free-trajectory minimum-time solution; with only drag and downforce coefficients tuned the model sits within 10 km/h of the 2025 Belgian GP top-10 qualifying traces. Estimand: the lap-time ratio of two model trajectories for one car on one circuit, no human and no measurement noise; the '4 s on a 100 s lap' gloss is a leg's arithmetic, not the paper's. Field correlate (CORRELATIONAL): Heilmeier et al. (Vehicle System Dynamics, June 2019) drove a minimum-curvature line with a forward-backward speed profile on the Roborace DevBot at the Berlin Formula E event (May 2018) to within a tenth of a second of a human driver at about 150 km/h and about 80 percent of the car's acceleration limits, which shows the surrogate is usable, not that it is time-optimal. Kapania et al.'s iterated two-step line (speed profile first, then a curvature-minimizing path update that knows the speed profile) drove 0.6 s faster than a time-minimizing clothoid search at Thunderhill. Boundary: the 4 percent figure belongs to one F1 model at Spa; the direction (minimum curvature loses) holds across the three legs and the opened sources, the magnitude does not transfer.

The physics underneath

Minimum-curvature optimization distributes curvature to minimize its integral or peak over the lap and so accepts local peaks where the track is tight; each peak fixes a minimum speed through a_n = v^2 kappa, and the time to rebuild speed from that minimum is bounded by the power and traction envelope, so time lost at a peak is not recovered on the following straight.

The sources
← Previous axiom: 4.3, On a fixed path the speed profile is the lower envelope of three propagations, so the braking point is set downstream; fixing the path makes the problem convex, freeing it makes every solver's line model-optimal onlyNext axiom: 4.5, The car reshapes the line →Link to this axiom: #ax-4-4
4.5 The car reshapes the lineVERIFIEDWhere power rather than grip caps forward acceleration the same track has a different minimum-time path, and the folk 'high power means a V-shaped line' is not settled by any opened source.Open this axiomClose this axiom
The axiom in full

The envelope a line is solved against is three-dimensional (longitudinal, lateral, speed): downforce expands it with speed and the engine truncates its forward edge at F <= P/v, so three different limits can govern three stretches of one corner (tire-limited, where easing curvature frees usable acceleration; power-limited, where easing curvature frees grip the engine cannot use; brake-limited, where downstream speed caps the present one). Because the optimizer trades curvature for acceleration only where acceleration is available, changing the car's envelope changes the optimal path on an unchanged track. Opened evidence, all MODEL: van den Eshof et al. 2025 reproduce an F1 car's g-g-v diagram and find the minimum-time line prefers a longer sustained curvature through the corners over the minimum-curvature line's short peaks (for the highest power-to-weight car in the set, the opposite of the folk V-shape); Zhang, Yuan and Zhu (Sci Rep, Oct 2025) simulate petrol and electric racecars with different driver models in CarSim and report that the minimum-time trajectories for different vehicles and drivers display significant difference; Velenis & Tsiotras (ISIE 2005) show that changing the cost from minimum time to maximum exit velocity, or the weight balance, reproduces different techniques including large-slip-angle solutions for one configuration. The U-versus-V bifurcation by power-to-weight stated by the gemini leg rests on ARGUMENT-class sources (a Rennlist forum thread, reddit, a TUM residual-policy paper) and on no opened model comparison that varies power at fixed track; it is recorded as unsettled. Population: vehicle models (F1, petrol and electric racecars, a bicycle model with suspension); outcome: optimal path curvature profile and lap time; no human population.

The physics underneath

The Hamiltonian minimum sits on the active envelope boundary and the speed costate prices a unit of lateral demand against the longitudinal acceleration it displaces; where the engine rather than the tire bounds longitudinal acceleration, releasing curvature buys nothing, so the optimizer holds curvature longer and the path changes shape with the car's power curve and downforce.

The sources
← Previous axiom: 4.4, A minimum-curvature line is slower than a minimum-time line because its curvature peaks fix corner minimum speeds that acceleration limits cannot recover: about 4 percent of lap time in the one F1 model opened, and a tenth of a second against a human in the one field run openedNext axiom: 4.6, Corners are coupled through the full exit state, not exit speed: a locally slower first corner is globally faster when it leaves position, heading, yaw rate and sideslip the next corner can use →Link to this axiom: #ax-4-5
4.6 Corners are coupled through the full exit state, not exit speed: a locally slower first corner is globally faster when it leaves position, heading, yaw rate and sideslip the next corner can useVERIFIEDEach corner's cost includes the minimum time from the state it leaves, so the exit's position, heading and yaw matter as much as its speed.Open this axiomClose this axiom
The axiom in full

The cost of a corner is its own time plus the minimum remaining time from the state it leaves, J = T_corner + V(z_exit), and that state includes lateral position, heading, yaw rate, sideslip and force distribution, not speed alone. Three consequences carry across the opened sources (MODEL plus CORRELATIONAL field). First, the value of exit speed is a downstream property: in the worked example a corner 0.2 s slower that exits 2 m/s faster (22 vs 20 m/s) wins only when the following straight exceeds about 76 m at 3 m/s^2 of available acceleration (illustrative arithmetic). Second, Kapania et al.'s field laps show sector-level trades, the two-step line predicted to gain from a to c, lose from c to e, gain from e to h, and winning the lap by 0.6 s; the same paper notes the planner deviating toward the outside of the track to create space for the upcoming right-handed corner. Third, Subosits & Gerdes (IEEE TIV 2021, autonomous VW GTIs on an icy circuit, mu about 0.26) found a point-mass-planned corner transition producing substantial unintended sideslip and yaw motion the planner never modeled, so a fast-looking exit left a dynamically expensive transition. Velenis & Tsiotras (ISIE 2005) show maximum-exit-velocity and minimum-time are different objectives producing different paths. Population: autonomous test vehicles; outcome: lap time and tracking; no human correlate opened.

The physics underneath

Bellman's principle: an optimal trajectory is optimal from every intermediate state, so the only lawful comparison of two corner paths is between common timing gates carrying the whole state through; exit speed is one coordinate of that state, and the yaw and sideslip coordinates set how soon the next corner's grip can be used.

The sources
← Previous axiom: 4.5, The car reshapes the lineNext axiom: 4.7, The planned line and the driven line differ by what the planner left out: omitting vertical curvature at a crest tripled an autonomous car's tracking error, and on ice the predicted ranking of planners reversed when driven →Link to this axiom: #ax-4-6
4.7 The planned line and the driven line differ by what the planner left out: omitting vertical curvature at a crest tripled an autonomous car's tracking error, and on ice the predicted ranking of planners reversed when drivenVERIFIEDDynamics a planner omits, such as crest unloading or yaw transients, reappear as tracking error that spends grip and track width, so the driven lap trails the planned one.Open this axiomClose this axiom
The axiom in full

A minimum-time plan is only as fast as its model is complete; omitted dynamics return as tracking error, which consumes grip and track width the plan assumed free. Subosits & Gerdes (ACC 2015, CORRELATIONAL field, Audi TTS at Thunderhill, mu 0.85, targeting 0.85 g, about 92 percent of the true limit) show that at a hill-crest left-hander including vertical curvature changed the intended speed by less than 2 m/s yet cut maximum path-tracking error from 1.30 m to 0.37 m; without it the front tires saturated and the car ran wide. Subosits & Gerdes (IEEE TIV 2021, autonomous VW GTIs, mu about 0.90 dry, 0.72 wet, 0.26 ice, ABS on, ESC off) report on the icy circuit that the point-mass planner's target lap (100.27 s) was faster than the single-track planner's (101.14 s) but drove slower (103.48 s vs 102.61 s); the double-track free-allocation planner targeted 98.02 s and drove 98.69 s, a 0.7 s gap against the point-mass planner's 3.2 s. Kapania et al. planned at mu 0.90 for safety, measured about 0.95, and both autonomous laps (138.6 s, 139.2 s) trailed the professional driver's 137.7 s while predicted times (136.4 s, 136.7 s) ran about 2 s under the driven ones. Kratochvil & Vomlel's Silverstone F1 model (83.95 s) undercut a test pilot (85.51 s) with constraints the authors call too cautious and acceleration a bit exaggerated. Estimands: differences between a planner's predicted lap and its driven lap, same car and day, one circuit each; the field population is autonomous research vehicles, not human drivers. The TIV 2021 figures were read by the openai-dr leg from the full text; the writer opened the DOI record only.

The physics underneath

Normal load varies with vertical curvature (lighter over a crest), so a horizontal radius plus a nominal mu overstates grip there; a point-mass model has no yaw or sideslip states, so the transition between corners it plans excites motion it cannot see, and the controller spends tire force correcting it.

The sources
← Previous axiom: 4.6, Corners are coupled through the full exit state, not exit speed: a locally slower first corner is globally faster when it leaves position, heading, yaw rate and sideslip the next corner can useNext axiom: 4.8, Robustness that backs off the force limit can cost a human driver no lap time while cutting steering activity about fivefold; robustness that backs off the track edge costs about 2 to 4 seconds a lap: a simulator result from two drivers →Link to this axiom: #ax-4-7
4.8 Robustness that backs off the force limit can cost a human driver no lap time while cutting steering activity about fivefold; robustness that backs off the track edge costs about 2 to 4 seconds a lap: a simulator result from two driversVERIFIEDBacking off the force limit keeps tires below the saturation knee so corrections shrink at little time cost; backing off the track edge lengthens every corner and always costs time.Open this axiomClose this axiom
The axiom in full

A nominal minimum-time line rides the friction boundary with zero margin, and a planner can add margin in two places: on the force constraint (a friction-limit back-off) or on the geometric constraint (a track-limit back-off). Masoni et al. (arXiv 2512.04917, submitted 4 Dec 2025; INTERVENTION, within-person, two drivers on the DriSMi dynamic driving simulator, one track, limited laps, sequencing not controlled) report median lap times: driver VP nominal 85.47 s, force-margin 87.32 s, geometric-margin 89.92 s; driver MM nominal 84.87 s, force-margin 84.65 s, geometric-margin 86.63 s. Steer energy (the integral of squared steering rate, rad^2/s) fell for VP from 552 under the nominal reference to 103 under the force-margin reference. The force-margin reference cost MM nothing (0.22 s faster) and VP 1.85 s, while the geometric-margin reference cost both drivers 1.8 to 4.5 s; the authors call the study exploratory. Autonomous correlates (CORRELATIONAL field and MODEL): Lew et al. (TRI, arXiv 2410.17183, Lexus LC 500) report that a deterministic MPC planning on one tire model may lose control in adverse road conditions while a CVaR risk-averse MPC over sampled friction parameters stays controllable; Werner et al. (GripMap, arXiv 2504.12115) make the constraint vary with position because surface grip varies by location while tires travel with the car; Bongard et al. (arXiv 2602.04744) tighten constraints online from a contraction metric so the robust line runs further from the kerbs with lower peaks. Population: two simulator drivers and autonomous test vehicles; outcome: lap time, steer energy, loss of control; PROBABLE because the human evidence is n = 2 and the direction is driver-dependent.

The physics underneath

Margin on the force envelope keeps the tires away from the saturation knee where small grip errors flip the sign of the correction, so steering corrections shrink and the lap can be held; margin on geometry shortens the usable track, lengthens the path and lowers attainable speed on every corner regardless of grip, so it always costs time.

The sources
← Previous axiom: 4.7, The planned line and the driven line differ by what the planner left out: omitting vertical curvature at a crest tripled an autonomous car's tracking error, and on ice the predicted ranking of planners reversed when drivenThis is the last axiom on the page.Link to this axiom: #ax-4-8

The easy answers this run refused

Each vector was told the common saying it must not settle for, and told to rebuild the answer from first principles instead.

Grip, weight and the driver's hands run on mechanisms, and every mechanism on this page can be checked again.