Security · provenance
Can a Machine Prove It Is Itself?
A drivetrain carries a measurable manufacturing signature. This report prices it in bits, and the price decides what it can be used for.
The Charlot Lab · Institute for Physical AI @ Bailey Military Institute
Companion to TR-2026-36, which states this question and does not answer it.
1. The question, and where it comes from
TR-2026-36 reduced the integrity of a metered system to custody of its measurement reference, and then observed that the reduction imports an assumption which embodiment breaks: legal metrology's apparatus of type approval, seals and periodic recertification assumes the instrument comes back for inspection. A deployed body may be stolen, altered in the field and returned to service, or simply leave the jurisdiction whose approval gives the seal meaning. That report named the question and declined to answer it.
This one answers as much of it as can be answered without hardware, which turned out to be all four of the questions it opened with. The result is more restrictive and more useful than expected, and the most important finding is that the question this work began with was not the binding one.
2. Scope, method and grading
Every figure carries a grade. Verified means a primary document was retrieved and the number read from it. Vendor means a manufacturer's published specification. Computed means our arithmetic over graded inputs, and the arithmetic is shown. Not located means this review searched for a quantity and did not find it, and the searches are named.
No hardware was used and none was needed. That is itself a finding: the first draft of this work opened with a request for five identical actuators, and every question it was meant to settle was already answerable from published specifications and computation.
3. Prior art, and what this report may not claim
| Field | What it owns | Granularity |
|---|---|---|
| Silicon PUF | Per-unit identity from manufacturing variance; the canonical form of this idea | individual unit, in silicon |
| Motor current signature analysis | That individual motors have learnable baselines, and systems that learn them per motor | individual unit, framed as condition monitoring |
| EMI fingerprinting | Counterfeit detection from electromagnetic emissions | make, model, configuration; patent-dense |
Table 1. Three fields own parts of this ground. The physics we lean on, that nominally identical motors differ enough to be modelled individually, is established by the second of them. Our contribution cannot be, and is not, "motors differ".
This report claims no fingerprinting method. It states a physical question, prices the answer in bits, and publishes both openly with the arithmetic attached. That is a deliberate position: a measurement method held behind a patent or a paywall is enclosure of something that ought to be common, and the answer to enclosure is to publish the free version.
What this review did not locate is a published test of whether a physics-structured energy model's identified parameters separate individual drivetrain units of the same make and model. Searches run were on physically unclonable functions combined with motor and drivetrain manufacturing variance, motor current signature analysis and unit-to-unit variation, and device authentication and counterfeit detection from motor electrical signatures. That is a statement about this review, not about the literature.
4. Is there a signal? Yes, and the number is on a datasheet
The model is the one identified in TR-2026-36 §5 for a humanoid arm 2: per joint, a mechanical term, a baseline-corrected copper-loss term, and Coulomb and viscous friction terms. The copper-loss coefficient is b = R / (n · ηg · Km)², so it carries phase resistance once and the torque constant twice.
Vendors publish unit-to-unit tolerance because design engineers need it to size a system that works with any unit off the line. For brushless DC motors, torque constant is quoted at ±10 %, of which magnet remanence contributes roughly 2.5 % and phase setting 7.5 %, and phase resistance at ±8 % 1. vendor
| Route | Implied spread in b | Against the 2.94 % floor |
|---|---|---|
| Worst case, linear | 28.0 % | 9.5x |
| In quadrature | 21.5 % | 7.3x |
| Combined datasheet figure alone | 18 % | 6.1x |
Table 2. The signal clears the model's own resolution floor by roughly an order of magnitude. computed from vendor tolerances
5. Is it identifiable? Only one coefficient, and excitation decides
A fingerprint must be a property of the hardware and not an artifact of the fitting procedure. If two coefficient values explain the data equally well, whichever one the optimiser returned is an accident, and it will be a different accident on the next fit. So identifiability precedes distinctiveness.
We built the per-joint regressor on a representative single-joint plant, using the source's own trajectory design, and computed each parameter's standard error at the model's 1.07 W identification residual.
| Excitation set | cond(ΦᵀΦ) | b copper | a mech | d viscous |
|---|---|---|---|---|
| 1 travel, 1 posture | 3.4e17 | 321 % | — | 3811 % |
| 3 travels | 2.4e07 | 1.0 % | 792 % | 121 % |
| 3 travels, 4 postures | 5.6e05 | 0.3 % | 54 % | 54 % |
| 4 travels, 6 postures | 1.1e06 | 0.2 % | 37 % | 36 % |
Table 3. Standard error as a fraction of each coefficient's value. Only b resolves tightly enough to be a property rather than a choice. computed
Two readings. First, only the copper-loss coefficient can carry an identity: 0.2 to 1.0 percent against the 21.5 percent it must resolve, while the mechanical and viscous terms sit at 36 to 54 percent and would contribute mostly noise. That it is the same coefficient the datasheet tolerances point at is not a coincidence, b contains both quantities that vary most between units, but the two arguments are independent and they agree.
Second, excitation is decisive. Adding two travel distances moved the conditioning by ten orders of magnitude. Identifiability is a property of the experiment as much as of the model, and a naive sweep set will not identify these parameters at any instrument quality.
6. Does it survive temperature? Not without a thermometer
Copper resistivity rises with temperature at +0.393 %/°C. NdFeB remanence, and therefore the torque constant, falls at −0.08 to −0.12 %/°C 3. Since b carries resistance in the numerator and the torque constant squared in the denominator, a rising numerator and a falling denominator push the same way and the effects compound rather than cancelling: +0.633 %/°C. computed from published coefficients
| Temperature rise | Drift in b | Against the 21.5 % spread |
|---|---|---|
| 20 °C | 12.7 % | 0.59x |
| 34 °C | 21.5 % | 1.00x, crossover |
| 60 °C | 38.0 % | 1.77x |
Table 4. Beyond roughly 34 °C the drift exceeds the entire difference between units, so a warm unit reads as a different cold one. A motor passes through that between cold start and steady running.
The remedy is cheap and it is sufficient. Correcting to a reference temperature leaves a residual set by how well winding temperature is known: ±5 °C leaves 3.2 %, about a seventh of the signal; ±1 °C leaves 0.6 %. The fix is a thermometer, not a better ammeter.
7. The binding constraint is entropy
The three sections above answer the questions this work opened with, and none of them is the constraint that decides the outcome. A fuzzy extractor, the standard construction for turning a noisy physical source into a stable identity, with an enrollment phase that computes public helper data through an error-correcting code and a reconstruction phase that recovers the identity from a noisy re-reading 7, is denominated in bits. For a continuous coefficient the usable bits are approximately log₂(spread / resolution).
| Condition | Effective resolution | Bits per coefficient |
|---|---|---|
| Model floor only | 2.94 % | 2.87 |
| Thermometer to ±5 °C | 3.16 % | 2.76 |
| No thermometer, 40 °C swing | 25.3 % | −0.24, none |
Table 5. The last row confirms §6 from an independent information-theoretic direction: uncompensated, the source carries negative usable entropy. Two arguments, one conclusion. computed
Because only one coefficient per joint is identifiable, identity scales with joint count, at about 2.76 bits each, less the entropy the public helper data leaks.
Figure 1. Drag the joint count. A seven-joint arm reaches about 13.5 bits, a full humanoid about 56, and the 128-bit line needed for key derivation is not reached below roughly 67 joints. Predict before dragging: most readers expect a robot to be able to generate a key from its own body. computed
8. The primitive splits in three, and only one is defensible
Entropy separates three security primitives that are easy to conflate, and the PUF literature is written mostly about the third of them.
| Primitive | What it asks | Cost | Verdict at these levels |
|---|---|---|---|
| Verification | Is this the unit I enrolled? | one hypothesis | supported, an arm's 13.5 bits is a ~1-in-11,600 false accept |
| Identification | Which of N units is this? | log₂(N) bits before anything else | marginal, 13.3 bits merely to name a unit among ten thousand |
| Key derivation | Generate a secret from the body | ~128 bits | not supported, needs ~67 joints |
Table 6. The claim this report makes is the first row and only the first row.
9. Wear: the question this work opened with, and why it was the wrong one
This report began by treating wear as the decisive experiment: if the coefficient drifts as a drivetrain ages, the mechanism dies. That framing was binary where the real answer is a budget. In a fuzzy-extractor construction, drift does not have to be zero, it has to sit inside the correction capacity, alongside every other noise contribution 7.
What was not located: a published coefficient for how a drivetrain's copper or friction terms drift with accumulated duty hours. Searches covered motor parameter drift and winding-resistance aging, harmonic-drive efficiency degradation over operating hours, and accelerated life testing of servo actuators. The direction is uncontroversial in the maintenance literature (resistance rises with thermal aging, gear friction rises with wear and lubricant degradation) but no magnitude usable here was found.
What was located is a single measured datum, and it is only weakly analogous: in a 57-hour gait-like life test of a 3D-printed open-source actuator, positive-work efficiency fell by 3.5 %, levelled off, and then rebounded to near its original value, which the authors attribute to transmission dust reaching saturation over an underlying break-in process 6. verified, weak analogy
Since b carries gear efficiency squared in its denominator, a 3.5 percent efficiency loss implies a 7 percent drift in the coefficient, a third of the manufacturing spread, and equivalent to an 11 °C temperature change on the thermal axis. Combined with the residual from a ±5 °C thermometer, the total sits at 10.2 percent, inside the 21.5 percent budget but consuming 47 percent of it, and costing 1.68 of the 2.76 bits per coefficient.
And the mitigation is already standard practice in the field this work started from. If the identity is re-enrolled periodically, wear drift only has to stay inside the correction capacity within the re-enrollment interval, not across a service life. That is the freshness bound TR-2026-36 §9.1 sketched from first principles, and it is what legal metrology already does under the name recertification. The physics converges on the operational practice.
10. Resolution
| Question | Answer | Binding constraint | Threshold that would move it |
|---|---|---|---|
| Is unit variance above the floor? | yes, 7.3x | none | — |
| Is any coefficient identifiable? | yes, b only | experiment design | an excitation set covering ≥3 travels and ≥4 postures |
| Does it survive temperature? | no uncompensated, yes at ±5 °C | engineering implementation | winding-temperature sensing as a build option |
| Does it survive wear? | open; one weak datum | material science | a long-duration efficiency curve on a representative drivetrain |
| Is there enough of it? | for verification, not for keys | entropy, the binding one | no instrument improves it; only joint count does |
Table 7. Four of five resolved from published coefficients and computation. The fifth needs hardware, and it is not the one this work expected.
11. Limitations
The identifiability analysis of §5 is a simulation of the regressor's structure on a representative plant, not a measurement of real units. It establishes which parameter can carry an identity given the model form and a stated residual; it does not establish that real units differ in that parameter by the datasheet amount. That step remains inferential.
The wear datum of §9 is from a 3D-printed actuator whose dominant wear mechanism is transmission dust, and it is not a harmonic drive, so it is used to size the term rather than to characterise it.
The entropy estimate treats log₂(spread / resolution) as usable bits and applies a nominal 30 percent helper-data leak. Both are first-order; a rigorous treatment would compute min-entropy of the actual coefficient distribution and the exact leakage of a chosen code, and would likely return a smaller number rather than a larger one.
Coefficients across joints on one machine are treated as independent. Shared supply voltage, a common thermal environment and correlated manufacturing batches would all reduce the effective independence, and therefore the joint-count scaling of §7 is an upper bound.
12. Conclusions
A drivetrain does carry a manufacturing signature large enough to see: about 21.5 percent unit-to-unit spread in the copper-loss coefficient against a 2.94 percent resolution floor. Only that one coefficient of four is identifiable well enough to use. It dissolves under ordinary thermal operation unless winding temperature is measured, and measuring it to ±5 °C is sufficient. And the constraint that actually decides the outcome is none of those: at about 2.8 bits per coefficient, a seven-joint arm carries roughly 13.5 bits and a full humanoid about 56.
That supports provenance verification and not cryptographic key derivation, and the distinction is the report's main contribution, because the literature this construction comes from is written mostly about the primitive a drivetrain cannot support.
Two of this report's own positions did not survive its own analysis: the wear question it opened with was not the binding constraint, and the linear extrapolation it attempted from the single available wear datum is contradicted by that datum's own source. Both are reported at the size they were claimed.
References
- Published brushless DC motor unit-to-unit tolerances: torque constant ±10 % (magnet remanence ≈2.5 %, phase setting ≈7.5 %), phase resistance ±8 %, combined electrical up to ±18 %. vendor specification
- N. N. Deniz, S. Vega, S. Parsons and F. A. Auat Cheein, "Identification of a Physics-Based Electrical Power Consumption Model for the Unitree G1 Humanoid Arm," arXiv:2606.15915v1, 14 June 2026. read in full
- Published NdFeB reversible temperature coefficients of remanence, −0.08 to −0.12 %/°C; torque constant tracks flux. vendor and materials literature
- Institute for Physical AI @ BMI, TR-2026-36, Energy Observability in Embodied Systems. Institute publication
- Institute for Physical AI @ BMI, PAI-290, Energy as Evidence, Module 3. Institute course
- Design and Characterization of 3D Printed, Open-Source Actuators for Legged Locomotion, arXiv:2202.12395. 57-hour life test; positive-work efficiency fell 3.5 %, levelled, then rebounded. verified for the cited figure
- A. Schaller, B. Škorić and S. Katzenbeisser, "On the Systematic Drift of Physically Unclonable Functions Due to Aging," in Proceedings of the 5th International Workshop on Trustworthy Embedded Devices (TrustED 2015), Denver, Colorado, ACM, pp. 15–20. ISBN 978-1-4503-3828-8, doi:10.1145/2808414.2808417. bibliographic record verified; full text not read
Run the instruments behind this report
The fingerprint that melts drives §6, two units, a temperature slider, and the crossover at 34 °C. What the meter can see drives the companion report. The lessons are PAI-290, Module 3.