The DP flow meter gives you R(x) for free

Supporting note for the anti-surge application, which leads with the results. It derives the sensor model that examples/surge_margin.py runs on, fixes its one calibration constant, and records what the demo does not model.

Nothing here is a citable result. The example gallery's numbers are showcases, and this note inherits that status.

The application comes from Sathish, Khader, Sayma and Shah, Centrifugal Compressor Design and Surge Simulation for Active Inference Based Control (ASME Turbo Expo 2024, GT2024-124905), which proposes active inference for anti-surge control. The sensor model and every choice below are mine. Nothing here is a claim about what that paper did.

The meter

Anti-surge control on a centrifugal compressor needs flow, and flow is almost always measured with a differential-pressure element: an orifice plate, a venturi, a flow nozzle. The element gives a pressure drop, and the flow follows the square-root law

Q = k·√ΔP

The transmitter's error is approximately flat across its span. Call it σ_ΔP ≈ const. That is a statement about the instrument, not about the process. Propagate it through the square root:

σ_Q = |dQ/dΔP|·σ_ΔP = (k / 2√ΔP)·σ_ΔP = k²σ_ΔP / (2Q)

That is a first-order propagation. It holds while σ_ΔP is small against ΔP, which is least true at low flow.

So the variance of the flow measurement is

R(Q) = σ_Q² = τ / Q²,        τ = k⁴σ_ΔP² / 4

The noise on the flow reading depends on the flow being read. Nobody chose that. It follows from the square root and a flat transmitter error, and it is why a DP meter has a turndown limit at all.

The element type sets the sign

That derivation is about a DP element. It is not a fact about flow metering, and the sign it produces is what the whole demo turns on:

Element	Error behaviour	R(Q)	ℓ' = R'/R	Epistemic term pushes
Orifice, venturi, nozzle	flat in ΔP span	∝ 1/Q²	−2/Q	away from surge
Coriolis, ultrasonic	roughly constant % of rate	∝ Q²	+2/Q	toward surge
Some thermal mass	constant absolute	constant	0	nowhere, it collapses

Read the middle row. A meter whose relative accuracy is flat has σ_Q = e·Q, so R = e²Q² and ℓ' = +2/Q. The sign flips, the information gain is now largest at low flow, and the agent is pulled toward surge instead of held off it. Same machinery, opposite conclusion, and nothing about the inference changed.

The demo runs that row. A Coriolis meter with the same accuracy at design flow puts the agent's first step below its preference, and in closed loop the library's own Agent settles past the surge line. --check asserts both, in rows 10 and 11.

The bottom row is the one that matters for whether any of this is worth doing. A constant-R meter puts ℓ' = 0, and then the epistemic term is constant across actions and the agent reduces to a controller.

There is a further wrinkle for anti-surge specifically. Conventional anti-surge control often works in reduced ΔP coordinates and never takes the square root at all, because the surge line is approximately a straight line through the origin there. The meter carries the same information about flow in either coordinate. What changes is where a model puts the square root. Here it sits in the sensor, as R(Q). In ΔP coordinates the sensor is linear with flat noise, and the square root moves into the dynamics because the valve acts on flow. A linear-Gaussian model with additive control linearises that away. Its R is then constant, and it collapses exactly like the fixed-R twin below. Whether an agent has an epistemic term at all depends on the coordinates its model is written in.

Calibration

One constant, and I want it in units an engineer recognises. Reparameterise by the relative error at design flow Q_ref:

σ_Q(Q) = e_ref·Q_ref² / Q          R(Q) = e_ref²·Q_ref⁴ / Q²

e_ref is the relative flow uncertainty at Q_ref. I take e_ref = 0.002, which is 0.2% at design flow.

This is the only free constant in the build, and it is an assumption rather than a derived quantity. ISO 5167 is where the computation of flow-rate uncertainty for a DP element is standardised, with orifice plates in its second part. A real value belongs to a specific installation rather than to a note like this one. The reasoning behind the value I use: instrumentation practice quotes orifice-plate flow uncertainty at roughly ±20% once you are down at 10% of maximum flow. Set Q/Q_ref = 0.1 in the expression above and the relative error is e_ref·(Q_ref/Q)² = e_ref·100. So e_ref = 0.002 reproduces the ±20% figure exactly.

That agreement is not independent evidence. The ±20%-at-10%-flow rule of thumb is the square-root law restated, so the two agreeing confirms the algebra and nothing more. Treat e_ref as a plausible calibration, and re-anchor it against a real datasheet before any number here is used for anything.

Coordinates

The demo's latent is the surge margin, not the flow:

m = Q − Q_s           Q = Q_s + m
R(m) = e_ref²·Q_ref⁴ / (Q_s + m)²

Q_s is the surge-line flow at the operating speed line. Positive m is safe, negative m is surge.

R is strictly decreasing in m on the whole operating range, so R'(m) ≠ 0 everywhere the demo evaluates it. That matters beyond tidiness. research/gate_d4_registration.md names R'(μ) ≠ 0 as a precondition on any fixture without a mean-moving action, in the amendments dated 2026-08-07. A fixture sitting at a stationary point of R looks like a state-dependent sensor and behaves like a fixed one. This one is nowhere near that degeneracy.

Near Q = 0 the expression blows up. The demo clamps Q at a floor well below Q_s, so R stays finite and positive-definite across the whole action grid. A non-positive- definite R does not raise. It surfaces as a NaN at action selection, which is a worse failure than an exception.

What the fixed-R twin does, and what it does not say

The twin's epistemic term is constant across actions, so its argmin G is its argmin pragmatic exactly. Koudahl, Kouw and de Vries proved this for fixed-noise linear Gaussian state space models, where EFE minimisation reduces to KL control and the exploratory drive does no work (Entropy 23(12):1565). DECISIONS.md records it as ADR-003, and the demo's first two rows measure it.

The scope condition is easy to overstate, so here it is precisely. The collapse is a property of the agent's generative model, not of the plant. It needs a linear-Gaussian model with additive control and a fixed R: control then enters the mean only, the covariance recursion P⁻ = APAᵀ + q, S = CP⁻Cᵀ + R, P⁺ is action-independent, and the epistemic term ½(ln det S − ln det R) cannot vary. q is the dynamics noise, written small to keep it apart from the flow Q. The plant the agent is pointed at may be arbitrarily nonlinear without changing any of that.

What it does not cover: an EKF linearised on a nonlinear plant has Jacobians A(x) and C(x) that depend on the linearisation point, which depends on the predicted mean, which depends on the action. Its covariance recursion is action-dependent, so its epistemic term is not flat, with no R(x) involved. Whether that variation is enough to move the argmin is an empirical question and this demo does not answer it.

Where the standoff sits

One EFE step moves only the predicted mean. With m the margin a valve step aims for, g the preferred margin and P⁻ the predicted variance, the two parts of G are

pragmatic(m) = ½Λ·[(m − g)² + P⁻ + R(m)]
epistemic(m) = ½·ln(1 + P⁻/R(m))

Differentiate and use R' = ℓ'·R. The epistemic term's slope is −½·κ·ℓ', where κ = P⁻/(P⁻ + R) is the Kalman gain. Setting dG/dm to zero gives

Λ·(m − g) = −ℓ'(m)·(κ(m) + Λ·R(m)) / 2

For this meter ℓ' = −2/Q, so

m − g = κ/(Λ·Q) + R/Q

The first part is epistemic. The second is pragmatic, and it comes from the R(m) inside the pragmatic term. The preference is over readings, a noisier reading costs expected pragmatic value, and this meter's reading is quieter at higher flow. One step from the demo's prior κ is large, near one at the default meter, and the second part is a few percent of the offset at most. Once the loop has settled, κ is small enough at the worse meters that it is not.

Λ sets the scale of the epistemic part, as 1/Λ. That is why sweeping it moves the standoff further than the meter ladder does. It needs no separate disclosure, because it is the formula.

A worse meter means a larger R. That lowers κ and shrinks the epistemic part, and it grows the pragmatic part. The two pull opposite ways. The epistemic part wins on every rung the demo visits, which is why a worse meter runs closer to surge. The check asserts that net direction for the first step and for the settled loop.

κ depends on the meter only through P⁻/R, and R scales as e_ref². One step from a fixed prior, a meter twice as bad and a prior four times as tight give almost the same standoff. Only the pragmatic part, which depends on R alone, separates them, and one step from the prior it is small. A ladder of first steps cannot tell them apart in practice.

The closed loop can. A loop that has run a while holds the filter's steady-state variance. For this scalar model, with q the dynamics noise, that variance solves

P⁻² − q·P⁻ − q·R = 0

at the margin the loop holds. That P⁻ comes from the meter, so the settled margin does too. The steady-state κ is far below one, so the settled offset is much smaller than the first step's. The demo finds the fixed point by iterating the two equations, and checks it against the library's own Agent run in closed loop.

The sign of the offset is the sign of −ℓ', since κ + Λ·R is positive. A meter with ℓ' > 0 puts the agent below its preference, on the surge side. The sign table above is that statement, and the demo's two Coriolis rows measure it.

Where this sits relative to c₂

The log-noise and its derivative are the quantities the programme's gap work is written in:

ℓ(m) = log R(m) = log(e_ref²Q_ref⁴) − 2·log(Q_s + m)
ℓ'(m) = −2 / (Q_s + m) = −2/Q

research/gate_d4_registration.md (RESULT 2026-08-07) and research/c4_hand_derivation.md give the leading coefficient of the inference gap as c₂ = (ℓ'(μ)/2)². For this sensor that is

c₂ = 1/Q²

A clean closed form, and the demo asserts ℓ' against it.

The demo does not measure an inference gap. Two different quantities travel under similar names, and conflating them is the easy mistake here:

• The EFE epistemic term, ½(ln det S − ln det R), is what expected_free_energy returns. It is what varies across the action grid under R(x) and goes flat under a frozen R. This is the demo's content.
• c₂ is the leading σ² coefficient of KL(q ‖ p(·|y)), the gap between the plug-in-R Kalman posterior and the exact non-Gaussian posterior. Measuring it needs the quadrature in research/checks/gap_kernel.py over the exact predictive. The demo runs none of that.

Both are driven by ℓ' = R'/R and both vanish when R' = 0. That shared root is why they look like the same object. They are not.

So the claim here is narrow: this compressor sensor lands in the same shape as the family the gate work is registered on, with ℓ' in closed form and nowhere zero. That is a statement about the sensor.

Two limits on reading c₂ as the whole gap. It is the same for forward and reverse KL only at order σ², and the third cumulant separates them above that. It is positive only locally, because c₄ < 0 turns the series over at larger spread. Both are recorded in research/gate_d4_registration.md.

What the checks are entitled to claim

Every row samples a sweep over a continuous valve range, which is a finite grid over an infinite domain. So each reports CORROBORATED and none is PROVED at any grid resolution. The monotonicity row is BOUNDED against the sweep's own step size: a standoff that moved by less than one grid cell did not measurably move. The ℓ'(m) = −2/Q row is EXACT, being autodiff against a closed form at machine precision, and it is a statement about the sensor rather than a measured inference gap. The law, closed-loop and Coriolis rows are BOUNDED by the same grid step.

--check prints both ladders, the probe_model verdict and the eleven falsifiers. It asserts that none of them fired and that Λ moves the standoff further than the meter does, with no plotting deps.

What this does not model

The demo and anything written about it quote the same list:

• No Greitzer dynamics. Surge is a dynamic instability of the compressor-plenum system (Greitzer 1976), and none of that is here.
• No cubic compressor characteristic.
• No speed line, and no travel along one.
• A local linearisation about a single operating point.
• R(Q) is a first-order propagation through the square root. The filter plugs in R at the predicted mean, which is the approximation the inference-gap work measures. The demo measures neither.
• Horizon-1 EFE only. No multi-step planning.
• The surge point Q_s is known and fixed. Treating it as an unknown latent is the interesting version of this problem and is not what the demo does.
• A DP element is assumed. The sign table above is what that assumption buys, and a different element buys a different sign or none.
• The Coriolis meter shares the DP meter's accuracy at design flow and nothing else. A real Coriolis meter adds a zero-stability term, constant in absolute flow, that dominates at low flow. There R stops falling and the pull toward low flow weakens. The demo leaves that term out.
• The meter model is derived from a standard that excludes the regime of interest. ISO 5167 does not apply to pulsating flow, and surge is a pulsating flow instability. So R(Q) describes the meter on the approach to surge and not in it. Anything the demo says about behaviour at the surge line inherits that.
• Reduced ΔP coordinates are not modelled. A controller working in them is reading a different quantity.
• No gas thermodynamics. No real-gas properties, no composition, no suction conditions.
• Not a controller benchmark. The comparison is against a fixed-R twin of itself, not against PID, MPC or a conventional anti-surge controller. Nothing here says the R(x) agent controls a compressor better than anything.
• The operating point is illustrative. Q_ref, Q_s and the design margin describe no machine and were not read off a performance map.
• Where the standoff sits is set by the economic weight Λ, which is a policy choice. The law makes the epistemic part of the offset scale as 1/Λ, and sweeping Λ over 10:1 moves the standoff further than the whole meter ladder does. The demo's claim is the direction of the offset, not its size.
• The ladders' first steps are taken from a prior with σ = 2% of design flow, and the prior's width scales them as much as Λ does.
• The closed loop is noise-free, and only the default accuracy runs through the library's Agent, once for each meter. A noisy loop fluctuates about the point it finds. The settled margins for the rest of the meter ladder come from the law.

The demo is a statement about what a state-dependent sensor does to action selection. It is not a compressor model.