Incoprea research · part 1 of 5 · audit and theory · 3 November 2026

Dimensionless laws, spreads, and what a law has to specify

Part A · the audit of eight laws

This page tests one question across eight standard dimensionless groups: at a fixed value of the group, how much do the accepted models disagree? If the spread is larger than the effect anyone is trying to measure, the group does not determine the answer, and a claim that it does is a relabelling.

The answer is the same in every case: the group is necessary and never sufficient. The spread at fixed value is the footprint of a second parameter that the group cannot contain — flow regime, inertia, accommodation, Prandtl number, back-mixing, boundary condition, disturbance level, or decoupling.

1. The audit

2. What each row means

Pipe friction — the transition is not in Re

At Re = 10⁴ the laminar law gives f = 0.0064 and the Blasius law gives 0.0316 — a factor 4.94. Both are correct in their own regime; the group is identical. The absent variable is the disturbance environment that decides whether the flow is laminar or turbulent at all. Any claim of the form "friction follows Re" is true only after the regime is specified, which is exactly what the group was supposed to supply.

Sphere drag — inertia is the second parameter

At Re = 1, Stokes and Oseen differ by a factor 1.19. Oseen is a first-order correction, not a solution, which is why the disagreement grows with Re. The group does not absorb the expansion.

Knudsen conductance — the wall model is missing

At Kn = 1 the slip expression and the free-molecular limit differ by 6.27. The accommodation coefficient is a surface property that no Knudsen number contains.

Natural convection — Prandtl enters the prefactor

The two standard correlations differ by 1.13 at Ra = 10⁹. The exponent ¼ is shared; the prefactor is not, and it carries Pr. This is the mildest case in the table — and still a factor 1.13.

Reactor conversion — back-mixing is not in Da

At Da = 1, a CSTR gives 0.500 and a PFR gives 0.632, a factor 1.26. The same Damköhler number describes two reactors that differ by the residence-time distribution. Da alone cannot be used to predict conversion.

Rayleigh–Bénard — boundary conditions set the threshold

Critical Rayleigh number is 1,707.76 for rigid boundaries and 657.51 for free ones: a factor 2.60. The dimensionless group is the same quantity in both; the threshold is not. This is the cleanest illustration in the set, because there is no measurement error and no fitting — the difference is exact.

Transition — three standard values

The laminar–turbulent transition is quoted as 2040, 2300 or 4000 depending on source: a factor 1.96. All three are "the Reynolds number at transition". The inlet condition and disturbance amplitude are not in the number.

Diffusion — the decoupling near a glass transition

The Stokes–Einstein relation is the group Dη/T = const. Near a glass transition the fractional form departs by a factor 125.89 across the viscosity range. The second parameter is the decoupling between viscosity and diffusion, which the relation assumes away.

3. The criterion this suggests

A claimed dimensionless law should report the spread of its predicted quantity at fixed value of the group. If that spread exceeds the effect being claimed, the group is a relabelling. The criterion is cheap, uses no new data, and is almost never reported.

Applied to the table above, no row survives as a standalone law. Applied honestly, that is not a criticism of the groups — Reynolds, Damköhler and Rayleigh numbers are indispensable. It is a criticism of using them as explanations rather than as bookkeeping.

4. New hypotheses from this page

5. Limits

• Eight laws, chosen by me. A different eight would give different numbers, though not a different pattern.
• Some "models" here are deliberately the crude and the refined versions of the same physics; the spread measures the cost of the crude one, not a disagreement between rival theories.
• No literature search. The claim that the spread is rarely reported is an impression, not a measurement.

Part B · theory notes: bounds, spreads, freedom

Theory notes arising from the batch. These are arguments, not measurements, and none has been checked against the literature.

1. Why a saturating channel cannot produce many crossings

Let the selectivity be α(P) = N(P)/D(P), where each of N and D is a sum of channels. Write each channel as K + A·bP/(1+bP). A pressure-independent term contributes no sign change. A saturating term contributes exactly one inflection: it rises from 0 to A, monotonically, with a single curvature change.

The difference α(P) − 1 has the sign of N(P) − D(P), which is a sum of terms each monotone with at most one inflection. Such a sum can change sign at most once per saturating term, so with S saturating channels the number of crossings is at most S. The observed test is consistent with a bound of S − 1 in the sampled cases, but the proof of S − 1 would require an argument about the cancellation of the leading terms, which has not been made.

Status: the S bound is argued informally; the S − 1 bound is observed but not proved. The 600-trial test found no violation of "zero saturating channels implies zero crossings", which is the part the argument actually secures.

2. The spread criterion

Let a claimed dimensionless law be f(Π) for some group Π. Define the spread

S(Π) = max over admissible systems of f(Π) / min over admissible systems of f(Π)

The law is informative only if the effect it claims to explain exceeds S(Π). For the eight laws audited here, S ranges from 1.13 to 125.89. In every case the effect being claimed — a 20% improvement, a factor of two, a sign change — is smaller than the spread.

Why this is not a triviality

It would be trivial if the spread were measurement noise. It is not: the spread is between correct models in different regimes, evaluated at the same group value. The group is doing its job — collapsing a family — but the family is parameterised by the omitted variables, and that parameterisation is what a law has to specify.

3. Free coefficients and the appearance of reversals

The base-rate result can be read as a statement about model capacity. A reversal requires the sign of N(P) − D(P) to change, which requires at least one saturating channel and at least one parameter free enough to place the crossing inside the observation window.

With four free coefficients, the modeller can place the crossing almost anywhere; with one, the crossing is either fixed or absent. The measured progression — 0%, 0%, 13.2%, 24.0% at 1, 2, 3, 4 free coefficients — is consistent with a simple counting argument: the number of ways to place a crossing grows combinatorially with the freedom, so the probability of landing in the window grows with it.

4. What would make these results wrong

• A demonstrated case where the spread at fixed group value is small but the omitted parameter varies widely — that would break the criterion.
• A reversal that survives full constraint, which would show the base-rate argument is not the whole story.
• Evidence that the eight audited laws are unrepresentative.

5. Open questions worth more than the results

Incoprea Research · batch 1 · 3 November 2026. Computational model results and one real-catalogue re-analysis. No new physical experiment was performed. No novelty claimed; no systematic literature search. Code, data and JSON accompany this page set.