Blending index: at fifty per cent the two rules disagree

Type two stream values and a limit. A mass fraction rule and a logarithmic index agree everywhere except at 50%, where they read 5 and 4 against a limit of 4.5.

Live result

Across the checked ladder, 1 ratio falls on opposite sides of the typed limit.

Ladder step10%
First ratio on opposite sides50%

A narrower ladder step can put more ratios on opposite sides.

Two blend rules and the checked sweep

Try a scenario

Typed blend case

First stream value

Second stream value8

This is the one control tracked across the curve and sweep rail.

Typed arithmetic limit

Ratio ladder step%

Declared index bounds

Lower bound

Upper bound

Both streams, the limit, and every swept value stay inside these positive bounds.

Second stream sweep

Sweep minimum

Sweep maximum

Sweep step

Tracked answer

Opposite side count: 1
First ratio: 50 %
All opposite side ratios: 50

50%: Straight rule5
Log index rule4
Readings against limitabove / below

Second stream	Straight rule	Log index rule	Readings against limit
50%	5	4	above / below

Adjacent checked changes

First change below: 6
First change above: 10

The boundary is two positive endpoint values, one typed arithmetic limit, one declared logarithmic index, and the checked ratio ladder. Both rules stay between the endpoints.

Here is the maths

The formula, in words

The straight rule weights the two endpoint values by mass fraction. The index rule takes each natural logarithm, weights those two indices, then applies exp to return to the value scale.

P_l = (1-r) P_1 + r P_2

where $P_l$ is the straight-rule value, $r$ the stream-two ratio, and $P_1$ and $P_2$ the endpoint values.

P_i = \exp((1-r) \ln P_1 + r \ln P_2)

where $P_i$ is the logarithmic-index value.

A worked example

At 50% with endpoint values 2 and 8, the straight rule gives 5.

The logarithmic index gives $\exp(0.5 \times \ln 2 + 0.5 \times \ln 8)$ , which equals 4.

Against a limit of 4.5, one reading is above and the other is below. The 40% and 60% points agree on their side, so the count is 1.

What this does not model

No real property is assigned to the logarithmic index. The model does not propagate uncertainty, read a specification, or return a conformity result.

Check it yourself

At 0% and 100% both rules equal the endpoint values. If the typed limit lies outside those endpoints, neither monotone rule can cross it.

Sources

Public logarithm, exponential, and convex combination mathematics.
Published chemical engineering literature on property specific nonlinear blending indices, described without reproducing a proprietary table.

Model version

1.0.0

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This comparison reads no specification, identifies no real property rule, and certifies no blend. Product identity, conformity, and automatic property rule selection remain outside the calculation.

Two rule blend record