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One sphere, two projections.
VMO × SKA = Emergentist Exergy. In the selected normalized profile, its maximum value is 1 at x = 1 and across the sphere equator.
The discovery
The selected model pairs two stereographic projections.
Let x = tan(θ/2) > 0 be one projection radius and 1/x its opposed radius. The selected normalization is VMO = 2x/(1+x) and SKA = 2/(1+x), so E_B = VMO × SKA = 4x/(1+x)². This is the chart's normalized Exergy profile; the names do not make it a measurement of a person or physical system.
For these declared functions, 1 − EB = (x−1)²/(x+1)² ≥ 0. Equality holds exactly at x = 1; the azimuth-independent sphere extension reaches the same maximum across the whole equator. The separate chart-balance coordinate is B = sin θ, with EB = 2B/(1+B). These are mathematical results for the selected chart, not scores for lives.
Keep the chart composition, its balance coordinate, and finite-node scoring in separate registers:
Sphere Exergy
EB = VMO × SKA
The selected profile reaches 1 at x = 1 and across the whole equator. It is a chart result, not an empirical score.
Chart balance
B = sin θ ≤ 1
A geometric coordinate with its maximum at the equator. It does not measure a person or day without separate mapping and calibration.
The chart profile, its balance coordinate, and a node's evidence profile answer different questions. The geometry alone establishes neither empirical measurement nor ethical direction. The separate ordinal node score is P_node := min(VMO₄, SKA₄) under a declared common normalization.
Why this matters to you
A maximum in the chart is not a verdict about a life.
The selected symmetric profile falls as the chart point moves away from the equator. That ordering is exact for this model; applying VMO and SKA to an organism, decision, or society requires independently defined factors, evidence, and calibration. The peak does not prove that balance is always ethically best.
The practical question is separate: what do the two factors mean in this application, and how would each be measured?
Held honestly
The selected profile's maximum is proved under its declared functions [A], and its normalization is a framework selection [S]. Its interpretation is [I]; applying it to a life or system remains a separately calibrated empirical question [C].
Held against adversarial audit — see the trial record.