The teaching chart declares I=(1−cosθ)/2, O=(1+cosθ)/2, and R=sinθ, with I+O=1.
Y TheoryThe Compass of RealityAlgebra · phase representation
The normalized Y phase closes on a unit circle
The declared I/O/R phase chart has an exact complex unit-circle address.
Exact question
What follows exactly from the normalized teaching coordinates without importing a physical claim?
Declared relation
p:=O−I=cosθ; p²+R²=1; zY:=p+iR=e^{iθ}Native mechanism → Y relation
Preserve the carrier before translating the structure.
The mapping is auditable only when both descriptions remain visible. Y language does not overwrite the equation, mechanism, units, boundary conditions, or uncertainty of the field being studied.
The signed polar difference p=O−I retains which side dominates; R retains the orientation of passage that equal pole weights alone lose.
Calibration and withheld return
The answer must not help write the question.
Calibration establishes measurement and allowed parameters. The held-out test remains unseen until the identity rule, model, uncertainty, rivals, scoring, and failure threshold are frozen.
- No empirical calibration is used; I, O, R, and θ are declared coordinates.
- The normalization I+O=1 is held fixed.
- Symbolically substitute the definitions and verify the identity for arbitrary θ.
- At I=O=1/2, verify that R=+1 and R=−1 remain distinct crossings.
Rivals · failure · present state
A claim becomes informative where it can lose.
No visual resemblance, repeated phrase, source density, or author confidence substitutes for a discriminating return.
- A one-coordinate polarity score that discards passage orientation.
The derivation fails if the declared chart does not satisfy I+O=1 or R²=4IO, or if a claimed downstream use silently changes those definitions.
Algebraically derived inside the declared normalized chart.
This does not show that every identity is circular, complex-valued, physical, conscious, seven-layered, or governed by one universal carrier.
Recoverable lineage