For normalized complex addresses z₁=e^{iθ₁} and z₂=e^{iθ₂}, phase subtraction is represented by a conjugate product.
Y TheoryThe Compass of RealityAlgebra · ordered phase relation · Matrix representation
Directed relative phase is invariant under a shared frame rotation
Two normalized phase addresses produce a reversible directed relation whose quarter-phase values are +i and −i under opposite references.
Exact question
What exactly do i and −i add to the relation between two phase-addressed identities?
Declared relation
q₂←₁:=z₂·conj(z₁)=e^{i(θ₂−θ₁)}; q₁←₂=conj(q₂←₁)=q₂←₁⁻¹; θ₂−θ₁=π/2 ⇒ q₂←₁=i, q₁←₂=−iNative 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 ordered relation q₂←₁ belongs to the pair; reversing the point of reference conjugates it, while a shared rotation of both identities cancels. Its real coordinate records alignment and its imaginary coordinate records signed phase direction.
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; both phase addresses are normalized and use one declared orientation and phase origin.
- The order of the pair is retained. Reversing the order is not silently treated as the same directed relation.
- For arbitrary θ₁, θ₂, and common shift χ, verify q(e^{iχ}z₂,e^{iχ}z₁)=q(z₂,z₁).
- Verify that swapping the pair conjugates and inverts q, and that positive quarter separation gives +i forward and −i in reverse.
- For normalized vector z, verify Q=zz† is Hermitian, Qaa=1, and QabQbc=Qac for every three-address cycle.
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.
- cos(Δθ) alone, which loses the sign of phase direction
- chord magnitude alone, which loses ordered orientation
- two absolute phase labels, which depend on an arbitrary common frame
The derivation fails if a common phase shift changes q, swapping the pair does not conjugate q, the ±π/2 signs disagree with the declared orientation, or the normalized one-phase relation Matrix fails its exact cycle closure.
Exactly derived by complex multiplication and conjugation under the declared normalized phase chart; the Mind Structure sine drive and cosine compatibility are its imaginary and normalized real coordinates.
This proves an internal coordinate relation. It does not show that every identity has one physical phase, that inverse algebra implies reciprocal causation, or that the pairwise phase Matrix replaces the governed Mind relation tensor.
Recoverable lineage