Complex Phase Cosmology 2nd: A Topological Attempt to Unify Cosmological Problems Using Euler's Identity and Juridical Structuring Methodology — Part 3: Dimension and Time — From the Reference State of Light to Effective Physics

Part 1 of the Second Series proposed a measurement grammar that converts the phase angle θ from an input value arranged on the observational coordinate into an output value produced from local conditions, and Part 2 established what that grammar describes at the limit as a chain running from collapse to the Big Bang. It did not, however, deal with what a phase state appears as in the ordinary region between measurement and the limit. If phase is not only a coordinate value but a state of existence, dimension, time, growth, heat, entropy, and the quantities measured in the real world must also be placed within the same structure. This study deals with that ordinary region. Dimension is not defined as the number of coordinate axes, and time is not defined as a quantity that simply flows. Starting from the reference state of light, seven objects are treated, and they form one connected sequence: reference state of light → 5D+ structure → dimension → time → growth → heat and entropy → effective physics. Building on the conceptual and mathematical structure of the First Series, and on the measurement grammar and limiting ontology established in Parts 1 and 2 of the Second Series, this study moves from a grammar that produces phase and the limit of that grammar to the structure in which phase is shown in the real world. Core definitions and structural results established in this study: Reference state of light = θₑₘ = 0.25, the central symmetric point at which the real and imaginary components are equally divided. At this point, cos²(πθₑₘ) = sin²(πθₑₘ) = 1/2 and tan(πθₑₘ) = 1. In the measurement grammar of Part 1, the same point therefore corresponds to the unit reference condition 𝓕(i,π,e)·T/V = 1 Dimension = the relational degrees of freedom that constitute existence and make different states distinguishable. Under this definition, the basic structure is three-dimensional real space + one-dimensional imaginary direction + one-dimensional time = 5D+, and the “+” indicates that the structure extends by the number of transition channels Directional distribution of dimension = setting the scale conserved within a phase representation as D, the component in the real direction is dᵣ = D cos(πθ), the component in the imaginary direction is dᵢ = D sin(πθ), and dᵣ² + dᵢ² = D². Here D is the scale conserved in the phase representation. This is the conservation of the representation, and is not by itself the physical total-energy conservation of A5 Integer dimension = an effective classification that appears when an observing instrument detects the components shown in the real world and reads those components as mutually independent axes. Non-integer values follow the same distribution rule and are set with the same standing Time = an open progressing dimension in which phase states and the history of events accumulate continuously without being automatically reduced to the same past. The imaginary direction and the phase angle itself are therefore distinguished from time. Phase can return while history goes only forward, so the same phase is not the same moment Growth = A(t), a dimensionless measure of how far R's form as shown in the real world has unfolded with time. d(θ) determines the distribution of the scale by direction, while A(t) determines the unfolding of the scale itself, so the two quantities measure different things and each has its own place Heat and entropy = the movement of arrangement under the conservation of total energy. Arrangement always moves to a new state because the time axis keeps accumulating history. Phase has a period and therefore can return, while history accumulates and therefore goes only forward. The direction of time is thereby placed not in the rotation of phase itself but in the accumulation of history Effective physics = for a physical quantity satisfying all three requirements — that its scale is conserved within the phase system, that the scale divides into real and imaginary directions, and that measurement catches only the real-direction component — the measured value appears as a phase-dependent effective quantity. For such quantities, the real-direction form is X(θ) = X₀ cos(πθ). Under the same three requirements, quantities treated as universal constants are likewise treated as values that appear phase-dependently according to how the conserved scale is distributed into, and measured along, the real direction Throughout, A5 is applied rather than re-derived. The unit norm |eⁱπθ|² = 1 remains a mathematical identity, and physical total-energy conservation remains the separate bridge axiom established in Part 1. The conservation of D in the phase representation is likewise kept distinct from A5, so the mathematical distribution of components is not identified with physical total-energy conservation. Two things remain to be done. First, to determine the upper bound of A(t) together with the relations between phase domains. Second, to make concrete the rule that connects phase-dependent effective quantities with the local phase field and with actual observable quantities. The latter leads directly to the local phase field, boundary form, directional geometry, and actual observation treated in Part 4. This research applies Juridical Structuring Methodology to cosmology, crossing traditional academic boundaries to propose a falsifiable scientific framework through explicit observational links.