4 The Conjectured D-ψ-G Trinity

4.1 Three Regimes of the Cost Functional

The exact results of Papers VII and VIII suggest, but do not yet establish, that the cost functional 𝒞_O[γ] naturally decomposes into three regimes:

(5)

The intuition is as follows. Any observer requires well-defined boundaries. The cost of specifying those boundaries depends on the representation:

• D (Discrete): Boundaries are subsets of a bitstring. Cheap but structureless.
• ψ (Spectral): Boundaries are subspaces of a function space. The spectral cost C_ψ suppresses high-frequency (rough) wavefunctions, producing the wave-like behaviour observed at quantum scales.
• G (Geometric): Boundaries are surfaces in a metric space. Closed surfaces in three dimensions admit compact, low-frequency descriptions. The geometric cost C_G is minimised for smooth manifolds.

The claim that these three regimes are forced by the boundary structure of any self-describing observer — rather than chosen by hand — is the key conjecture of this paper. A derivation of the decomposition (5) from equation (1) alone has not yet been carried out.

4.2 Scale Dependence

Under this conjecture, the three regimes dominate at different scales:

The evidence that C_ψ produces quantum mechanical structure is solid (Paper VII). The evidence that C_G produces gravitational structure is solid for toy models (Paper VIII). The claim that the crossover between C_ψ and C_G corresponds to the Planck scale, and that C_D governs the discrete regime, is physically motivated but not yet derived.