Mode Identity Theory

Scaling Law Calculator

A / A_P ≈ C(Θ) · (√Ω)⁻ⁿ

Preset

Θ numerator Θ denominator n numerator n denominator

Scale

Phase Operator C(Θ)

The Mass Formula

m(ρ, σ) = μ_Λ × C_geom(ρ) × (√Ω_Λ)^dist/30 × T²(ρ ⊗ σ)

Four factors. Four sources. Each traced to the postulate. The only operation is multiplication.

Factor	Value	Source
μ_Λ	ρ_Λ^1/4	Vacuum energy floor
C_geom(ρ)	Geometric mean of C(e/D) over Kostant exponents	Phase operator on sunflower
(√Ω_Λ)^dist/30	√Ω_Λ	McKay graph distance
T²(ρ ⊗ σ)	24 vacuum-twisted values	Reidemeister torsion

Mass Calculator

Preset

Irrep ρ Vacuum σ

predicted mass

The 24 Predictions

Formula A applied to 8 nontrivial irreps × 3 vacua. Sorted by predicted mass.

#	ρ	dim	dist	σ	T₃	T²	Mass (GeV)	SM	Ratio

Dead zone: Ranks 4-9 (R₃ at dist 2, R₆ at dist 3) produce masses between 10⁻⁹ and 10⁻⁶ GeV. No known SM fermions occupy this range. Overlaps sterile neutrino and warm dark matter search windows.

Locked Data

C_geom values (Kostant sunflower, computed live from exponents)

Irrep	Spin	D	Kostant exponents	C_geom

Vacuum torsion T²(ρ ⊗ σ) (24 entries)

ρ	T²(triv)	T²(std)	T²(gal)

The Postulate

S¹ = ∂(Möbius) ↪ S³, ∂S³ = ∅

The temporal edge bounds the Möbius surface embedded in hypersphere space. The space has no boundary. Everything derived traces to this topology.

Five Foundations

Foundation	Statement
Wave-matter identity	λ = h/p is identity. Matter is wave, sampled.
Surface origin	Time is phase on the boundary of a 2D manifold.
Topology	ψ(y + L) = −ψ(y). A field traversing the manifold returns to minus itself.
Bounded evaluation	R_H / ℓ_P has definition at the limit ∞/0. The observer is the structural midpoint.
Embedded sampling	The observable's character selects the manifold (n = 1, 2, 3). Dimension follows the measurement.

The Cosmic Standing Wave

Ψ = cos(t/2)

Anti-periodicity requires period 4π. Cosine is inherited because the universe initiates at maximum amplitude Ψ = +1. The present epoch sits at t ≈ 5.22 rad, approximately 2.8 Gyr before turnaround.

The 120 Domain

The binary icosahedral group 2I is the largest exceptional discrete subgroup of SU(2) ≅ S³, with |2I| = 120.

Term	Resolution	Source
120 domain	120 positions	\|2I\| = 120 native to S³
Grid	120 (fermionic)	Half-integer modes on the domain
60R-grid	60 (even only)	\|ψ\|² projects 2I → I

Manifold Assignment

n	Manifold	Observables	(√Ω)⁻ⁿ
1	Temporal edge S¹	H₀, a₀	~10⁻⁶¹
2	Möbius surface	Λ	~10⁻¹²²
3/2	Gauss-Codazzi interface	Gravity
3	Space S³	Null "dark matter"	~10⁻¹⁸³

Derivation Chain

The calculator derives every scale from first principles. No hardcoded outputs.

Quantity	Formula	Value