TauLib.Kernel.Diagonal

The Diagonal Discipline: why exactly 4 orbit channels, and why the iterator ladder saturates at tetration (level 3).

Registry Cross-References

• [I.D03] Diagonal Discipline — diagonal_discipline

Mathematical Content

The free diagonal on {α, π, γ, η} × {α, π, γ, η} has 12 off-diagonal pairs (g, h) with g ≠ h. These 12 pairs organize into 3 rewiring levels:

• Level 1 (addition): consumes the π channel

• Level 2 (multiplication): consumes the γ channel

• Level 3 (exponentiation): consumes the η channel

The α channel is the counting scaffold (τ-Idx) and cannot be consumed. Since K6 closes the universe at exactly 4 non-omega generators, no 4th rewiring level exists. The iterator ladder saturates at 3 rewirings (4 operations: ρ, +, ×, ^).

Tau.Kernel.diagonal_channel_count

source theorem Tau.Kernel.diagonal_channel_count :nonOmegaGenerators.length = 4

[I.D03] There are exactly 4 non-omega generators. This is the source of the diagonal discipline: 4 generators yield 3 rewiring levels.

Tau.Kernel.nonOmega_complete

source **theorem Tau.Kernel.nonOmega_complete (g : Generator)

(hg : g ≠ Generator.omega) :g ∈ nonOmegaGenerators**

The complete list of non-omega generators is [α, π, γ, η].

Tau.Kernel.solenoidalGenerators

source def Tau.Kernel.solenoidalGenerators :List Generator

The 3 solenoidal generators that serve as rewiring targets. α is the counting scaffold and is NOT a rewiring target. Equations

• Tau.Kernel.solenoidalGenerators = [Tau.Kernel.Generator.pi, Tau.Kernel.Generator.gamma, Tau.Kernel.Generator.eta] Instances For

Tau.Kernel.solenoidal_count

source theorem Tau.Kernel.solenoidal_count :solenoidalGenerators.length = 3

Exactly 3 solenoidal generators.

Tau.Kernel.solenoidal_ne_alpha

source **theorem Tau.Kernel.solenoidal_ne_alpha (g : Generator)

(hg : g ∈ solenoidalGenerators) :g ≠ Generator.alpha**

The solenoidal generators are distinct from α.

Tau.Kernel.solenoidal_ne_omega

source **theorem Tau.Kernel.solenoidal_ne_omega (g : Generator)

(hg : g ∈ solenoidalGenerators) :g ≠ Generator.omega**

The solenoidal generators are distinct from ω.

Tau.Kernel.rewiring_levels_eq_solenoidal

source theorem Tau.Kernel.rewiring_levels_eq_solenoidal :solenoidalGenerators.length = nonOmegaGenerators.length - 1

[I.D03] The diagonal discipline: exactly 3 rewiring levels exist because exactly 3 solenoidal generators are available as targets. Each rewiring level consumes one generator:

• Level 1 (addition) ↔ π

• Level 2 (multiplication) ↔ γ

• Level 3 (exponentiation) ↔ η No 4th level: α is the counting scaffold, ω is the beacon.

Tau.Kernel.alpha_unique_scaffold

source theorem Tau.Kernel.alpha_unique_scaffold :¬Generator.alpha ∈ solenoidalGenerators ∧ Generator.alpha ≠ Generator.omega

Alpha is the unique non-omega, non-solenoidal generator: the counting scaffold.