TauLib.Denotation.GrowthEscape

Tetration escapes the primorial tower: an independent quantitative argument that tetration values eventually exceed any fixed primorial, making them impossible to capture modularly.

Registry Cross-References

• [I.L02] Growth Escape — growth_escape

Mathematical Content

The primorial tower (M_1 = 2, M_2 = 6, M_3 = 30, M_4 = 210, M_5 = 2310, …) grows polynomially-like, while tetration grows super-exponentially. For any tower depth d, there exists a tetration height c such that 2↑↑c > M_d, meaning the tetration value “escapes” the primorial modulus.

This provides an independent arithmetic reason — beyond the algebraic degradation and channel exhaustion arguments — for why tetration cannot be canonically integrated into the primorial tower framework.

Concrete witness: 2↑↑4 = 65536 > 2310 = M_5.

Tau.Denotation.GrowthEscape.tetration_exceeds_primorial

source theorem Tau.Denotation.GrowthEscape.tetration_exceeds_primorial (d : TauIdx) :∃ (c : Nat), Orbit.tetration 2 c > Polarity.primorial d

For any primorial depth d, tetration 2 eventually exceeds primorial d.

Tau.Denotation.GrowthEscape.growth_escape

source theorem Tau.Denotation.GrowthEscape.growth_escape (d : TauIdx) :∃ (c : Nat), Orbit.tetration 2 c % Polarity.primorial d ≠ Orbit.tetration 2 c

[I.L02] Growth Escape: Tetration escapes the primorial tower.

For any tower depth d ≥ 1, there exists a tetration height c such that 2↑↑c mod M_d ≠ 2↑↑c — the tetration value cannot be represented faithfully within the primorial modulus.

This is the quantitative shadow of saturation: the 4th hyperoperation level produces values that outrun the finite primorial approximations, no matter how deep the tower extends.