Cantor Transport Exponent

Abstract

The Cantor exponent converts every positive triadic scale to its binary scale and defeats every Lipschitz constant.

Definition 1.1 (Cantor exponent).

Lean statement: D5/S0/Tower/MetricGeometry/CantorTransportExponent.cantorExponent

Formalization. D5/S0/Tower/MetricGeometry/CantorTransportExponent.cantorExponent (✓ std3).

Source. Repository-derived.

Commentary.

The transport exponent is log two divided by log three. Both logarithms are evaluated at positive arguments greater than one, so the quotient does not enter Lean’s totalized nonpositive logarithm branch.

Definition 1.2 (Positive-depth triadic scale).

Lean statement: D5/S0/Tower/MetricGeometry/CantorTransportExponent.triadicScale

Formalization. D5/S0/Tower/MetricGeometry/CantorTransportExponent.triadicScale (✓ std3).

Source. Repository-derived.

Commentary.

At depth Q this source scale is three to the negative Q-plus-one power.

Definition 1.3 (Positive-depth binary scale).

Lean statement: D5/S0/Tower/MetricGeometry/CantorTransportExponent.binaryScale

Formalization. D5/S0/Tower/MetricGeometry/CantorTransportExponent.binaryScale (✓ std3).

Source. Repository-derived.

Commentary.

At depth Q this transported scale is two to the negative Q-plus-one power.

Theorem 1.4 (Exact Hölder conversion and Lipschitz obstruction).

$$\left(0<alpha\wedge alpha<1\right)\wedge \left(\left(\forall Q\in N,\mathrm{rpow}\left(d3\left(Q\right),alpha\right)=d2\left(Q\right)\right)\wedge \left(\forall K\in R,0<K\Rightarrow \left(\exists Q\in N,K\cdot d3\left(Q\right)<d2\left(Q\right)\right)\right)\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/MetricGeometry/CantorTransportExponent.cantor_transport_exponent (✓ std3). ∎

Source. Repository-derived.

Commentary.

The exponent lies strictly between zero and one. At every positive depth, raising the triadic scale to that exponent gives exactly the corresponding binary scale.

The exact scale identity uses Mathlib’s logarithm of a real power and real-power multiplication laws. Positivity of the bases is proved before applying those laws.

For every proposed positive Lipschitz constant K, geometric divergence of three-halves supplies a depth where the binary scale exceeds K times the triadic scale. Thus the exponent change is not merely a symbolic logarithm identity.

This theorem records the metric-scale part of the source claim. It does not assert the separate measure-pushforward statement for the Cantor function.

References

• Truth anchor: D5/S0/Tower/MetricGeometry/CantorTransportExponent.binaryScale
• Truth anchor: D5/S0/Tower/MetricGeometry/CantorTransportExponent.cantorExponent
• Truth anchor: D5/S0/Tower/MetricGeometry/CantorTransportExponent.cantor_transport_exponent
• Truth anchor: D5/S0/Tower/MetricGeometry/CantorTransportExponent.triadicScale