Step 019 — Cylinders and ultrametric topology

The local domains of ${\tau }^3$ are not open sets chosen from an external space.

Book / layer. Book II · E0
Macro step. CS-02 — Recover Core Mathematics
Short routes. https://prrp.site/s19 · https://prrp.site/s019

Status tags. formal

Construction role

The local domains of ${\tau }^3$ are not open sets chosen from an external space. They are ABCD-prefix cylinders: refinement classes of address tails. Two points are close when their generated address data agree for a long initial segment; their first-disagreement depth δ(t,t’) defines the ultrametric

d(t,t’)=2^-δ(t,t’).

Thus locality is earned from prefix agreement. The topology is the topology of address resolution, not a topology imposed on an already spatial object.

What this step earns

Cylinders and ultrametric topology is made addressable as Step 019 in the fine-grained Construction Spine routing layer.

• Routing scaffold. Detailed earned-output extraction is pending the source-map pass.

What this step does not yet establish

This route exposes the RC3 construction step for inspection. It does not by itself establish peer review, empirical adequacy, or final manuscript-level source closure.

Source anchors

• Full Construction Spine RC3 whitepaper source (papers/whitepapers/panta-rhei-construction-spine-e0-e3/main.tex:305)

Manuscript extraction state: Detailed source anchors pending extraction.

Anchor classification: whitepaper source resolved; manuscript extraction pending; Registry extraction pending; TauLib extraction pending.

Formalization and verification

Registry links: Registry extraction pending.

TauLib modules: TauLib module extraction pending.

Dependencies

• Previous: Step 018 — τ³ as geometric readout τ¹×_f T²
• Next: Step 020 — Stone/profinite shape and two-readout principle

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• 100-Step Routing Ledger
• Ten-step Construction Spine