Step 014 — Four-valued τ-topos

The four boundary atoms now acquire a second role: they classify internal truth.

Book / layer. Book I · E0
Macro step. CS-01 — Build the τ-Kernel
Short routes. https://prrp.site/s14 · https://prrp.site/s014

Status tags. hinge

Construction role

The four boundary atoms now acquire a second role: they classify internal truth. The subobject classifier Ω_τ=B_σ(𝔻)=Neither,True,False,Both turns the boundary algebra into a four-valued internal logic with paraconsistent negation σ and Belnap – Dunn structure. Self-referential propositions stabilize by Cauchy iteration; the value Both is not a metaphorical “both” but the multiplicative unit of 𝔻 – - algebraic identity, not metaphor. Thus the same idempotent algebra that separates boundary channels also carries semantic load: truth, failure, overlap, and underdetermination become internal values rather than external interpretive bookkeeping.

What this step earns

Four-valued τ-topos is made addressable as Step 014 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:252)

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 013 — τ-holomorphy as ω-germ transformer
• Next: Step 015 — Lemniscate characters

Navigate

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