Four-valued τ-topos

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.