Executable Cognitive Geometry

This paper develops the current mathematical carrier used by the Prime-Generated Extensible Space (PGES) research programme and sharply separates three layers that earlier drafts partially conflated: carrier mathematics; factor-growth governance; cognitive interpretation. For an arbitrary finite factor-dimension sequence: d₁,d₂,…,dₖ ≥ 2 define: Fᵢ = ℝᵈⁱ and: Hₖ = ⊗ᵢ₌₁ᵏ Fᵢ. Then: dim Hₖ = ∏ᵢ₌₁ᵏ dᵢ. A distinguished unit reference vector uₖ₊₁ in each newly added factor defines conservative extension: ιₖ(x)=x⊗uₖ₊₁ with retraction: ρₖ=id⊗⟨uₖ₊₁,·⟩ satisfying: ρₖ∘ιₖ=id. Thus already represented structure can be embedded exactly while the ambient carrier gains a new orthogonal complement. The PGES lineage specialises this generic construction to the increasing prime sequence: p₁=2,p₂=3,p₃=5,… so: Hₖ = ⊗ᵢ₌₁ᵏ ℝᵖⁱ and: Dₖ = dim Hₖ = ∏ᵢ₌₁ᵏ pᵢ = pₖ#. The resulting primorial sequence: 1,2,6,30,210,2310,… is therefore a special growth schedule, not a theorem establishing prime necessity. The standard tensor basis may be addressed by a mixed-radix tuple. Under the prime specialisation: Qₖ = ℤ₂×ℤ₃×…×ℤₚₖ. CRT additionally supplies a ring isomorphism: Qₖ ≅ ℤ/Dₖℤ because the moduli are pairwise coprime. No current cognitive operator requires that ring structure; CRT is therefore optional structure rather than a demonstrated source of computational advantage. The prime sequence is retained for a narrower architectural hypothesis. If a newly admitted representational factor is required to have an integer dimension that is not itself expressible as a nontrivial tensor-product dimension: d=ab, a,b>1, then prime dimensions are precisely the allowed dimensions, and successive primes give the minimal increasing such sequence. This closes one specific route to covert tensor-product subdivision by dimension arithmetic. It does not prevent direct-sum recoding, subspace decomposition, or semantic reinterpretation. Those require separate governance. Accordingly: the prime skeleton records governed representational births; it does not detect, define, or measure novelty. Whether a new degree of freedom is warranted is determined upstream by lawful re-expression failure. The current P2 form is: β(d,Gₜ,ℛ,c)=max Preserve(d,g) over: g∈Legal(Gₜ,ℛ,c). A finite toy gate now demonstrates genuine search-and-refuse mechanics. Independent validation over 300 random five-vertex graphs agreed with brute-force ground truth in 300/300 cases. Under a deliberately starved search budget, the gate returned UNEVALUATED_REDUCIBILITY 251 times and issued zero wrongful expansion proposals on representable cases. The real Orchard grammar Legal(Gₜ,ℛ,c) remains to be built. Separate companion experiments have also moved the programme beyond basis addressing. Non-basis ensemble states: ψ_q=√p(q) are split by a cross-factor orthogonal admissibility projector into: ψ_ADMIT=P_Aψ and: ψ_HOLD=(I−P_A)ψ. The operation creates genuine cross-factor tensor correlation and commutes with an independently defined semantic split at machine precision. The HOLD complement has further been decomposed by exact failure signature: H_HOLD = ⊕S≠∅ H_S, so the reason a component was held is recoverable by projection rather than added as an external tag. The current evidence therefore supports a more precise research objective than earlier drafts: Can a conservatively extensible structured carrier host cognitive state in such a way that constraints, refusal reasons, provenance, typed uncertainty, and lawful transforms increasingly become constitutive operator geometry rather than metadata bureaucracy beside a payload? No computational advantage, prime-specific performance advantage, native consciousness, or general semantic compiler is established. The paper's contribution is the cleaned mathematical carrier, the explicit separation between generic tensor facts and the candidate prime growth skeleton, and a falsifiable route from representational inadequacy to governed dimensional growth.