Homotopical Apophasis: Bilattice Symmetry, the Progenitor as Eilenberg–MacLane Space, and the Shadow of Classical HoTT

Changes Brought in the current June 2026 version: 1. Correction of the Bilattice Symmetry Group The previous draft erroneously claimed that the progenitor $𝐀₀$ was defined by the full automorphism group $𝐀𝐮𝐭(𝐋) ≅ 𝐙/𝟐𝐙$. This has been mathematically corrected. Trivial Designation-Preserving Group: The manuscript proves that the designation-preserving subgroup is actually trivial: $𝐀𝐮𝐭_𝐝𝐞𝐬(𝐋) = 𝟏$. Label-Level De Morgan Symmetry: The correct group defining the progenitor is the label-level De Morgan symmetry group $𝐒𝐲𝐦_𝐃𝐌^𝐥𝐚𝐛(𝐋) = \{𝟏, \sim\} ≅ 𝐙/𝟐𝐙$, generated by the coordinate swap $\sim(𝐚, 𝐛) = (𝐛, 𝐚)$ which fixes the glut value $𝐛 = (⊤, ⊤)$ but fails to preserve general designated values. 2. Strict Valuation Obstruction and Passive Metadata The prior draft treated paths as natively equipped with bilattice valuations. The new version proves this is topologically impossible without destroying identity elimination ($𝐉$). Strict Valuation Obstruction (Theorem 5.1): It formally proves that no strict function $𝐯 : (𝐱₀ = 𝐱₀) → 𝐋$ can satisfy $𝐯(\mathbf{refl}) = 𝐭$ and the multiplicative rule $𝐯(𝐩 · 𝐪) = 𝐯(𝐩) \wedge_𝐋 𝐯(𝐪)$ while simultaneously mapping a loop $𝛚$ (where $𝛚 · 𝛚 = \mathbf{refl}$) to the glut value $𝐛$. This would force $𝐛 \wedge_𝐋 𝐛 = 𝐭$, which contradicts the bilattice structure. Passive Metadata Layer: Consequently, labels are strictly decoupled from identity types. They are now defined entirely as passive metadata on displayed path representatives $\mathbf{Rep}_𝐀(𝐱, 𝐲)$, leaving the underlying HoTT paths and the $𝐉$-eliminator completely unchanged. 3. The Universal Involution Field The definition of the progenitor has been generalized and embedded into a larger universe of involutions. Universal Involution Field (Theorem 4.2): The manuscript defines $𝐀_𝐭𝐨𝐭 := \sum_{𝐗:𝐔} \sum_{𝐞:𝐗 ≃ 𝐗} (𝐞 ∘ 𝐞 ≃ \mathbf{id}_𝐗)$, classifying all type-level involution data. Centralizer Component: It proves that the underlying progenitor $𝐀₀ \simeq 𝐁(𝐙/𝟐𝐙)$ is exactly the 1-truncated connected component of $𝐀_𝐭𝐨𝐭$ at the point $(𝟐, \mathbf{swap}, \mathbf{swap}² = \mathbf{id})$, identifying $𝐀₀$ as the delooping of the centralizer $𝐂_{\mathbf{Aut}(𝟐)}(\mathbf{swap})$. 4. Componentwise TAE Collapse (Replacing Global Duality) The broad "Duality Theorem" from the old draft, which supposedly collapsed all TAE types to $𝟏$, has been refined into a strict, componentwise 0-truncated HIT quotient. Generated Non-Real Quotient (Theorem 7.1): It defines the effective set-quotient $𝛒(𝐀) := 𝐀 / 𝐑̂_𝐀$, where $𝐑̂_𝐀$ is the equivalence relation generated by representatives carrying non-real labels ($𝐛$ or $𝐧$). Collapse Criterion: It proves the collapse $𝛒(𝐀) \simeq 𝟏$ holds if and only if $𝐑̂_𝐀$ is total on an inhabited $𝐀$. This isolates the classical point-generation boundary and prevents unwarranted global collapses. 5. Shadow Determinization as Orbit Closure The "Shadow HoTT" framework has been mathematically formalized into finite orbit mechanics (Section 8). Shadow Closure Formula: Determinization at the shadow metadata level is now given the explicit closure formula $\mathbf{Det}^𝐬𝐡(𝐓) = 𝐓 \cup 𝛕(𝐓)$, where $𝛕$ is the nontrivial $𝐙/𝟐𝐙$-action. Boolean Orbit Lattice: It proves that the unit-fixed subobjects of this monad exactly form the Boolean orbit-power-set lattice $𝐁_𝐧$, cementing the poset isomorphism $𝐁_𝐧 \simeq \mathbf{Fix}(\mathbf{Det}^𝐬𝐡)^{≤𝐧}$ for finite free-action shadow TAE types. 6. Lindenbaum Homomorphism Obstruction & Torsor Requirement The prior draft claimed a direct classifying geometric morphism from Gödelian undecidability. The June 2026 version identifies this as a category leap and strictly bounds it. Boolean-Homomorphism Obstruction (Theorem 9.1): It proves that the bilattice status valuation $𝐯_𝐓([\varphi]) = 𝐛$ for an independent sentence $𝐆$ cannot be extended to a complement-preserving Boolean homomorphism $𝐡 : \mathbf{Lind}(𝐓) → 𝐋$. Torsor-Data Boundary: Consequently, while independence yields a clopen Stone resolution split $\mathbf{Stone}(\mathbf{Lind}(𝐓)) = 𝐔_𝐆 \sqcup 𝐔_{\neg 𝐆}$, the classifying morphism to $\mathbf{Set}^{𝐙/𝟐𝐙}$ requires the explicit external injection of genuine principal $𝐙/𝟐𝐙$-torsor data $𝐏_𝐆$.