Twisted Higher Obstruction Towers for Identity-Eliminator-Conserved n-Truncated Graded Types
Year: 2026
Type: preprint
Abstract: This is Paper 2 in the 20 Paper PHHT Series. This paper constructs the twisted higher obstruction tower for identity-eliminator-conserved n-truncated graded types. Starting from a selected reduced presentation f:XβBπ’β€n, the primary obstruction is the pullback parity class ΟβΛ£ = f*Οβα΅βΏβ±α΅ = [Οβdisp_X] β HΒΉ(X; π½βα΅), so primary vanishing is always interpreted relative to the selected presentation, not as universal vanishing on Bπ’β€n. For each kβ₯2, a partial identity-eliminator-conserved solution through the (kβ1)-skeleton, a cellular local coefficient system π_k, transported identity-eliminator defect data Def_Idβ½α΅βΎ, boundary coherence, and a boundary-compatible filling condition determine a twisted cellular cochain c_k β Cα΅(X;π_k). Boundary coherence makes c_k closed, and lower-dimensional coherence modifications change c_k by a twisted coboundary. Hence Ο_k = [c_k] β Hα΅(X;π_k) is the k-th obstruction class. Vanishing of Ο_k is necessary for extension, and sufficient under the stated boundary-compatible filling hypotheses. The formal choices killing c_k form the corresponding torsor before passage to semantic fillers. The paper develops the tower in absolute, relative, skeletal, gluing, naturality, finite detected cochain-calculation, and minimal-cellular-model forms. It also separates representative-level vanishing, class-level vanishing, nullity data, and realized fillers. Naturality is class-level unless representative and filler transport data are supplied, and minimality is relative to cellular obstruction data realizing the declared classes. The non-collapse theory is given in two forms. Axiomatic cellular non-collapse shows that prescribed closed representatives yield towers with all lower stages zero and a nonzero top class. Semantic non-collapse is realized through central-extension and Postnikov realization packages: the Heisenberg central extension realizes the dimension-two class, while for every kβ₯3 a connected Postnikov stage with Οβ=β€α΅, Οβββ=β€, and primitive k-invariant [u_k]βHα΅(β€α΅;β€) realizes Οβ=β―=Οβββ=0, Ο_k=1βHα΅(Tα΅;β€). Thus primary parity conservation is only the first layer of identity-eliminator conservation; the full theory is a genuinely higher obstruction tower.
Author: Betzer, David
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166533487
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