PostingEnrichedPathClass
plain-language theorem explainer
Exact path classes of complexity n, product-enriched by an external Fin-8 posting-phase coordinate from the RS eight-tick transaction cocycle. Gap2 carrier work cites this as the honest shell-plus-phase type on which forgetful descent and non-descent theorems live. Pure structure definition: underlying exact path class paired with a Fin 8 phase.
Claim. For each $n \in \mathbb{N}$, a posting-enriched path class is a pair $(c,\phi)$ where $c$ is an exact path class of complexity $n$ (combinatorially distinct exact complexes of shell complexity exactly $n$) and $\phi \in \mathrm{Fin}\,8$ is an adjoined eight-tick posting-phase coordinate.
background
The exact complexity shell ExactPathClass n is the set of combinatorially distinct exact complexes of complexity exactly $n$: a disjoint union over shell signatures of the quotient of the exact labeled class by global equivalence. It carries only incidence data (edge and tetrahedron vertices), not transaction state.
Recognition Science forces an eight-tick octave (T7, period $2^3$). The eight-tick phases are $k\pi/4$ for $k=0,\ldots,7$. The RS recognition-posting cocycle already exists as a period-8 transaction on a different carrier (PairKernel phase-bearing transaction residual). Certified Gap2 Fin-8 phase close wants a Fin-8 tick on exact path classes; those live on different carriers.
This module implements the carrier half of the Gap2 posting-cocycle design: product enrichment adjoining an external Fin-8 posting phase, rather than inventing an incidence hash or fake descent from the shell alone.
proof idea
No proof body: a structure definition. Two fields only. The first is the underlying exact path class of complexity $n$. The second is a bare Fin 8 posting-phase coordinate. Accessors and the forgetful map are defined separately by projection onto each field.
why it matters
This is the carrier type for the Gap2 posting-cocycle half of D-qg-gap2-posting-cocycle. Downstream, the forgetful map drops the phase and retains the exact path class; the phase reader returns the adjoined coordinate. The theorem that any family factoring through forget is constant in the phase, and the theorem that the phase coordinate itself does not factor through forget, both quantify over this type. Together they close the STOP A residual (carrier forgets posting phase) and refuse a fake product-section cocycle.
Framework landmark: T7 eight-tick octave. The enrichment is the honest product of the Gap2 shell carrier by the transaction phase type. An open residual remains: the non-forgetful GE-invariant antipodal bridge still required for certified close. Does not flip the continuum-and-measure gap flag.
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