twoLoopsClass
plain-language theorem explainer
Packages the two-loop exact complex (two self-loops at one vertex, shell signature (2,2,0)) as a point of the exact path class of complexity 2. Gap-2 gravity workers cite it as the concrete quotient class whose self-loop count is invisible to shell signature alone. Construction is a one-line pair: signature plus the global-equivalence class of the labeled complex.
Claim. The exact path class of complexity $2$ contains the pair consisting of shell signature $(V,E,T)=(2,2,0)$ and the global-equivalence class of the labeled exact complex with two self-loops at vertex $0$ (and empty tetrahedron data).
background
The ambient object is the exact complexity shell: for each $n$, the type of combinatorially distinct exact complexes of complexity exactly $n$. Formally it is a dependent sum over shell signatures $s$ of the quotient of the exact labeled class by global equivalence; no bounded-complexity cap appears.
This module banks the enriched-carrier API for the continuum R5 residual (existence of a nonzero oscillatory-tail phase). Route A (eventual mass balance / identical-zero late amplitudes) stalled; route B (Fin-8 signature blocker) was refused. The terminal is route C: sharper typed residual plus characterization lemmas. R5 itself stays open.
The shell signature used here is $(2,2,0)$. The labeled complex places two loops at vertex $0$ with empty tet data. The definition lifts that labeled complex into the exact path class by quotienting under global equivalence.
proof idea
One-line definitional constructor. Pair the shell signature $(2,2,0)$ with the image of the two-loop labeled complex under the quotient map for the exact setoid at that signature. No lemmas or tactics; pure structure introduction.
why it matters
Witness class for the enriched-carrier phase attack below the exact path class. Downstream, the self-loop class tick evaluates to $2$ on this point, and the non-factorization theorem uses it against the two-bridges class of the same signature: any purported shell-signature tick would assign equal values to both, but the self-loop counts differ. That shows the tick uses quotient-internal incidence data, not signature alone.
In the Seven Gaps gravity program this is scaffolding for the continuum oscillatory-tail residual after the Fin-8 signature attack stalled. It does not close gap 2 or inhabit the continuum residual; it supplies the concrete carrier class those characterization lemmas need.
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