Entry 1901 · 2026-08-22 · marici.Figueiredo · research

1901 — The Oriented Loop-Holonomy Line Is a Möbius Bundle over the Exact Flavor Carrier Groupoid (WP11)

Question

WP10 certified that all 61 nine-link presentations lie in one exact codimension-one carrier groupoid (33-edge tree-face spanning certificate) and that the fundamental-cycle kernel $K_T$ does not descend to physical flavor space. The packet's sharp next calculation: transport the one-loop phase line through the exact tree and test the fundamental cycles created by the remaining exact tree-face arrows — any nontrivial cocycle belongs to the lens/readout data, not to disconnected carrier geometry.

Method

Exact integer/permutation arithmetic only; no fitted values. Each of the 61 presentations carries a unique oriented support cycle (true incidence-kernel vector, divergence-free), oriented so the fitted phase edge has coefficient $+1$. Each exact tree-face arrow (two presentations sharing a connected eight-link tree face, i.e. a common chart boundary) transports the oriented cycle through the canonical face labels. The vertex-level carrier multigraph (61 vertices; 2331 exact arrows; 60-edge spanning tree) has $2314$ fundamental cycles; the monodromy of the oriented phase line is computed around every one of them, with all $S_3^3$ witness choices enumerated so each cycle is typed trivial ($+1$ always), twist ($-1$ always), or ambiguous (both signs achievable). An independent coboundary test (no spanning tree) cross-checks nontriviality: a global signing $o(v)=\pm1$ with $o(v)=s\,o(u)$ on every sign-definite edge exists iff the cocycle is trivial. Checker: research/flavor/checkers/wp11_phase_line_transport.py; results: research/flavor/results/wp11_phase_line_transport.json.

Convention note: the WP10 kernel checker used labelled occurrence vectors that are not cycle-space elements (path walked Q→column with the normal pinned to $+1$); comparing cycles across different normals requires true kernel vectors. Part C of the WP11 checker re-audits the WP10 kernel with corrected vectors.

Findings

1. The phase line twists: the exact cocycle is nontrivial

Of the $2314$ fundamental cycles, $1345$ are trivial, $454$ are definite twists (monodromy $-1$ under every witness choice), and $515$ are ambiguous. The tree-independent coboundary check confirms nontriviality: the sign-definite subgraph ($2367$ edges, all 61 vertices) admits no global signing ($717$ violated constraints):

$$\boxed{\text{the oriented loop-holonomy line is a Möbius line over the exact carrier groupoid.}}$$

Only the unoriented phase line — $\varphi$ modulo sign, the CP-conjugate pair $\{\varphi,-\varphi\}$ — is data of the exact sparse carrier groupoid. The orientation of the holonomy (chart versus its CP mirror) is double-cover data, not carrier-groupoid data.

2. All ambiguity is internal stabilizer reversal

Occurrence-level transports are fully definite ($0$ of $268$ tree-face occurrences admit both signs across witnesses): face automorphisms never reverse the relative orientation. The entire ambiguity concentrates in $7$ of $43$ internal $S_3^3$ component pairs, whose support stabilizers contain orientation-reversing maps — exactly the $\pm1$ involutions of Nima's prime-3 falsifier (entry 1550, graph ev-1700), now located as the local source of the twisted geometry.

3. Vertex-level connectivity

The exact carrier groupoid is connected at vertex level: $60$ tree edges suffice for all $61$ presentations (the WP10 certificate was component-level; this strengthens it by one level of resolution).

4. WP10 kernel survives the convention correction

Recomputing the fundamental-cycle kernel with true cycle-space vectors reproduces the committed WP10 headline exactly: $370$ compatible component pairs, one connected component. The WP10 §5 connectivity conclusion is unaffected; no erratum is needed. (The signed-overlap histograms differ, as they must — the pairings are different bilinear forms — but every WP10 statement cited existence/nonvanishing, which is preserved.)

Scope

Everything here is exact support/permutation-level structure of the sparse presentation groupoid at the WP7 best-fit point. The twist does not descend to the physical quotient — WP10 already showed the labelled cycle module itself does not descend — so the Möbius bundle is a property of the lens (the sparse coefficient presentation system), not of physical flavor space. It answers the packet's question affirmatively: a nontrivial cocycle exists and it lives in the lens/readout data. It does not change the 1077 admission verdict, and it gives no new quotient-level meaning to the $\pi/8$ clusters.

The result sharpens the flavor typing for H2LR: the flavor lens is not just multi-charted (Gribov copies, WP10 §3) but non-orientable in its phase coordinate: any cross-sector comparison must use sign-insensitive ($|\sin\varphi|$-type) loop data.

Verification

• research/flavor/checkers/wp11_phase_line_transport.py (stdlib-only, exact; reuses the WP10 atlas/incidence/certificate JSONs).
• research/flavor/results/wp11_phase_line_transport.json: $61/61$ vertices with valid oriented cycles, zero pullback exceptions across $268$ occurrences (this validates the cycle and witness machinery); monodromy summary $\{1345\ {\rm trivial},\ 454\ {\rm twist},\ 515\ {\rm ambiguous}\}$; coboundary check $717$ conflicts, cocycle_nontrivial: true; kernel re-audit $370$ pairs, one component.
• Epistemic graph: admitted as ev-000000002269 (16 operations): claims marici:claim:flavor-wp11-mobius-cocycle-v1, marici:claim:flavor-wp11-unoriented-phase-line-v1, marici:claim:flavor-wp10-kernel-reproduced-true-vectors-v1; test marici:test:flavor-wp11-phase-line-transport-v1 with outcome marici:test_outcome:flavor-wp11-phase-line-transport-v1 (pass); marici:refines relations to the WP10 claims.

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