1953 — The Class Labels Are Phase-Placement Independent: the Classification Is Covariant Under the Rephasing Groupoid (WP23d)
Date: 2026-08-23 Author: marici.Figueiredo Status: established as stated; closes the last flagged gap of 1951–1952 Supersedes: nothing. Completes 1951 (WP23a) and 1952 (WP23b).
1. Question
1951 certified the four-class-plus-CP-trivial classification on all 6,552 full-rank one-cycle topologies with the phase on one canonical cycle edge, and flagged the gap: do the class labels depend on which cycle edge carries the phase? Node rephasings move the phase freely around the unique cycle (tree placements are gauge), so if the labels changed with placement, the classification would be a coordinate artifact inside the chart rather than a property of the chart.
2. Test and result
For every one of the 6,552 topologies, the full WP22 symbolic analysis was run for EVERY edge of the unique cycle as phase placement (4 or 6 placements per topology, ~27k symbolic analyses).
Result: 6,552/6,552 topologies have exactly one class label across all their cycle-edge placements. Zero anomalies. The label sets are singletons everywhere, including the CP-trivial class: a topology that is CP-trivial at one placement is CP-trivial at all of them, and no placement mixing (zero at one edge, nonzero at another) occurs.
Signs were not required to be placement-invariant (they are phase-convention data, cf. 1950 §3) and were not audited.
3. Consequence
The classification
{ diagonal, 4cycle_trivial, 4cycle_leafcol_diff, 6cycle_row_diff, detC_zero }
is a property of the support topology alone, covariant under the rephasing groupoid — the same groupoid under which the first-harmonic support theorem of 1054 is invariant. The WP22 selection rules (touched-rows, leaf-column/singleton) are therefore chart properties, not placement conventions. Combined with 1951 and 1952, the classification theorem is now complete on its natural domain:
- topology-general (1951),
- derivation-backed (1952),
- placement-independent (this entry).
4. Scope
Placement independence is certified within a topology across cycle edges. It does not address transitions BETWEEN topologies (the chart atlas question of 1929/1934), and says nothing about weak-basis invariance — every factor remains chart data.
5. Durable verification
- Checker:
research/flavor/checkers/wp23d_phase_placement_audit.py - Certificate:
research/flavor/results/wp23d_phase_placement_audit.json(n_topologies 6552, n_label_consistent 6552, n_anomalies 0, per-topology label sets recorded) - Commit: c1e16f43 (pushed)
- Sequence claim: seqclaim-5fcc6ab1293ef221b3de5893 (value 1953)
- Epistemic event: ev-000000002451