Skip to main content

A registered result in its source context

Original ledger classification: Conformal-symmetry/deck face of SEAM.EQUIV.01 [O] (downstream per v323) + [E] (equivariant uniformisation + the weakening)

Recorded statement: A COROLLARY OF SEAM.EQUIV.01 (no independent open premise; reclassified 2026-06-22, v335): per v323/QGEO.BW.01 it is downstream of SEAM.EQUIV.01 + a rotation-invariant vacuum, and the rotation-invariant vacuum is a conformal-NET AXIOM (a…

Statement, assumptions and sources

Paper context: TFPT — Introduction and reading guide

The exact identifier, original classification and full claim are available below.

The source wording and classification below are preserved as recorded. A paper link, shared script or authored topic relation does not upgrade the claim or report a new proof run. Read later reviewed findings separately.

Read the exact registered claim and its dependencies

QGEO.SYM.01

Original ledger classification: Conformal-symmetry/deck face of SEAM.EQUIV.01 [O] (downstream per v323) + [E] (equivariant uniformisation + the weakening)

Original claim text

A COROLLARY OF SEAM.EQUIV.01 (no independent open premise; reclassified 2026-06-22, v335): per v323/QGEO.BW.01 it is downstream of SEAM.EQUIV.01 + a rotation-invariant vacuum, and the rotation-invariant vacuum is a conformal-NET AXIOM (a chiral net's Moebius-covariant vacuum is the unique invariant positive-energy vector; rotations U(1) < Moebius), so SEAM.EQUIV.01 => QGEO.SYM.01 with NO extra premise -- machine-pinned in Lean as qgeoSymIsCorollary (FORM.QGEO.BW.01). It is therefore NOT a second open bedrock item; the conformal-symmetry/deck face below is its (now corollary) content: the isometry premise QGEO.ISO.01 (v180) is weakened one more genuine step to a CONFORMAL-symmetry premise. (1) EQUIVARIANT UNIFORMISATION (Nielsen realisation; for cyclic groups Kerekjarto + uniformisation): a finite-order orientation-preserving action on a genus-0 surface preserves a complex structure for which it acts by Moebius maps, so a CONFORMAL automorphism -- NOT a full isometry -- already suffices to put the order-4 clock in the form z->iz (the order-4 rotation class is unique up to inverse). (2) STRICT WEAKENING: every isometry is conformal (scale 1) while a conformal map allows any positive scale, so 'conformal automorphism' is strictly weaker than 'isometry' yet (by (1)) still implies the full realisation; hence QGEO.ISO.01 is replaced by the weaker QGEO.SYM.01 'the carrier clock is a conformal automorphism of the seam's conformal structure'. (3) THE CLOCK IS THE mu4 DECK: it is the Coxeter element of the carrier monodromy W(A3)=S4 (v117), order h(A3)=4=|mu4|; a deck transformation preserves the pulled-back conformal structure by definition, so 'the clock is a conformal automorphism' is the DEFINITIONAL property of the mu4 deck once the seam conformal structure is taken to be the mu4-deck structure. (4) BEDROCK [O]: QGEO.SYM.01 reduces to the single identification 'the seam's conformal structure IS the mu4-deck structure of the carrier clock' (the carrier mu4 glue is the conformal monodromy of the seam) -- a definitional/physical statement tying the carrier (P2) to the seam's conformal geometry, NOT a finite computation and NOT reducible further without relabeling. Per v323/QGEO.BW.01 it is DOWNSTREAM of the keystone SEAM.EQUIV.01 (+ a rotation-invariant vacuum, Bisognano-Wichmann/Hislop-Longo), NOT an independent residual: it is the definitional/symmetry face of the single named open arrow, left honestly OPEN only modulo SEAM.EQUIV.01. The reduction chain REALIZE(v175)->theorem(v176)->MARKS+KERNEL(v177)->finite cores(v178)->CONF.01(v179)->ISO.01(v180)->SYM.01(v181) ends here, and the keystone SEAM.EQUIV.01 is the single named open arrow that closes both

Original classification detail

Face of keystone [O]: QGEO.SYM.01 = the definitional conformal-deck face of SEAM.EQUIV.01; per v323/QGEO.BW.01 downstream of SEAM.EQUIV.01 + rotation-invariant vacuum, not an independent residual

Dependencies as recorded

SEAM.EQUIV.01; QGEO.BW.01; QGEO.ISO.01; QGEO.CONF.01; AX.P2.01; v117_monodromy_weyl_a3.py; v180_clock_is_mobius.py

Open the original ledger source, row 261

Location recorded in the ledger

These document locations are recorded directly on the ledger row. They are document level navigation, rather than a newly inferred section citation.

Sections that explicitly cite this claim

A literal claim identifier is present in these paper sections. The section and its premises determine what the citation supports.

All 83 linked sections

Registered verification modules

  • v181_clock_is_conformal_symmetry.pyRegistered verification module · registry
    Read the description recorded in the module registry

    the FINAL honest reduction + the explicit identification of BEDROCK. The isometry premise QGEO.ISO.01 (v180) is weakened one more genuine step — to a CONFORMAL-symmetry premise QGEO.SYM.01 — via equivariant uniformisation (Nielsen realisation, finite cyclic on genus-0): a CONFORMAL automorphism (strictly weaker than an isometry — every isometry is conformal, conformal allows any positive scale) already suffices to put the order-4 clock in the form z↦iz. The clock is the Coxeter element of W(A3)=S4 (v117), order h(A3)=4=|μ4| = the μ4 deck, and a deck transformation preserves the pulled-back conformal structure by definition. So below QGEO.SYM.01 the residual is DEFINITIONAL, not a missing theorem: 'the seam's conformal structure IS the μ4-deck structure of the carrier clock' (the carrier μ4 glue is the conformal monodromy of the seam) — a physical/definitional identification tying P2 to the seam's conformal geometry, NOT reducible further without relabeling. The chain REALIZE(v175)→theorem(v176)→MARKS+KERNEL(v177)→finite cores(v178)→CONF.01(v179)→ISO.01(v180)→SYM.01(v181) has reached bedrock; we stop here honestly.

A registered module is a source reference. Its presence does not assert that the module was executed in this snapshot.

Search later reviewed research for this identifierRead the coverage boundaries