Skip to main content

A registered result in its source context

Original ledger classification: Identity [I] (exact matrix + exact group enumeration) + Numerical [N] (ODE identification) + Physical [P] (status)

Recorded statement: THE MONODROMY IS THE WEYL GROUP OF A3 (the U_wall holonomy is an EXACT 24-element representation; the harmonic diagonal becomes a theorem): (1) THE EXACT MONODROMY [I]: M0 = [[0, -(1+i)/2, (1-i)/2], [-(1+i)/2, -i/2, -1/2], [(1-i)/2, -1/2,…

Statement, assumptions and sources

Paper context: Research contracts

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

GATE.UWALL.09

Original ledger classification: Identity [I] (exact matrix + exact group enumeration) + Numerical [N] (ODE identification) + Physical [P] (status)

Original claim text

THE MONODROMY IS THE WEYL GROUP OF A3 (the U_wall holonomy is an EXACT 24-element representation; the harmonic diagonal becomes a theorem): (1) THE EXACT MONODROMY [I]: M0 = [[0, -(1+i)/2, (1-i)/2], [-(1+i)/2, -i/2, -1/2], [(1-i)/2, -1/2, i/2]] (entries in (1/2)Z[i]) is unitary with det 1, tr 0, char poly lam^3 - 1 (the cusp class {1,w,w^2} EXACT), M0^3 = 1, and diag M0 = (0, -i/2, +i/2) - d1 = 0 AND delta = 1/2 are exact matrix entries: the v20 hand value, the v40/v41 observation, the GATE.UWALL.06 torsion value and the GATE.UWALL.07 anchor slice meet in ONE exact matrix. (2) TORSION/FLATNESS [I]: (M0 U)^4 = 1 with tr(M0 U) = 1 (the {1,i,-i} branch, d1 = 0 slice - exactly where GATE.UWALL.06/07 located the physical point) and prod_k U^k M0 U^{-k} = 1 exactly. (3) THE GROUP IS W(A3) = S4 [I]: exact enumeration of <U, M0> gives order 24 with order statistics (1, 9, 8, 6) and character values (3, -1, 0, 1) - uniquely S4 in its standard x sign representation = the (twisted) reflection representation of the WEYL GROUP OF THE FAMILY LATTICE A3: the flavor-wall monodromy realises the A3 factor of the glue D5 + A3 + mu4 as its Weyl group; U = image of a 4-cycle (the mu_4 deck), M0 = a 3-cycle (the family rotation), 24 = |W(A3)| = 4!. (4) IDENTIFICATION [N]: the unitarised ODE monodromy of the exact A0* system equals the exact M0 to 3.5e-10 and the invariant form H is U-invariant to 2.5e-11 (the harmonic frame preserves the equivariance). (5) STATUS [P] (recorded): holonomy VALUES are now [I] (exact properties of the W(A3) representation); only the identification with the analytic monodromy of A0* stays [N] (ODE matching); the delta thread v20 -> v40/v41 -> v114 -> v115 -> v117 is closed end to end

Original classification detail

Theorem: U_wall monodromy group = W(A3) = S4 (order 24 exact); diag M0 = (0 -i/2 i/2) exact - delta = 1/2 and d1 = 0 are matrix entries; only ODE identification stays [N]

Dependencies as recorded

GATE.UWALL.08; GATE.UWALL.07; GATE.UWALL.06; E8.GLU.01 (A3 factor); CONTRACT.U.01

Open the original ledger source, row 201

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.

Navigation through a shared verification script

These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.

All 24 linked sections

Registered verification modules

  • v117_monodromy_weyl_a3.pyRegistered verification module · registry
    Read the description recorded in the module registry

    THE MONODROMY IS THE WEYL GROUP OF A₃ (the U_wall holonomy is an EXACT 24-element representation; the harmonic diagonal becomes a theorem): THE EXACT MONODROMY [I] — M̃₀ = [[0, −(1+i)/2, (1−i)/2], [−(1+i)/2, −i/2, −1/2], [(1−i)/2, −1/2, i/2]] (entries in ½ℤ[i]) is unitary, det 1, tr 0, char poly λ³−1 (cusp class EXACT), M̃₀³ = 1, diag M̃₀ = (0, −i/2, +i/2) ⇒ d₁ = 0 AND δ = ½ are exact matrix entries — the v20 hand value, the v40/v41 observation, the v114 torsion value and the v115 anchor slice meet in ONE exact matrix; TORSION/FLATNESS [I]: (M̃₀U)⁴ = 1, tr(M̃₀U) = 1, ∏ₖ Uᵏ M̃₀ U⁻ᵏ = 1 exactly; THE GROUP IS W(A₃) = S₄ [I]: exact enumeration of ⟨U, M̃₀⟩ gives order 24, order statistics (1,9,8,6), characters (3,−1,0,1) — uniquely S₄ in standard⊗sign = the (twisted) reflection rep of the WEYL GROUP OF THE FAMILY LATTICE A₃: the flavor wall realises the A₃ glue factor as its Weyl group (U = 4-cycle/μ₄ deck, M̃₀ = 3-cycle/family rotation, 24 = |W(A₃)| = 4!); IDENTIFICATION [N]: ODE monodromy of the exact A₀* system = exact M̃₀ to 3.5e−10, H U-invariant to 2.5e−11; the δ thread v20 → v40/v41 → v114 → v115 → v117 is closed end to end

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