A registered result in its source context
Original ledger classification: Identity [I] (character algebra + Schur rigidity + grading transport) + Physical [P] (one Z_3 choice)
Recorded statement: THE CANONICAL MAP EXISTS AND IS RIGID (the GATE.QGEO naturality residue collapses to ONE Z_3 choice): (1) THREE mu_4 ACTIONS ONE MATCH [I]: the deck U = diag(1,i,-i) has characters (0,1,3) (invariant line, misses 2 - cannot match H^1); the…
Statement, assumptions and sources
Paper context: Standard Model and flavor structure
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FLAV.QGEO.05
Original ledger classification: Identity [I] (character algebra + Schur rigidity + grading transport) + Physical [P] (one Z_3 choice)
Original claim text
THE CANONICAL MAP EXISTS AND IS RIGID (the GATE.QGEO naturality residue collapses to ONE Z_3 choice): (1) THREE mu_4 ACTIONS ONE MATCH [I]: the deck U = diag(1,i,-i) has characters (0,1,3) (invariant line, misses 2 - cannot match H^1); the sheet-twisted deck -U has (2,3,1) and the cusp exponential i^{Q+} has (1,2,3) - both with character SET {1,2,3} = the H^1 set (FLAV.QGEO.04): only the SIGN-TWISTED actions match, the same sheet twist that builds the v118 hexagon (FLAV.H2.02). (2) RIGIDITY (Schur) [I]: both sides are multiplicity-free mu_4-modules with equal character sets, so the equivariant isomorphism exists and is unique up to one scalar per character (commutant = diagonal algebra); the explicit map is the character-matching permutation - the canonical map is WELL-DEFINED; existence and uniqueness are no longer open. (3) GRADING TRANSPORT [I]: under the i^{Q+}-equivariant map the H^1 Euler degree pulls back to Q+ itself (assignment (1,2,3)); under the -U-map to Q+ o rho with rho = the (123) family rotation (assignment (2,3,1)) - the two candidate actions differ by EXACTLY one Z_3 rotation. (4) RESIDUE [P] (recorded): GATE.QGEO collapses from 'prove naturality' to ONE discrete choice - which sign-twisted action is the geometric boundary deck on generation space (-U monodromy sheet twist vs i^{Q+} cusp exponential)
Original classification detail
Identity: only sign-twisted actions match H^1; equivariant iso Schur-rigid; i^Q+ assignment = Q+ and -U = Q+ o rho; residue = one Z_3 choice
Dependencies as recorded
FLAV.QGEO.04; FLAV.H2.02 (sheet twist); GATE.UWALL.09 (U); GATE.QGEO.01
Open the original ledger source, row 225Location recorded in the ledger
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- Standard Model and flavor structure
- Research contracts
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.
- 2026-06-24 (The [formula in source] dual keystone + the flavor residuals it homes: v405– v407)Research changelog; TFPT — Introduction and reading guide
- The Sheet Diamond and the Winding Line: masses and transport from one surfacetfpt 2 standard model
- The unique attractor: a gapped transport selects ONE fixed pointorigin theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- Front matter / supportStandard Model and flavor structure
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- 2026-06-12Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Registered verification modules
- v140_canonical_map.pyRegistered verification module · registry
Read the description recorded in the module registry
THE CANONICAL MAP EXISTS AND IS RIGID (the GATE.QGEO naturality residue collapses to ONE ℤ₃ choice): THREE μ₄ ACTIONS, ONE MATCH [I]: the plain deck U = diag(1,i,−i) has characters (0,1,3) — an invariant line, missing 2 — and CANNOT match H¹; the sheet-twisted deck −U (characters (2,3,1)) and the cusp exponential i^{Q₊} (characters (1,2,3)) both carry the character SET {1,2,3} = the H¹ set (v137) — only the SIGN-TWISTED actions match, by the same sheet twist that builds the v118 hexagon; RIGIDITY (Schur) [I]: both sides are multiplicity-free μ₄-modules with equal character sets, so the equivariant isomorphism exists and is unique up to one scalar per character (commutant = diagonal algebra) — the canonical map is WELL-DEFINED, existence and uniqueness are no longer open; GRADING TRANSPORT [I]: under the i^{Q₊}-equivariant map the H¹ Euler degree pulls back to Q₊ itself (assignment (1,2,3)); under the −U-map to Q₊∘ρ with ρ = the (123) family rotation (assignment (2,3,1)) — the two candidate actions differ by EXACTLY one ℤ₃ rotation; RESIDUE [P]: GATE.QGEO collapses from 'prove naturality' to ONE discrete choice — which sign-twisted action is the geometric boundary deck on generation space (−U monodromy sheet twist vs i^{Q₊} cusp exponential) (the choice since derived, v141)
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