Proof Obligations For The Faddeev / Efimov Bridge

The bridge can feed back into the core Spin(2,3) framework only after these gates are completed or explicitly marked as partially complete with limits.

Gate 1: Define The Spin-Derived Operator

Define the SO(2,1) generators and Casimir from Spin(2,3) threshold data without importing the Faddeev answer.

Acceptance:

  • the relevant subgroup is specified;
  • the carrier space is specified;
  • normalization conventions are fixed;
  • the operator is computable in a basis.

Gate 2: Embed Three Transport States Into Faddeev Channel Data

Show that the three near-boundary transport states map to the three Faddeev spectator/pair channels.

Acceptance:

  • the channel basis is explicit;
  • the role of T1 projection is stated;
  • Bethe-Peierls boundary data are matched or identified as an external input;
  • symmetrization for identical bosons is handled without extra fitting.

Gate 3: Compute The Restricted Casimir Matrix

Restrict the Spin-derived SO(2,1) Casimir to the proposed three-state collective subspace.

Acceptance:

  • the resulting matrix is computed;
  • diagonal and off-diagonal terms are derived;
  • the symmetric eigenvalue is written explicitly;
  • no Faddeev kernel value is inserted by hand.

Gate 4: Match Faddeev Kernel Normalization

Compare the restricted Spin-side matrix with the Faddeev 3x3 channel-coupling matrix.

Acceptance:

  • normalization of visible/quaternionic generators is fixed;
  • the Faddeev recoupling constants are recovered or the mismatch is quantified;
  • the Langer shift is accounted for separately;
  • assumptions are recorded in CLAIM_LEDGER.md.

Gate 5: Recover Or Fail To Recover s_0

Determine whether the Spin-side calculation gives the Efimov exponent.

Acceptance:

  • the calculation recovers s_0 ~= 1.00624, or
  • it fails cleanly and identifies which assumption breaks.

Level Rule

Only after Gates 1-5 are satisfied may casimir-faddeev-conjecture.md move from Level 5 to Level 4 or stronger. Partial progress should be recorded as Level 4 only when the assumptions and limits are explicit; theorem-level language requires Level 3 or better.