Conjecture: Faddeev Collective Mode as an SO(2,1) Casimir Eigenstate

Claim Level

This file is a conjectural bridge target, not a theorem. It merges the earlier theorem-sketch material with the stronger Faddeev embedding draft, but demotes the claim until the Spin-derived Casimir calculation and the Faddeev kernel normalization are explicitly matched.

Current level: 5, plausible but future work.

Target Claim

The symmetric collective eigenvalue

[ \lambda_{\rm sym} = -s_0^2 - \frac{1}{4}, \qquad s_0 \approx 1.00624, ]

isolated from the three-identical-boson Faddeev channel-coupling problem at unitarity may equal the eigenvalue of an SO(2,1) Casimir restricted to a three-state collective transport mode inside the Spin(2,3) ~= Sp(4,R) framework.

If true, the Faddeev recoupling algebra would be a concrete realization of the threshold SO(2,1) structure already suggested by the Spin(2,3) transport classification.

Inputs Being Compared

On the Spin(2,3) side:

  • Spin(2,3) spinors split as T1 + T2 under J^{01}.
  • Near persistence/locking boundaries, the reduced dynamics linearize as dot R ~= epsilon R.
  • That scaling form suggests an SO(2,1) conformal quantum-mechanics sector.
  • Three near-boundary transport states are proposed to form a collective three-state mode.

On the Faddeev/Efimov side:

  • The wave function decomposes into three pair channels.
  • Bethe-Peierls boundary conditions at unitarity produce a symmetric 3x3 channel-coupling problem.
  • The symmetric channel yields the Efimov exponent through the known transcendental equation.
  • Langer reduction produces the supercritical inverse-square potential.

Proposed Identification

The target identification is:

[ C_{\rm SO(2,1)}\big|{\rm collective} \quad \longleftrightarrow \quad K{\rm Faddeev}^{(3\times 3)}\big|_{\rm symmetric}. ]

In the simplest symmetric channel basis, the restricted object is expected to have the all-to-all form

[ \begin{pmatrix} a & b & b
b & a & b
b & b & a \end{pmatrix}, ]

with symmetric eigenvalue a + 2b. The conjecture is that the Spin-derived values of a and b, after the correct normalization and Langer shift, reproduce -s_0^2 - 1/4.

Assumptions To Make Explicit

  • The three Faddeev channels can be embedded into a three-state transport-sector construction rather than merely compared by analogy.
  • The relevant SO(2,1) subgroup is fixed by the Spin(2,3) threshold dynamics, not chosen after seeing the Faddeev answer.
  • The off-diagonal Faddeev recoupling terms are matrix elements of Spin-derived generators, with no fitted normalization.
  • Bethe-Peierls boundary data can be matched to the proposed T1-like threshold/boundary traversal without changing the physical problem.
  • Finite T1/T2 mixing is a perturbation of the bridge, not part of the unitarity-limit identification.

What Would Promote This

This conjecture becomes a target theorem only if the gates in proof-obligations.md are completed:

  1. define the Spin-derived operator;
  2. embed the three transport states into Faddeev channel data;
  3. compute the restricted SO(2,1) Casimir matrix;
  4. match Faddeev kernel normalization;
  5. recover, or decisively fail to recover, s_0 ~= 1.00624.

Until those steps are done, the safe statement is structural:

The Faddeev/Efimov construction is a promising quantitative test of the Spin(2,3) threshold SO(2,1) idea.