🧭 Handoff Document — Completed Work (Phase I)

Relational Decomposition Framework

→ Operator Formulation of Efimov Universality

1. Objective (What has been achieved)

Establish a first-principles, representation-independent formulation of Efimov physics based on:

cross-decomposition consistency and the spectrum of the coupling operator

This replaces:

• coordinate-specific derivations
• scalar decompositions ((\varepsilon_0, W_0))

with:

• operator-level invariants
• collective modes as physical objects

2. Core Principle (Final Form)

Universality arises when zero-range consistency across competing decompositions produces a scale-invariant collective mode of the cross-channel coupling operator, whose induced hyperradial channel is supercritical.

3. What is Fully Established

3.1 Decomposition Structure

• Few-body wavefunction decomposes into pair-spectator charts
• Charts are coupled by Bethe–Peierls boundary conditions
• This induces a linear operator acting across decompositions

3.2 Operator Structure at Unitarity

• Self-coupling vanishes: [ K_{\text{self}} = 0 ]

• Dynamics is entirely cross-channel

• The physical object is: [ \hat K \quad \text{(coupling operator)} ]

3.3 Scale-Invariant Sector

At unitarity:

• no length scale exists
• operator is homogeneous

⇒ admits power-law eigenfunctions: [ t(p) \sim p^{s-1} ]

or equivalently: [ x^\nu ]

3.4 Spectral Condition (Invariant Object)

All physics reduces to:

existence and structure of solutions to [ \hat L, x^\nu = \lambda(\nu), x^{\nu-1} ]

• (\lambda(\nu)) = operator eigenvalue function
• roots determine scaling behavior

3.5 Efimov Condition

Efimov physics occurs when:

• roots become complex: [ \nu = -1 \pm i s_0 ]

• induced potential: [ U_{\text{eff}}(\rho) = -\frac{s_0^2 + \tfrac{1}{4}}{\rho^2} ]

U_{\text{eff}}(\rho) = -\frac{s_0^2 + \tfrac{1}{4}}{\rho^2}

• supercritical condition: [ \lambda < -\frac{1}{4} ]

3.6 Representation Independence

The following are not invariant:

• (\varepsilon_0), (W_0)
• any scalar decomposition of the operator

They depend on:

• coordinate choice
• variable transformations (e.g. Langer correction)

3.7 Invariant Statement

Only the spectrum of the coupling operator is physical.

4. Equal-Mass Case (Benchmark)

Established Results

• symmetric decomposition → single collective mode
• STM kernel derived from kinematic recoupling
• Mellin transform gives transcendental equation

Result:

[ s_0 \cosh!\left(\frac{\pi s_0}{2}\right) ========================================

\frac{8}{\sqrt{3}} \sinh!\left(\frac{\pi s_0}{6}\right) ]

• solution: [ s_0 \approx 1.00624 ]

Structural Interpretation

Efimov effect = symmetric collective mode of a fully connected decomposition graph

5. Heteronuclear (HHL) Case — Completed

5.1 Key Structural Correction

If only heavy–light interactions are resonant:

• system reduces to two channels
• further reduces by symmetry to one amplitude

5.2 Exact Operator

[ \hat L, x^\nu = \lambda_\ell(\nu), x^{\nu-1} ]

• (\lambda_\ell(\nu)) known analytically

• depends on:

  • mass ratio (\alpha = M/m)
  • angular momentum (\ell)
  • statistics

5.3 Bosonic HHL

• symmetric sector allowed
• always admits: [ \nu = -1 \pm i s_0 ]

⇒ Efimov effect exists for all (\alpha)

5.4 Fermionic HHL (Critical Case)

• s-wave forbidden by symmetry
• lowest allowed channel: (\ell = 1)

Efimov condition:

[ \lambda_{\ell=1}(\nu)=0 ]

with:

[ \nu = -1 \pm i s_0 ]

5.5 Threshold

Efimov effect appears when:

[ \frac{M}{m} > 13.6069 ]

5.6 Structural Interpretation

Universality requires a symmetry-compatible collective mode.

• bosons → always allowed
• fermions → only in higher angular momentum
• threshold = emergence of allowed supercritical mode

5.7 Subcritical Regime

Below threshold:

• roots real
• still bound states exist (Kartavtsev–Malykh)

⇒ same operator, different spectral regime

6. Unified Interpretation

All cases fit:

7. What This Framework Adds

New Contributions

1. Operator-level formulation

• replaces coordinate-dependent derivations

2. Decomposition viewpoint

• universality = failure of simultaneous chart consistency

3. Spectral classification

• systems classified by (\lambda(\nu)), not wavefunctions

4. Symmetry-driven universality

• explains when Efimov is allowed or forbidden

8. Minimal Working Procedure (Reusable)

For any system:

Step 1

Define decomposition channels

Step 2

Construct coupling operator

Step 3

Apply symmetry/statistics constraints

Step 4

Reduce to minimal sector

Step 5

Solve: [ \hat L x^\nu = \lambda(\nu)x^{\nu-1} ]

Step 6

Check:

• real roots → no Efimov
• complex roots → Efimov
• critical transition → threshold

9. What NOT to Do

Avoid:

• interpreting (\varepsilon_0, W_0) as physical invariants
• relying on specific coordinate systems
• treating matrix elements as fundamental objects

Always:

focus on operator spectrum

10. One-Line Summary

Efimov universality is the appearance of a supercritical, scale-invariant eigenmode of the symmetry-reduced cross-channel coupling operator.

11. Immediate Continuation (Phase II)

Best next directions:

1. Mixed interaction graphs

→ minimal conditions for universality

2. Dimensional crossover (2D/3D)

→ why operator never becomes supercritical in 2D

3. 4-body systems

→ higher-order decomposition graphs

4. Finite-range deformation

→ how scale invariance breaks

12. Status

✅ Operator formulation complete ✅ Equal-mass derivation complete ✅ HHL bosonic + fermionic structure complete ✅ Threshold mechanism derived and interpreted

🚧 Ready for generalization (Phase II)