🧭 Handoff Document β€” Phase II

Relational Decomposition Framework β†’ New Physics Program

1. Objective

Extend the decomposition inconsistency framework beyond Efimov physics to:

  • identify new universal regimes
  • classify few-body systems structurally
  • generate testable predictions

Core Principle (carry forward)

Physical behavior emerges from consistency constraints across competing decompositions. Universality arises when a scale-independent collective mode forms.

2. What is Already Established (Do NOT redo)

  • 3-body identical bosons β†’ 3 decomposition charts
  • Bethe–Peierls β†’ cross-chart coupling
  • symmetric collective mode β†’ Efimov channel
  • scale independence β†’ (1/\rho^2)
  • only invariant quantity: [ \lambda_{\text{sym}} = -s_0^2 - \frac{1}{4} ]

3. New Program: General Method

Step 1 β€” Identify decomposition space

For a given system:

  • list all valid factorizations / pairings / channels
  • treat them as basis states

Step 2 β€” Build coupling graph

Construct:

  • nodes = decompositions
  • edges = coupling strength (from physics)

This replaces the simple 3Γ—3 matrix.

Step 3 β€” Apply constraints

From underlying physics:

  • symmetry (bosonic, fermionic, mass imbalance)
  • boundary conditions (zero-range, finite-range)
  • kinematics

These determine:

  • which couplings exist
  • whether they are equal / suppressed

Step 4 β€” Look for collective modes

Solve:

  • eigenmodes of coupling structure
  • identify symmetric or dominant collective modes

Step 5 β€” Check scale structure

Key question:

does the effective operator become scale-independent in some regime?

If yes:

  • expect (1/\rho^2)-like behavior
  • candidate for universality

Step 6 β€” Check criticality

Determine if eigenvalue crosses:

[ \lambda = -\frac{1}{4} ]

If:

  • above β†’ no instability
  • below β†’ Efimov-like / new regime

4. First Target Systems (Priority Order)

πŸ₯‡ Target 1 β€” Mass-imbalanced systems

System:

  • 2 heavy + 1 light (HHL)

Known:

  • Efimov scaling changes with mass ratio
  • disappears below threshold

Task

  1. Build decomposition graph:

    • (HH)+L
    • (HL)+H (Γ—2)

  2. Note asymmetry:

    • couplings not equal

  3. Compute:

    • how symmetry breaking affects eigenvalues

Goal

Derive mass ratio threshold for Efimov effect from decomposition asymmetry

Expected payoff

  • structural explanation of known threshold (~13.6 for fermions)
  • possible prediction of new regimes

πŸ₯ˆ Target 2 β€” Fermionic systems

System:

  • identical fermions + one distinguishable particle

Key effect:

  • Pauli exclusion removes some channels

Task

  1. Remove forbidden decompositions
  2. Build reduced coupling graph
  3. Analyze eigenmodes

Goal

Show Efimov disappears because symmetric mode is not allowed

Big insight

universality requires symmetry-compatible collective mode

πŸ₯‰ Target 3 β€” 2D vs 3D

Known:

  • Efimov exists in 3D, not 2D

Task

  1. Rebuild decomposition structure in 2D

  2. Check:

    • scale invariance?
    • kernel structure?
    • eigenvalue behavior?

Goal

explain dimensional dependence via decomposition structure collapse

5. Second Layer (Harder, Higher Impact)

Target 4 β€” 4-body systems

System:

  • 4 identical bosons

Structure:

  • many decompositions:

    • 2+2
    • 3+1
    • etc.

Task

  1. Enumerate decomposition graph
  2. Identify dominant modes
  3. Look for scale-independent sectors

Goal

explain tetramer states as higher-order collective modes

Target 5 β€” Finite-range corrections

System:

  • real interactions with scale (r_0)

Task

  1. introduce scale into coupling
  2. track breakdown of scale invariance

Goal

understand how universality dies

6. Deliverables for Each Case

Each system should produce:

A. Decomposition graph

  • nodes + couplings
  • symmetry constraints

B. Reduced operator

  • matrix or integral form

C. Eigenmode structure

  • symmetric / broken modes

D. Scaling behavior

  • does (1/\rho^2) appear?

E. Critical condition

  • does eigenvalue cross (-1/4)?

F. Physical prediction

  • bound states?
  • scaling law?
  • absence of universality?

7. Minimal Working Example (Template)

For any system:

  1. define decomposition basis
  2. write coupling matrix (symbolic OK)
  3. impose symmetry
  4. diagonalize
  5. identify dominant mode
  6. check scaling
  7. compare to known physics

8. What Counts as Success

You have real new physics if you can:

  • predict when Efimov appears/disappears
  • derive thresholds from structure
  • explain known anomalies
  • identify new scaling regimes

9. Biggest Pitfalls

Avoid:

  • trying to define Ξ΅β‚€, Wβ‚€ uniquely
  • overfitting to known results
  • assuming symmetry where it doesn’t exist

Always:

  • focus on eigenvalues, not components
  • separate representation vs invariant

10. One-Line Program Summary

Use decomposition graphs and their collective modes to identify when scale-independent physics β€” and therefore universality β€” must emerge.

11. Immediate Next Step

Start with:

πŸ‘‰ mass-imbalanced 3-body system

because:

  • simplest deviation from known case
  • already rich physics
  • high chance of publishable result