G2-SSM — G₂ Snap as Spontaneous Symmetry Making
| Field | Value |
|---|---|
| Domain | Mathematical Physics / ISA Theory |
| System | β-deformed G₂ spider at β = 1/7 |
| Group | G₂ (broken) → PSL(2,7) (crystallised) |
| H^k tier | H² |
| ISA | Forge (β near 1/7) |
| Status | Theoretical |
| Opcodes | BIND |
| Papers | Papers 515, 572 |
The standard picture and why it is incomplete
Conventional spontaneous symmetry breaking (SSB): a system in a symmetric phase (high temperature, disordered) passes through a critical point and enters a broken phase (low temperature, ordered) with lower symmetry. The order parameter is zero above the transition and non-zero below.
At the G₂ snap β = 1/7, something different happens: a continuous symmetry (G₂) breaks, but simultaneously a discrete symmetry (PSL(2,7)) crystallises — it was not present at generic β and appears only at the critical point and below. The transition is symmetry breaking and symmetry making simultaneously, depending on which symmetry you are measuring.
What actually happens at β = 1/7
At generic β, the β-deformed G₂ spider has:
- A simple representation category (semisimple; all modules decompose into simples)
- The fundamental 7-dimensional simple module V₇ with quantum dimension [7]_β ≠ 0
- G₂-equivariant structure: the full continuous G₂ acts on morphisms
At β = 1/7 exactly ([7]_β = 0):
- V₇ becomes a ghost: it still exists algebraically but has zero quantum dimension (zero trace in the categorical sense)
- The representation category becomes non-semisimple: V₇ has a non-trivial self-extension; the indecomposable projective cover P(V₇) emerges
- P(V₇) is a new object not present at generic β: it is the tilting module at the 7th root of unity
- The quotient category (killing the ghost sector) carries an action of PSL(2,7) ≅ GL(3,𝔽₂) = Aut(Fano plane) ⊂ G₂ that is absent at generic β
From the G₂ perspective: symmetry breaking (the 7-rep collapses). From the PSL(2,7) perspective: symmetry making (a discrete Fano symmetry crystallises).
The order parameter
The order parameter for this transition is [7]_β = sin(7πβ)/sin(πβ) itself:
- Generic β: [7]_β ≠ 0 — G₂ phase, continuous symmetry active
- β = 1/7: [7]_β = 0 — transition point, G₂ collapses, PSL(2,7) forms
- β < 1/7 (toward tropical): [7]_β oscillates through zero — multiple snap events, cascading symmetry-making at each root of unity
This is the ISA analogue of the Landau order parameter, but it measures continuous symmetry content rather than broken symmetry magnitude. Vanishing order parameter = maximum discrete symmetry, not minimum order.
Spontaneous symmetry making (SSM) in the ISA
Paper 515 (Protein Folding ISA) coined SSM for the H¹ → H² transition in protein folding: the native fold creates the folding symmetry group G_fold de novo — it does not break a pre-existing symmetry, it makes a new one.
The G₂ snap at β = 1/7 is the algebraic realisation of the same phenomenon:
| Protein folding (Paper 515) | G₂ snap (Paper 572) | |
|---|---|---|
| Before transition | H¹ unfolded ensemble | Generic β; continuous G₂ |
| Transition | H¹ → H² step | β → 1/7; [7]_β → 0 |
| After transition | G_fold crystallises de novo | PSL(2,7) crystallises |
| Order parameter | Contact map / NOON spectrum | [7]_β |
| Type | SSM (new symmetry created) | SSM (discrete Fano from continuous G₂) |
The key distinction from SSB: in SSB, the symmetry group of the ordered phase is a subgroup of the disordered phase’s symmetry, and it was present (unbroken) in the Hamiltonian all along. In SSM, the symmetry group of the ordered phase is not present at generic coupling — it only becomes visible at (and below) the critical point.
Physical instances across domains
| Domain | SSM transition | Symmetry made |
|---|---|---|
| Chemistry (β=1/7) | FeMoco coherence transition | PSL(2,7) selection rules |
| Chemistry (β=1/5) | Mott transition | ℤ₂ particle-hole symmetry at NOON=1/2 |
| Chemistry (β=1/3) | Spin-crossover | ℤ₂ HS/LS discrete symmetry |
| QEC (β=1/7) | Steane code threshold | PSL(2,7) transversal gate group |
| QEC (β=1/5) | 5-qubit perfect code | ℤ₅ perfect syndrome symmetry |
| Protein folding | Native fold transition | G_fold (Paper 515) |
| Turbulence (speculative) | 3D→2D inverse cascade | Planar G₂ symmetry (Paper 613) |
Connection to the non-associative ladder
The SSM at β = 1/7 is the H² analogue of the H¹ SSM in protein folding. The ISA H^k ladder provides the general framework:
- H⁰ snap: classical phase transition (Ising, Potts) — SSB of discrete symmetry
- H¹ snap: quantum phase transition (Mott, SCO) — SSB of continuous U(1)/SU(2) → SSM of ℤ_n
- H² snap: exceptional phase transition (G₂ → PSL(2,7)) — SSM of discrete Fano symmetry
The H² snap is exceptional in the technical sense: G₂ is an exceptional Lie group (not part of the classical ABCD series), and PSL(2,7) is an exceptional simple group (the second-smallest non-Abelian simple group). Both are related to the Fano plane. The SSM at β = 1/7 is the meeting point of these two exceptional objects.
Part of the ISA Zoo. See also: G2-CHEM snap events in chemistry, G2-QEC snap events in QEC, Paper 572, Paper 515.