G2-CHEM — G₂ Snap Events in Chemistry

Field	Value
Domain	Chemistry / Condensed Matter
System	β-deformed G₂ spider at roots of unity
Group	G₂ → PSL(2,7) at β=1/7
H^k tier	H²
ISA	Forge (β near 1/3, 1/5, 1/7)
Status	Predicted
Opcodes	BIND · ORBIT · TWIST
Papers	Papers 488, 491, 563, 572

Overview

The β-deformed G₂ spider has quantum dimensions [n]_β = sin(nπβ)/sin(πβ). At β = 1/3, 1/5, 1/7, the dimensions [3], [5], [7] vanish in sequence — snap events where one interaction channel freezes out. Each snap corresponds to a real chemical phase transition.

The cascade runs top-down: from the highest-dimensional channel (7) to the lowest (3), as β decreases from 1/3 toward 0 (the tropical/classical limit).

The three chemical snap events

β = 1/3 → spin-crossover critical point

What collapses: [3]_β = 0 — the rank-1 (triad) sector of G₂. In R3, the (*φ)-coefficient vanishes: the three-body exchange channel that couples triad angular momenta switches off.

Chemical identity: the spin-crossover (SCO) critical point in transition-metal complexes. At SCO, spin-orbit coupling that mediates H¹ exchange between d-electrons decouples: the high-spin (HS) to low-spin (LS) transition occurs because the triad exchange term reaches zero. Papers 488/491 identify SCO as a β* snap; the G₂ spider now specifies β* = 1/3.

Prediction: SCO compounds have β_eff ≈ 1/3. The Weyl c₂ parameter (measurable from CASSCF NOONs, Paper 596) should peak at β_eff = 1/3, not at a generic β. Compounds with β_eff further from 1/3 should show less sharp SCO transitions.

β = 1/5 → Mott metal-insulator transition

What collapses: [5]_β = 0 — the bigon self-composition vanishes: BIND∘BIND† = [5]·id → 0. A BIND pair can no longer self-compose — double occupancy costs zero extra energy to break.

Chemical identity: the Mott metal-insulator transition. The Mott condition U = W (on-site Coulomb repulsion equals bandwidth) is exactly the condition that double occupancy is energetically neutral — neither favoured nor penalised. Paper 563 (experiment x563c) found the Mott β* snap at U/t ≈ 1.8; the G₂ spider identifies this as β* = 1/5.

Prediction: at the Mott critical point, NOONs (natural orbital occupation numbers) equal exactly 1/2 — the maximally mixed state. This follows from BIND self-annihilation: when BIND∘BIND† = 0, neither the doubly-occupied nor the empty orbital configuration is preferred, so the NOON splits exactly at 1/2. This is a quantitative, testable prediction distinguishing the Mott transition from other strongly-correlated crossovers.

β = 1/7 → FeMoco coherence transition / PSL(2,7) crystallisation

What collapses: [7]_β = 0 — the full Fano loop evaluation vanishes. Classical counting of 7 Fe spin configurations gives trace = 0: the 7-dimensional continuous G₂ representation collapses.

What emerges: the discrete symmetry PSL(2,7) ≅ GL(3,𝔽₂) = Aut(Fano plane) ⊂ G₂ crystallises in the quotient category. This is spontaneous symmetry making (SSM): the continuous G₂ breaks, the discrete Fano symmetry forms.

Chemical identity: the FeMoco quantum coherence transition (Paper 488). At β = 1/7, the 7-iron FeMoco cluster transitions from a classical magnetic configuration (describable by pairwise Heisenberg exchange, H¹) to a PSL(2,7)-symmetric quantum state where all 7 Fe-Fe exchange paths are maximally entangled. This is the regime where FeMoco is “most quantum” — where room-temperature quantum coherence is possible.

Broader prediction: any 7-centre cluster with Fano connectivity (7 metal atoms, exchange paths along the 7 Fano lines) should show:

A sharp EPR or Mössbauer anomaly near β_eff = 1/7
PSL(2,7) selection rules in spectroscopic transitions (forbidden lines become allowed; allowed lines split into PSL(2,7) multiplets)
Anomalous magnetic susceptibility not explainable by pairwise (H¹) exchange

The cascade as a chemical phase diagram

β = 1/3   Spin-crossover        [3]=0   Triad exchange freezes → HS/LS transition
β = 1/5   Mott transition       [5]=0   BIND self-pair degeneracy → NOON = 1/2
β = 1/7   Fano coherence        [7]=0   G₂ → PSL(2,7)  → room-T quantum coherence
β → 0     Classical limit        [n]→n   Fano combinatorics; DFT works

Moving from β = 1/3 toward β = 0 (increasing temperature or decreasing correlation strength) passes through the Mott transition and then the FeMoco point. Moving from β = 1/3 toward β = 1/7 (decreasing temperature or increasing correlation) passes through increasing quantum coherence until the Fano symmetry crystallises.

ISA programme

INIT:    LABEL[n_d electrons in d-manifold]     -- set up d-electron register
SCREEN:  ORBIT[DFT/HF ground state]             -- H0 reference
CORR:    BIND[Weyl c2 diagnostic]               -- is H2 present?
SNAP?:   LABEL[beta_eff from NOON spectrum]     -- which snap are we near?
SCO:     LABEL[beta near 1/3 → HS/LS boundary] -- spin-crossover
MOTT:    LABEL[beta near 1/5 → NOON near 1/2]  -- Mott criticality
FANO:    BIND[PSL(2,7) selection rules]         -- FeMoco / 7-centre clusters
OUTPUT:  LABEL[phase + predictions]

Validation status

Snap	Chemical system	Evidence	Status
β=1/3	Fe(phen)₂(NCS)₂ SCO	Papers 488/491; c₂ peak at SCO	Predicted
β=1/5	1D Hubbard chain	x563c: D collapse at U/t≈1.8	Partially validated
β=1/7	FeMoco (nitrogenase)	Paper 488: 7-qubit G₂ programme	Predicted

Part of the ISA Zoo. See also: CM01 Hubbard-Mott, C02 Spin-Crossover, GA02 FeMoco Galois, G2-QEC snap events.