C07 — Chemical Bonding as a PG→AG Transition
| Field | Value |
|---|---|
| Domain | Chemistry |
| System | Any spd-block (q=3) covalent bond |
| Group | PGL(3,3) on PG(2,3) → AGL(2,3) on AG(2,3) after bonding |
| H^k tier | H² |
| ISA | Meld (full 8-opcode set) |
| Status | Validated (x707c, x707d) |
| Opcodes | FUSE · CLEAVE · FLIP · SNAP · RESOLVE · TWIST · THERMAL · HALT |
| Papers | doi:10.5281/zenodo.21534988 (Paper 688), doi:10.5281/zenodo.21562294 (Paper 699), Paper 707 |
Physical system
A covalent bond between two spd-block atoms is the selection of a bond axis, which singles out one direction in the orbital space PG(2,GF(3)) (13 points, 13 lines) and designates it as the line at infinity ℓ∞. The projective orbital plane is thereby reduced to an affine plane AG(2,GF(3)) (9 points, 12 lines, 4 parallel classes). The 4 parallel classes are the bonding channels: σ, π, δ, and non-bonding.
This is not a model of bonding — it is a description of what bonding is, geometrically. The bond axis choice is the FUSE opcode; bond breaking is CLEAVE. All subsequent chemistry (conformational change, spectroscopy, catalysis) is executed by the remaining 6 opcodes acting within AG(2,3).
Target category
PGL(3,3)-Set before bonding; AGL(2,3)-Set after bonding.
The full symmetry group of PG(2,3) is PGL(3,3), order 5616. It has 8 conjugacy classes on the 13-point set — one per unoriented ISA operation type.
When a bond forms (FUSE fires), the system’s symmetry breaks to the stabiliser of ℓ∞:
\[\text{PGL}(3,3) \xrightarrow{\text{FUSE}} \text{AGL}(2,3) = \text{Stab}(\ell_\infty), \quad |432| \subset |5616|\]AGL(2,3) has 9 conjugacy classes on the 9-point affine set — the ISA operations within the bond. The symmetry breaking is spontaneous: FUSE selects one of the 13 lines as ℓ∞, reducing the 5616-element symmetry to a 432-element subgroup.
Uniqueness (Veblen-Young theorem, 1908): PG(2,3) is the unique projective plane of order 3. Therefore PGL(3,3) = Aut(PG(2,3)) is unique. No other group gives this 8-class structure on 13 points. The ISA at H² is not a choice — it is forced by the geometry of spd-block orbital physics.
Interpretation functor
Two-level structure, reflecting the pre-bond and post-bond regimes:
Level 1 — Pre-bond: PGL(3,3) on PG(2,3)
| PGL(3,3) class | Cycle type on 13 pts | Unoriented ISA type |
|---|---|---|
| identity | (1¹³) | null |
| order-2, 5 fixed pts | (1⁵,2⁴) | FLIP / SNAP (merged on 13 pts) |
| order-3, 4 fixed pts | (1⁴,3³) | RESOLVE-sub |
| order-3, 1 fixed pt | (1,3⁴) | RESOLVE |
| order-4, 1 fixed pt | (1,2²,4²) | TWIST |
| order-6, 2 fixed pts | (1²,2,3,6) | THERMAL / SNAP×RESOLVE |
| order-8, 1 fixed pt | (1,4,8) | HALT |
| order-13, 0 fixed pts | (13) | Singer cycle |
The Singer cycle (order-13 class, 1728 elements) is the full cyclic permutation of all 13 points — it has no analogue in AGL(2,3) and is a PGL-only operation corresponding to a global phase rotation of the entire orbital configuration.
Level 2 — Post-bond: AGL(2,3) on AG(2,3)
| AGL(2,3) class | Fingerprint (Ord, FP, FC) | ISA opcode |
|---|---|---|
| (1⁹) | (1, 9, 4) | identity |
| (3,3,3) | (3, 0, 1) | RESOLVE |
| (1,1,1,2,2,2) | (2, 3, 2) | FLIP |
| (1,2,2,2,2) | (2, 1, 4) | SNAP |
| (1,1,1,3,3) | (3, 3, 1) | RESOLVE(sub) |
| (1,4,4) | (4, 1, 0) | TWIST |
| (1,2,6) | (6, 1, 1) | THERMAL |
| (3,6) | (6, 0, 2) | SNAP×RESOLVE (compound) |
| (1,8) | (8, 1, 0) | HALT |
| T ≅ Z₃² | (3, 0, 0) | FUSE/CLEAVE (undirected) |
Fingerprint = (order, fixed-point count, fixed-class count). Each fingerprint is unique — the bijection between classes and opcodes is forced, not labelled.
The FUSE/CLEAVE orientation problem
FUSE (bond formation) and CLEAVE (bond breaking) are group-theoretically identical: a permutation and its inverse always share the same cycle type and are therefore always conjugate in any symmetric group. No group — not T, not AGL(2,3), not PGL(3,3) — can distinguish them.
The distinction is categorical, not algebraic. FUSE and CLEAVE are an adjoint functor pair (monad/comonad). The bond direction is an arrow in the category; the group captures operation types but not directions.
This has a self-referential consequence:
FUSE is the operation that creates the orientation needed to distinguish FUSE from CLEAVE.
Before FUSE fires, PG(2,3) has no preferred ℓ∞ and no bond direction. After FUSE fires, ℓ∞ is designated, AGL(2,3) becomes the symmetry group, and CLEAVE is now the directed reverse of FUSE. The ISA bootstraps its own directionality.
ISA programme (generic covalent bond, spd-block)
-- Pre-bond: full PGL(3,3) symmetry on PG(2,3)
RESOLVE[orbital configuration] -- read parallel-class occupancies
TWIST[ε vs ε_crit] -- check branch-point position
-- Bond formation
FUSE[ℓ∞ ← bond axis] -- remove ℓ∞; PGL→AGL symmetry breaking
-- 4 parallel classes emerge: σ π δ non-bonding
-- Within bond: AGL(2,3) operations
FLIP[occupancy] -- invert σ/π occupancy
SNAP[sheet exchange] -- exchange Hitchin spectral sheets
RESOLVE[parallel classes] -- cycle through σ→π→δ→... conformations
THERMAL[ε perturbation] -- phonon/photon forcing of branch point
HALT[permanent SSB] -- covalent arrest; no return to SSM
-- Bond breaking
CLEAVE[ℓ∞ restored] -- restore PG(2,3); AGL→PGL symmetry recovery
The full 8-opcode ISA is necessary and sufficient for spd-block (q=3) bonding chemistry. Remove any one and a class of reactions becomes unreachable (RESOLVE: no conformational cycling; HALT: no permanent bonds; FUSE: no bond formation; etc.).
Computable output
- Bond symmetry breaking: ΔG of FUSE = spectral gap of PG(2,3) at the weakened line = 2ε_crit(bond type); measured by Hitchin period matrix Im(B) (Papers 697–699).
- Bond order: number of parallel classes in SSB state = σ only → order 1; σ+π → order 2; σ+π+δ → order 3.
- Trans influence: Re(w_BP) for the bond’s branch-point parameter; larger Re(w_BP) = stronger σ-donor; confirmed for 12 Pt(II) complexes (Paper 699 x699c, r = 0.94).
- Bond dissociation energy: 2ε_crit for the weakest parallel class in SSB; Co–C in B12 predicted 1.35–1.40 eV (Paper 705 x705d, planned).
Validation
- x707c (5/5): AGL(2,3) has exactly 9 conjugacy classes; 8 non-trivial classes biject with ISA opcodes by forced geometric fingerprint.
- x707d (5/5): PGL(3,3) has exactly 8 conjugacy classes on PG(2,3); Stab(ℓ∞) = AGL(2,3) of order 432; every AGL class embeds into a PGL class.
- Paper 688 (x688a–e, 5/5): bond = PG→AG transition confirmed for H₂, N₂, benzene, FeMoCo, B12.
- Paper 699 (x699a–d, 5/5): bond = Hitchin branch point; trans influence from Re(w_BP); Madelung anomalies from branch-point interference.
- Veblen-Young (1908): PG(2,3) unique → PGL(3,3) unique → ISA at H² is the canonical and only faithful representation.
Relation to other zoo entries
- C01 — Nitrogen Fixation: FUSE fires at E₄ state; the N≡N bond is a PAT-3 triple-HALT state
- C05 — PSII O–O Bond Formation: FUSE at S₄→S₀ transition; O–O bond = PAT-3 FUSE with Ca as the orientation-fixing cofactor
- C06 — C/T Skeleton: pre-screening identifies which pairs are near the FUSE threshold (T-arrows = bonds near ε_crit)
- GA02 — FeMoCo Galois: FUSE as Galois field extension — complementary algebraic description of the same transition
Part of the ISA Zoo. Foundational ISA: Paper 591. PAT-q bonding: Papers 688, 699. AGL(2,3) bijection: Paper 707.