Close atomic pairing and match published bonding surfaces

Occupancy-2 electrons pair inside the closed atom. Leftover unpaired
(nucleus, e_i) incidences are the bonding surfaces molecules may match.

Author: erinepshovel-code

research/epac/README.md

@@ -20,10 +20,10 @@ until a canonical `The-Interdependency/epac` source is created and pinned in
 - [`docs/constraints.md`](docs/constraints.md) — standing constraints and
   resolved contradictions for this candidate.
 - [`docs/preregistration-molecular-geometry-from-element-gonols.md`](docs/preregistration-molecular-geometry-from-element-gonols.md) —
-  provisional candidate: Z=1–18 element gonols from atomic structure only, then
-  affixiation of H₂, H₂O, NH₃, CH₄, CO₂ through unpaired valence electrons,
-  declared dimensional couplings, and UCNS Möbius observation. Known chemistry
-  is sealed until after construction.
+  provisional candidate: Z=1–18 element gonols from nucleons, nucleus–electron
+  couplings, occupancy-2 pairing, and leftover unpaired `(nucleus, e_i)`
+  bonding surfaces, then affixiation of H₂, H₂O, NH₃, CH₄, CO₂ by matching
+  those surfaces. Known chemistry is sealed until after construction.
 
 ## Usage
 

research/epac/docs/arity.md

@@ -22,19 +22,26 @@ their affixiation. Neutrons couple to protons as `(proton_j, neutron_i)` with
 slot charges `(+1, 0)`. Proton-proton and neutron-neutron are not inferred.
 Hydrogen-1 is one proton and no neutrons.
 
-At atomic scale the hub is that closed nucleus and every electron instance has
-its own `(nucleus, electron_i)` with slot charges `(Z, -1)`.
+At atomic scale there are two declared relations:
 
-Molecular scale has two declared relations:
+- 3-structure: hub is the closed nucleus; every electron instance has
+  `(nucleus, electron_i)` with slot charges `(Z, -1)`.
+- pairing: occupancy-2 `(n, l, m_l)` electrons couple as
+  `(e_ms+1, e_ms-1)` with charges `(-1, -1)`.
 
-- 3-structure: hub is the closed center atom; every ligand unpaired valence
-  electron has `(center, ligand_e_i)`.
-- bond: unpaired valence electrons are arity-coupled as
+Leftover unpaired valence `(nucleus, electron_i)` incidences are the atom's
+bonding surfaces. Pairing is not a proof of `(nucleus, e_i, e_j)`.
+
+Molecular scale matches those published surfaces. Two declared relations:
+
+- 3-structure: hub is the closed center atom; every ligand electron of a
+  matched surface has `(center, ligand_e_i)`.
+- bond: the electrons of two matched surfaces couple as
   `(center_e_i, ligand_e_i)` with charges `(-1, -1)`.
 
-Nucleons and core electrons stay inside closed atoms. Water's 3 is
-`(O#2, H0_e)` and `(O#2, H1_e)`, degree 2 on oxygen. That is not `(O#2, H#0)`
-atom-atom and not a ternary `(x,y,z)`.
+Nucleons, core electrons, and paired electrons stay inside closed atoms.
+Water's 3 is `(O#2, H0_e)` and `(O#2, H1_e)`, degree 2 on oxygen. That is not
+`(O#2, H#0)` atom-atom and not a ternary `(x,y,z)`.
 
 See `constraints.md`.
 

research/epac/docs/constraints.md

@@ -18,11 +18,13 @@ now obeys the listed rule. Unresolved remains `hmmm`.
 | C9 | protons and neutrons are precursor gonols of the nucleus | resolved |
 | C10 | electrons couple to the closed nucleus; nucleons stay inside it | resolved |
 | C11 | closed gonols are atomic at the next scale; promotion is allowed | resolved |
-| C12 | molecular attachment consumes closed unpaired valence electrons, not `atomic_of` | resolved |
+| C12 | molecular attachment consumes published bonding surfaces, not `atomic_of` | resolved |
 | C13 | atomic s→p promotion is an atomic excited gonol, not VSEPR | resolved |
 | C14 | H₂ is a single binary and is not a 3 | resolved |
 | C15 | sealed chemistry shapes stay out of construction | resolved |
 | C16 | nothing bigger than atoms is an *input* to element construction | resolved |
+| C17 | occupancy-2 `(n,l,m_l)` electrons pair inside the atom as `(e_ms+1, e_ms-1)` | resolved |
+| C18 | bonding surfaces are leftover unpaired valence `(nucleus, e_i)` after pairing | resolved |
 
 ## Contradictions
 
@@ -33,8 +35,8 @@ now obeys the listed rule. Unresolved remains `hmmm`.
 
 **Resolution:** two declared relations, not one inferred from the other.
 
-1. **3-structure:** hub `z` is the closed center atom; instances are ligand unpaired valence electrons: `(center, ligand_e_i)`.
-2. **Bond:** unpaired valence electrons are arity-coupled to each other: `(center_e_i, ligand_e_i)`.
+1. **3-structure:** hub `z` is the closed center atom; instances are ligand electrons of matched bonding surfaces: `(center, ligand_e_i)`.
+2. **Bond:** matched bonding-surface electrons are arity-coupled: `(center_e_i, ligand_e_i)`.
 
 Water: `deg(O#2)=2` on `(O#2, H0_e)` and `(O#2, H1_e)`; quaternion `(ε, 8, −1, −1)`. Bonds remain the two electron–electron couplings. Not `(x,y,z)`.
 
@@ -44,7 +46,14 @@ Water: `deg(O#2)=2` on `(O#2, H0_e)` and `(O#2, H1_e)`; quaternion `(ε, 8, −1
 
 ### X3 — consume atomic 3-structure vs formula-count geometry
 
-**Resolution:** composition still names which atoms. Attachment sites and charges come from closed gonols. Molecular 3 uses the closed center as hub and ligand valence electrons as instances.
+**Resolution:** composition still names which atoms. Attachment sites are the leftover unpaired valence `(nucleus, e_i)` published as bonding surfaces. Molecular 3 uses the closed center as hub and the ligand electrons of matched surfaces as instances.
+
+### X5 — pairing vs nucleus–electron 3-structure
+
+- C5/C6 already make `(nucleus, e_i)` the atomic 3.
+- Occupancy-2 pairing is a second declared relation `(e_ms+1, e_ms-1)`, not a proof of `(nucleus, e_i, e_j)`.
+
+**Resolution:** two declared atomic relations, same shape as X1. Pairing does not install a ternary. Leftover unpaired valence incidences remain the bonding surfaces.
 
 ### X4 — internucleon `(every proton, every neutron)` vs no all-pairs authority
 

research/epac/docs/preregistration-molecular-geometry-from-element-gonols.md

@@ -16,6 +16,8 @@
 | affixiation | `metapat.affixiation_harmonics.affixiation` | METAPAT | identity-preserving higher-order relation | UCNS topology selection |
 | UCNS coupling | `ucns.native-mobius-root-loop` | UCNS | established 360° frame flip / 720° restore | invented 3-space arrangement |
 | molecular gonol | `epac.molecular.affixiated_whole` | epac candidate | closed recursive gonol of element-gonol participants | known chemistry shape names |
+| electron pairing | `epac.atomic.occupancy_pair` | epac candidate | occupancy-2 `(n,l,m_l)` electron-electron coupling inside the closed atom | chemistry spin-pairing as a sealed shape rule; `(nucleus, e_i, e_j)` ternary |
+| bonding surface | `epac.atomic.bonding_surface` | epac candidate | leftover unpaired valence `(nucleus, e_i)` after pairing | VSEPR domain count; reopened core or paired electrons |
 | predicted geometry | `epac.molecular.construction_invariants` | epac candidate | atom count, center valence, slot occupancy, Möbius frame sequence | sealed comparison labels |
 
 Collision check: physics/chemistry own empirical molecular shapes. This candidate
@@ -28,9 +30,11 @@ scopes) until comparison.
 proton gonols (charge +1) and neutron gonols (charge 0)
   -> nucleus = affixiation of those nucleons; (proton_j, neutron_i)
   -> every electron instance: (nucleus, electron_i) with charges (Z, -1)
-  -> close that atomic 3-structure inside the element gonol
-  -> unpaired-valence electron gonols already closed inside each atom
+  -> occupancy-2 (n,l,m_l) pairing: (e_ms+1, e_ms-1) with charges (-1, -1)
+  -> leftover unpaired valence (nucleus, electron_i) published as bonding surfaces
+  -> close that atomic structure inside the element gonol
   -> if needed, a closed promoted atomic gonol (same n s→p); no table re-lookup
+  -> match published bonding surfaces only
   -> declared oriented (center_electron, ligand_electron) bond couplings with charges (-1, -1)
   -> declared oriented (closed_center, ligand_electron_i) 3-structure; one hub, every instance
   -> molecular three-dimensional structure = those atom-instance couplings + charge states + degree
@@ -47,8 +51,9 @@ proton gonols (charge +1) and neutron gonols (charge 0)
 - every electron: n, l, m_l, m_s, shell, subshell
 - hydrogenic angular identity Y_l^m, radial node count n-l-1
 - Slater atomic Z_eff and hydrogenic Rydberg energy -Z_eff²/n²
-- unpaired valence electrons from Hund filling
-- atomic s→p promotion in the same n when more unpaired sites are required
+- unpaired valence electrons leftover after occupancy-2 pairing
+- published leftover `(nucleus, electron_i)` bonding surfaces
+- atomic s→p promotion in the same n when more bonding surfaces are required
 - caller-supplied stoichiometric formula (element counts only)
 
 ## Inputs forbidden in construction

research/epac/epac_dimensional_arity.py

@@ -11,10 +11,11 @@
 occurrence is a second instance, not a reuse of the first coupling.
 ``(x_i, z)`` does not satisfy ``(z, x_i)``. Letters and abbreviations are
 not this domain. At atomic scale the hub is the nucleus and the instances
-are electrons. At molecular scale the hub is the closed center atom and the
-instances are ligand unpaired valence electrons. Bond couplings between
-those valence electrons are a second declared relation, not a replacement
-hub.
+are electrons; occupancy-2 pairing is a second declared electron-electron
+relation. Leftover unpaired valence ``(nucleus, e_i)`` incidences are bonding
+surfaces. At molecular scale the hub is the closed center atom and the
+instances are ligand electrons of matched surfaces. Bond couplings between
+those electrons are a second declared relation, not a replacement hub.
 
 The three-dimensional structure is the combination of declared oriented
 couplings, their arity charge states, and degree. That span can involve three