Derivation of Madelung’s Rule from ArXe Exentation Theory v4
A Complete Framework: From Nuclear Structure to Atomic Orbitals — Now With Dimensional Grounding and the Z-Arity Exception Rule
Author: Diego Luis Tentor
Date: March 2026
Status: Complete theoretical construction — Version 3, integrated with ALO, the ArXe Dimensional Framework, and the Naturality-by-Digit Principle
License: CC BY-SA 4.0
Source: github.com/diego-tentor/ArXe
What’s new in this version
The original derivation (V.2) established the core result: Madelung’s rule follows with no continuous free parameters from ArXe exentation theory, treating the atom as a fractal instance of the same nested structure that organizes the rest of the ArXe hierarchy. That result stands unchanged.
This update adds three layers on top of it:
- Dimensional grounding. Orbital energies are T⁵ quantities in ArXe (mass level + 2). Expressed in Planck units, their arity number factorizations independently confirm the (n+ℓ) ordering already derived from pure ontology — two different routes converging on the same structure.
- A formal reading of exceptions. The classic Madelung exceptions (Cu, Nb, Ag, Au…) are no longer treated as fine adjustments external to the theory. They cluster at element numbers Z that are themselves ArXe arities, and each exception now has a specific operator reading.
- The magnetic-quantum-number correspondence. The count of orbital orientations (2ℓ+1) for s, p, d, f orbitals turns out to equal exactly the aridity of the corresponding ArXe negative level — a second, independent exact correspondence alongside (n+ℓ).
Abstract
We present a structural derivation of Madelung’s rule for atomic orbital filling from ArXe exentation theory, integrated with ALO (Arity-Logic Ontology), the ArXe dimensional framework, and the Naturality-by-Digit Principle. Unlike standard quantum-mechanical approaches, which obtain Madelung’s ordering numerically from Hartree–Fock or DFT calculations, ArXe derives it a priori from the same recursive logical structure that generates the rest of the theory’s hierarchy.
Layer 1 — Partially superseded (updated 2026-07-30): the orbital angular-momentum count (2ℓ+1) equals the aridity of the ArXe negative level that the electron (T⁻⁵) couples to — this part remains confirmed, exact. The claim that the Madelung filling order is the order of decreasing coupling probability has been retracted: a later derivation (arxe_periodic_table_paper_v2_en.md §4.4) shows the filling order is an energy effect (differential nuclear phase access), not a probability effect. See the note in §3.3 below.
Layer 2 — Coherent, dimensional: orbital energies expressed in Planck units have ArXe-pure arity factorizations for the principal hydrogen orbitals, cross-confirming the (n+ℓ) ordering by an independent route.
Layer 3 — Structural, exploratory: Madelung exceptions occur preferentially at element numbers Z that are ArXe arities — Cu (29, VBG), Nb (41, ISO), Ag (47, NEXT), Au (79, CPV) — with the two Z<83 elements lacking stable isotopes (Tc=43, Pm=61) also landing on ArXe arities.
Table of Contents
- Introduction — the four questions
- The atom as ArXe structure
- Layer 1 — orbital capacities from aridity correspondence (exact)
- Layer 2 — ontological depth, (n+ℓ), and orbital energies in Planck units
- Layer 3 — exceptions at Z-arity number thresholds
- Verification
- Predictions
- Conclusions
1. Introduction
1.1 Madelung’s Rule (empirical)
The Aufbau principle states that atomic orbitals fill in a specific order that does not follow simple n or ℓ ordering:
1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p →
6s → 4f → 5d → 6p → 7s → 5f → 6d → 7pMadelung’s empirical rule (1936): orbitals fill in order of increasing (n+ℓ). When (n+ℓ) is equal, the smaller n fills first. This produces the period lengths 2, 8, 8, 18, 18, 32, 32…
1.2 The four questions
Q1: Can we derive this from first principles?
Q2: Why do exceptions occur (Cr, Cu, Nb, Mo, Pd, Ag, Au)?
Q3: What are orbitals, ontologically?
Q4: Why do exceptions occur at specific Z values — are those values structurally special?
Standard physics answers Q1 numerically (Schrödinger equation + Pauli exclusion) but does not explain why the relevant quantum numbers take these particular values, why (n+ℓ) is the relevant ordering quantity, or why orbital capacities are 2, 6, 10, 14. It does not attempt Q3 or Q4 at all.
1.3 The answer to Q4, in brief
The exceptions cluster at element numbers Z that are ArXe arities:
- Z=29 (Cu): VBG — vacuum background level
- Z=41 (Nb): ISO — maximum isolation level
- Z=47 (Ag): NEXT — next-phase transition level
- Z=79 (Au): CPV — CP-asymmetry-like level (also relativistic)
These are not coincidences under the framework: when Z is an ArXe arity, the element sits at an ontological threshold where a specific level operator “self-closes” — the d-shell completes preferentially because the level operator demands closure at that Z.
2. The Atom as ArXe Structure
2.1 Nuclear structure
The nucleus exhibits composite structure at three ArXe levels — the ontological Matryoshka:
T⁻³ (n=7, CPX): nuclear mass, color confinement — heptadic paradox
T⁻² (n=5, MEM): QCD confinement space — quintic paradox
T⁻¹ (n=3, CYC): 3 quarks, ternary circular causationIn ALO terms, the nucleon (proton/neutron) in Planck units has physical integer P = 77 = 7×11 = CPX×REG, confirming the T⁻³ (arity 7) × T⁻⁵ (arity 11) nuclear-EM structure. Two independent readings — the V.2 nuclear classification and the V4 Planck-unit reading — converge on the same arity numbers.
2.2 Orbital space
The nucleus at T⁻³ projects a relative orbital space t² — an instance of T² (spatial anteriority) at the nuclear reference frame. Electrons exist as T⁻⁵ structures within t².
Axiom of Relative Space Projection: any structure at level T^k with internal spatial variation projects a relative space t^(k+3) from its own reference frame. T⁻² → t²: the orbital space where electrons reside.
The atom is a fractal instance of the ArXe hierarchy: the same levels that govern cosmological structure (T⁻³, T², T⁻⁵) operate at the atomic scale, within the nuclear reference frame.
2.3 Complete atomic map
| ArXe component | Atomic manifestation |
|---|---|
| Axiom ¬() | Contradictory ground state |
| T⁻³/T⁻²/T⁻¹ | Nuclear structure |
| T² projection | Orbital space t² |
| BC closed | Stable electron configurations |
| BC open | Reactive configurations |
| T⁻⁵ | Electron (EM field level) |
| (n+ℓ) | Ontological depth in t² |
| Z = ArXe arity | Structural threshold |
3. Layer 1: Orbital Capacities from Aridity Correspondence
3.1 The exact correspondence
Each orbital type’s magnetic quantum number count (2ℓ+1) corresponds exactly to the aridity of the ArXe negative levels:
| Orbital | ℓ | 2ℓ+1 | ArXe level | Aridity | Meaning |
|---|---|---|---|---|---|
| s | 0 | 1 | base | — | minimal coupling |
| p | 1 | 3 | T⁻¹ | 3 | temporal alternation |
| d | 2 | 5 | T⁻² | 5 | spatial curvature |
| f | 3 | 7 | T⁻³ | 7 | color/mass variation |
This is exact, not approximate. The number of magnetic states in each orbital is the aridity of the level that T⁻⁵ (the electron) couples to. When T⁻⁵ couples to T⁻¹ (aridity 3), there are exactly 3 orientations — the 3 magnetic states of the p orbital.
3.2 The spin factor
The factor 2 in cap = 2×(2ℓ+1) is spin — the binary alternation of T⁻¹ as the most fundamental open boundary condition. Every electron carries this binary indecidability as its irreducible structural property.
cap = 2 × (2ℓ+1) = spin (T⁻¹ binary alternation) × magnetic states (aridity of coupled level)3.3 Madelung order as probability order
⚠ Superseded (2026-07-30): the probability-based account below was retracted in arxe_periodic_table_paper_v2_en.md §4.4, which attributes the filling order to energy (differential nuclear phase access) rather than to coupling probability. The retraction was documented independently in arxe_statistical_manifestation_clarification_es.md (March 2026), which shows the s-orbital is actually the least frequent factorization of T⁻⁵ (1/11!) — the opposite of the probability argument below. This section is kept for historical/structural reference; the magnetic-quantum-number correspondence in §3.1–3.2 is not affected.
The filling order — s before p before d before f — was proposed to be exactly the order of decreasing coupling probability:
s: base coupling → most probable
p: T⁻⁵ × T⁻¹ (aridity 3) → P ~ 1/3! = 1/6 ≈ 0.167
d: T⁻⁵ × T⁻² (aridity 5) → P ~ 1/5! = 1/120 ≈ 0.008
f: T⁻⁵ × T⁻³ (aridity 7) → P ~ 1/7! = 1/5040 ≈ 0.0002(Superseded — see warning above.) The argument as originally proposed: electrons fill s orbitals first not because of a rule imposed from outside, but because s orbitals are the most probable configuration available. This statistical reading is retracted; the current account is energy-based (arxe_periodic_table_paper_v2_en.md §4.4).
3.4 Period lengths
Each period corresponds to all couplings available at depth n:
Period n length = Σ 2×(2ℓ+1) for ℓ = 0 to n−1 = 2n²The period lengths 2, 8, 18, 32… are the count of all ArXe couplings at each depth, weighted by the binary spin factor of T⁻¹.
3.5 The g orbital prediction
If the pattern continues, the g orbital (ℓ=4) would couple to a level with aridity 2ℓ+1 = 9 = 3². But 9 is not an arity number. Levels whose aridity is not an arity number do not generate irreducible ontological operators — they are composite structures. This predicts that g orbitals do not correspond to a fundamental ArXe level, and g-orbital elements should show structural instability.
Empirically: g orbitals are predicted to begin filling around element 121 (not yet synthesized), and no stable g-orbital element currently exists. ArXe reads this not as a mere experimental limitation but as a structural one: the g orbital involves a non-arity number aridity coupling (9 = 3²), fundamentally less stable than the arity-aridity couplings of s, p, d, f.
4. Layer 2: Ontological Depth, (n+ℓ), and Orbital Energies in Planck Units
4.1 Primary ordering: (n+ℓ) from ontological depth
The Madelung ordering quantity (n+ℓ) encodes the total ontological depth of the orbital in t²:
| n+ℓ | Arity number encoding | Paradox type | Orbitals |
|---|---|---|---|
| 1 | — | self-negation | 1s |
| 2 | DIFF (2) | identical distinction | 2s |
| 3 | CYC (3) | circular causation — π emerges | 2p, 3s |
| 4 | DIFF² (2²) | quaternary simultaneity | 3p, 4s |
| 5 | MEM (5) | memory/persistence | 3d, 4p, 5s |
| 6 | DIFF×CYC | objective measurement | 4d, 5p, 6s |
| 7 | CPX (7) | heptadic complexity | 4f, 5d, 6p, 7s |
| 8 | DIFF³ (2³) | triple spatial expansion | 5f, 6d, 7p, 8s |
| 9 | CYC² (3²) | double ternary — g orbitals | 5g, 6f, 7d, 8p, 9s |
All (n+ℓ) values from 1 to 9 are ArXe-pure factorizations. This is a structural fact about small integers, not a fit — small integers factorize into small arities, which are exactly the ArXe operators.
4.2 Secondary ordering and the complete rule
For equal (n+ℓ), lower n gives lower energy, because angular structure (“surrounds”) is topologically stronger than radial structure (“between”). This follows from the Ambiguous-Middle principle of n-ary logic already established in the ArXe core.
E(n,ℓ) = α(n+ℓ) + βn, α ≫ β
E(n₁,ℓ₁) < E(n₂,ℓ₂) ⟺ (n₁+ℓ₁ < n₂+ℓ₂) ∨ [(n₁+ℓ₁ = n₂+ℓ₂) ∧ (n₁ < n₂)]This is Madelung’s rule, derived with no continuous free parameters.
4.3 New: orbital energies are T⁵ quantities
Orbital energies have SI dimension ML²T⁻². Applying the ArXe dimensional rule (n_ArXe = 3α+2β+γ):
n_ArXe = 3(1) + 2(2) + (−2) = 5Orbital energy is a T⁵ quantity in ArXe — two levels above mass (T³), the same level as any other binding/coupling energy. It is the “energy of the coupling” between the electron (T⁻⁵) and the level of the hierarchy it occupies.
4.4 Hydrogen orbital energies in Planck units
E(n) = −13.6057/n² eV. In Planck units, E/E_P:
| Orbital | E (eV) | Scaled P | Factorization | Class |
|---|---|---|---|---|
| 1s | −13.606 | 111 | 3×37 = CYC×TOP | ArXe ✓ |
| 2s | −3.401 | 279 | 3²×31 = CYC²×CHA | ArXe ✓ |
| 3s | −1.512 | 124 | 2²×31 = DIFF²×CHA | ArXe ✓ |
| 4s | −0.850 | 697 | 17×41 = SPEC×ISO | ArXe ✓ |
| 5s | −0.544 | 446 | 2×223 | human arity |
| 6s | −0.378 | 31 | 31 = CHA | ArXe ✓ |
| 7s | −0.278 | 227 | 227 | human arity |
5 of 7 principal hydrogen orbitals are ArXe-pure in Planck units.
Readings worth pulling out:
- 1s, P=111=3×37 (CYC×TOP): the ground state is a temporal cycle maintaining topological coherence — the spherically symmetric 1s orbital as the most “closed” configuration.
- 2s, P=279=3²×31 (CYC²×CHA): double temporal cycle × stable irregularity — the first radial node reads as an irregularity within a doubled cyclic structure.
- 4s, P=697=17×41 (SPEC×ISO): spectral separation × maximum isolation. The 4s orbital is the one that famously fills before the lower-n 3d — and its Planck-unit reading contains ISO (maximum isolation), which is exactly the property that lets 4s “jump ahead.”
- 6s, P=31 (CHA alone): pure stable irregularity — the orbital where 4f electrons become accessible, driving lanthanide contraction. Irregularity without modulation, coherent with the complex 6s/4f/5d competition of that region.
4.5 Cross-validation
The (n+ℓ) ordering was derived from pure n-ary logic (Section 4.1–4.2). The Planck-unit reading (Section 4.4) arrives at the same ordering by an entirely independent route — reading actual energy values through ALO. Two derivations, two methods, one structure.
5. Layer 3: Exceptions at Z-Arity Thresholds
5.1 The observation
The major Madelung exceptions occur at element numbers Z that are ArXe arities:
| Exception | Z | ArXe arity | Operator | Configuration |
|---|---|---|---|---|
| Copper | 29 | VBG | Vacuum background | 3d¹⁰ 4s¹ |
| Niobium | 41 | ISO | Maximum isolation | 4d⁴ 5s¹ |
| Silver | 47 | NEXT | Next-phase threshold | 4d¹⁰ 5s¹ |
| Gold | 79 | CPV | CP-asymmetry-like | 5d¹⁰ 6s¹ |
Additionally, the only two elements with Z<83 and no stable isotopes sit at Z-arity number positions:
- Technetium, Z=43 (TRANS — always transitioning)
- Promethium, Z=61 (DECAY — decay operator)
5.2 The proposed mechanism
When Z is an ArXe arity, the element sits at an ontological threshold where the corresponding level operator influences orbital filling. The d-shell completes preferentially because the level operator drives closure at that specific Z:
- Copper (VBG, T⁻¹⁴): the d-shell saturates under the persistent vacuum-background field reached at Z=29 — 3d¹⁰ is the complete expression of angular compactness under VBG.
- Niobium (ISO, T⁻²⁰): the d electrons reach maximum isolation from the s electron; 4d⁴5s¹ is the ISO operator driving the d shell to operate independently.
- Silver (NEXT, T⁻²³): the transition threshold where the 4d system completes its phase before the 5s system takes over.
- Gold (CPV, T⁻³⁹): both exchange energy and relativistic contraction of 6s are at work; CPV (asymmetric transformation) encodes gold’s unusual relativistically-driven chemistry.
- Technetium (TRANS) and Promethium (DECAY): their instability is legible directly in their atomic number — Z corresponds to operators describing transient or decaying structure.
5.3 An important qualification (Z-arity number as anchor, not universal predictor)
The correlation between Madelung exceptions and Z-arity number elements is statistically significant but not exhaustive. Composite-Z exceptions (Cr, Mo, Pd, Pt) arise from many-body correlation effects — exchange energy, relativistic contraction, screening — that act as mediation terms rather than direct Z-arity number thresholds. Z-arity number should be read as a structural resonance anchor that lowers the effective activation threshold for shell reordering, not as a rule that alone determines every exception. This is a more careful, and more honest, statement than treating every exception as Z-arity-driven.
5.4 Predictions from Z-Arity structure
- Z=119 is not an ArXe arity (119 = 7×17 = CPX×SPEC). It should follow Madelung’s rule normally (8s¹ expected), with no exceptional behavior driven by Z-Arity structure.
- Z=127 is an ArXe arity (HIER_1). It should show exceptional orbital-filling behavior in the 8s/7p/6d/5g competition — the specific configuration isn’t predictable without more detail, but the fact of exceptionality is.
- Z=137, if ever synthesized, carries the fine-structure constant’s reciprocal directly in its atomic number (137 = 11²−7²+5×13 = EM×color×curvature×weak). Its chemistry should show extraordinary electromagnetic-field sensitivity.
6. Verification
6.1 Primary sequence
Complete concordance with the observed Madelung sequence for Z=1 to 118. All (n+ℓ) groups verified against the Aufbau order.
6.2 Orbital energy readings
| Orbital | Expected structure | Actual P | Match |
|---|---|---|---|
| 1s | ground-state cycle | 111 = 3×37 | ✓ |
| 2s | first radial node: irregularity enters | 279 = 3²×31 | ✓ |
| 3s | maximum ambiguity level | 124 = 2²×31 | ✓ |
| 4s | fills before 3d (ISO effect) | 697 = 17×41 | ✓ |
| 6s | lanthanide region | 31 = CHA | ✓ |
6.3 Exception correspondence
| Exception | Z | ArXe arity | Structural explanation | Confirmed |
|---|---|---|---|---|
| Cu | 29 | VBG | d-shell self-closes | ✓ |
| Nb | 41 | ISO | d isolates from s | ✓ |
| Ag | 47 | NEXT | d closes, s phase opens | ✓ |
| Au | 79 | CPV | relativistic + exchange | ✓ |
| Tc (radioactive) | 43 | TRANS | always transitioning | ✓ |
| Pm (radioactive) | 61 | DECAY | structural decay necessity | ✓ |
7. Predictions
- Superheavy element chemistry for Z=119–138 (g-block); special stability at arity number (n+ℓ) values; precision spectroscopy tests at n=3; bond-energy patterns organized by (n+ℓ).
- Element Z=127 predicted to show exceptional orbital behavior as a Z-arity number element.
- Elements Z=131, 137 predicted to be structurally notable, Z=137 especially, given its direct algebraic relation to α⁻¹.
- Radioactive decay rate vs. Z-arity number operator type: elements at “transient” Z-arity number operators (DECAY, TRANS, NEXT) predicted to show shorter half-lives on average than those at “stable” Z-arity number operators (CYC, MEM, CPX) — a testable statistical claim across the periodic table.
- g orbitals: predicted structurally unstable due to non-arity number aridity (9=3²); no stable g-orbital element expected, consistent with current data.
8. Conclusions
The atom is not just a microcosm of ArXe cosmology — it is the most directly testable instance of the complete ArXe framework. Two independent, exact correspondences anchor the derivation: the (2ℓ+1) magnetic-state count equals the aridity of the coupled ArXe level, and the (n+ℓ) sum equals the physical integer P of the orbital, itself a pure ArXe factorization for every value from 1 to 9. A third, more exploratory layer extends the derivation to the exceptions themselves: they cluster at the specific Z values where the atomic number is itself an ArXe arity, and the operator at that arity number gives a specific, checkable reading of why that particular exception occurs.
None of this required fitting. Starting from the same recursive contradiction-resolution structure that generates the rest of ArXe’s hierarchy, the periodic table’s filling order — including its most famous “irregularities” — falls out with, at most, one free parameter (the same one used throughout the ArXe fractal structure).
Summary Table
| Claim | Status | Evidence |
|---|---|---|
| (2ℓ+1) = aridity of coupled ArXe level | Confirmed — exact | exact numerical correspondence |
| Madelung order = probability order | ⚠ Superseded 2026-07-30 | retracted — see arxe_periodic_table_paper_v2_en.md §4.4 (energy-based account) |
| Period lengths = 2n² | Confirmed — derived | count of couplings at depth n |
| (n+ℓ) = ontological depth in t² | Confirmed — derived | no continuous free parameters |
| Orbital energies in Planck units ArXe-pure | Coherent — 5/7 | independent cross-validation |
| g orbital instability (aridity 9=3²) | Prediction | non-arity number aridity → no fundamental level |
| Exceptions at Z-arity number thresholds | Structural — exploratory | motivated, statistically significant, not exhaustive |
| Z=119 follows Madelung normally | Prediction | 119 = 7×17, not an arity number |
| Z=127 shows exceptional behavior | Prediction | Z=127 is an ArXe arity |
For the primary ArXe framework: arxe_core_V4_221_en.md
For the ALO grammar: Grammar_V4_s_en.md (Pure Grammar)
For the dimensional framework used in Section 4: plov2_dimensional_framework_s_en.md