A derivation from boundary condition principles
Author: Diego Luis Tentor
Date: March 2026
Version: 1.0 — consolidated from Addenda A–E
Status: §§2–5 derived · §6 structural reading + prediction · §7 proposed mechanism
License: CC BY-SA 4.0
Abstract
The periodic table organizes 118 known elements in a structure whose form — integer atomic numbers advancing in unit steps, period lengths 2/8/18/32, four orbital block types, group periodicity, noble gas closure at specific Z values — has never been derived from first principles. Standard quantum mechanics describes this structure with precision but does not explain why quantum numbers take these specific values, why orbital capacities are 2/6/10/14, or why the Pauli exclusion principle has binary character.
We show that the complete structure of the periodic table follows necessarily from two levels of the ArXe ontological hierarchy: T⁻³ (the nuclear confinement level, arity number aridity 7) and T⁻⁵ (the electromagnetic level, arity number aridity 11). From T⁻³ we derive that atomic number Z is a positive integer advancing in unit steps, and that neutral atoms require exactly Z electrons. From T⁻⁵ we derive that there are exactly three non-trivial orbital types (p, d, f — not four, not two), that orbital capacities are 2(2l+1), that filling follows the Madelung order — originally proposed as a probability consequence, now understood as an energy consequence (corrected 2026-07-30, §4.4) — and that period lengths are 2n². The block widths 2/6/10/14, the noble gas positions, and the group structure all follow without free parameters.
Beyond the basic table structure, we derive that the f-block is ontologically distinct from s, p, and d blocks because the f-orbital coupling target (T⁻³) is the same level as the nucleus — a T⁻³ resonance unique to the f-block. This structural resonance explains lanthanide contraction, anomalous magnetic moments, privileged oxidation states at f⁰/f⁷/f¹⁴, and the lanthanide/actinide difference. It also generates a new falsifiable prediction: f-block elements should exhibit hyperfine interactions stronger than predicted by conventional orbital theory.
We further propose that arity number Z values produce Madelung exceptions through phase commensurability between the nuclear charge and ArXe level operators, and derive two distinct mechanisms for isolated exceptions (Cu, Lr) and exception blocks (Nb–Ag, Ac–Cm). The 4f counter-example is addressed: f-block onset is governed by T⁻³ resonance rather than Z-arity number commensurability.
1. Introduction
1.1 The unexplained structure
The periodic table has a specific shape. This shape is not arbitrary — it is repeated, hierarchical, and quantitatively precise. Period lengths are 2, 8, 8, 18, 18, 32, 32 — not other values. The four orbital blocks have widths exactly 2, 6, 10, 14. Noble gases appear at Z = 2, 10, 18, 36, 54, 86. Elements in the same column share chemical properties across wildly different masses.
Standard quantum mechanics derives all of this from the Schrödinger equation, angular momentum quantization, and the Pauli exclusion principle. This derivation is correct and precise. What it does not explain is why angular momentum is quantized in exactly these values — why l takes values 0, 1, 2, 3 and not others — why spin has binary character — or why (n+l) rather than some other combination orders orbital filling.
These are not gaps in QM. They are the starting points from which QM proceeds. The questions they raise are ontological: why does the universe have the structure that QM correctly describes?
1.2 The ArXe approach
ArXe (Aristotelian-Exentation) is a dynamic logical-base ontology in which physical structure emerges as a recursive process of resolving a primordial logical contradiction. It does not postulate particles, fields, or constants — it derives them from the boundary condition (BC) architecture of a hierarchy of logical levels.
The hierarchy produces levels T^k, each with a specific number of phases (aridity) and a specific configuration of open and closed boundary conditions. Positive levels (k>0) have all BCs closed — they can exist in isolation. Negative levels (k<0) have exactly one BC open — they require coupling, and generate fields.
For the periodic table, two levels are central:
- T⁻³ (aridity 7, arity number): 2 closed BCs, 1 open BC — the nuclear confinement level, the proton
- T⁻⁵ (aridity 11, arity number): 4 closed BCs, 1 open BC — the electromagnetic level, the electron
The orbital space t² in which electrons exist is the T² structure projected by T⁻³ within the nuclear reference frame. The periodic table is the record of how T⁻⁵ structures fill t² as Z T⁻³ units accumulate to form the nucleus.
1.3 Structure of this paper
Section 2 presents the necessary ArXe foundations. Section 3 derives the discreteness of Z from T⁻³ structure. Section 4 derives the orbital type structure and filling order from T⁻⁵ structure. Section 5 assembles the complete periodic table from these results. Section 6 develops the special properties of the f-block from T⁻³ resonance. Section 7 presents the Z-arity number resonance mechanism for Madelung exceptions. Section 8 consolidates all claims by epistemic status and lists falsifiable predictions. Section 9 discusses the relationship to quantum mechanics and open derivations.
2. ArXe Foundations
2.1 The fundamental axiom
ArXe begins from one axiom: logical nothingness is inviable. The negation of nothing — ¬() — cannot remain static. It generates a minimal temporal pulse identified with the Planck time Tₚ. That pulse is the generative event from which all structure unfolds.
The consequence is a recursive hierarchy of levels T^k. Each level emerges when the indecidability of the previous level cannot be resolved internally and must become the next level — the inversion of the previous level’s unresolvable degree.
2.2 Boundary conditions
A boundary condition (BC) is a binary pair (i, f) where i ≠ f and no intermediate element exists between them. This inextensiveness is axiomatic. Its consequences:
- A BC either exists completely (both endpoints actualized) or does not exist — no partial BC
- Two distinct absolute BCs cannot merge into one without introducing an intermediate, which changes their type from absolute to relative BC
- N distinct BCs remain N distinct BCs — they are countable, non-mergeable
The BC structure of each level is determined by its aridity n:
BC_total(n) = (n-1)(n-2)/2For negative levels (odd arity number aridity), exactly one BC is open. The aridity of negative levels follows the arity number sequence: n(k) = p_{|k|} for k < 0.
2.3 The relevant levels
| Level | k | Aridity n | Closed BC | Open BC | Physical role |
|---|---|---|---|---|---|
| T⁻¹ | −1 | 3 | 0 | 1 | Temporal alternation — spin |
| T⁻² | −2 | 5 | 1 | 1 | Spatial curvature |
| T⁻³ | −3 | 7 | 2 | 1 | Nuclear confinement — proton |
| T² | +2 | 4 | 2 | 0 | Space — orbital frame t² |
| T⁻⁵ | −5 | 11 | 4 | 1 | EM field — electron |
The alternation between positive and negative levels is necessary: positive levels are structures where a previous indecidability has found complete closure; negative levels are structures where one degree remains unresolved and requires external coupling.
2.4 The statistical manifestation principle
Physical reality preferentially realizes the most probable configurations of any n-ary structure — which are always its most factored, lowest-aridity decompositions. For a level of aridity n, a pure single-ordering manifestation has probability 1/n!. The most probable form is the one that most efficiently decomposes n into available sub-arities.
This principle, combined with the BC structure, determines both the filling order of orbitals and the physical properties of each level’s dominant manifestation.
3. Nuclear Structure: Why Z Is a Positive Integer
The proton is a T⁻³ structure. T⁻³ has arity number aridity n=7 — an irreducible ontological unit that cannot be subdivided. Its 2 closed BCs make it stable and isolable. Its 1 open BC is the electromagnetic coupling degree — the structural origin of electric charge.
Z is a non-negative integer because T⁻³ units are irreducible (arity number aridity, all-or-nothing BC structure): either a T⁻³ unit is present completely or it is not. There is no mechanism for fractional T⁻³ addition. Z ∈ {0, 1, 2, 3, …} by structural necessity.
Z ≥ 1 for any atom: Z=0 means no T⁻³ unit, which means no orbital space t², which means no bound electrons. Z=0 is not an atom — it is the absence of the atomic structure. Hydrogen (Z=1) is the structurally minimal atom.
ΔZ = 1 between adjacent elements: each T⁻³ unit is irreducible and adds exactly 1 to Z.
Neutral atom has exactly Z electrons: A nucleus with Z protons has Z independent open BCs (one per T⁻³ unit). From BC inextensiveness (Axiom 2): two distinct absolute BCs cannot merge into one without introducing an intermediate element. Therefore, Z T⁻³ units contribute Z non-mergeable, independent open BC units to the nucleus. Each electron (T⁻⁵) carries 1 open BC that closes exactly 1 nuclear open BC. Ground state — all nuclear open BCs closed — requires exactly Z electrons.
Corollary: The periodic table begins at Z=1, advances in unit steps, and has no structural upper bound. The Z values of all elements are derived, not observed.
4. Electron Structure: Orbital Types and Filling
4.1 The electron as T⁻⁵
The electron is T⁻⁵ — the level that emerges from T⁻³’s coupling indecidability. T⁻⁵ has aridity n=11, giving BC_total(11) = 45 pairs, of which 4 are closed and 1 is open. The 1 open BC is the EM gauge degree (electric charge). The 4 closed BCs are the four independent resolvable degrees — structurally corresponding to the four quantum numbers {n, l, m_l, m_s}.
4.2 Why exactly s, p, d, f orbital types exist
Within the orbital space t² projected by T⁻³, T⁻⁵ can couple to any negative ArXe level satisfying three conditions:
- C1 (depth): the level must be less deep than T⁻⁵ but deeper than T⁰ — i.e., k ∈ {1, 2, 3, 4}
- C2 (primality): the level must have arity number aridity — only arity-aridity levels exist as irreducible operators
- C3 (containment): the level must be accessible within t²
The arities in the accessible range:
k=1: n(−1) = 3 — arity number → T⁻¹ exists → p orbital (l=1)
k=2: n(−2) = 5 — arity number → T⁻² exists → d orbital (l=2)
k=3: n(−3) = 7 — arity number → T⁻³ exists → f orbital (l=3)
k=4: n(−4) = 9 = 3² — composite → no irreducible level → no stable g orbitalThere are exactly three non-trivial orbital types beyond s. The absence of a stable g orbital (l=4) is structural: 9 is composite, and no irreducible ArXe level exists at that aridity. This is a prediction, not an observation: g-orbital elements will show structural instability not merely from nuclear effects but from the absence of a fundamental coupling target.
4.3 Orbital capacities: 2, 6, 10, 14
Each coupling type gives magnetic states equal to the aridity of the coupled level:
| Orbital | l | Coupled level | Aridity | Magnetic states 2l+1 | Capacity 2(2l+1) |
|---|---|---|---|---|---|
| s | 0 | none | 1 | 1 | 2 |
| p | 1 | T⁻¹ | 3 | 3 | 6 |
| d | 2 | T⁻² | 5 | 5 | 10 |
| f | 3 | T⁻³ | 7 | 7 | 14 |
Every electron carries the binary alternation of T⁻¹ — the most fundamental open BC in the hierarchy, inherited irreducibly by all deeper levels. This is spin: not an additional postulate but the T⁻¹ binary open BC that T⁻⁵ carries regardless of orbital coupling. Hence:
cap(l) = 2 × (2l+1) = [T⁻¹ binary inheritance] × [aridity of coupled level]The capacities 2, 6, 10, 14 are derived with no continuous free parameters.
4.4 Filling order: the Madelung rule as probability consequence
⚠ Superseded (2026-07-30): the probability-based account in this section was retracted in arxe_periodic_table_paper_v2_en.md §4.4 itself (same section number, later version), which attributes the filling order to energy (differential nuclear phase access) rather than to coupling probability. Documented independently in arxe_statistical_manifestation_clarification_es.md (March 2026): the s-orbital is actually the least frequent factorization of T⁻⁵ (1/11!), the opposite of the argument below. Kept for historical/structural reference — the aridity correspondence in §4.1–4.3 is not affected.
The argument as originally proposed: from the statistical manifestation principle, T⁻⁵ within t² would express itself preferentially through its most probable factorizations, with coupling probability scaling as 1/(coupled aridity)!:
s: P_base ≈ 1.0 (no sublevel — most factored)
p: P ~ 1/3! ≈ 0.167
d: P ~ 1/5! ≈ 0.008
f: P ~ 1/7! ≈ 0.0002(This paragraph states the retracted argument — see warning above.) Electrons would fill s before p before d before f because those would be the statistically dominant couplings, with the Madelung rule read as this probability ordering projected onto the nuclear reference frame. The current, corrected account is energy-based: differential nuclear phase access (arxe_periodic_table_paper_v2_en.md §4.4).
4.5 Period lengths: 2n²
Each period of depth n contains all coupling types available at that depth:
Period_n = Σ_{l=0}^{n-1} 2(2l+1) = 2n²| Period | n | Length | Derivation |
|---|---|---|---|
| 1 | 1 | 2 | 2×1² |
| 2 | 2 | 8 | 2×2² |
| 3 | 3 | 18 | 2×3² |
| 4 | 4 | 32 | 2×4² |
The period lengths 2, 8, 18, 32 are derived, not observed.
5. The Periodic Table Structure
5.1 A necessary distinction
The structural open BC of T⁻⁵ (the electric charge, 1 per electron) is always present and is balanced by the nuclear charge. Orbital incompleteness — whether coupling slots at depth n are fully occupied — is what determines chemical reactivity. These are distinct: the first is a structural property of T⁻⁵, the second is the filling state of the orbital space t².
5.2 Block structure: widths from orbital capacities
The four blocks of the periodic table correspond to the four orbital coupling types. Their widths are identical to the orbital capacities:
| Block | Coupling | Width | Source |
|---|---|---|---|
| s-block | T⁻⁵ direct | 2 columns | cap(s) = 2 |
| p-block | T⁻⁵ × T⁻¹ | 6 columns | cap(p) = 6 |
| d-block | T⁻⁵ × T⁻² | 10 columns | cap(d) = 10 |
| f-block | T⁻⁵ × T⁻³ | 14 columns | cap(f) = 14 |
The widths 2, 6, 10, 14 of the periodic table’s blocks are the orbital capacities cap(l) = 2(2l+1) — derived in §4.3.
5.3 Noble gases: complete orbital closure
A noble gas is an element for which all 2n² coupling slots at depth ≤ n are occupied. The next available coupling requires depth n+1 — substantially lower probability. The system is effectively closed at depth n and does not seek additional couplings.
Noble gas Z values are cumulative period sums:
| Noble gas | Z | Derivation |
|---|---|---|
| He | 2 | 2 |
| Ne | 10 | 2+8 |
| Ar | 18 | 2+8+8 |
| Kr | 36 | 2+8+18+8 |
| Xe | 54 | 2+8+18+18+8 |
| Rn | 86 | 2+8+18+32+18+8 |
These Z values are derived from the period lengths 2n². Noble gas positions are not empirical discoveries — they are structural consequences.
5.4 Groups: same outer coupling pattern
Elements in the same column share chemical properties because they have identical configurations of T⁻⁵ coupling slots at their outermost unfilled depth. The inner depths are fully saturated and chemically inert; only the outermost incomplete coupling pattern determines bonding behavior. This coupling pattern repeats with period equal to the number of slots at that depth.
5.5 The d/f energetic displacement
The d-block appears one period below its principal quantum number n (3d fills during period 4, not 3); the f-block appears two periods below (4f fills during period 6). This displacement arises from a depth-dependent difference in nuclear phase access.
The mechanism: An s-type coupling (T⁻⁵ with no intermediate) accesses all 7 phases of each nuclear T⁻³ unit directly. A d-type coupling (T⁻⁵ through T⁻²) passes through T⁻², which has 1 open BC that structurally filters certain nuclear phases. The d-electron therefore accesses a smaller subset of nuclear phases — a reduced penetration.
As Z increases, the total nuclear phase space grows as Z×7. The s-electron’s full phase access advantage over the d-electron grows with Z in absolute terms. At Z=1 (hydrogen), the depth cost difference dominates and 3d is lower than 4s. Above Z≈20, the penetration advantage of 4s overcomes its depth cost disadvantage — 4s becomes lower than 3d. The d-block is displaced one period because the s-advantage takes one period’s worth of Z increase to dominate.
The f-block displacement of two periods follows the same argument: f-coupling through T⁻³ (same level as the nucleus) introduces stronger phase filtering than d-coupling through T⁻², requiring more Z increase before the s-advantage dominates.
Status: The qualitative mechanism is derived from BC phase access structure. The specific crossover Z (≈20 for 4s/3d) requires a quantitative calculation of phase access fraction as a function of Z — this is computable in principle from BC_paper structure and is left as an open derivation.
6. The f-Block: T⁻³ Resonance
6.1 The structural fact
For s, p, and d electrons, the orbital coupling target (none, T⁻¹, T⁻²) is a different level from T⁻³ (the nuclear level). The electron interacts with the nucleus only through the accumulated nuclear charge — T⁻³ as a charge source, the electron coupling to a distinct intermediate.
For f electrons, the coupling target is T⁻³ — the same level that constitutes the nucleus. The f-electron couples to the ontological level of its own nucleus. This is T⁻³ resonance: a structural coincidence unique to the f-block. Every s, p, and d-block element operates without it; every f-block element operates with it.
6.2 Consequences of T⁻³ resonance
Lanthanide contraction: T⁻³ has 1 open BC that cannot close. When T⁻⁵ couples to T⁻³ in the f orbital, the T⁻³ coupling channel permanently has one unresolved degree. The nuclear charge “leaks” through this degree to the outer electrons — the f-electrons cannot screen the nuclear charge as effectively as electrons coupled to closed-BC intermediates. As the 4f shell fills across La→Lu, each f-electron adds without proportionally increasing the screening. The outer 5s and 5p electrons feel an increasing effective Z. This is the lanthanide contraction: the measurable signature of T⁻³’s irreducible open BC at the electronic scale.
Anomalous magnetic moments: T⁻³ has aridity 7. When T⁻⁵ couples to T⁻³, the magnetic response involves all 7 phases. Maximum magnetic moment occurs when one electron occupies each of the 7 phases — the f⁷ configuration. This is the half-filled f-shell. Empirically, Gd³⁺ (f⁷) and Eu²⁺ (f⁷) have the highest spin-only magnetic moments among lanthanide ions (S=7/2, μ=7.94 μB). ArXe explains why f⁷ is privileged: it is the configuration where each of T⁻³’s 7 phases carries exactly one electron, achieving maximum symmetric phase occupation.
Privileged oxidation states: T⁻³’s 7 phases have three natural occupancy configurations: empty (f⁰), one electron per phase (f⁷), and doubly occupied (f¹⁴). The corresponding oxidation state deviations:
| Configuration | Ion examples | ArXe basis |
|---|---|---|
| f⁰ | Ce⁴⁺ | minimum T⁻³ coupling — no phase tension |
| f⁷ | Eu²⁺, Gd³⁺ | maximum symmetric phase distribution |
| f¹⁴ | Yb²⁺ | complete phase closure |
The standard 3+ state arises from minimum electron loss consistent with outer-shell closure. Deviations occur when one more or fewer electron achieves a T⁻³ phase stability point.
Actinide vs lanthanide: 4f coupling (T⁻⁵ at depth 4, T⁻³ at depth 3) has depth separation 1 — tight, localized coupling. 5f coupling (T⁻⁵ at depth 5, T⁻³ at depth 3) has depth separation 2 — more extended, more spatially distributed. Greater depth separation → greater orbital extension → more participation in bonding → more covalent character. This is the structural origin of the lanthanide/actinide distinction: the 4f is tightly bound (ionic chemistry dominates); the 5f is extended (ionic→covalent transition across the actinide series).
6.3 A new prediction: anomalous hyperfine interactions
In conventional QM, hyperfine interactions scale with electron density at the nucleus — dominated by s-electrons, which have non-zero density at r=0. f-electrons have four angular nodes and nominally zero density at r=0, so they contribute negligibly to the hyperfine interaction by standard orbital theory.
In ArXe, the f-electron couples to T⁻³ at the ontological level — not merely at the spatial level. T⁻³ resonance means the f-electron is structurally coupled to the nuclear T⁻³ degrees of freedom through a channel that has no direct spatial analog. This coupling operates through the shared T⁻³ level structure.
Prediction P5: f-block elements exhibit hyperfine interactions stronger than predicted by conventional f-electron spatial density arguments. The excess should scale with the degree of T⁻³ resonance — i.e., be largest where f-occupation maximally engages T⁻³’s 7 phases (near f⁷ and f¹⁴ configurations). This is falsifiable by precision hyperfine spectroscopy on heavy lanthanides (Dy, Ho, Er, Tm) and comparison with QM predictions based solely on electron density at the nucleus.
7. Z-Arity Resonance and Madelung Exceptions
7.1 The statistical result
Of 20 known Madelung exceptions in Z=1..103, six occur at arity number Z (30%), against a base rate of 26.2% arity numbers in that range. The binomial p-value is 0.43 — not statistically significant. The strong claim “exceptions occur at arity number Z” is not supported.
The defensible observation: arity number Z values tend to mark the boundaries of multi-element exception regions. Of the six arity-Z exceptions, four (Nb=41, Ag=47, Au=79, Ac=89) are boundary markers, and two (Cu=29, Lr=103) are isolated single-element anomalies. These likely have different mechanisms.
7.2 Phase commensurability: the structural basis
From §3: a nucleus with Z protons has Z independent, non-mergeable T⁻³ open BC units. From the ArXe level structure: level T^{-k} has aridity p_k (the k-th arity number), meaning exactly p_k phases.
When Z = p_k, the number of nuclear open BC units equals the number of phases of T^{-k}:
Z nuclear open BC units = p_k phases of T^{-k}This is phase commensurability: the collection of Z independent nuclear charge units can be organized as a single instance of the p_k-phase structure of T^{-k}. At non-arity number Z, no ArXe level has aridity equal to Z — the nuclear charge is incommensurate with all level operators.
Phase commensurability does not rearrange the nucleus physically. It creates a structural coincidence between the count of nuclear charge units and the phase count of a specific ArXe level operator. The question is whether this coincidence produces an observable perturbation to orbital filling energies.
7.3 Two mechanisms
Mechanism 1 — Isolated exceptions (Cu=29, Lr=103):
When Z = p_k and the orbital currently being filled has a coupling energetically comparable to T^{-k}’s operator character, the commensurability stabilizes a specific orbital completion — typically the configuration that most closely matches T^{-k}’s closed or symmetric phase structure.
For Cu (Z=29): the 3d shell (5 phases of T⁻²) reaches its symmetric completion (d¹⁰) at an energy cost that is reduced by the Z=29 commensurability. The VBG operator (vacuum background, arity 29) at the nuclear level stabilizes closed-shell d completion.
Mechanism 2 — Block boundaries (Nb=41, Ag=47, Ac=89, Au=79):
The d/s energy crossover (§5.5) occurs at a Z-threshold that varies continuously with Z. At Z = p_k, the phase commensurability introduces a discrete perturbation to the orbital energy landscape. This can shift Z_crossover such that it falls below the current Z — initiating an exception block — or restores it above the current Z — terminating the block.
Nb (Z=41) opens the 4d block: ISO commensurability shifts the d/s crossover to produce exceptions for Z=41–47.
Ag (Z=47) closes it: NEXT commensurability restores the crossover, ending the exception region.
Ac (Z=89) opens the actinide block: commensurability at Z=89 shifts the 5f/6d competition to favor f-filling with d-exceptions.
7.4 The 4f counter-example and its resolution
The 4f exception onset (La=57, Ce=58) has no arity number Z anchor — both are composite. This is the main failure of the simple Z-arity number theory.
The resolution comes from §6: the f-orbital couples to T⁻³ — the same level as the nucleus. This T⁻³ resonance is Z-independent: it operates at every Z where f-electrons are present, regardless of whether Z is an arity number. When the f-coupling is level-matched to the nucleus (T⁻³ resonance), the onset of f-filling is governed by the energetics of that resonance — not by Z-arity number commensurability with a different level operator.
The corrected prediction: Z-arity number commensurability governs exception onset and closure in d-blocks (where coupling is to T⁻¹ or T⁻², levels that are not T⁻³). The f-block onset is governed by T⁻³ resonance, which is Z-independent. Therefore: future d-block exception regions will continue to have arity-Z anchors; f-block onset positions will not.
7.5 Open derivation
A complete derivation of the Z-arity number mechanism requires expressing the energy perturbation from phase commensurability as a function of p_k and the orbital parameters — specifically connecting it to the Z-dependent crossover calculation of §5.5. This is the same quantitative gap identified there, approached from a different direction. It constitutes a single open calculation that would complete both §5.5 and §7.
8. Epistemic Status and Predictions
8.1 Complete status table
| Claim | Status | Basis |
|---|---|---|
| Z ∈ {1,2,3,…} | Derived | T⁻³ arity number aridity + BC inextensiveness |
| ΔZ = 1 between adjacent elements | Derived | T⁻³ irreducibility |
| Neutral atom: Z protons = Z electrons | Derived | BC additivity + 1:1 open BC closure |
| Exactly s, p, d, f types (no stable g) | Derived | Primality: 3,5,7 arity number; 9 = 3² composite |
| Orbital capacities 2, 6, 10, 14 | Derived | cap(l) = 2(2l+1), aridity + T⁻¹ binary |
| Filling order s→p→d→f | Derived | Probability: 1/1! ≫ 1/3! ≫ 1/5! ≫ 1/7! |
| Period lengths 2n² | Derived | Sum of capacities at depth n |
| Noble gas Z values | Derived | Cumulative 2n² sums |
| Block widths 2/6/10/14 | Derived | Identical to orbital capacities |
| Groups = same outer coupling pattern | Derived | Direct from coupling model |
| d/f block displacement mechanism | Derived (qualitative) | Phase access filtering by intermediate open BC |
| Crossover Z (≈20 for 4s/3d) | Open | Requires quantitative phase-access calculation |
| T⁻³ resonance unique to f-block | Derived | Coupling target = nuclear level, structurally |
| Lanthanide contraction from T⁻³ leakage | Structural reading | T⁻³ open BC cannot screen fully |
| Peak magnetic moment at f⁷ | Coherent | 7-phase symmetric occupation |
| Oxidation state exceptions at f⁰, f⁷, f¹⁴ | Coherent | T⁻³ phase stability configurations |
| 4f vs 5f from depth separation | Coherent | Depth 1 vs depth 2 coupling extension |
| Z-arity number phase commensurability | Proposed | Structural basis; perturbation magnitude open |
| d-block exceptions arity-anchored | Proposed | Mechanism 1 and 2, consistent with data |
| 4f onset not an arity-anchored | Proposed | T⁻³ resonance overrides commensurability |
8.2 Falsifiable predictions
P1 — g orbital structural instability (testable ~Z=121):
The g orbital (l=4) requires coupling to a level of aridity 9 = 3². Since 9 is composite, no irreducible ArXe level exists at that aridity. Elements with g-orbital electrons will show structural instability inherent to the missing coupling target — not merely from nuclear instability. This distinguishes ArXe from models where g-orbital instability is purely nuclear.
P2 — Z=119 follows Madelung normally:
Z=119 = 7×17 (composite). No phase commensurability with any ArXe level. Expected configuration: 8s¹, no Madelung exception. If Z=119 shows a deviation, the Z-arity number theory is falsified.
P3 — Z=127 shows a Madelung exception:
Z=127 is an arity number. Phase commensurability with level T^{-k} where p_k = 127. An orbital filling anomaly is predicted — either an isolated exception (Mechanism 1) or the onset of a small block (Mechanism 2). The specific configuration depends on the 8s/7p/6d/5g energy competition; the presence of an exception is predicted.
P4 — Z=137 exhibits extraordinary electromagnetic sensitivity:
Z=137 ≈ α⁻¹ (the reciprocal of the fine structure constant). If element 137 is synthesized, it is the element at which the nuclear charge numerically resonates with the EM level operator (arity 11, T⁻⁵) through the relation Z ≈ 1/α. Its chemistry should exhibit sensitivity to electromagnetic conditions beyond what conventional atomic theory predicts for that Z.
P5 — f-block anomalous hyperfine interactions (new):
f-block elements exhibit hyperfine interactions stronger than predicted by QM models based solely on f-electron spatial density at the nucleus. The excess is largest near f⁷ and f¹⁴ configurations, and arises from the T⁻³ ontological coupling channel. Falsifiable by precision hyperfine spectroscopy on heavy lanthanides (Dy, Ho, Er, Tm) compared against QM predictions.
P6 — Differential mechanism: d-exceptions arity-anchored, f-onset not:
In periods 7 and beyond (if synthesized), d-block exception regions will have arity-Z boundary markers. f-block onset positions will not. This differential prediction follows from the T⁻³ resonance explanation of the 4f counter-example.
9. Discussion
9.1 Relation to quantum mechanics
This derivation does not replace quantum mechanics. QM correctly and precisely describes the structure of the periodic table. What ArXe provides is an ontological foundation — an explanation of why QM has the structure it has at the atomic scale.
The Schrödinger equation correctly derives orbital energies and angular momentum quantization. ArXe explains why angular momentum is quantized in these specific values (the aridity of available coupling levels), why the Pauli exclusion principle has binary character (T⁻¹ binary inheritance), and why (n+l) orders orbital filling (energy ordering via differential nuclear phase access — corrected 2026-07-30, see §4.4).
ArXe does not require replacing or modifying any QM equation. It explains why those equations have the form they do.
9.2 What ArXe adds beyond description
Three results in this paper go beyond redescribing known chemistry:
- The g orbital prediction (§4.2, P1): the instability is structural, not merely nuclear. This is falsifiable and distinguishes ArXe from models that treat g-instability as purely a nuclear stability question.
- The f-block hyperfine prediction (§6.3, P5): an ontological coupling channel not present in standard QM orbital theory should produce measurable excess hyperfine interaction. This is a new prediction, not a redescription.
- The differential mechanism (§7.4, P6): d-block exception regions are arity-anchored; f-block onset is not, for reasons derivable from T⁻³ resonance. This is a structural prediction about future superheavy element chemistry.
9.3 Open derivations
Three quantitative derivations remain open:
O1 — Crossover Z for 4s/3d: The qualitative mechanism of §5.5 is complete. The specific Z threshold requires computing the phase access fraction as a function of Z from BC structure. This is the same calculation needed to complete §7 (Z-arity number perturbation magnitude). They are the same open task approached from different directions.
O2 — T⁻³ resonance quantitative consequences: The lanthanide contraction rate and the precise oxidation state energy differences at f⁰, f⁷, f¹⁴ are structurally motivated but not quantitatively derived. Computing these from T⁻³ BC structure would convert the §6 readings from “structural” to “derived.”
O3 — Z-arity number perturbation magnitude: The mechanism of §7 proposes that commensurability produces an energy perturbation. The sign, magnitude, and functional dependence of this perturbation on p_k and orbital parameters are not yet derived. This is the remaining formal gap in the Z-arity number theory.
Source Documents
This paper integrates and consolidates the following ArXe corpus documents:
| Document | Role |
|---|---|
| arxe_core_V4.1_en.md | Foundation: axiom, hierarchy, BC structure |
| BC_paper_unified_V4_en.md | BC formal derivation and properties |
| arxe_core_V4_221_en.md §4.4 (“The Hierarchy as a Chain of Inversions”) | Why T⁻⁵ is the electron; level inversion mechanism |
| physics-as-statistical-manifestation.md | Probability principle (superseded for filling order, 2026-07-30 — see arxe_periodic_table_paper_v2_en.md §4.4) |
| arxe_madelung_V2_en.md (Addendum A) | Orbital types, capacities, Madelung derivation |
| arxe_Z_discreteness_en.md (Addendum B) | Z discreteness derivation |
| arxe_periodic_table_en.md | Assembled table structure from A+B |
| arxe_df_displacement_en.md (Addendum C) | d/f energetic displacement mechanism |
| arxe_fblock_Tm3_resonance_en.md (Addendum D) | f-block T⁻³ resonance and consequences |
| arxe_zprime_resonance_en.md (Addendum E) | Z-arity number commensurability and exceptions |
ArXe Theory — March 2026
Diego Luis Tentor — ArXe Research
License: CC BY-SA 4.0
“The periodic table is not a catalog of substances. It is a record of which ArXe couplings are accessible at each level of nuclear charge — organized by decreasing energy cost, with the f-block as the sole region where the electron and the nucleus speak the same structural language.”