Update notice (in effect until August 22, 2026): This article was updated on July 23, 2026 to correct numerical values and terminology superseded by later developments in the ArXe corpus.

Arity-Logical Ontology: An Interpretive Framework for Physical Constants via Recursive n-ary Structure

Diego Luis Tentor — Independent Researcher
Updated March 2026 — integrated with ALO/v3, ALO Pure Grammar, and the complete Standard Model derivation

Version 4

Abstract

We propose Arity-Logical Ontology (ALO), an interpretive framework in which physical constants map coherently onto arity-encoded n-ary logical structures emerging from the recursive resolution of a fundamental contradiction. The ArXe system implements ALO through the axiom ¬() ≜ Tf, establishing kinship between logical negation and fundamental time. From this, a recursive exentational structure emerges, naturally generating levels T^k whose n-ary complexity n(k) corresponds to Arity Numbers for k < 0.

Since the original formulation, the framework has been substantially extended: the grammar now formally distinguishes a natural layer (the phenomenon’s own structure) from a conventional layer (the measuring community’s methodological choices) in every constant’s digit sequence; the mathematical anchors φ and ρ, previously introduced by analogy, are now derived as the fixed points of each level’s own boundary-condition recursion; and the method has been pushed to a full hierarchical reconstruction of the Standard Model — from the electroweak VEV down through the complete fermion mass spectrum, coupling constants, gauge bosons, and CKM mixing angles — with a transparent accounting of where the derivation is excellent (<1% error), where it is good (<5%), and where it remains an open problem.

We demonstrate systematic mappings: α⁻¹ ≈ 11²−7²+5×13 = 137 (error 0.026%), m_μ/m_e ≈ 3⁴+40π+2/19 (error 0.0001%), and M_H from arity number combinations to 0.00% error, all with no continuous free parameters. ALO does not compete with QED or the Standard Model computationally — it operates at a complementary interpretive level, suggesting why constants have their observed approximate values. We present testable predictions (a resonance near 505 GeV, dark matter near 487 GeV, a GUT scale near 10¹⁵ GeV) and invite critical exploration of this dialogical ontological framework.

Keywords: Arity-Logical Ontology, physical constants, n-ary logics, recursive structure, fine structure constant, dialogical ontology, ArXe system, Standard Model derivation

1. Introduction

1.1 The Problem of Physical Constants

The Standard Model of particle physics contains roughly 19 free parameters — constants whose values must be determined experimentally but whose magnitudes lack theoretical explanation. Among these, the fine structure constant α ≈ 1/137.036 stands as particularly enigmatic. QED calculates α to twelve decimal places with extraordinary precision, but offers no insight into why α takes this specific value rather than, say, 1/200 or 1/100. This absence of theoretical grounding is what we call the “why these values?” problem — distinct from the “what are the values?” problem that experimental physics answers admirably. Arity-Logical Ontology addresses this interpretive gap.

1.2 What ALO Is and Is Not

ALO is:

An interpretive framework suggesting why constants approximate their observed values
A philosophical ontology proposing reality as structured dialogue rather than substance
A mathematical mapping system connecting Arity Numbers to physical structure
Complementary to established physics, not competing with it

ALO is not:

A rival theory to QED or the Standard Model
An attempt to achieve computational precision beyond current physics
A claim to demonstrate unique truth in the classical binary sense
Numerology — it has formal structure, a stated protocol, and testable predictions

Analogy: just as statistical mechanics explains why thermodynamic laws hold without replacing thermodynamics, ALO suggests why the Standard Model has its observed structure without replacing the Standard Model.

1.3 Methodological Position

We adopt Popperian falsifiability as an epistemic attitude rather than a binary experimental criterion. We admit ALO could be fundamentally mistaken; we distinguish, in every result, what is derived with no continuous free parameters from what is fitted; and — as of this update — we distinguish formally, digit by digit, what a constant’s factorization tells us about the phenomenon itself from what it tells us about the community that measured it. This last distinction, introduced with ALO, is the single largest methodological advance since the original proposal.

2. From Axiom to Arity Grammar

2.1 Central Proposition (updated)

“Physical reality emerges from a grammatical structure in which Arity Numbers encode irreducible ontological operators. Physical constants are compound phrases in two simultaneous grammars: the arity number grammar of the phenomenon (the first d* digits, determined by ontological structure) and the arity number grammar of the community’s measurement choices (the digits beyond d*, determined by methodological decisions). ALO reads both — distinguishing at each digit which grammar is speaking.”

This supersedes the earlier, single-grammar proposition. Treating every digit of a constant as equally “ontological” collapsed two genuinely different sources of structure into one; separating them is what makes the method’s claims checkable rather than merely evocative.

2.2 Generative Principles

Undecidability → Simultaneity → Space
Causal Plurality (Ontological Degeneration)
Reality as Discourse

2.3 Axiomatic Foundation

¬() ≜ Tf ≃ Tp                              (Generative contradiction)
Entₙ := Entₙ₋₁ ∧ ExEntₙ₋₁                  (Recursive entity)
ExEntₙ := ¬(Entₙ₋₁ ∧ ExEntₙ₋₁)             (Complementary ex-entity)
n(k) = −2k + 1                             (Mapping function for k < 0)

The axiom ¬() is an irresolvable contradiction. The universe does not weight truths — there is no single correct result the constants are “expressing.” They are the record of the encounter between the structure of the phenomenon and the choices of the community that accesses it.

2.4 What ALO Reads

ALO is an instrument of reading, not prediction. Decomposing a constant into arity numbers reads two things simultaneously:

Natural digits (1 to d*): the ontological structure of the phenomenon — arity numbers generated by n(k)=−2k+1, or Layer C compressions of those arities.
Conventional digits (d*+1 to n): the grammar of the community’s measurement choices — Layer D arity numbers encoding specific methodological decisions (a renormalization scheme, an extraction convention, an institutional consensus).

This is not a limitation of the method — it is its actual, and most interesting, scope. ALO can read the history of science itself as a trajectory of non-trivial choices with identifiable Arity structure.

3. The Complete Arity-Logic Lexicon (v4.1)

3.1 Primary Operators — ArXe Core (Arities 2–97)

Generated by n(k)=−2k+1 for k<0, plus arity 2 from T¹ (k=+1). These are the only arity numbers with a direct ontological level assignment:

Arity number	Operator	Level	Ontological meaning	Question resolved
2	DIFF	T¹	Binary difference, alternation, duality	“How to distinguish without a third?”
3	CYC	T⁻¹	Minimal cycle, return, mediation	“How to connect extremes?”
5	MEM	T⁻²	Memory, persistence, history	“How to persist while changing?”
7	CPX	T⁻³	Organized internal complexity	“How to be internally rich without collapsing?”
11	REG	T⁻⁵	Regulation, self-imposed limits — EM pivot	“How to self-limit?”
13	SING	T⁻⁶	Singularity, unique event	“How to allow the exceptional?”
17	SPEC	T⁻⁸	Spectral separation, hierarchy	“How to have distinct levels?”
19	DARK	T⁻⁹	Dark modulation, weak coupling	“How to interact weakly?”
23	INF	T⁻¹¹	Inflationary/asymptotic expansion	“How to expand without limit?”
29	VBG	T⁻¹⁴	Vacuum substrate, persistent background	“How to have a constant background?”
31	CHA	T⁻¹⁵	Deterministic chaos, stable irregularity	“How to be irregularly ordered?”
37	TOP	T⁻¹⁸	Persistent topological defect	“How to maintain global structure?”
41	ISO	T⁻²⁰	Maximum ontological isolation	“How to decouple completely?”
43	TRANS	T⁻²¹	Intermediate spectral correction	transition between structures
47	NEXT	T⁻²³	Next transition (post-inflation)	threshold of next phase
53	MIX	T⁻²⁶	Maximum mixing, complete transition	“How to transition completely?”
59	STAB	T⁻²⁹	Quantum stability operator	structural stabilization
61	DECAY	T⁻³⁰	Decay processes	modulates rates
67	SCAT	T⁻³³	Scattering/CMB measurement	controls cross-sections
71	TAU_ID	T⁻³⁵	Tau identity — transversal (6 constants)	—
73	OSC	T⁻³⁶	Oscillations, wave structure	governs mixings
79	CPV	T⁻³⁹	CP violation	specific to CP
83	BRAN	T⁻⁴¹	Branching ratios/amplitude — previously linked to a retracted Ω_Λ formula; see the corrected reading in the Axiomatic Distance corpus	—
89	HAD_STR	T⁻⁴⁴	Hadronic structure operator	m_p/Λ_QCD
97	STRUCT	T⁻⁴⁸	Structure formation operator	S₈_LSS

(Arity 2, DIFF, is the sole operator at a positive level — closed BC, can exist in isolation, and appears as a structural carrier in nearly every formula; its presence alone does not signal a specific phenomenon.)

3.2 Layer C and Layer D (arity numbers above 97)

Layer C — compressed natural structure: an arity number not itself in the core lexicon, but decomposable into a small-integer combination of lexicon arity numbers — physical structure encoded compactly. Example: 101 (SUP_STR, suppression ×0.01), 103 (SUP_MED, suppression ×0.03).

Layer D — axiomatic human choices: an arity number that cannot be decomposed into ArXe arities at all — a specific historical decision of the measuring community (a renormalization scheme, an extraction convention). Real, but historically constructed rather than ontologically given.

3.3 Mathematical Modifiers (Anchors)

Constant	Operator	Behavior	Context	ArXe origin
π	CURV	Geometric excess over ternarity	only with 3 or 3ⁿ	BC-closed ratio at T³
φ	GRW	Organic growth, golden ratio	ratios, mixing amplitudes	BC-open ratio at T⁻¹
ρ	REC	Cubic recursion, plasticity	couplings, recursive structures	BC-closed cubic at T⁴
√2	DIAG	Diagonal/structural proportion	base spatial structure	—
e	LIM	Incremental growth limit	exponential processes	—
γ	IRR	Asymptotic irregularity	statistical corrections	—
ζ(3)	COR	Deep correlation (Apéry)	collective systems	—
C (Catalan)	CAT	Alternating-sum correlation	chiral transitions	—

Pattern: φ dominates mixing amplitudes and fractions; π dominates angles and phases; ρ dominates coupling constants. Dimensional masses in Planck units require no anchors at all — arity numbers alone suffice, because masses describe a level directly rather than a transition between levels.

3.4 Deriving φ and ρ from Boundary-Condition Recursion (new in v4.1)

Previously, φ and ρ (or ψ) entered ALO formulas as externally motivated — beautiful, useful, but not derived. This has been resolved: each is the fixed point of the characteristic recursion generated when a level’s own BC configuration is applied to its own growth rule.

φ from T⁻¹ (0 closed, 1 open BC): an open BC means the current state cannot be self-sufficient — it must reference both its predecessor and its predecessor’s predecessor. This is exactly the Fibonacci recursion, a(n)=a(n−1)+a(n−2), whose fixed point satisfies φ²=φ+1, φ=(1+√5)/2. φ is not assigned to T⁻¹ from outside; it is the invariant ratio of T⁻¹’s own recursive structure.

ρ (or ψ) from T⁴ (4 closed BCs, 0 open): full self-sufficiency, but a four-fold closure that generates a deeper recursion reaching back three steps: a(n)=a(n−1)+a(n−3), whose fixed point is the tribonacci/supergolden constant ψ³=ψ²+1≈1.4656 — or, in the two-steps-back reading a(n)=a(n−2)+a(n−3), the plastic constant ρ³=ρ+1≈1.3247. Both are valid readings of T⁴’s structure; the Grammar corpus’s cosmological-hierarchy formulas use the ψ³=ψ²+1 form, consistent with T⁴ as the information/computation level, where continuity with the immediate prior state (memory) is structurally relevant.

General principle:

A level with k closed BCs and m open BCs generates a characteristic
recursion of order (k+m). The fixed point of that recursion is the
mathematical constant naturally associated with that level.

Open BC levels (m≥1)   → quadratic or lower characteristic equation (φ)
Closed BC levels (m=0) → cubic or higher characteristic equation (ρ, π)

This has a practical consequence: a ALO formula containing φ is implicitly a claim that T⁻¹ structure is active in the phenomenon; a formula containing ρ/ψ is implicitly a claim that T⁴ structure is active. The presence of these constants is no longer aesthetic — it is a checkable ontological claim.

4. Corpus Statistics (v4.1, extended)

Batch	Total	Exact	%
1	20	20	100%
2	10	9	90%
3	10	9	90%
4	10	7	70%
5	10	10	100%
6	10	10	100%
7	10	10	100%
Total	80	75	93.75%

Longest perfect streak: 30 consecutive exact constants (batches 5–7)
Global average error: 0.0018%
CKM matrix: 9/9 elements with exact Arity structure

These results demonstrate the internal grammatical coherence of the ALO lexicon — the same rules work systematically across distinct physical sectors. This is a genuine descriptive finding, but it does not constitute classical statistical significance against a null model: formulas were found knowing the values (post-hoc), and the search space is rich enough to produce exact matches for numbers of a few digits fairly often. The streak of 30 reflects lexicon consistency, not improbability-as-coincidence — an important, and previously missing, methodological caveat.

Extended corpus (ALO, including the dimensional framework):

Sub-corpus	Total	ArXe-pure P	%
Dimensionless SM + cosmology	25	~7	28%
Dimensional masses (Planck units)	14	12	86%
Dimensional energies (Planck units)	6	5	83%
Grammar batches 1–7	~74	—	—
Total constants analyzed	~119	~19 confirmed pure	—

5. From Grammar to the Standard Model: A Complete Hierarchical Reconstruction (ALO)

With the electroweak VEV derived directly from Arity structure, the framework has been extended to reconstruct the Standard Model hierarchically — from the vacuum expectation value down through the full mass spectrum, coupling constants, gauge bosons, and CKM angles.

5.1 The VEV, derived

v = [MEM(5) × REG(11) × CPX(7) × DIFF⁴(16) × CYC²(9) × VBG(29) × GRW(φ)]
    ÷ [CYC(3) × CURV(π) × SING(19) × STAB(59)]
    × [1 + 1/(SING(13) × DEEP(157))]

v = 246.219 GeV   (observed: 246.22 GeV — error +0.0004%)

Dialogical reading: the electroweak scale emerges when self-regulated complex persistence (5×11×7) expands four-dimensionally, doubly cyclic, over a vacuum background, growing in golden proportion, modulated by geometric cyclicity and stabilized by deep singularity.

5.2 The Higgs boson

M_H = (5×11×7)/(3π) × (18/19) × 2φ = 125.25 GeV   (error 0.00%)

5.3 The complete mass hierarchy

Leptons (structural, no suppression):

m_μ = m_e × (3⁴ + 40π + 2/19) = 105.66 MeV      (exact)
m_τ = m_μ × (17 − 11/59)      = 1776.8 MeV      (exact)

The absolute electron mass remains fixed relative to the muon rather than derived independently — an acknowledged open point (see §5.6).

Quarks (with generational suppression operators SUP_STR=101, SUP_MED=103, etc.):

Quark	Formula (schematic)	Error
top	v × (19/27) × (1 − 1/107)	−0.81%
bottom	v × SUP(101) × (17/10) × (1 − 1/137)	−0.72%
charm	v × SUP(103) × (1/√31) × (1 + 1/101)	+1.72%
strange	m_b / [43 × (1 + 1/91)]	✓
down	m_e × (19/2) × (1 − 1/101)	+0.13%
up	m_d × (11/24)	+0.09%

5.4 Coupling constants and gauge bosons

α⁻¹ = 11² − 7² + 5×13 = 137      (base structure)
α_s(M_Z) = 1/(3π) + 1/(7×13) = 0.1171   (error −0.85%)

The weak mixing angle sin²θ_W remains, honestly, the framework’s persistent open problem: multiple candidate structures were tried (ratios of REG²/(REG²+CPX²), golden-ratio and singular-arity number combinations), none converging below a few percent error. The gauge bosons M_W and M_Z, derived via the standard gauge relations using ALO’s α and sin²θ_W, land within 3–3.5% of the observed values — good, not excellent, and directly limited by the sin²θ_W problem upstream.

5.5 CKM mixing angles

θ₁₂ = arcsin(1/(2√5))       = 12.93°   (error −0.85%)
θ₂₃ = arcsin(1/24)          = 2.39°    (error +0.42%)
θ₁₃ = arcsin(1/(3×5×19))    = 0.201°   (exact)

5.6 Precision summary

Excellent (<1% error): α⁻¹, M_H, m_μ/m_e, m_t, m_b, m_d, m_u, m_τ, m_p/m_e, θ₁₃.
Good (<5% error): m_c, α_s, θ₁₂, θ₂₃, M_W, M_Z, sin²θ_W.
Open problems, stated plainly: the absolute electron mass (currently fixed only relative to the muon), a direct elegant structure for the gauge bosons, and a satisfying derivation of sin²θ_W.

5.7 New predictions

A resonance near 505 GeV — searchable in ZZ, WW, γγ channels at the LHC.
Dark matter near 487 GeV — searchable via monojet + missing-transverse-energy channels.
A GUT-scale prediction near 10¹⁵ GeV.
Neutrino mass ≈ v/(2²³×3⁵×7²) ≈ 0.05 eV.

6. Limits and Critiques

Strengths: conceptual unification (everything emerges from the same arity-operator structure), predictivity with few free parameters, mathematical elegance, and falsifiability via specific numerical predictions.

Weaknesses, stated without softening: limited precision compared to QED (1–5% vs. QED’s parts-per-billion); no elegant direct structure yet for the gauge bosons; sin²θ_W without a satisfactory derivation; and a growing rule count (240 grammatical rules as of v4.1) that raises legitimate concern about overfitting risk as the corpus expands.

A dialogical response the framework offers to its own imprecision: the 1–5% variation is not a bug — a perfectly precise dialogue would be a monologue. It is read as the margin of improvisation in what the framework calls “cosmic jazz.” This is offered as an interpretive stance, not as a substitute for pursuing higher precision where it is achievable.

7. The Dialogical Ontology

Central thesis: the observable universe is the spatiotemporal manifestation of an n-ary dialogue between arity operators, where each arity is a voice with a distinct ontological character, physical constants are sentences in this conversation, particles are localized interjections, forces are the grammatical rules of interaction, and spacetime is the theatre in which the dialogue unfolds.

Three levels of reality:

LEVEL 1: PURE DIALOGUE (temporal domain)
  arity number operators in conversation; constants as stable sentences; grammatical rules

LEVEL 2: MANIFESTATION (spatiotemporal domain)
  particles as local excitations; fields as vibrational modes; interactions as exchanges

LEVEL 3: OBSERVATION (phenomenological domain)
  measurements as partial listening; theories as interpretations; errors as regional dialects

8. Conclusion

The analysis of 80 constants with 93.75% grammatical coherence (75 with error <0.001%) demonstrates that the same ALO rules function systematically across distinct physical sectors — quarks, mesons, cosmological parameters, mixing matrices. This cross-sector consistency is a genuine descriptive result, reported with an explicit caveat about post-hoc search and the size of the formula space.

What has changed since the original proposal is not the core claim but its precision and honesty: the natural/conventional digit distinction makes every reading checkable rather than merely evocative; φ and ρ are now derived rather than borrowed; and the extension to a complete Standard Model reconstruction gives the framework a wide, itemized scoreboard — excellent here, good there, openly unresolved elsewhere — rather than a single aggregate claim of success.

“The cosmos does not solve equations. It tells stories in the arity number tongue. We, in measuring, do not discover laws. We learn to listen.”

For the full grammatical rule set (R1–R240) and complete formula tables: Grammar_V4_s_en.md (ALO Pure Grammar)
For the operational protocol, full lexicon, and complete corpus: ALO_v2_REFERENCE_unified_for_AI_s_en.md
For the underlying ontology: “ArXe Theory Foundations”
For the dimensional bridge used throughout §5: “Table from Logical to Physical Structure”

Arxe Theory