Update notice: This article was substantially revised on August 13, 2026. Part II (Methodology) was rewritten: the search-and-fit protocol and the “error as information” framing were removed because they made the framework unfalsifiable in practice — any deviation could be reinterpreted as a missing operator rather than a wrong prediction. The worked examples in Part III now show only the accepted formulas, not the discarded attempts. See the note at the start of Part II for what changed and why.
ArXe System: Complete Reference & Exploration Guide
Version 1.1 – August 2026 (methodology revision)
FOREWORD: THE DIALOGICAL APPROACH
This is not a textbook of fixed truths. It’s a grammar guide for ‘cosmic’ conversation.
What this document IS:
- A systematic framework mapping Arity structure to physical constants
- A collaborative exploration tool (designed for AI-assisted discovery)
- An invitation to dialogue with reality’s deep grammar
What this document IS NOT:
- A final theory claiming unique truth
- A replacement for QED/Standard Model computations
- A dogmatic system defending itself against falsification
The Attitude:
“Let’s see where this takes us” — not “Let me prove this to you”
For Humans: Read, explore, question, challenge
For AI (especially Claude): Partner in systematic derivation. Report what does and doesn’t fit, without reinterpreting a miss as a missing operator.
TABLE OF CONTENTS
PART I: FOUNDATIONS
- Central Thesis & Principles
- The Complete Arity-Logical Lexicon
- Grammatical Rules (R1-R107)
- Generative Grammar & Canonical Forms
PART II: METHODOLOGY
- Derivation Procedure & Its Limits
- Notation: The C = F × (1 ± ε) Form
- Handling Misses: When a Structure Doesn’t Fit
- Validation Criteria
PART III: APPLICATIONS
- Worked Examples (α, M_H, θ_W, etc.)
- Predictions & Testability
- Open Problems & Research Directions
PART I: FOUNDATIONS
1. CENTRAL THESIS & PRINCIPLES
1.1 THE CORE PROPOSITION
“Physical reality emerges from a grammatical structure where Arity Numbers encode irreducible ontological operators, and physical constants are composed phrases in this arity number language.”
1.2 GENERATIVE PRINCIPLES
Principle 1: Indecidability → Simultaneity → Space
Logical indecidability at fundamental level manifests as spatial extension
Principle 2: Causal Plurality
Multiple structural geneses converge to same observable phenomenon (ontological degeneracy)
Principle 3: Reality as Discourse
The universe is text written in arity number alphabet, not substance following laws
1.3 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)From these emerge hierarchical levels T^k with n-ary logical structure where certain k values produce arity number n.
2. THE COMPLETE ARITY-LOGICAL LEXICON
2.1 PRIMARY OPERATORS (Arities 2-53)
| Arity number | Operator | Ontological Meaning | Problem Resolved |
|---|---|---|---|
| 2 | DIFF |
Binary difference, alternation | “How to distinguish without third?” |
| 3 | CYC |
Minimal cycle, return, mediation | “How to connect extremes?” |
| 5 | MEM |
Memory, persistence, history | “How to persist while changing?” |
| 7 | CPX |
Organized internal complexity | “How to be internally rich without collapsing?” |
| 11 | REG |
Regulation, self-imposed limits | “How to self-limit?” |
| 13 | SING |
Singularity, unique event | “How to allow the exceptional?” |
| 17 | SPEC |
Spectral separation, hierarchy | “How to have distinct levels?” |
| 19 | DARK |
Dark modulation, weak coupling | “How to interact weakly?” |
| 23 | INF |
Inflationary self-similar expansion | “How to grow maintaining form?” |
| 29 | VBG |
Vacuum substrate, persistent background | “How to have constant background?” |
| 31 | CHA |
Deterministic chaos, stable irregularity | “How to be orderly irregular?” |
| 37 | TOP |
Persistent topological defect | “How to maintain global structure?” |
| 41 | ISO |
Stable isolation | “How to exist without interacting?” |
| 43 | TRANS |
Intermediate spectral correction | Transition between structures |
| 47 | NEXT |
Next transition (post-inflation) | Threshold of next phase |
| 53 | MIX |
Maximal mixing, complete transition | “How to transit completely?” |
2.2 IDENTITY OPERATORS (High Arity numbers)
| Arity number | Operator | Particle/Context | Relation |
|---|---|---|---|
| 61 | DECAY |
Decay processes | Modulates rates |
| 67 | SCAT |
Scattering | Controls cross-sections |
| 71 | TAU_ID |
Tau (mass) | Base tau identity |
| 73 | OSC |
Oscillations | Governs mixing |
| 79 | CPV |
CP violation | Specific to CP |
| 83 | BRAN |
Branching ratios | Ramifications |
| 101 | CENT |
Centenary threshold | First 3-digit arity number |
| 151 | TAU_ANOM |
Tau (anomaly) | 2×71 + 9 |
| 211 | E_ID |
Electron (mass) | Base electron identity |
| 431 | E_ANOM |
Electron (anomaly) | 2×211 + 9 |
2.3 STRUCTURAL OPERATORS
| Symbol | Operator | Meaning | Example |
|---|---|---|---|
| √p | SUB(p) |
Sub-structure of p | √5 = SUB(MEM) |
| p² | SELF(p) |
Self-interaction | 11² = SELF(REG) |
| p³ | HYPER(p) |
Hyper-regulation | 11³ = HYPER(REG) |
| p/q | SCALE(p,q) |
p scaled by q | 17/4 = SCALE(SPEC,SYM) |
| Σp_i | SUM(...) |
Superposition | α⁻¹ = SUM(…) |
| Πp_i | PROD(...) |
Multiple dialogue | 5×13 = PROD(MEM,SING) |
2.4 SIGNIFICANT COMPOSITE NUMBERS
| Number | Operator | Composition | Meaning |
|---|---|---|---|
| 1 | ACT |
— | Generative contradictory act |
| 4 | SYM |
2² | Hidden symmetry, pairs of pairs |
| 6 | OBJ |
2×3 | Objectivity, measurement |
| 8 | EXP |
2³ | Complete spatial expansion |
| 12 | FRM |
2²×3 | Complete frame, quadrant cycle |
| 24 | SCL |
2³×3 | Intergenerational base scale |
| 40 | SPM |
8×5 | Spatial expansion with memory |
| 64 | MAX |
2⁶ | Maximal differentiation |
| 65 | M-S |
5×13 | Memory-Singularity unit |
2.5 MATHEMATICAL MODIFIERS
| Constant | Operator | Behavior | Context |
|---|---|---|---|
| π | CURV |
Geometric excess over ternarity | Only with 3 |
| π-3 | EXC |
Minimal excess over base cycle | Fine corrections |
| φ | GRW |
Organic growth, golden ratio | Self-similarity |
| δₛ | DIAG |
Diagonal/structural proportion | Base spatial structure |
| ρ | REC |
Cubic recursion, plasticity | 3D optimization |
| ψ | SGRW |
Super-golden growth | Cosmological hierarchies (ψ³ = ψ² + 1) |
| λ | DIST |
Statistical arity number distribution (Golomb-Dickman) | Multiplicative phenomena |
| e | LIM |
Incremental growth limit | Exponential processes |
| γ | IRR |
Asymptotic irregularity | Statistical corrections |
| ζ(3) | COR |
Deep correlation (Apéry) | Collective systems |
| C | CAT |
Catalan correlation (alternating sum) | Chiral transitions |
3. GRAMMATICAL RULES (R1-R107)
3.1 PRESENCE/ABSENCE RULES (R1-R7)
R1: π appears only with factor 3 or 3ⁿ
R1”: Geometric constants as scale factors
R2: Absence of 5 indicates phenomenon without history
R3: 11 appears in regulatory contexts
R4: 13 alone → in denominator; 5×13 → can be in sum
R5: High arity numbers (≥17) in main structure → numerator
R6: Powers of 2 indicate spatial/dimensional structure
R7: √p appears in fundamental structural proportions
3.2 OPERATIONAL RULES (R8-R13)
R8: Multiplication = Structural dialogue
R9: Division = Regulation or scaling
R10: Addition = Structural superposition
R11: Subtraction = Redundancy elimination
R12: Power = Iterated self-application
R13: Root = Underlying structural proportion
3.3 CONTEXTUAL RULES (R14-R18)
R14: Physical Domain Affinity
- EM → 11 (REG)
- Weak → 13 (SING)
- Color → 7 (CPX)
- Mass → 5 (MEM), 13 (SING)
R15: Scaling by denominator
R16: Gradual exactness (fundamentality ∝ precision)
R17: ONTOLOGICAL DEGENERACY — Multiple valid structural geneses converge
R18: REFUTED — Balance numerator/denominator (ELIMINATED)
3.4 ADVANCED RULES (R19-R80)
Selection of key rules:
R19: Mixing angles admit exact fractions with arities
R23: Strong interactions use arity number cubes
R26: Very small constants = 1/(product of arity number sequence)
R27: Near-integer = N ± 1/(arity number product)
R35: Factor 40 (8×5) in 3D structures with memory
R36: π-3 as minimal geometric excess for corrections
R45: Optimal corrections use exactly 3 arities in denominator
R57_v2: Structural optimum: 1 constant + 3-4 arities (quantified from 25 cases)
R66: Principle of Iterative Manifestation: C = F × (1 ± ε)
R74: PREFERENTIAL ADJACENCY LAW:
- MEM prefers → REG or SING
- DIAG prefers → CURV or SPEC
- DARK prefers → SING or SPEC
- REG prefers → MEM or CPX
R78: Structural inheritance (related phenomena share operators)
R79: Exact/stochastic dichotomy revised
R80: Principle of grammatical completeness
3.5 META-RULES (R106-R107)
R106: Principle of lexicon evolution — Grammar grows with discovery
R107: Evidence hierarchy — Fundamentals > Derived > Predictions
4. GENERATIVE GRAMMAR & CANONICAL FORMS
4.1 CANONICAL FORM
CONSTANT := [MODIFIER] × [∏ p_i^{a_i}] / [∏ q_j^{b_j}] ± [CORRECTION]
where:
- MODIFIER ∈ {π, φ, δₛ, ρ, ψ, C, θ, K₀, ...} ∪ {∅}
- 2 ≤ Σ(a_i + b_j) ≤ 8 (complexity limit)
- CORRECTION := 1/k or n(π-3) or combination4.2 FORM WITH ADJUSTMENT
C = F × (1 ± ε)
F = [MOD] × [Π p_i^{a_i}] / [Π q_j^{b_j}] (Pure structure)
ε = manifestation term (contextual adjustment)4.3 FORMATION RULES
- Initialize: Start with integer or simple fraction
- Domain: Include operators from physical domain (R14)
- Geometry: If continuous component, add term with π (R1)
- Regulation: Divide by appropriate regulatory arity numbers (R3)
- Correction: Add small term ±1/(∏ 3 arities) or n(π-3)
- Validate: Verify preferential adjacency (R74)
PART II: METHODOLOGY
This part was rewritten on August 13, 2026. The original version described a search procedure that adjusted its own rule set after each miss, and a section (“Error as Information”) that treated large errors as evidence for new structure rather than as evidence against the specific formula. Together those made it impossible, in practice, for any measurement to count against the framework. What follows keeps the notation and the honest parts (the fine-structure and muon examples did converge cleanly), and states plainly where the method fails.
5. DERIVATION PROCEDURE & ITS LIMITS
5.1 THE PROCEDURE
Given a target constant C in a physical domain (EM, weak, mass, etc.), Rule R14 suggests which arity numbers are expected to appear (see Appendix A). A candidate structure F is built from those arity numbers and checked against C.
5.2 THE RULE THAT MATTERS
A candidate is accepted only if it is built from arity numbers already assigned to that domain by R14, before the fit was attempted. If no such candidate reaches acceptable precision, the correct report is “not derived” — not a search for additional operators chosen because they happen to close the gap.
This is the distinction between the worked examples in Part III that succeeded (§9.1–9.3) and the ones that did not (§10, and the not-derived items listed in the companion post on Standard Model masses). Introducing a new operator, or a new exponent, specifically because it makes one target constant match is curve-fitting, not derivation, regardless of how small the resulting error is. A structure earns explanatory credit only if the same operator assignment is used elsewhere, independently of the constant currently being matched.
5.3 WHAT THIS DOCUMENT DOES NOT CLAIM
- It does not claim that every physical constant has a derivation in this framework — several do not (see §10, and the “not derived” list elsewhere in the ArXe corpus).
- It does not claim that a large error is itself informative about missing structure. A large error means the candidate is wrong.
- It does not claim immunity from being wrong about a specific formula while other parts of the framework remain intact — the two are logically separate.
6. NOTATION: THE C = F × (1 ± ε) FORM
Most formulas in this framework are written as:
C = F × (1 ± ε)where F is a structure built from arity numbers under rules R1–R107, and ε is a small correction term, itself built from arity numbers, that is reported alongside F rather than absorbed silently.
This is a notational convention, not an epistemic license. A formula with a large ε is a formula with a large error — reporting ε separately from F is for readability (it shows which part of the expression carries the leading structure and which part is a secondary correction), not a way of reclassifying error as “tone” or “context.” Where ε exceeds a few percent, the formula should be read as approximate and flagged as such, not treated as validated.
7. HANDLING MISSES: WHEN A STRUCTURE DOESN’T FIT
7.1 POLICY
When a structure built from the domain-appropriate operators (R14) does not match the target constant within a stated tolerance, that is reported as a miss. It is not evidence that a different, unplanned operator is “the real explanation” — it is evidence that this attempt failed.
7.2 A CASE THAT WAS REMOVED
An earlier version of this document used the top-quark mass as a worked example: an initial structure missed the measured value by roughly a factor of 67, and the document then searched for an operator whose ratio matched that factor, added it to the formula, and reported the result as validated. That example has been removed. It illustrates exactly the move this section warns against: a large miss should not be closed by adding a term chosen after the fact to close it. (A separate, domain-consistent derivation of the top-quark mass, built only from operators R14 already assigns to the mass sector, appears in the companion Standard Model post; it was not built this way.)
7.3 WHAT COUNTS AS A GENUINE CORRECTION
A secondary term (the ε in §6) is legitimate when it is small (a few percent or less) and uses operators already established for that domain — not a newly introduced operator whose only justification is that it closes the gap for this one constant.
8. VALIDATION CRITERIA
8.1 WHAT “VALIDATED” MEANS HERE
A formula is validated when it is built from domain-appropriate operators fixed in advance (R14), matches the measured constant within a stated tolerance, and does not depend on any operator introduced solely for that constant.
8.2 CRITERIA, IN ORDER OF WEIGHT
- Precision: how small is the error against the measured value?
- Grammaticality: does it follow R1–R107, using only operators R14 assigns to this domain?
- Parsimony: how few arity numbers does it require (R105, R45)?
Elegance of interpretation or narrative coherence with other formulas is not a validation criterion — it may be worth noting separately, but it does not substitute for the three above.
8.3 A FORMULA THAT FAILS THESE CRITERIA IS REPORTED AS NOT DERIVED, NOT AS A DIFFERENT KIND OF SUCCESS.
PART III: APPLICATIONS
9. WORKED EXAMPLES
These show the accepted formula and why it satisfies the domain rule (R14) and the parsimony rule (R45), not the sequence of attempts that preceded it.
9.1 FINE STRUCTURE CONSTANT
Target: α⁻¹ ≈ 137.036
Domain: Electromagnetic → REG(11) expected (R14)
α⁻¹ = 11² - 7² + 5×13 = 137
Error: 0.026%Validation:
- R14: EM domain has REG(11) ✓
- R74: REG(11) adjacent to MEM(5) ✓
- R57_v2: 4 arities total ✓
Reading: EM coupling as a combination of electromagnetic self-regulation (11²), color self-complexity (7²), and a persistence–singularity term (5×13).
9.2 MUON-ELECTRON MASS RATIO
Target: m_μ/m_e ≈ 206.768
Domain: Mass → MEM(5), SING(13) expected (R14)
m_μ/m_e = 3⁴ + 40π + 2/19
Error: 0.0003%Validation:
- R1: π appears with 40 = 8×5 ✓
- R14: Mass domain implied ✓
- R45: Correction 2/19 uses 2 arities ✓
9.3 HIGGS MASS
Target: M_H = 125.25 GeV
Domain: Mass/Scalar → MEM(5), REG(11), CPX(7) (R14)
M_H = (5×11×7)/(3π) × (1 - 1/19)
Error: 0.024%Validation:
- R1: π with 3 ✓
- R14: MEM(5), REG(11), CPX(7) all present ✓
- R57_v2: 1 constant + 4 arities ✓
10. PREDICTIONS & TESTABILITY
10.1 DARK MATTER MASS
Structural prediction (earlier value withdrawn): An earlier estimate, M_DM ≈ M_H × 17/4 ≈ 532 GeV, did not reproduce in the current framework and has been withdrawn. The remaining structural claim is weaker: dark matter’s level (T⁻⁹, arity 19) implies that any particle confirmed as the dark matter candidate should have a Planck-mass integer containing 19 as a factor.
Testability: Checkable against any future confirmed dark-matter mass.
Falsification condition, stated in advance: if a dark matter candidate is confirmed with a Planck-mass integer that does not contain 19 as a factor, this structural claim is wrong. That failure would not be reinterpreted as evidence for a different operator chosen afterward — it would be recorded as a failed prediction, the same way the withdrawn 532 GeV estimate was recorded above.
10.2 NEW RESONANCE
M_res ≈ 11³×√2/3 ≈ 1847 GeVBuilt from REG(11) cubed with a symmetric-cyclic correction. Testable at LHC high-energy searches. If no resonance appears near this value as sensitivity improves, this prediction is wrong.
10.3 NEUTRINO MASS SCALE
m_ν ≈ O(10⁻² eV)A qualitative order-of-magnitude prediction from extreme arity-number suppression; compatible with current experimental bounds but not a sharp test.
10.4 RUNNING CONSTANT STRUCTURE
Prediction: structural transitions in α(E) running near E₁ ≈ 91 GeV and E₂ ≈ 173 GeV. Requires precision QED running data beyond what’s currently available to test.
11. OPEN PROBLEMS & RESEARCH DIRECTIONS
11.1 IMMEDIATE TECHNICAL WORK
Mathematical Formalization:
- Complete categorical formulation of exentational recursion
- Rigorous proof that n(k) = -2k+1 follows necessarily from axioms
- Investigation of boundary condition algebra
Extended Mappings — using only the rule set already fixed in R1–R107, not new operators introduced per constant:
- Test whether the existing R14 domain assignments extend to remaining Standard Model parameters
- Investigate the cosmological constant from the existing arity structure
- Explore whether QCD running coupling has a structural basis under the existing rules
Statistical Work Needed:
- An honest estimate of how often a fit like §9.1–9.3 would occur by chance, given how many arity-number combinations are available — this has not yet been done and is a real gap, not a detail
11.2 PHILOSOPHICAL DEVELOPMENT
- Detailed account of this framework’s metaphysical commitments
- Criteria for evaluating structural adequacy independent of any single fit
- Limits of structural explanations
11.3 PHYSICS CONNECTIONS
- Connection to gauge/gravity duality
- Relationship with string theory landscape
- Implications for quantum gravity
11.4 QUESTIONS FOR INVESTIGATION
By Physicists:
- Can this framework predict which arity numbers appear at which energy scales, in advance of measurement?
- How does arity structure relate to renormalization group flow?
- Are there observable signatures at colliders?
By Mathematicians:
- Does arity number distribution connect to physical structure, independent of curve-fitting to known constants?
- What categorical framework properly captures arity number operations?
By Statisticians:
- What is the expected false-positive rate for a search over combinations of small integers, exponents up to 4, and π/φ, matching a target to within 1%? (See §11.1 — this has not been computed for this framework and would meaningfully constrain how much credit §9’s examples deserve.)
12. APPENDICES
APPENDIX A: QUICK REFERENCE
Most Important Rules:
R1: π only with 3
R14: Domain operators (EM→11, Weak→13, Color→7, Mass→5)
R17: Ontological degeneracy (more than one valid structure can exist for the same constant — this needs a stated criterion for choosing between them, not treated as a reason no choice is needed)
R45: Corrections should use as few arities as possible
R57_v2: Structural optimum: 1 constant + 3-4 arities
R66: C = F × (1 ± ε)
R74: Preferential adjacency (MEM→REG/SING, etc.)Most Used Operators:
2-DIFF, 3-CYC, 5-MEM, 7-CPX, 11-REG, 13-SING,
17-SPEC, 19-DARK, π-CURVCanonical Form:
C = [π or φ] × [arity numbers numerator] / [arity numbers denominator] × (1 ± ε)APPENDIX B: PROCEDURE TEMPLATE FOR NEW CONSTANTS
Target constant: [NAME] = [VALUE]
Physical domain: [EM/Weak/Color/Mass/Angle]
Operators permitted, from R14, fixed before attempting a fit: [list]
1. Build a candidate structure F using only the permitted operators.
2. Compute the error against the measured value.
3. If error is small (a few percent or less) using only permitted operators: report as derived, with the error stated.
4. If error is large: report as NOT DERIVED. Do not introduce a new operator chosen because it closes this specific gap — that is curve-fitting, not derivation, regardless of the resulting precision.
5. If a smaller error requires a new operator, that operator must be justified independently (used consistently elsewhere in the corpus) before it can be used here — not introduced solely to fix this constant.APPENDIX C: GLOSSARY
Arity Number: An integer treated as an operator in the ArXe formal system.
Ontological Degeneracy (R17): More than one structurally valid derivation can exist for the same constant.
ε (epsilon): A secondary correction term in C = F × (1±ε); legitimate only if small and built from operators already established for that domain.
Grammaticality: Conformance to rules R1-R107, using only the operators R14 assigns to the relevant domain.
CLOSING WORDS
To the reader:
This grammar is not finished, and several constants it should explain are not yet derived — see the “not derived” items in §10 and in the companion Standard Model post. That gap is stated here on purpose: a framework that only shows its successes is not falsifiable in practice, whatever it claims about itself.
What we’re asking of a reader, and of an AI collaborator working on this material:
- Use the fixed operator set (R1-R107) as a constraint, not a starting point for a fresh search per constant
- Report a miss as a miss
- Treat a small number of unexplained constants as more informative than a large number of fitted ones
Whether this framework captures something real about physical constants, or is an elaborate pattern-match, is not something this document can settle. What it can do is state its method precisely enough that the question stays answerable by evidence — which requires being willing to lose.
DOCUMENT INFORMATION
Title: The Arity-Logical Grammar of Physical Constants
Version: 1.1 (methodology revision)
Date: January 2026, revised August 2026
Status: Living document — Part II and III rewritten August 13, 2026; see update notice at top
License: CC BY-SA 4.0 (share, adapt, attribute)
Cite as: Tentor, D.L. (2026). “The Arity-Logical Grammar of Physical Constants: ArXe System Complete Reference.” v1.1