Arxe Theory

Categories

Annex Fundamental Particles

Working draft. Complements the “Particles and Phenomena”. Numbering and definitions follow the foundations document.

Fundamental Particles

Abstract. This article places the particles of the Standard Model, and the hypothetical graviton, in the levels of ArXe. Each particle is a tuple of aspects (mass, color, charges, spin). The sum of the arities of the aspects that coexist gives a level. The text records the conventions adopted, analyzes which phenomenon is the one called Higgs, and compares two ways of counting (by aspects and by phases). It makes no numerical predictions.

Author: Diego Luis Tentor · Independent researcher
Date: October 2026
Status: draft under review
License: CC BY-SA 4.0

Base text: ArXe — Foundations and Framework. The references “Foundations §n” point to that document. This article continues ArXe — Particles and Phenomena by Level, where OBS, EXP and the phenomena of each level are explained in more detail.

0. How to read this article

Label Meaning
Verified Exact consequence of the definitions, or a checked calculation.
Input datum A fact taken from physics. ArXe does not derive it.
Consistent with The framework matches a known datum, but did not anticipate it.
Working decision A convention adopted in order to move forward. It is not derived.
Hypothesis A plausible proposal, not demonstrated.
Reading A resonance with an external concept. It is evocation, not derivation.
Open An unresolved question.

Scope. This text derives no physical value: no masses, no charges, no mixing angles. It places structure.

Warning about coincidences. The levels are integers, and any sum of arities lands on one of them. A coincidence with an already-named level is not evidence, unless the assignment was fixed before summing. What was fixed beforehand is indicated in each case.

1. The minimum of the framework

  • Level Tᵏ and arity n. For k > 0, n = 2k. For k < 0, n = 2|k| + 1. For k = 0, n = 1. T¹ is time, T² is space, T³ is mass, T⁶ is action.
  • Sign. Positive levels are closed (field, vacuum). Negative ones have one open BC, the unknown. A particle has one open BC and does not exist in isolation (Foundations §7.1).
  • Mode. This is the partition of the phases into terms. Terms of equal size cannot be ordered (Foundations §9.53).
  • Operators (Foundations §20). PRODUCT adds exponents. FUSION adds arities. COC-INV is the inverse. CHOICE (ELECC in the Spanish original) is a choice. OBS removes a phase and EXP adds one (see the particles and phenomena article).
  • Dimensional rule (Foundations §12.1). A quantity M^a·L^b·T^c is at level 3a + 2b + c.

2. Accounting rules

2.1 Properties are distinct OBS of the same level

(Hypothesis)

  • A level of n phases, when observed, leaves n − 1 phases.
  • Those n − 1 phases are grouped into a mode. Each mode is a property.
  • Properties of the same level exclude one another: they cannot be determined at the same time. This is complementarity (Foundations §9.4).
  • The number of properties of a level is the number of modes of n − 1 phases. For example: T⁻¹ has 2, T² has 3, T⁻² has 5, T³ has 7.

2.2 What coexists adds arities

(Verified the sum. The physical assignment is a hypothesis.)

If two properties coexist without collapsing, the joint arity is the sum: n = Σ nᵢ (FUSION).

Proposed physical criterion:

  • Coexisting are the intrinsic properties: mass, magnitude of spin, charges. They are the labels that define the type of particle and do not change. All of them can be fixed at once.
  • Mutually exclusive are the state properties: position, momentum, spin projection, phase.

2.3 What is excluded is chosen

(Working decision)

A level admits several valid readings, in different theories or at different moments. At any one instant only one holds. This is CHOICE: they are not added.

Example, which is not an assignment: T³ can be read as mass or as space. Both are valid, and they cannot hold at the same time. This applies to any pair of readings of the same level.

2.4 A level that allows two readings at once

(Verified the sum. Working decision the reading.)

T⁶ = T³·T³ has the budget for two readings of T³ to coexist (Foundations §21.4). In the sum of arities: 6 + 6 = 12. The structure allows it.

Reading note on Foundations §21.5. The principle says that the propagation of a choice applies “when CHOICE resolves an ambiguity in a level with readings that were mutually exclusive in another level”. That is, it depends on the choice having been exercised. The example in the same section ends with “the same holds for T⁶”, which, read in isolation, could demand a single reading in T⁶. Here it is read as follows: the propagation holds for a T⁶ built from an already-resolved T³. A T⁶ without a prior choice in T³ admits both readings. (Proposed wording)

Generalization. A level T²ᵏ = Tᵏ·Tᵏ would admit two readings of Tᵏ at once. It remains to be proved in T⁴ = T²·T². (Hypothesis)

2.5 From the total n to the level

  • n even: k = n/2.
  • n odd: k = −(n − 1)/2.

A particle has one open BC for each open charge it carries. If it has two or more, the level label may be positive and the structure is not closed (Foundations §9.53).

3. Spin and charge

3.1 Spin

The unit of spin is the action: T⁶ (n = 12). (Verified + Hypothesis)

  • Spin has the dimension of action: ħ = M·L²/T, and 3 + 4 − 1 = 6.
  • Angular momentum = mass · velocity · radius = T³·T¹·T² = T⁶. (Verified)
  • The three partitions of T⁶ are products that give ħ (Foundations §21.7). (Verified)
    • 1 + 5 = time · energy.
    • 2 + 4 = length · momentum.
    • 3 + 3 = mass · (L²/T).
  • The first two are the canonically conjugate pairs (Δt·ΔE and Δx·Δp). Only the third uses T³.
  • The conjugate variable of spin is an angle, which is dimensionless (T⁰). T⁰ is the level of the observer’s act. Choosing the axis is that act. (Reading)

The value of the spin is the size of a term. (Reading)

Spin s 0 ½ 1 2
States (2s+1) 1 2 3 5
  • The phases of a term are indistinguishable until an axis is chosen. Choosing it is CHOICE.
  • A check fixed beforehand. A massive vector has 3 states. A massless one has 2 (helicities ±1). The difference is the longitudinal polarization, which in standard physics is the Goldstone scalar absorbed by the W or the Z. 2 + 1 = 3. (Consistent with)

The spin projection is an exclusive OBS. Sx, Sy and Sz are not fixed at the same time. They are distinct properties of the same level. (Verified, with §2.1)

Counterpart. The action (T⁶) and the weak charge (T⁻⁶) are counterparts. The weak interaction acts only on left-handed chirality, which is an orientation of spin. (Reading)

3.2 Charge

Charge is the open phase (BCO) of a negative level. (Hypothesis)

  • It is a term of size 1: the unknown. Each negative level has exactly one (Foundations §7.3).
  • Sign. An open BC leaves one end undetermined: the start or the end. These are two possibilities, the source and the sink. Positive and negative charge.
  • Neutral. The ends are paired. No free unknown remains.
  • Conservation. A BCO does not appear alone. It is created together with its partner and annihilated in pairs. This matches Gauss’s law: the net flux comes out of the enclosed unknowns. (Reading)
  • Unit. There is a single unknown per level, which gives one unit of charge per level. Fractional values (⅓, ⅔) are not explained. (Open)
Charge Level n Structure
Electric T⁻⁵ 11 T⁵ (energy, 10) + 1
Weak T⁻⁶ 13 T⁶ (action, 12) + 1

In each case, n(T⁻ᵏ) = n(Tᵏ) + 1. The charge is the “+1”. (Verified)

Dimension. Charge has no level of its own independent of units. In Gaussian units q² = energy · length = T⁷, and q is at level 3.5, a half-integer. In natural units it is dimensionless. That is why it is placed by its structural role and not by dimension. (Verified)

4. Aspects and arities

Aspect Level n Class Note
Mass T³ 6 Intrinsic, closed Excludes 3D space and color
3D space T³ 6 Closed Excludes mass
Color (structure) T³ 6 Symmetry, closed Excludes mass
Electric charge T⁻⁵ 11 Open (BCO) = T⁵ + 1
Weak charge T⁻⁶ 13 Open (BCO) = T⁶ + 1. Its placement did not pass the reading rule (Foundations §19.2)
Spin (action) T⁶ 12 Intrinsic See §3.1
Curvature (metric) T⁻² 5 Open (BCO) = T² + 1. It is the graviton’s

(Hypothesis, except where indicated)

On the exclusions of T³. The base document writes two: space-time versus color (§21.1), and mass versus color (§21.3 and §21.5). The working rule is more general: any pair of readings of a level excludes each other at an instant. Mass and 3D space are one example. The names in §19 are mnemonics (§18). (Open: which readings are treated as mutually exclusive)

5. Placement of the particles

The 17 particles of the standard table and the graviton (hypothetical) are listed. Antiparticles and color multiplicities are ignored.

Particle Spin Coexisting aspects (n) Σn Level Unknowns Status
u, c, t ½ mass (6) + color (6) + electric (11) + weak (13) + spin (12) 48 T²⁴ 2 Hypothesis
d, s, b ½ idem 48 T²⁴ 2 Hypothesis
e, μ, τ ½ mass (6) + electric (11) + weak (13) + spin (12) 42 T²¹ 2 Hypothesis
νₑ, ν_μ, ν_τ ½ mass (6) + weak (13) + spin (12) 31 T⁻¹⁵ 1 Hypothesis
γ (photon) 1 electric (11) + spin (12) 23 T⁻¹¹ 1 Hypothesis
g (gluon, ×8) 1 color (6) + spin (12) 18 T⁹ 0 Hypothesis
W± 1 mass (6) + electric (11) + weak (13) + spin (12) 42 T²¹ 2 Hypothesis
Z 1 mass (6) + weak (13) + spin (12) 31 T⁻¹⁵ 1 Hypothesis
H (Higgs) 0 mass (6) + weak (13) 19 T⁻⁹ 1 Reading (open, see §7)
G (graviton) 2 curvature (5) + spin (12) 17 T⁻⁸ 1 Hypothesis (hypothetical)

5.1 How to read the table

  • The level encodes which aspects coexist. It does not encode their values. That is why the electron and the W share T²¹, and the neutrino and the Z share T⁻¹⁵. What distinguishes them lives in the OBS: the term size (2s+1) and the value of each charge.
  • Unknowns = open charges. Only the electric and the weak contribute a BCO. Quarks, charged leptons and the W have two. Their level label is positive, but they are not closed. The gluon has none.
  • Quarks. They are the only particles with mass and color at the same time. The sum mass (6) + color (6) = 12 = T⁶ is visible in their row. It was fixed before summing, and it rests on §2.4. (Hypothesis)
  • Photon and gluon. They have no mass, and neither sums that aspect. It is an input datum that the framework reproduces. (Consistent with)
  • Coincidences that are not interpreted. With sums of small numbers, it is to be expected that two rows coincide.

6. Conventions adopted and open decisions

6.1 Spin does not contain mass

(Working decision)

The unit of spin is T⁶, the action as a level, with no fixed partition and no reading of mass. Mass is added separately to every massive particle with spin. This is the option called B here. The alternative (A) was that T⁶ = T³·T³ included mass as one of its two copies.

Clues that tipped the decision:

  • The photon and the gluon have spin and no mass. Leptons have spin and no color. If spin contained both readings of T³, everything with spin would have mass and color. This contradicts the data.
  • FUSION does not retain components (Foundations §20): an arity of 12 does not “contain” 6 + 6 or 5 + 1. Saying it contains mass requires declaring a partition.
  • T⁶ has three partitions, and only 3 + 3 uses T³. If spin is read through 1 + 5 or 2 + 4, it is free of the exclusion.
  • Against: for massive particles, spin is defined in the rest frame, through mass (the square of the Pauli–Lubanski vector is proportional to m²·s(s+1)·ħ²). For massless ones it is defined by helicity. This suggests that the 3 + 3 partition could occur there.

Consequence: the Σn in the table do not change. (Open: choose the partition that reads spin, or leave it unfixed.)

6.2 Color is counted closed

(Working decision)

Color is the closed structure (n = 6). It is not counted as a BCO of T⁻³ (n = 7).

  • Only electric and weak charge are BCO.
  • Gluon. Color (6) + spin (12) = 18 = T⁹, with zero unknowns. A structure without an open BC belongs to the closed face (field), not an isolable particle. It is consistent with the gluon not being observed free. (Reading)
  • Limit. The same argument does not explain the confinement of quarks, which do have unknowns (electric and weak). (Open)
  • What, then, is color charge? It would be the symmetry structure of three equal valences, not an open phase. (Open)

6.3 Action, compactification and the electroweak sector

(Pending confirmation)

T⁶ houses several terms: the action (Foundations §12.1), a compactification mnemonic (§19) and, according to the analysis of the Higgs (§7), the symmetric electroweak sector, with 12 phases. They are not mixed within a single derivation (§18 and §21.5). For spin and the phase of T⁻⁵ the action is used.

6.4 Open decisions

  1. Higgs. Whether or not to count the weak charge. See §7.
  2. Neutrinos. Massless, with Majorana mass or with Dirac mass. Massless, the table row would become 25 = T⁻¹². The phase count (§8) makes it visible: massless or Majorana, 118 phases; Dirac, 124.
  3. γ–Z mixing. The photon and the Z are not states of definite isospin. The table treats them as if they were.
  4. Manifestation principle. The table gives the content of each particle (Σn). The manifested particle would be EXP(content) = Σn + 1, because no particle is isolated without the observer’s phase. It would resolve the gluon and the Higgs without the weak charge. Cost: all labels shift by +1 and parity changes. Alternative: the table stays as content, and the manifestation goes in a separate column. For now the table is read as content, without that +1.

7. The Higgs: three phenomena

The three accounting options for the Higgs are not rival hypotheses about a single thing. They are three phenomena with properties in common. The question is which of them is what experiments observe and call Higgs.

Option Content Σn Level Unknowns Standard Model phenomenon
H2 mass 6 T³ 0 The field and its vacuum value, v
H3 mass + the observer’s phase 7 T⁻³ 1 The boson h: the variation of mass
H1 mass + weak charge 19 T⁻⁹ 1 The doublet sector: Goldstones and longitudinal W, Z

(Reading)

7.1 Relations and measurements

  • The scale of the field. G_F = 1/(√2·v²). With the CODATA value, v = 246.2197 GeV. It is determined without producing any Higgs boson. It is the permanent manifestation of the field. (Verified + Input datum)
  • The boson couples to mass. m(h) = m·(1 + h/v), whence g_Hff = m_f/v and g_HVV = 2·m_V²/v. The boson is, literally, the local variation of masses. ATLAS reports that the measured couplings are proportional to mass for fermions and to its square for bosons, and agree with the Standard Model. (Input datum)
  • Spin and parity. The 125 GeV boson is a scalar, neutral, colorless, spin 0 and CP-even. None of the three options includes spin, electric charge or color. (Input datum)
  • Lifetime. Γ·τ = ħ, the 1 + 5 partition of T⁶. With the Standard Model width (about 4.1 MeV), τ ≈ 1.6·10⁻²² s. The boson itself is not seen, only its products. Width measurements are made through off-shell production: CMS obtained 3.2 (+2.4 −1.7) MeV and ATLAS 4.6 (+2.6 −2.5) MeV.
  • The relation between the field and the boson. m_H² = 2·λ·v². With m_H ≈ 125.1 GeV, λ ≈ 0.129. λ has not been measured: self-coupling searches are only beginning. It is the only bridge between the field (H2) and the boson (H3) that has not been tested.

7.2 The phase count of the doublet

(Verified the arithmetic. Reading the rest.)

  • The Higgs field is a complex doublet: 4 real components.
  • After breaking, 3 are absorbed by W⁺, W⁻ and Z (they become their longitudinal polarization). 1 remains: the boson h.
  • In modes: (2,2) → (3,1). It is the same transition as entanglement: an ambivalence goes to different sizes upon choosing. The breaking is a choice: the vacuum takes a point of S³, the sphere of equivalent directions of field space.
  • The boson h is the term of one phase. A term of size 1 is an unknown, the open phase. That is why h is open by structure: neither the +1 of the manifestation principle nor the weak charge is needed.

The complete electroweak sector (4 phases of the doublet + 6 of W¹, W², W³ + 2 of B) is 12 phases, all in pairs: the canonical mode of T⁶ (2⁶), with no unknowns. After breaking they are W⁺ (3), W⁻ (3), Z (3), γ (2) and h (1): the mode (3,3,3,2,1). The total is conserved. The three terms of 3 are equal and not orderable, which matches the custodial triplet (W⁺, W⁻ and Z are degenerate when g′ → 0, and ρ = 1 at tree level with a single doublet).

7.3 What each option is in this picture

Option In the picture Phenomenon
H3 The term of one phase, or the 1 of (3,3,3,2,1) The boson h
H1 The three terms of 3 phases, with the phase of one Goldstone each W⁺, W⁻, Z with longitudinal polarization
H2 It is not a term. It is the choice: the point of S³ that the vacuum takes The vacuum value, v

(Reading) H2 is the only option with no place in the partition, because it is not a degree of freedom: it is the selection that reorganizes the 12.

7.4 Result

  • H2 is the field. It is what is always measured, in the vacuum. Science calls it the “Higgs field”. It is not the boson.
  • H3 is what is observed and called the Higgs boson. Its content matches the measurement: it couples to mass. Its open character comes out of the structure.
  • H1 is another phenomenon. It is the doublet sector, which is observed in the scattering of longitudinal W and Z. It is not what was detected in 2012, and its weak aspect does not match the physical h, which has no definite weak isospin.

(Reading. It is the best match among the three options, not a demonstration.)

7.5 The relation with the weak charge and T⁶

The base document places the weak charge in T⁻⁶ (13 phases), the open counterpart of T⁶. The symmetric electroweak sector is exactly T⁶: 12 phases. 13 = 12 + 1. It is consistent with the T⁶ ↔ T⁻⁶ pairing.

Caution. That placement was fixed before this count and did not pass the reading rule (Foundations §19.2). The 12 comes from summing 4 + 6 + 2, and with three fields the range of totals is small. It is not used as proof. It is, however, a reason not to discard the placement. If the weak sector were T² (the doublet) or T³ (the triplet), the weak charge would weigh 5 or 7. That decision remains open.

7.6 Why this is not decided with numerical formulas

A numerical criterion (that some formula reproduces m_H/v = 0.508) does not discriminate. In the family √(a/b)·(1 + 1/c), with a < b ≤ 20 and c ≤ 20, there are 3,800 expressions. With the current uncertainty of m_H, 2 fall within one standard deviation of m_H/v. For a random number, the average number of hits in that family is 2.4, and the probability of at least one hit is 56%. A formula chosen after seeing the number has no probative value.

8. Phase accounting

Phases are counted as real on-shell degrees of freedom, including spin states and the two faces (particle and antiparticle).

8.1 Count by field type

(Input datum + Verified)

Field Phases Term
Real scalar 1 (1)
Complex scalar 2 (2)
Massless vector 2 (2)
Massive vector 3 (3)
Weyl fermion 2 (2)
Dirac fermion 4 (2,2)
Massless graviton 2 (2)

Sizes 1, 2 and 3 are the only ones that appear among the known particles. They coincide with the arities of T⁰, T¹ and T⁻¹. That adds no information about the physics: the known particles have spin ≤ 1 (plus the massless graviton). (Reading)

8.2 Before and after breaking

Sector Before breaking After breaking Phases
Gluons (8 × 2) 8 massless vectors 8 massless vectors 16
Bosonic electroweak Higgs (doublet), W¹, W², W³, B h, W⁺, W⁻, Z, γ 12
Fermions (3 generations) 45 Weyl fields Dirac (quarks and charged leptons) and Weyl (neutrinos) 90
Total 118
  • Per generation, the fermions are 15 Weyl fields: 30 phases, 15 pairs. With three generations, 90.
  • Bosons: 28. Fermions: 90. Total: 118 phases, or 59 pairs. With the graviton, 120.
  • The total is conserved. Only the bosonic electroweak sector changes partition: from (2⁶) to (3,3,3,2,1).
  • Fermions do not change size. The mass of a Dirac fermion brings together two Weyl pairs (left and right) into (2,2). It adds no phases. The mass of fermions is a pairing. That of vectors is a transfer of phases. (Reading)

8.3 Neutrinos

Option Phases per flavor Model total
Massless 2 118
Majorana mass 2 118
Dirac mass (with the right-handed neutrino) 4 124

A Dirac mass adds 3 pairs and a new field. A Majorana one adds no phases, but identifies particle and antiparticle.

8.4 Generations

(Reading)

Each generation is 30 phases. With three generations, the mode is (30, 30, 30): three terms of equal size, not orderable. The tension is that the generations are in fact ordered by mass.

One way out: the three generations are identical in phases and structure. The only thing that distinguishes them are the Yukawa couplings, which are dimensionless numbers of level 0. It is the level that the dimensional rule does not see (see the exact bridges article, in preparation). Without the Higgs, the model has an exact symmetry among the three.

8.5 Two accountings, two questions

Accounting What it counts What it answers Example (Z)
Aspects (Σn) Which properties coexist and their arity What type is it? Its level 31 = T⁻¹⁵
Phases How many states it has and how they are grouped How many possibilities are there in its structure? 3

There is no numerical relation between the two, and none is needed: the phase one is the reading of the value of spin (the size of a term). Proposal: keep both under distinct names, content (Σn) and multiplicity (phases). (Hypothesis)

8.6 Coincidences that are not interpreted

  • 118 phases (28 + 90) are the standard counts.
  • The bosonic electroweak sector has 12 phases. The gauge group of the Standard Model has dimension 12 (8 + 3 + 1). They are two different 12s: one counts phases, the other generators. The arity of T⁶ is also 12.
  • Fermion multiplets have multiplicities of 3, 2 and 1 (color, isospin and singlet).

9. Graviton and mixings

  • Graviton. Curvature is an aspect of T⁻² (n = 5, metric tensor). With spin (12), the sum gives 17 = T⁻⁸. It does not coincide with any level named in §19. With 2 phases, it enters the mode (2) of the sector. Physical geodesics are in T⁻³. (Open: T⁻², T⁻³ or T⁻⁸)
  • CKM and PMNS. They describe how mass states and weak states mix. The framework holds a qualitative pattern (Foundations §16): the open-BC anchors would be present in one matrix and absent in the other. No values are derived. (Reading)
  • γ–Z. The Weinberg angle is not derived. A proposed formula for sin²θ_W was withdrawn, and remains an open problem.
  • Fractional charge. In standard physics, electric charge modulo 1 accompanies color triality. That links color with electric charge. (Open)

10. What this article does not cover

  • Statistics (fermion or boson). It has no place yet. A candidate is the principle of “one choice at a time”.
  • Values of mass, charge and angles. None is derived.
  • Input data taken from outside ArXe. The dimension of the gauge algebras: U(1) = 1, SU(2) = 3, SU(3) = 8. It is not made to coincide with the number of BCs of a level (Foundations §7.5).
  • Hadrons and bound states. They are composites and are treated with the dialogical operators.
  • Higgs mass. There are two candidate formulas, with no ranking, awaiting HL-LHC data. They are not treated here.

11. Open questions

  1. Choose the partition of T⁶ that reads spin (1+5, 2+4), or leave it unfixed.
  2. Make precise the last sentence of the example in Foundations §21.5 so that it does not contradict §21.4 (proposal in §2.4).
  3. An independent falsifier of the mass–color rule. Today the classification in the table is an input datum.
  4. Why quarks confine if they have unknowns and color is counted closed.
  5. Fractional values of charge (⅓, ⅔).
  6. Graviton: T⁻², T⁻³ or T⁻⁸. Resolve together with the relation between geodesics and variation of mass.
  7. Placement of the weak charge: T⁻⁶, T² or T³ (§7.5).
  8. Higgs: with or without the weak charge (§7), tied to the manifestation principle (§6.4).
  9. Neutrinos: massless, Majorana or Dirac.
  10. Statistics: fermion versus boson.
  11. Generations: is the difference by level 0 enough, or is an exact relation among the three blocks missing?
  12. Fix the method of extracting constants before any numerical assignment.

External sources