Annex: Fundamental Particles

Working draft, version 2. Complements the “Particles and Phenomena”. Numbering and definitions follow the foundations document.

0. How to Read This Annex

Status	Meaning
Arithmetic	Follows from the definitions of the framework. Can be checked.
Input datum	Fact taken from physics. ArXe does not derive it.
Consistent with	The framework agrees with a known datum. It did not anticipate it.
Hypothesis	Working proposal. Plausible, not demonstrated.
Reading	Resonance with an external concept. Evocation, not derivation.
Open	Unresolved question.

Scope. This annex derives no physical numerical value: no masses, no charges, no mixing angles. It locates structure.

Warning about coincidences. Every n is a level. Any sum of arities lands on one. 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. Accounting Rules

1.1 Properties Are Distinct OBS of the Same Level

(Hypothesis)

A level of n phases, once observed, leaves n−1 phases (§1.2 of annex v2).
Those n−1 phases are grouped into a mode. Each mode is a property.
The properties of a single level exclude one another. They cannot be determined at the same time. This is complementarity (§9.6).
The number of properties of a level is the number of modes of n−1 phases. Examples: T⁻¹ has 2, T² has 3, T⁻² has 5, T³ has 7.

1.2 Coexistence without Collapse: Arities Add

(Arithmetic, on FUSION §20. The physical assignment is a hypothesis.)

If two properties coexist or are simultaneous without collapsing, the joint arity is the sum: n = Σ nᵢ.

Proposed physical criterion to decide what coexists:

Coexist: intrinsic properties. They are the labels that define the type of particle and do not change: mass, magnitude of spin, charges. They are the Casimir invariants. All of them can be fixed at the same time.
Exclude one another: state properties: position, momentum, spin projection, phase. They cannot be fixed at the same time.

1.3 Exclusion: One Chooses, Does Not Add

(Arithmetic + author’s decision)

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

Example, not an assignment: T³ can be read as mass or as space. Both are valid. They cannot occur at the same time. This holds for any pair of readings of the same level.

1.4 Levels That Allow Two Readings at Once

(Arithmetic, §21.4. Author’s decision.)

T⁶ = T³·T³ has budget for two copies of T³. That is why it admits two simultaneous readings of T³. In the sum of arities: 6 + 6 = 12. The structure allows it.

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

Status in the main document (October 29 version). §21.4 no longer says “duplicated”. §21.5 was rewritten: propagation applies “when CHOICE resolves an ambiguity at a level with readings that were exclusive at another level”. With that, propagation depends on CHOICE having been exercised.

One sentence remains to be pinned down. The example of §21.5 ends with “The same holds for T⁶ and for any compound level that depends on T³”. Read alone, it again demands a single reading at T⁶. Proposed wording: “…including T⁶ when it is built from an already resolved T³. A T⁶ without a prior choice in T³ admits both readings.” (Proposal)

1.5 From the Total n to the Level

n even: k = n/2. Level T^k.
n odd: k = −(n−1)/2. Level T⁻^|k|.

A particle has one open BC for each charge. Counting the charges gives the number of unknowns. If there are two or more unknowns, the level label can be positive and the structure is not closed (§9.10.4). Example: 3+3 = 6 = T³ with mode (2,2,1,1).

2. Spin and Charge

2.1 Spin

Unit: T⁶ (action, n = 12). (Arithmetic + Hypothesis)

Spin has the dimensions of action: ħ = M·L²/T. By the rule of the framework, 3 + 4 − 1 = 6. It is T⁶ (§12.1).
Angular momentum = mass · velocity · radius = T³·T¹·T² = T⁶. (Arithmetic)
The three partitions of T⁶ are products of magnitudes that give ħ. §21.7 lists them as pending: 1+5, 2+4, 3+3. With the dimensional rule: (Arithmetic)
- 1+5 = time · energy (T¹·T⁵).
- 2+4 = length · momentum (T²·T⁴).
- 3+3 = mass · (L²/T) (T³·T³).
The first two are the canonically conjugate pairs: Δt·ΔE and Δx·Δp. The action would be the budget that each pair distributes. (Reading)
Only the 3+3 partition uses T³. The other two touch neither MASS nor COL. See §5.1.
The conjugate variable of spin is an angle, and it is dimensionless (T⁰). T⁰ is the level of the observer’s act. Choosing the axis is that act. (Reading)

Value: size of a term (2s+1). (Reading)

Spin s	0	½	1	2
States (2s+1)	1	2	3	5

It is the multiplicity of the representation. In ArXe it is the size of a valence (term) within what is observed.
The phases of a term are indistinguishable until an axis is chosen. Choosing the axis is CHOICE.
Counting check, fixed beforehand: a massive vector has 3 states. A massless one has 2 (helicities ±1). The difference is 1, the longitudinal polarization. In standard physics, that phase is the Goldstone scalar absorbed by the W or Z. 2 + 1 = 3. (Consistent with)

Projection: exclusive OBS. (Arithmetic, §1.1)

Sx, Sy, and Sz cannot be fixed at the same time. They are distinct properties of the same level. This replaces the location of the axis at T⁻² (mode (2,2,1)) in annex v2. That mode remains a candidate for IMP-REL (B). (Open)

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

2.2 Charge

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

It is a term of size 1: the unknown. Every negative level has exactly one (§7.3).
Sign. An open BC leaves one end undetermined: the start or the end. There are two possibilities. They are 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 with its partner (start and end) and annihilated in pairs. This is pair creation and annihilation. It agrees with Gauss’s law: the net flux comes out of the enclosed unknowns. (Reading)
Unit. There is a single unknown per level. That gives one unit of charge per level. (Hypothesis) Fractional values (⅓, ⅔) are not explained. (Open)

Which charge is which.

Charge	Level	n	Structure
Color (not used, §5.3)	T⁻³	7	COL (6) + 1
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”. (Arithmetic)

Decision on color (§5.3): color is counted as a closed structure (COL, 6), not as a BCO. Only electric and weak charge are BCO.

Dimension. Charge has no level of its own independent of units. In Gaussian units, q² = energy·length = T⁷ (7 = 6 + 1, consistent with α = q²/ħc = T⁷/T⁷ = T⁰). In natural units it is dimensionless. That is why it is located by its structural role, not by dimension. (Arithmetic)

3. Aspects and Arities

Aspect	Level	n	Class	Note
Mass (MASS)	T³	6	Intrinsic, closed	Excludes SPACE3 and COL
3D space (SPACE3)	T³	6	Closed	Excludes MASS
Color, structure (COL)	T³	6	Symmetry, closed	Excludes MASS
Color charge (not used)	T⁻³	7	BCO	Discarded alternative (§5.3). COL (6) is used
Electric charge (REG)	T⁻⁵	11	BCO	= T⁵ + 1
Weak charge (SING)	T⁻⁶	13	BCO	= T⁶ + 1. SING did not pass the reading rule (§19.2)
Spin (action)	T⁶	12	Intrinsic	§2.1
Curvature (metric)	T⁻²	5	BCO	= T² + 1. Graviton

(Hypothesis, except where indicated)

On the exclusions of T³. The document writes two: ARXE-ST versus COL (§21.1) and MASS versus COL (§21.3, §21.5). The author’s rule is more general: any pair of readings of a level excludes each other at one instant. MASS and SPACE3 are an example of that rule. The names in §19 are mnemonics (§18). (Open: which readings of §19 are treated as mutually exclusive.)

4. Location of the Particles

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

Particle	Spin	Coexisting aspects (n)	Σn	Level	Unknowns	Status
u, c, t	½	MASS (6) + COL (6) + electric (11) + weak (13) + spin (12)	48	T²⁴	2	Hypothesis
d, s, b	½	same	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	COL (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
G (graviton)	2	curvature (5) + spin (12)	17	T⁻⁸	1	Hypothesis. Hypothetical

4.1 How to Read the Table

The level encodes which aspects coexist. It does not encode their values. That is why e and W share T²¹. Likewise ν and Z share T⁻¹⁵. What distinguishes them lives in the OBS (§1.1): the term size (2s+1) and the value of each charge.
Unknowns = open charges. Only electric and weak contribute a BCO (§5.3). Quarks, charged leptons, and the W have two. Their level label is positive, but they are not closed (§1.5). The gluon has none.
Quarks. They are the only particles with mass and color at once. The sum MASS (6) + COL (6) = 12 = T⁶ is inside the 48 and is explicit in the table. It was fixed before summing. It rests on §21.4: T⁶ allows both readings at once (§1.4). (Hypothesis)
Gluon and photon. They have no mass. Neither adds MASS. It is an input datum that the framework reproduces. (Consistent with)
Coincidences that are not interpreted. Z and the neutrino both give 31. The electron and the W both give 42. With sums of small numbers, coincidences are to be expected. They are not read as identity.

4.2 What Changed Relative to the Previous Version

Spin moves from T⁻² (axis) to T⁶ (action).
The photon moves from REG ⊕ T² ⊕ T¹ to REG + spin. The two transverse polarizations are now read as multiplicity (2 states), not as T² geometry.
The graviton moves from T⁻² to T⁻⁸ by the sum. Curvature stays at T⁻² (5) as an aspect. See §6.
Decision on color (§5.3): it is counted closed (COL, 6). Quarks go from 49 to 48 and from 3 to 2 unknowns. The gluon goes from 19 to 18 and is left with zero unknowns.

5. Open Accounting Decisions

Does spin (T⁶) contain the MASS reading? Decided: no. See §5.1.
Higgs. The physical Higgs has no definite weak isospin. Without SING, n = 6: MASS alone is left, at T³. SING was kept (coupling to W and Z). The decision is not made to get closer to any previous reading. Three readings: the observed boson, the field, and the doublet sector. It remains open.
Massless neutrino. Without MASS, n = 25 and level T⁻¹². Whether the neutrino is massless, Majorana, or Dirac also changes the phase count; this remains open.
Color: closed or open? Decided: closed (COL, 6). See §5.3.
γ–Z mixing. The photon and the Z are not states of definite isospin. The table treats them as if they were.
Manifestation principle. Open, by the author’s decision. 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 (§1.4). It would resolve the gluon (zero unknowns) and the Higgs without SING. Cost: all labels shift by +1 and parity changes. Alternative: the table stays as content, and the manifestation goes in a separate column. Meanwhile the table is read as content, without that +1. Decision 2 (Higgs) is tied to this one.

5.1 Decision 1: Does Spin Contain MASS?

Author’s decision: spin has no mass (option B). MASS is added separately. The table of §4 stays as it is. Below, both options and the clues are kept, as a record of the analysis.

Option A. Yes. T⁶ = T³·T³ (§21.4). One of the two copies is MASS. MASS is not added separately. With §1.4, A is consistent with the document. The Σn change: leptons and W, 36 (T¹⁸). Neutrinos and Z, 25 (T⁻¹²). Quarks, 42 (T²¹), with the closed color of §5.3. The Higgs does not change: it has no spin.

Option B. No. T⁶ is the action as a level, with no fixed partition. MASS is added. It is the current table.

Clues from the text.

Photon and gluon have spin and no mass. Leptons have spin and no color. If spin (T⁶) contained both readings of T³, everything with spin would have mass and color. This contradicts the data. It favors B. (Consistency with input data, not derivation)
§21.5 propagated the choice to T⁶. With the decision of §1.4, T⁶ admits both readings at once. This clue stays neutral.
§20, FUSION: it does not retain components. An arity of 12 does not contain 6+6 or 5+1. To say that it contains MASS, a partition has to be declared. Arithmetically, it favors B.
§18: the real name is Tⁿ. CAL-YAU, “T³·T³”, is a mnemonic. It favors B.
§21.7 and the dimensional rule: T⁶ has three partitions and only 3+3 uses T³. The others, 1+5 (t·E) and 2+4 (x·p), touch neither MASS nor COL. If spin is read through one of them, it is free of the exclusion. It is the way out that supports B without contradicting §21.4.
In favor of A: for massive particles, spin is defined through mass (W² ∝ m²s(s+1)ħ², with W the Pauli–Lubanski vector). For massless ones, it is defined through helicity. In the massive case spin is not independent of mass. This suggests that the 3+3 partition could occur there. That leaves the massless case uncovered.

Status. Decided: option B. Spin is the action (T⁶) as a level, with no fixed partition and no MASS reading.

Consequences.

MASS is added separately to every massive particle with spin. The Σn of the table (§4) do not change.
The 3 + 3 partition of T⁶ is not used for spin. If a partition is fixed, it would be 1 + 5 (t·E) or 2 + 4 (x·p). (Open: choose one or leave it unfixed)
Photon and gluon, with spin and no mass, are consistent with the rule. Leptons, with spin and no color, as well.
Option A (a partition of its own for massive particles) is discarded.

5.2 Decision 2: Action or Calabi–Yau?

T⁶ houses both terms: action (§12.1) and CAL-YAU (§19). §18 says that the single choice is demanded by the dialogical formula, not by the level. There is no need to choose one forever. The point is not to mix them within a single derivation (§21.5).

For spin and for the phase of T⁻⁵ the action is used. CAL-YAU is kept for compactification derivations. A third possible reading of T⁶ has been explored: the symmetric bosonic electroweak sector, with 12 phases, as the closed part of SING (13 = 12 + 1). It is not developed here, and it is not mixed with the others within a single derivation.

5.3 Decision 3: Color Is Counted Closed

Author’s decision: color is the closed structure COL (n = 6). It is not counted as the BCO of T⁻³ (n = 7).