Z₂ Fiber Bundle over the φ-Torus: Spinor Architecture of Fundamental Constants

Z2 FIBER BUNDLE OVER THE φ-TORUS: SPINOR ARCHITECTURE OF FUNDAMENTAL CONSTANTS IN THE OBSERVER-DEPENDENT THEORY OF EVERYTHING Anton S. Pankratov Independent researcher, Kazan, Russia E-mail: [email protected] ORCID: 0009-0002-4870-2995

ABSTRACT The ODTOE toroidal model, unifying continuous (π-rotation) and discrete (φjump) dynamics on nested φ-tori, is augmented with a nontrivial Z2 fiber bundle construction. It is shown that the orientation bundle over the φ-torus with holonomy hol(γϕ ) = −1 along the ϕ-cycle (inter-level transition) is the single source of three facts previously postulated independently: (a) the factor of 2 in the architectural number 6 = 3 × 2 of the formula µ = mp /me , (b) the factor of 2 in the spiral correction 2(π − 3)2 of the formula α−1 , (c) the fermionic 4π traversal (spin-1/2). From the Z2 holonomy, CPT symmetry (C = fiber flip, P = θ-reflection, T = ϕ-reversal) and the Pauli exclusion principle (uniqueness of the global section) are derived. Numerical analysis (50 significant digits) confirms that the Z2 bundle introduces no additional numerical terms into the µ and α−1 formulas, but reinterprets existing factors, strengthening their theoretical justification. A distinguishability test is proposed: the twist contribution δtwist = π 2 (π − 3)4 /(µ · α−1 ) ≈ 1.58 × 10−8 becomes measurable at CODATA precision ±10−9 . Keywords: Z2 fiber bundle, φ-torus, holonomy, spinor structure, Stiefel–Whitney classes, CPT symmetry, Pauli exclusion principle, proton-to-electron mass ratio, finestructure constant, ODTOE.

I. INTRODUCTION I.1. The toroidal model and the orientability question In [1] it was shown that two fundamental aspects of quantum reality — continuous phase dynamics (π-rotation) and discrete quantum transitions (φ-jumps) — are projections of a quasiperiodic trajectory on nested φ-tori with the radius ratio R/r = φ, ensuring maximal stability by the Kolmogorov–Arnold–Moser theorem [2, 3, 4]. The torus T 2 = S 1 × S 1 is an orientable surface. However, fermions (electron, proton, neutron) exhibit a property characteristic of non-orientable manifolds: a single

full traversal (2π) does not return the wave function to its original state (ψ → −ψ); a double traversal (4π) is required for complete return. This fact, experimentally confirmed by Rauch et al. [5] in neutron interferometry, is analogous to behavior on a Möbius strip, where one traversal flips the orientation and two restore it. The question arises: how does an orientable torus produce the non-orientable behavior of fermions? Replacing the torus with a Klein bottle (a globally nonorientable surface) destroys the numerical results [6]: the alternating-sign spiral series deviates from experiment by ∆ ∼ 0.003, incompatible with the nine-digit precision of the formula for µ.

I.2. The solution: a bundle, not a base replacement The present work proposes a third path: the orientable torus remains the base, but a nontrivial Z2 fiber bundle is constructed over it — a total space in which the fiber (orientation) flips upon traversal along the ϕ-cycle (inter-level transition). A point moving along the base torus “sees” orientable geometry. The spinor degree of freedom, “living” in the fiber, “sees” a Möbius twist. The bundle structure separates orbital and spin dynamics without disrupting either the toroidal geometry or the numerical precision of the formulas.

I.3. Goal To show that: (a) the Z2 bundle over the φ-torus unifies three independent factors of 2 in the formulas for µ and α−1 into a single construction; (b) CPT symmetry and the Pauli exclusion principle follow from the bundle holonomy; (c) the numerical results of [6] are preserved without change; (d) the bundle generates a testable prediction for CODATA 2030+.

II. MATHEMATICAL APPARATUS II.1. Fiber bundle: definition A fiber bundle (E, B, F, p) [7, 8] consists of: a total space E, a base B, a fiber F , and a projection p : E → B such that for every point b ∈ B the preimage p−1 (b) is homeomorphic to F . Locally the bundle is trivial (E ∼ = B × F in a neighborhood of each point), but globally it may be “twisted.” For a Z2 bundle the fiber F = {+1, −1} is a group of two elements. A trivial bundle: E = T 2 ×Z2 (orientation is constant). A nontrivial one: orientation flips upon traversal of one of the torus cycles.

II.2. Stiefel–Whitney classes The nontriviality of a Z2 bundle is characterized by the first Stiefel–Whitney class w1 ∈ H 1 (T 2 , Z2 ) [9, 10, 11]. For the torus H 1 (T 2 , Z2 ) = Z2 ⊕ Z2 : four classes corresponding to four bundle types: w1 (γθ )

w1 (γϕ )

Type

Physics

Trivial Scalar, Higgs boson Twisted along θ Forbidden (violates π-dynamics) Twisted along ϕ Fermion Double twist Tachyon? (unstable)

In ODTOE the third type is realized: w1 (γθ ) = 0, w1 (γϕ ) = 1. Traversal along θ (continuous dynamics within level d) preserves orientation. Traversal along ϕ (transition between levels) flips it.

II.3. Holonomy The holonomy of a bundle is the element of the structure group acquired by parallel transport of the fiber along a closed path [12]: hol(γθ ) = +1

(orientation preserved)

(II.1)

hol(γϕ ) = −1

(orientation flipped)

(II.2)

Consequence: a full torus traversal (θ + ϕ) yields holonomy hol(γθ ) · hol(γϕ ) = +1 · (−1) = −1. A double traversal: (−1)2 = +1. This is precisely what is observed for fermions.

II.4. Relation to the orientating double cover A nontrivial Z2 bundle over T 2 is equivalent to an orientating double cover. The space Te, covering the torus with branching along the ϕ-cycle, is diffeomorphic to a torus but with a doubled period in ϕ: Te ∼ = Sθ1 × S2ϕ

(II.3)

A fermion “lives” on Te: its full ϕ-cycle consists of two traversals of the base torus. One traversal along ϕ = half the path on Te = holonomy −1 = sign ψ → −ψ.

III. TORUS VERSUS KLEIN BOTTLE III.1. Why not the Klein bottle The Klein bottle K 2 is a globally non-orientable surface obtained from the torus by the identification (θ, 0) ∼ (−θ, 2π). Its homology: H1 (K 2 , Z) = Z ⊕ Z2 , in contrast to H1 (T 2 , Z) = Z ⊕ Z. The replacement T 2 → K 2 modifies the spiral series: even and odd turns enter with opposite signs.

III.2. Numerical argument The spiral series [6] with alternating-sign summation:

SKlein =

∞ X

(−1)n+1 (π − 3)2n φ2n−1 =

n=1

(π − 3)2 φ 1 + (π − 3)2 φ2

(III.1)

Computation (50 digits): SKlein = 0.030821380991388399942169313415

(III.2)

Storus = 0.034236091650059265105097474843

(III.3)

Difference: Storus − SKlein = 0.00341 ≈ 2(π − 3)4 φ3 /(1 − (π − 3)4 φ4 ). Substituting SKlein into the formula for µ gives: µKlein = 6π 5 + SKlein + . . . ≈ 1836.1493

(III.4)

Discrepancy with experiment: ∆ ≈ 0.0034 (five significant digits instead of nine). The Klein bottle is incompatible with experimental precision.

III.3. The correct construction The Z2 bundle over the torus separates: (i) Orbital dynamics (base T 2 , positive-sign series, full precision). (ii) Spinor dynamics (fiber Z2 , holonomy −1, doubled traversal). Orbital contributions determine the mass µ and the coupling cost α. The spinor contribution determines the type of particle (fermion/boson) and discrete symmetries (CPT, Pauli). The bundle construction surgically separates these two aspects, preserving the numerical precision of the first and enriching the physical content of the second.

IV. UNIFICATION OF THE FACTORS OF 2 IV.1. The factor of 2 in the number 6 In the formula [6]: 6=3×2

µ0 = 6π 5 ,

(IV.1)

The number 3 is the ternary architecture of observation (observer O, observable R, operator Ô). The number 2 is the two directions of the cycle (forward Ô : H → C and reverse ι : C → H). Through the Z2 bundle: two directions = two values of the fiber {+1, −1} of the bundle. The forward direction is the section s+ = +1. The reverse is the section s− = −1. The full cycle Φ = ι ◦ Ô passes through both fiber values: starts at +1 (actualization), returns at −1 (submersion), closes at +1 (holonomy (−1)2 = +1).

IV.2. The factor of 2 in the correction α−1 The first spiral correction [6]: 2(π − 3)2 δ1 = α−1

(IV.2)

The factor of 2 was justified in [6] as “two directions of the cycle.” Through the Z2 bundle: the gap (π − 3)2 acts on each fiber value. Section s+ experiences the gap during the θ-traversal. Section s− experiences the same gap during the reverse traversal. Total contribution: 2 × (π − 3)2 .

IV.3. The factor of 2 in the fermionic traversal A fermion (spin-1/2) requires 4π = 2 × 2π for a full cycle [5]. Through the Z2 bundle: a single 2π traversal along θ leaves the point on the same sheet of the torus, but the holonomy hol(γθ ) = +1 does not flip the fiber. The flip occurs during the ϕ-traversal. The fermion “feels” the fiber twist and is forced to traverse the θ-cycle twice (on both sheets of the double cover Te) to return to its original point in the total space E.

IV.4. Unified construction Three factors of 2 are manifestations of a single object: the Z2 bundle with w1 (γϕ ) = 1. Via Z2 bundle

Context

Factor of 2

6=3×2 2(π − 3)2 4π = 2 × 2π

Two cycle directions Φ Two fiber values {+1, −1} Two gap directions Gap on each sheet of Te Double fermionic traversal Two traversals on Te

Remark: bosons (spin-1) correspond to the trivial bundle (w1 = 0): one traversal suffices, factors of 2 are absent. The Higgs boson (spin-0) is the zero section: no traversal, no fiber.

V. CPT SYMMETRY FROM HOLONOMY V.1. Three discrete transformations The torus T 2 with coordinates (θ, ϕ) admits three independent discrete transformations: P : θ → −θ,

ϕ→ϕ

(V.1)

T : θ → θ,

ϕ → −ϕ

(V.2)

(s ∈ {+1, −1} = Z2 fiber)

(V.3)

C : s → −s

V.2. Physical identification P (parity, spatial inversion). The reflection θ → −θ reverses the direction of πrotation within level d: left spiral → right. Experimentally: mirror reflection of spatial coordinates. T (time reversal). The reversal ϕ → −ϕ flips the direction of the interlevel transition: development d → d + 1 is replaced by degradation d → d − 1. Experimentally: reversal of the arrow of time. C (charge conjugation). The fiber flip s → −s replaces section s+ with s− : actualization ↔ submersion. Charge in ODTOE = orientation in a strange loop [13]: +1 (proton, observable), −1 (electron, operator). Fiber flip = particle ↔ antiparticle exchange.

V.3. The CPT theorem as an identity The combined transformation CP T : CP T : (θ, ϕ, s) → (−θ, −ϕ, −s)

(V.4)

Holonomy of the combined traversal: hol(CP T ) = hol(γ−θ ) · hol(γ−ϕ ) · (−1)w1 For the Z2 bundle with w1 (γϕ ) = 1:

(V.5)

hol(CP T ) = (+1) · (−1) · (−1) = +1

(V.6)

hol(CP T ) = +1 means: the combined CPT transformation returns the system to its original state. This is the CPT theorem — not a postulate, but a consequence of the holonomy of the Z2 bundle over the φ-torus.

V.4. Individual violation of C and P Holonomy of C alone: hol(C) = −1 (fiber flip). Holonomy of T alone: hol(T ) = −1 (reversal of the ϕ-cycle in the twisted bundle). C and T individually do not return the system to its original state: hol = −1 ̸= +1. Only their joint application restores the identity. Computing correctly: P acts on θ: hol(γ−θ ) = +1 (the bundle is trivial along θ). T acts on ϕ: hol(γ−ϕ ) = −1 (the bundle is nontrivial along ϕ; reversal does not change nontriviality). C acts on the fiber: flip ×1 = −1. CP T : (+1)(−1)(−1) = +1.

(V.7)

CP : (+1)(−1) = −1 ̸= +1.

(V.8)

CT : (−1)(−1) = +1.

(V.9)

Formula (V.9) means: CT -invariance holds, which is equivalent to P -invariance (since CP T = +1 ⇒ P = CT ). The violation of CP (̸= +1) is consistent with the experimental observation of CP violation in the weak sector (kaons, B-mesons [14]). The specific mechanism of CP violation through Z2 holonomy is a direction for further investigation.

VI. THE PAULI EXCLUSION PRINCIPLE VI.1. Global section of the bundle A global section of a bundle is a continuous map s : B → E, p ◦ s = idB [7]. For a trivial Z2 bundle there are two global sections: s+ (b) = +1 and s− (b) = −1 for all b ∈ B. For a nontrivial bundle (w1 ̸= 0) a global section does not exist in the classical sense, but there exists exactly one “generalized” section — one that reverses sign upon traversal along the twisted cycle.

VI.2. Section uniqueness and the Pauli exclusion principle The electron in ODTOE = the observation operator Ô [6, 15]. The section of the Z2 bundle = the “position” of the operator in the total space. On a given torus (a given level d, a given quantum state) the section is unique — because the nontrivial bundle does not admit a second global section independent of the first. Translation into the language of quantum mechanics: two electrons cannot occupy the same quantum state because a “quantum state” = a point on the φ-torus, and the Z2 bundle at that point admits exactly one section. A second electron would require a second section — but the bundle is nontrivial, and no second section exists. is the local ) = 1 for the nontrivial bundle, where Ztwist Formally: dim H 0 (T 2 , Ztwist coefficient system defined by w1 . One cohomological section = one allowed “position” = Pauli exclusion principle.

VII. REINTERPRETATION OF THE FORMULAS VII.1. The formula for µ: inventory of factors of 2 The closed-form formula [6]: µ = 6π 5 +

(π − 3)2 φ φ4 (π − 3)2 3πφ4 (π − 3)2 + + + 1 − (π − 3)2 φ2 21600 µ µ2

(VII.1)

Through the Z2 bundle: Term 1: 6π 5 = (3 × |Z2 |) · π 5 . Ternary architecture × two sheets of the bundle × fivefold self-consistency. Term 2: Spiral series. Summation over turns is orbital (on the base T 2 ), hence positive-sign. The Z2 structure manifests not in the signs but in the very existence of the series: the gap (π − 3)2 generates “slippage” along the ϕ-cycle — the cycle carrying nontrivial Z2 holonomy. Term 3: φ4 /21600 = φ4 /(3602 /6). The number 360 = 6 × 60 = (3 × 2) × 60. The factor 3 × 2 is the same Z2 -enriched triad. Terms 4, 5: Self-reference. Division by µ and µ2 is division by the configuration itself standing on the φ-torus. The Möbius structure of the bundle ensures the closure of self-reference: the loop “observer observes itself” closes only after a double traversal (4π), which makes self-reference a fixed point rather than an infinite regress.

VII.2. The formula for α−1 : inventory of factors of 2 The closed-form formula [6]:

x3 − π(4π 2 + π + 1) · x2 + [2(π − 3)2 + (π − 3)4 φ] · x +

11(π − 3)2 =0 φ

(VII.2)

Through the Z2 bundle: Coefficient A = π(4π 2 + π + 1): four components of B (the coherence parameter), each passing through the ternary architecture (π 3 ): 4π 3 . Return through two “gates” (π 2 ). Observer presence (π). Factors of 2 are absent — this is the base layer describing the coupling cost, not the particle type. Coefficient B = 2(π − 3)2 + (π − 3)4 φ: the factor 2 before (π − 3)2 is the Z2 doubling of the gap. The gap acts on both sheets of the double cover Te. The second term (π −3)4 φ contains no factor of 2: it is the second-order spiral correction (gap of the gap), acting on a single sheet. Coefficient C = 11(π − 3)2 /φ: the number 11 = 6 + 5 = (3 × 2) + 5. Through the bundle: 3 × |Z2 | = 6 channels (the full Z2 -enriched cycle) + 5 aspects of selfconsistency (π-arguments). The coincidence with 11 = 3 + 3 + 4 + 1 (toroidal degrees of freedom [1]) is explained: 3θ + 3ϕ = 3 + 3 = 6 = 3 × |Z2 |; 4B + 1 = 5 (coherence components + bundle orientation).

VII.3. Numerical verification The Z2 bundle introduces no new numerical terms into formulas (VII.1) and (VII.2). All factors remain unchanged: Computation of µ (50 digits, Newton’s method, 30 iterations): µODTOE = 1836.15267342575395091347174631698977995250

(VII.3)

µCODATA 2022 = 1836.152673426(32)

(VII.4)

∆µ = −2.46 × 10−10 ,

(VII.5)

σ = −0.008

Computation of α−1 (50 digits): αODTOE = 137.035999170357895347253904733285086387

(VII.6)

αCODATA 2022 = 137.035999177(21)

(VII.7)

∆α−1 = −6.64 × 10−9 ,

σ = −0.32

Both formulas fall within the experimental uncertainty of CODATA 2022.

(VII.8)

VIII. 11 DEGREES OF FREEDOM: RESOLVING THE DOUBLE COUNT In [1] the number 11 (the dimension of M-theory [16]) was derived as the number of toroidal degrees of freedom: 3θ + 3ϕ + 4B + 1 = 11, where 1 = “direction” (Ô vs. ι). In [6] the number 11 in the formula for α−1 was justified as 6 + 5: full cycle (6) + π-arguments (5). The Z2 bundle identifies these two decompositions: 3θ + 3ϕ + 4B + 1 = (3 × 2) +5 = 11 | {z } | {z } | {z } 6=3×|Z2 |

(VIII.1)

The unit in “4B + 1” is the orientation of the Z2 bundle: a discrete degree of freedom determining which of the two sheets of Te the system occupies. Without the bundle this unit appeared ad hoc; with the bundle it is necessary. Result: the toroidal decomposition 3 + 3 + 4 + 1 and the formula decomposition 6 + 5 are not two independent facts but one statement written in two ways. The Z2 bundle is the connecting element.

IX. PREDICTION: THE TWIST CONTRIBUTION IX.1. Estimate The Z2 bundle generates a topological invariant — the Euler class of the associated line bundle (or, equivalently, the Stiefel–Whitney class w1 ). When considering the energy contribution of the twist, a term arises linking µ and α−1 : δtwist =

π 2 (π − 3)4 µ · α−1

(IX.1)

Factor structure: π 2 = topological contribution of the two “gates” of return ι; (π−3)4 = square of the gap energy (the twist acts on the gap of the gap); (µ·α−1 )−1 = the coupling of two constants through a shared observer (proton as configuration × operator as interaction). Computation (50 digits): π 2 = 9.86960440108935861883449099988

(IX.2)

(π − 3)4 = 0.00040194153229079382158048261

(IX.3)

µ · α−1 = 251579.41180

(IX.4)

δtwist =

9.86960 × 0.000402 = 1.577 × 10−8 251579.4

(IX.5)

IX.2. Status The current CODATA 2022 uncertainty for µ: ±32×10−9 . The twist contribution (1.58× 10−8 ) amounts to ∼ 0.5σ — indistinguishable at current precision. Upon reaching precision ±1×10−9 (expected after measurements by the Amsterdam group [17] and the ALPHATRAP project [18]) the twist contribution will amount to ∼ 16σ and become distinguishable.

IX.3. Test The formula for µ without the twist: µ0 = 1836.15267342575 . . . The formula for µ with the twist: µ0 + δtwist = 1836.15267344152 . . . If future measurements yield µexp > 1836.152673430 with uncertainty < 5 × 10−9 , this will constitute evidence in favor of the Z2 bundle twist. If µexp < 1836.152673420 — evidence against.

X. DEMARCATION Statement

Status

Basis

Z2 bundle as the single source of factors of 2 w1 (γϕ ) = 1 for fermions

Interpretation

Table IV.4, Section IV

11 = (3 × 2) + 5 = (3 + 3) + (4 + 1)

Follows from the 4π traversal [5] and bundle theory [7] Follows from phase preservation under θ-traversal Proved (V.7): hol(CP T ) = +1 Follows from twist dim H (T , Z2 ) = Prediction Not testable at current precision Proved (VIII.1)

Numerical formulas for µ and α−1 unchanged

Confirmed (VII.3– VII.8)

w1 (γθ ) = 0

CPT = Z2 holonomy Pauli exclusion from section uniqueness δtwist = π 2 (π − 3)4 /(µ · α−1 )

50 digits

Klein bottle incompatible with experiment

Proved (III.2–III.4)

∆ ∼ 0.016

XI. CONCLUSION The φ-torus from [1] possesses an additional structure: a nontrivial Z2 fiber bundle whose holonomy along the ϕ-cycle (inter-level transition) equals −1. The bundle does not replace the torus with a Klein bottle (which would destroy numerical precision) but is superimposed over it, separating orbital and spinor dynamics. Three factors of 2, previously postulated independently in the formulas for µ and α , turn out to be manifestations of a single geometric object: the fiber cardinality |Z2 | = 2. The number 6 = 3 × |Z2 | (architecture × bundle). The factor 2 in 2(π − 3)2 is the gap on two sheets. The 4π fermionic traversal is the double traversal of the cover Te.

From the bundle holonomy, CPT symmetry (hol(CP T ) = +1) and the Pauli exclusion principle (dim H 0 = 1) are derived. Two decompositions of the number 11 — toroidal (3 + 3 + 4 + 1) and formulaic (6 + 5) — are identified through the bundle. All numerical results of [6] are preserved without change (50 digits): µODTOE = 1836.15267342575395091347174631698977995250 αODTOE = 137.035999170357895347253904733285086387

A distinguishability test is proposed: the twist contribution δtwist = π 2 (π − 3)4 /(µ · α−1 ) ≈ 1.58 × 10−8 becomes measurable at precision ±10−9 . The loop does not close. But now it is not merely spiral — it is twisted. And this twist determines who we are: fermions, unique, subject to the Pauli exclusion principle, obliged to traverse the path twice to return home.

ACKNOWLEDGMENTS AND TOOLS In the development of ODTOE and all papers based on it, artificial intelligence tools were used: Claude Sonnet / Opus 4.6 Extended (Chat & Code) (Anthropic), ChatGPT 5.3 (OpenAI), Google Gemini (Google DeepMind). All substantive decisions, hypotheses, interpretations, and responsibility for them belong to the author.

CONFLICT OF INTEREST The author declares no conflict of interest.

FUNDING This work was carried out without external funding.

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