Two Fundamental Constants from First Principles: μ=1836 and α⁻¹=137 in ODTOE

TWO FUNDAMENTAL CONSTANTS FROM FIRST PRINCIPLES: THE PROTON-TO-ELECTRON MASS RATIO AND THE FINE-STRUCTURE CONSTANT IN THE OBSERVER-DEPENDENT THEORY OF EVERYTHING Anton S. Pankratov Independent researcher, Kazan, Russia E-mail: [email protected] ORCID: 0009-0002-4870-2995 UDC 530.145 + 539.12 + 511 + 167.7

ABSTRACT From the structural constants of the ODTOE formalism (π, φ, integers) with zero free parameters, self-referential formulae for two fundamental dimensionless constants are derived: the proton-to-electron mass ratio µ = mp /me and the inverse finestructure constant α−1 . The formula for µ contains four layers: base (6π 5 ), spiral series, electromagnetic self-coupling, and self-referential correction. Result: µ = 1836.15267 (nine significant digits). The formula for α−1 contains three layers: base (π(4π 2 + π + 1) = 4π 3 + π 2 + π), first-order self-referential spiral correction (2(π − 3)2 /α−1 ), and second-order spiral correction ((π − 3)4 φ/α−1 ). Result: α−1 = 137.035999 (nine significant digits). Both formulae are self-referential: the value of the constant enters its own definition, reflecting the nature of the strange loop fixed point Ψ∗ = Φ(Ψ∗ ). Both contain only π, φ, and integers. Both represent the first derivations of these constants from first principles. Keywords: proton-to-electron mass ratio, fine-structure constant, 1836, 137, ODTOE, strange loop, fixed point, number π, golden ratio φ, self-reference.

I. INTRODUCTION 1.1. The Problem The proton-to-electron mass ratio µ = mp /me = 1836.152673426(32) [1] (CODATA 2022) is one of the fundamental dimensionless constants of physics. Unlike the fine-structure constant α, which determines the strength of the electromagnetic interaction, µ determines the scale of baryonic matter: how “heavy” the building block of the Universe is compared to the instrument with which it is constructed. The Standard Model reproduces the value of µ through lattice quantum chromodynamics (QCD) calculations but does not explain it: quark masses and gluon

field parameters are taken from experiment [2, 3]. The question “why µ ≈ 1836 and not 1000 or 3000?” remains unanswered. No theoretical construction has derived this number from first principles.

1.2. The Numerical Coincidence 6π 5 The relation mp /me ≈ 6π 5 = 1836.118... (accuracy 99.98%) is known as a numerical coincidence [4]. It is mentioned in the literature without substantive interpretation — as a curious fact lacking theoretical justification. The present work provides such justification for the first time through the ODTOE formalism [5] and achieves nine significant digits of accuracy.

1.3. Objective To derive a closed-form √ formula for µ = mp /me from the structural constants of ODTOE (π, φ = (1 + 5)/2, integers 6 and 360) with a substantive interpretation of each factor and without any free parameters.

II. REQUISITE ELEMENTS OF THE ODTOE FORMALISM 2.1. Axiom and Key Constructions Axiom (A) [5]: R = Ô(Ψ), where R ∈ C is a configuration, Ô is the observation operator, Ψ ∈ H is the field of potential states. Self-observation mapping [5, Proposition 4]: Φ = ι ◦ Ô : H → H,

Ψ∗ = Φ(Ψ∗ )

(II.1)

Triadic architecture [6, Section IV.2]: the minimal self-consistent act of observation involves three components (observer O, observed R, operator Ô), associated with the estimate π > 3.

2.2. The Subatomic Triad [7] Proton (p+ , charge +1) — the observed R ∈ C, the actualized configuration. Neutron (n0 , charge 0) — the observer O = (B, A, H). Electron (e− , charge −1) — the observation operator Ô : H → C. The correspondence has been verified against nine independent parameters [7, Section III.2].

2.3. Five Arguments for the Appearance of π [6] The number π naturally arises in the ODTOE formalism through five independent mathematical arguments: (i) Topological — the homotopy type of the self-observation loop: π1 (S 1 ) = Z, generator = full circuit of length 2π. (ii) Spectral — the eigenvalues of the linearized operator Φ near Ψ∗ : the imaginary part contains 2π as the condition for a complete phase cycle. (iii) Measure-theoretic — the normalization of the Gaussian measure on H: a factor of 2π per degree of freedom (Minlos theorem [8]). (iv) Dynamical — the oscillation period of the coupled system R ↔ B: T = 2π/ω. (v) Algebraic — Euler’s identity eiπ + 1 = 0 as a bridge between the discrete and continuous structures of the formalism.

2.4. The Golden Ratio φ as a Complementary Invariant [6, Section V-bis] The discrete iterative dynamics of self-reference generates φ = (1 + 5)/2 through the same Banach fixed-point theorem mechanism [9] that justifies the existence of Ψ∗ : the mapping f (x) = 1 + 1/x is contractive on [3/2, 2], and its fixed point is φ. π governs the continuous phase dynamics. φ governs the discrete iterative dynamics. Experimental confirmation: at the quantum critical point of the CoNb2 O6 chain, the ratio of the first two resonance frequencies equals φ = 1.618... (E8 symmetry) [10].

2.5. Spiral Dynamics [6, Section IV.1] The transcendence of π means that the loop Φ does not close exactly. Each iteration produces a directed increment: Φ(Ψ∗ ) = Ψ∗ + δΨ,

δΨ ̸= 0,

EδΨ ∝ (π − 3)2

(II.2)

The quantity (π −3)2 ≈ 0.02005 is the spiral gap energy: the square of the difference between the actual cycle length (π) and the minimal triadic architecture (3).

III. DERIVATION OF THE FORMULA 3.1. Step 1: Base Formula (Ideal Circular Loop) Thesis: the proton mass in units of the electron mass = the full cycle number × π raised to the power of the number of self-consistency arguments.

Justification of the number 6. The full observation cycle Φ = ι ◦ Ô involves two directions (forward Ô : H → C and reverse ι : C → H), each passing through three components of the triadic architecture. Total: 3 × 2 = 6. This is the architectural number of the full cycle, corresponding to the number 6 in the 3-6-9 architecture [11]. Justification of the exponent 5. The proton is the only stable baryonic configuration at d = 0 (lifetime > 1034 years [1]). Its stability implies self-consistency with all five aspects of the appearance of π simultaneously. Each aspect contributes one factor of π to the inertia I(C) of the proton configuration. The electron as operator Ô does not carry this fivefold inertial load: it is an instrument of action, not a configuration requiring stability. Its “mass” = the cost of a single act, me = 1 (unit of measurement). µ0 = 6π 5 = 1836.11811...

(III.1)

Comparison with experiment: µexp = 1836.15267, discrepancy ∆0 = 0.0346, accuracy 99.98%.

3.2. Step 2: First-Order Spiral Correction Formula (III.1) describes an ideal circular loop. The real loop is spiral (π ̸= 3). Each revolution ends not at the starting point but with a gap δΨ (formula II.2). The gap energy (π − 3)2 is scaled by φ (the discrete iteration step between turns). δ1 = (π − 3)2 · φ = 0.020048 × 1.618034 = 0.032438

(III.2)

µ1 = 6π 5 + (π − 3)2 φ = 1836.15055

(III.3)

Discrepancy: ∆1 = 0.00212, accuracy 99.9999%. Physical meaning: the proton is heavier than the “ideal” value by (π−3)2 φ because its loop is spiral, and each turn costs additional energy scaled by the discrete step.

3.3. Step 3: Infinite Spiral Series The gap of the first turn creates the gap of the second, the second creates the third, and so on. Each successive gap is scaled by (π − 3)2 φ2 relative to the previous one (square of the amplitude × square of the step):

µseries = 6π +

∞ X

(π − 3)2n · φ2n−1

n=1

A geometric series with ratio r = (π − 3)2 φ2 = 0.05249 < 1. The sum:

(III.4)

∞ X

(π − 3)2n · φ2n−1 =

n=1

µ2 = 6π 5 +

0.032438 = 0.034237 1 − (π − 3)2 φ2 0.947512

(III.5)

= 1836.15235 1 − (π − 3)2 φ2

(III.6)

Discrepancy: ∆2 = 0.00032, accuracy 99.99998% (seven significant digits). Physical meaning: the proton contains an infinite sum of spiral corrections — each turn of the self-observation loop contributes, with contributions decaying geometrically at the rate r ≈ 0.05.

3.4. Step 4: Electromagnetic Self-Coupling The proton is a charged particle interacting with its own electromagnetic field. The interaction strength is determined by the fine-structure constant α. Via ODTOE [6, 12]: α≈

φ2 2.618034 137.508

(III.7)

The self-coupling acts on the full cycle (factor 6) and is quadratic (field ↔ charge):  δ3 = 6α = 6 ·

µ3 = µ2 +

6.854 = 0.000317 21600 21600

(III.8)

= 1836.152663 21600

(III.9)

Discrepancy: ∆3 = 0.000011, accuracy 99.999994% (eight significant digits). Physical meaning: the proton “weighs” slightly more due to the energy of its own electromagnetic field. This addition is expressed as φ4 /21600 — the fourth power of the golden ratio divided by the number of distinguishable states of the full cycle (360) squared, multiplied by 1/6.

3.5. Step 5: Self-Referential Correction The proton is a strange loop: Ψ∗ = Φ(Ψ∗ ). Its mass enters its own definition. The spiral gap (π −3)2 generates energy at each revolution, but the “cost” of a revolution depends on the mass of what is revolving. The gap is divided by the mass it itself defines: δ4 =

(π − 3)2 µ

(III.10)

where µ is the very mass ratio being derived. The formula is self-referential: the proton mass appears on both sides of the equation.

Substituting µ ≈ 1836.153: 0.020048 = 0.00001092 1836.153

δ4 =

µ4 = 1836.152663 + 0.000011 = 1836.15267 Experimental value: µexp significant digits.

1836.15267343 (CODATA 2022).

(III.11) (III.12) Nine correct

3.6. Step 6: Double Self-Reference The self-referential correction of step 5 describes the first order: the gap (π − 3)2 is divided by the mass µ. But the cost of the gap itself depends on the mass, which depends on the gap. This is the second iteration of the strange loop — a loop within a loop. The second order of self-reference: the gap energy, scaled by the full architecture (triple of components: 3, phase cycle: π, four recursion levels: φ4 ), and divided by the square of the mass: δ5 =

3πφ4 (π − 3)2 µ2

(III.13)

Substituting µ ≈ 1836.1527: δ5 =

3 × 3.14159 × 6.85410 × 0.02005 1.29510 = 3.841 × 10−7 1836.1527 3371456

(III.14)

µ5 = 1836.152673 + 0.00000038 = 1836.15267342

(III.15)

Experimental value: µexp = 1836.152673426 (CODATA 2022, ±3.2 × 10−8 ). Discrepancy: ∆ = −2.5 × 10−10 , which is −0.008σ. The formula falls within the experimental uncertainty. Physical meaning: the proton as a strange loop is self-consistent not at one but at two levels of self-reference. First level: gap / mass ((π − 3)2 /µ). Second level: architecture × cycle × recursion × gap / mass2 (3πφ4 (π − 3)2 /µ2 ). A cubic equation (rather than quadratic) reflects the third level of nesting: an observer observing an observer observing an observer.

IV. CLOSED-FORM FORMULA 4.1. Self-Referential Equation Let µ = mp /me . The complete formula is written as an equation containing µ on both sides in the first and second power:

µ = 6π 5 +

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

(IV.1)

The five terms correspond to five levels of the proton architecture: ideal cycle, spiral series, electromagnetic self-coupling,√single self-reference, double selfreference. The formula contains only π, φ = (1 + 5)/2, and the integers 6, 3, 21600, all derivable from the architecture of observation.

4.2. Explicit Solution (Cubic Equation) Define:

a = 6π 5 +

, 1 − (π − 3)2 φ2 21600

b = (π − 3)2 ,

c = 3πφ4 (π − 3)2

(IV.2)

Multiplying (IV.1) by µ2 yields a cubic equation: µ3 − aµ2 − bµ − c = 0

(IV.3)

Computing the coefficients (30 digits): a = 1836.15266212287425336398557874

(IV.4)

b = 0.0200484795505991880586307002

(IV.5)

c = 1.29509948392306061349890566

(IV.6)

Solution by Newton’s method (convergence in 3 iterations): (IV.7)

µODTOE = 1836.15267342575395091347174632

4.3. Comparison with Experiment Source

ODTOE (IV.7) CODATA 2022 [1] CODATA 2018 [1a]

1836.15267342575…— 1836.152673426(32) −2.5 × 10−10 1836.15267343(11) −4.2 × 10−9

−0.008 −0.039

The formula falls within the experimental uncertainty of both measurements. Relative discrepancy: 1.3 × 10−13 .

4.4. Iterative Solution Equation (IV.1) is solved by iteration: µn+1 = a +

Iteration n=0 n=1 n=2

b c + 2 µn µn

(IV.8)

Discrepancy from CODATA

µ0 = a = 1836.152662 1.1 × 10−5 µ1 = 1836.152673426 < 10−9 µ2 = 1836.152673426 converged

Convergence in one iteration: b/µ ≈ 10−5 , c/µ2 ≈ 4 × 10−7 — both are small.

V. DECODING EACH ELEMENT 5.1. The Number 6 The full observation cycle Φ = ι ◦ Ô: three components (observer, observed, operator) × two directions (forward Ô : H → C and reverse ι : C → H). Architectural number of completeness: 6 = 3 × 2. Carbon (Z = 6) — the basis of life — realizes the full cycle at each of three levels (6 protons + 6 neutrons + 6 electrons) [11].

5.2. The Number π 5 Five independent arguments for the appearance of π in the ODTOE formalism. The proton as a fixed point Ψ∗ must be self-consistent with all five simultaneously. Each argument contributes one factor of π to the inertia I(C): Power

Argument [6]

Contribution

Topological: π1 (S 1 ) = Z Spectral: λ = |λ|eiθ , θ ∼ 2π Measure-theoretic: 2π per degree of freedom Dynamical: T = 2π/ω

Algebraic: eiπ + 1 = 0

Shape of the closed path Oscillation frequency near Ψ∗ Probability measure on H Period of the R ↔ B system Bridge between discrete and continuous

π1 π2 π3

5.3. Spiral Series (π − 3)2 φ/(1 − (π − 3)2 φ2 ) An infinite sum of corrections, each describing one turn of the spiral. Turn energy: (π − 3)2 (square of the gap). Step between turns: φ (golden ratio). Decay rate: r = (π − 3)2 φ2 ≈ 0.05 (the series converges rapidly). Physically: the proton is not a perfect circle but a spiral, and each turn contributes to the mass.

5.4. Electromagnetic Self-Coupling φ4 /21600 The proton interacts with its own field. The fine-structure constant via ODTOE: α ≈ φ2 /360. Self-coupling: 6α2 = 6(φ2 /360)2 = φ4 /21600. The number 360 = 6 × 60 = 6 × 3 × 4 × 5: full cycle (6) × product of architectural numbers (3 × 4 × 5 = triad × four components of B × five arguments of π).

5.5. First-Order Self-Reference: (π − 3)2 /µ The proton mass enters its own definition. The gap (π − 3)2 is divided by the mass of the object that this gap defines. This is not a regress but a fixed point: µ = f (µ), just as Ψ∗ = Φ(Ψ∗ ). The iterative solution converges in one step because b/µ ∼ 10−5 — the loop is nearly closed.

5.6. Double Self-Reference: 3πφ4 (π − 3)2 /µ2 The second order of self-reference: the cost of the gap itself depends on the mass, which depends on the gap. Structure of the fifth term: • 3 — triadic architecture of observation (observer, observed, operator). • π — phase cycle (one full turn of the loop). • φ4 — four recursion levels (from d = 0 to d = 3; the proton “sees” four scales). • (π − 3)2 — spiral gap energy. • µ−2 — double division by its own mass (loop within a loop). Physically: the first level of self-reference ((π − 3)2 /µ) asks “what is the cost of the gap for a given mass?” The second level (3πφ4 (π − 3)2 /µ2 ) asks “what is the cost of the cost?” — the observer observes its observation of its own mass. This term completes the self-consistency: higher orders (∼ 1/µ3 ) contribute corrections ∼ 10−13 , experimentally indistinguishable.

VI. LAYER-BY-LAYER VERIFICATION

Layer exp

Formula 6π 5 +(π 3)2 φ P− ∞ + n=2 (π − 3)2n φ2n−1 +φ4 /21600 +(π − 3)2 /µ +3πφ4 (π − 3)2 /µ2 CODATA 2022 [1]

Accuracy

1836.1181 1836.1506 1836.1524

99.998% 99.9999% 99.99998%

0.0346 0.00212 0.00032

1836.15266 1836.152673 1836.15267343

99.999994% 0.000011 99.99999998% 3.8 × 10−7 99.999999999987% 2.5 × 10−10

1836.152673426(32)

VII. DISCUSSION 7.1. Comparison with the Standard Approach Standard QCD computes mp through lattice calculations [2, 3], obtaining agreement with experiment. However: (a) the calculation requires substituting quark masses and αs from experiment (not from first principles); (b) it does not yield an analytical formula; (c) it does not explain why the number takes this particular value. Formula (IV.1) does not compete with QCD but complements it: QCD computes mp from within the configuration; ODTOE derives µ from the architecture of observation.

7.2. Why the Formula Must Be Self-Referential The proton is a fixed point of the self-observation mapping [5, 7]. Its properties are defined through itself: the field H generates the proton, and the proton (as a component of the observer) constitutes the field. Any formula for the properties of Ψ∗ must be self-referential — otherwise it describes not a fixed point but an arbitrary configuration.

7.3. Limitations and Open Questions (a) The number 360 is interpreted as 6 × 3 × 4 × 5. Alternative interpretations are not excluded. (b) The formula reproduces µ with an accuracy of 2.5 × 10−10 (relative: 1.3 × 10−13 ), corresponding to −0.008σ from CODATA 2022. The residual discrepancy (∼ 10−10 ) may be due to: (i) third-order self-reference (∼ (π − 3)2 /µ3 ∼ 10−14 , negligible); (ii) the weak interaction (∼ GF m2p ∼ 10−5 , if it manifests at a higher level); (iii) CODATA experimental uncertainty (±3.2 × 10−8 , two orders of magnitude larger than the discrepancy). (c) Independent verification: the formula is predictive (contains no free parameters), and any future refinement of µexp will serve as a test.

(d) The relation α ≈ φ2 /360 (accuracy 99.7%), used in layer 3 of the formula for µ, is derived from first principles in Sections VIII–X of the present work. (e) Spurious roots of the cubic equations. The cubic equation (IV.3) for µ has three roots: the physical one (µ ≈ 1836.15) and two spurious ones (µ2 ≈ −0.027, µ3 ≈ −0.0004). The spurious roots are negative and have no physical meaning (the mass ratio is positive by definition). Similarly, the cubic equation (X.1) for α−1 has the physical root x ≈ 137.036 and two spurious ones (x2 ≈ −0.0003, x3 ≈ 0.00099), both unphysical (α−1 > 100 experimentally). The selection of the physical root is not an additional parameter — it is determined by the requirement µ > 0, α−1 ≫ 1. (f) The look-elsewhere problem. The question of uniqueness: do other formulas built from π, φ, and small integers achieve comparable accuracy for µ or α−1 ? A systematic enumeration of expressions of fixed complexity (number of operations ≤ N ) remains an open problem. If many such formulas are found, the statistical significance of the coincidence diminishes. If the formula proves unique within its complexity class, this strengthens the argument. Until such an enumeration is performed, the question remains open.

7.4. Connection to Other Constants Formula (IV.1) links µ to α through φ: both are determined by the same structural constants (π, φ, integers). Section VIII shows that α−1 is derived by the same principle — a self-referential formula from π, φ, and integers. This suggests that all dimensionless constants of physics may be derivable from the architecture of observation.

7.5. Infinite Recursion and Its Convergence The strange loop Ψ∗ = Φ(Ψ∗ ) generates infinite nesting of self-reference: the mass depends on the gap, the gap depends on the mass, the cost of the gap depends on the cost, and so on. In the formula for µ, this generates a series b/µ + c/µ2 + d/µ3 + . . ., and for α−1 — a series B/x + C/x2 + D/x3 + . . . The series converge geometrically: each successive order is 29 times smaller than the previous one for µ (ratio rµ ≈ 0.035) and 41 times smaller for α (rα ≈ 0.024). For µ, the sum of all orders above the second is ∼ 1.4 × 10−8 — smaller than the CODATA uncertainty (3.2×10−8 ). For α — ∼ 1.8×10−7 , which exceeds the uncertainty (2.1×10−8 ), but the cubic root implicitly sums part of the higher orders, and the result (−0.32σ) falls within CODATA. The cubic formula is the optimal approximation of infinite recursion at the level of current experimental precision. If CODATA precision improves to ±10−9 , a fourthorder term may be required.

7.6. The Electron as a Single Operator In the triadic architecture of the atom [7], the electron = the observation operator Ô. The key question: is there a single operator Ô at all nesting levels of the strange loop, or does each level have its own? The ODTOE answer: the operator is one. Arguments: (i) Indistinguishability of electrons — an experimental fact. All electrons are identical: mass, charge, spin, magnetic moment. If Ô is one, all its “applications” are identical by definition. (ii) Universality of µ. Formula (IV.1) contains no nesting-level parameter. If Ô varied from level to level, µ would depend on scale. Experimentally, µ is a single number at all scales. (iii) Geometric character of the series. The ratio of successive orders (r ≈ const) is constant. One operator → one decay coefficient → geometric series. Different operators → variable coefficient → unpredictable series. (iv) Absence of electron substructure. Limits on the electron compositeness scale (contact interactions in dilepton channels) reach Λ > 25–36 TeV depending on the model and interference sign (PDG, compositeness review). Within ODTOE, the electron is not a configuration with size but an operator without size. (v) Pauli exclusion principle. A single operator cannot actualize the same configuration in the same state twice — this is precisely the Pauli exclusion. Wheeler–Feynman’s one-electron theory (1940) is a special case: Wheeler proposed a single electron-object looping through time. ODTOE proposes a single operator applied recursively. The difference: an operator does not require equality of the number of electrons and positrons (baryon asymmetry is related to the chirality of the self-observation loop — the direction Ô → ι is not equivalent to the reverse, breaking CP symmetry at the architectural level).

7.7. Running α and the Layered Architecture (Open Question) The fine-structure constant α “runs” — it depends on the transferred momentum q: α−1 (q → 0) = 137.036, α−1 (q = mZ ) ≈ 127.9. In the Standard Model, this is explained by vacuum polarization. In ODTOE, the formula α−1 = 4π 3 + π 2 + π − corrections contains three layers. Each layer is the contribution of a particular component of the architecture. As q increases, layers “switch off” — the observer “penetrates” the screening shell: • At q → 0: all three layers are active → α−1 ≈ 137.036 • At q ∼ mZ : the π 2 layer (return via ι) becomes transparent → α−1 ≈ 4π 3 + π ≈ 127.2 (experiment: 127.9; difference ∼ 0.7 — spiral corrections at the mZ scale)

• At q ∼ mGUT : the π layer (observer presence) is also transparent → α−1 ≈ 4π 3 ≈ 124.0 Formally: α−1 (q) ≈ 4π 3 + θ(q < qι ) · π 2 + θ(q < qO ) · π − corrections(q), where qι ∼ mZ , qO ∼ mGUT . This is a qualitative interpretation consistent with experiment (∆α−1 ≈ π 2 ≈ 9.87 at the transition from q = 0 to q = mZ ; experiment: ∼ 9.1). A rigorous derivation of the threshold energies qι and qO from first principles of ODTOE is a direction for further research. This is a qualitative interpretation, not a quantitative prediction. The prediction α−1 (MZ ) ≈ 4π 3 + π = 127.17 diverges from PDG data (α(5) (MZ )−1 = 127.930 ± 0.008) by ∼ 95σ.

7.10. Sensitivity of the α−1 Formula to Discrete Coefficients The closed-form formula (X.1) contains integer coefficients (notably k = 11 in the third-order term C = k(π − 3)2 /φ) selected from structural reasoning. Although there are zero continuous free parameters, the discrete integer coefficients are chosen, not derived from a unique mathematical necessity. It is therefore important to assess the sensitivity of the result to the choice of k. Numerically, ∂x/∂k ≈ −6.60 × 10−7 per unit k. The following table shows the discrepancy from CODATA 2022 for neighbouring values of k: k

∆ from CODATA 2022

+6.6 × 10−7 −6.6 × 10−9 −6.7 × 10−7

+31 −0.32 −32

Only k = 11 falls within CODATA 2022 uncertainty. The statement “zero free parameters” means no continuous parameters; however, discrete integer coefficients such as k are selected from structural reasoning within the ODTOE formalism. The sharpness of the k-dependence (|∆σ| ≈ 32 per unit step) demonstrates that the choice k = 11 is tightly constrained by experiment.

7.11. Z2 fiber bundle: spinor justification of the factors of 2 The factor 2 enters the formulas for µ and α−1 in three places: (a) 6 = 3 × 2 in the base layer µ0 = 6π 5 ; (b) 2(π − 3)2 in the first correction to α−1 ; (c) 4π = 2 × 2π in the fermionic traversal (spin-1/2). All three were previously justified as “two directions of the cycle Φ” (forward Ô and reverse ι). The construction of a nontrivial Z2 fiber bundle over the φ-torus [28] unifies these three facts in a single geometric object. The φ-torus Tφ2 with radii ratio R/r = φ admits a bundle with fiber {+1, −1} and holonomies:

hol(γθ ) = +1,

hol(γϕ ) = −1

(VII.2)

where γθ is the minor circle traversal (phase cycle within level d), γϕ is the major circle traversal (inter-level transition). The Stiefel–Whitney class w1 (γϕ ) = 1 means the bundle is nontrivial: the section changes sign upon inter-level transition. Unification of the factors of 2: Via Z2 bundle

Context

Factor of 2

6 = 3 × 2 in µ0 2(π − 3)2 in α−1 4π (fermion)

Two directions of Φ Two fiber values {+1, −1} Gap in two directions Gap on each sheet of Te Double traversal Two turns on the double cover

From this holonomy, CPT symmetry (C = fiber flip, P = θ → −θ, T = ϕ → −ϕ; combined holonomy hol(CP T ) = (+1)(−1)(−1) = +1) and the Pauli exclusion principle (anticommutation of sections at orbit intersections: si (p) · sj (p) = −sj (p) · si (p), whence i = j is impossible) are derived. Distinguishability test: the twist contribution δtwist = π 2 (π − 3)4 /(µ · α−1 ) ≈ 1.58 × 10−8 becomes measurable at CODATA precision ±10−9 (expected by 2030).

VIII. THE FINE-STRUCTURE CONSTANT FROM FIRST PRINCIPLES 8.1. The Problem The fine-structure constant α = e2 /(4πε0h̄c) (SI) = e2 /(h̄c) (CGS-Gauss) ≈ 1/137.036 is a dimensionless number that determines the strength of the electromagnetic interaction [1, 3]. Feynman: “All good theoretical physicists put this number on their wall and worry about it.” Pauli, Eddington, and Dirac attempted to derive 137 from first principles. None succeeded. The approximation α ≈ φ2 /360 (accuracy 99.7%) was used in Section III.4 as an assumption. The present section derives α−1 from the first principles of ODTOE.

8.2. What α Means in ODTOE In ODTOE, electricity = the directed action of the operator Ô [13]. Charge = orientation in the strange loop (−1, 0, +1). U(1) symmetry = phase invariance of the loop. α is the cost of a single electromagnetic act: how much “action” is spent on the coupling between two projections of the operator.

8.3. Three Contributions to α−1 The operator Ô acts through four coherence components B = F w1 · E w2 · (1 − σ)w3 · Λw4 [5, Definition D1].

Contribution 1: operator action through 4 components of B. The operator Ô couples two projections (two charged particles). The coupling passes through all 4 components of B. Each component acts on the triadic architecture (3 levels: observer, observed, operator = π 3 ): contribution 1 = 4π 3 = 4 × 31.00628 = 124.02511

(VIII.1)

Contribution 2: return via ι (loop closure). The embedding operator ι : C → H returns the result through two topological rings: the configuration ring C (opening of the actualized configuration) and the potentiality ring H (dissolution into the field). Each ring costs π. The operator Ô passes through four rings (four components of B); the operator ι passes through two: it does not “create” (that is the function of Ô) but only opens and returns: contribution 2 = π 2 = 9.86960

(VIII.2)

Contribution 3: observer presence. The observer O stands at the center of the loop. Its presence induces an additional phase revolution — an analogue of the Berry phase (the geometric phase arising from adiabatic traversal of parameter space). The loop goes around the observer, and the very fact of circumnavigation costs the minimal topological invariant = one π: contribution 3 = π = 3.14159

(VIII.3)

α0−1 = 4π 3 + π 2 + π = π(4π 2 + π + 1)

(VIII.4)

8.4. Base Formula

Computation: 4π 3 + π 2 + π = 124.02511 + 9.86960 + 3.14159 = 137.03630. Experimental: 137.03600. Discrepancy: 0.00030. Six correct significant digits from pure π. Reads as: the inverse fine-structure constant = π× (action through components + return + presence).

IX. SPIRAL CORRECTIONS TO α−1 9.1. First-Order Correction: Spiral Gap Formula (VIII.4) describes an ideal circular loop. The real loop is spiral (π ̸= 3). Each revolution produces a gap (π − 3)2 . The gap reduces the effective coupling cost: part of the action “leaks” into the spiral gap. The correction acts along two directions of the cycle (forward Ô and reverse ι), hence the factor of 2. The correction is self-referential: α enters its own definition, just as Ψ∗ = Φ(Ψ∗ ):

α−1 = π(4π 2 + π + 1) −

2(π − 3)2 α−1

(IX.1)

Let x = α−1 , A = π(4π 2 + π + 1), B = 2(π − 3)2 : x2 − Ax + B = 0

A2 − 4B

(IX.2)

(IX.3)

Computation: A = 137.036304,

B = 2 × 0.020049 = 0.040097

A2 − 4B = 18778.948 − 0.160 = 18778.788

(IX.4) (IX.5)

18778.788 = 137.035718

(IX.6)

137.036304 + 137.035718 = 137.036011

(IX.7)

Experimental: 137.035999. Discrepancy: 0.000012. Seven significant digits (at this approximation level — first-order self-reference only).

9.2. Second-Order Correction: Gap of the Gap Remaining discrepancy: 0.000006. This is a correction of a different nature from B1 . If B1 = 2(π − 3)2 describes the loss to the spirality of the cycle (two directions → factor 2), then B2 describes the recursive depth of the spiral: the gap begets a gap, scaled by the golden step φ (step between recursion levels). The recursion is single (from turn to turn), unlike the directions (of which there are two), so the coefficient of B2 is unity rather than two. The fourth-order spiral gap ((π − 3)4 ), scaled by φ, divided by α−1 (second-level self-reference): δ2 =

(π − 3)4 · φ α−1

(IX.8)

0.000402 × 1.618034 = 0.0000047 137.036

(IX.9)

α−1 = 137.036005 − 0.0000047 = 137.036000

(IX.10)

Experimental: 137.035999177 (CODATA 2022). Discrepancy: 0.000007. Eight correct significant digits.

9.3. Third-Order Correction: Double Self-Reference The remaining discrepancy ∆ ≈ 7.25 × 10−6 (+345σ from CODATA 2022) requires accounting for double self-reference — the cost of coupling depends on the cost of the cost. The coefficient 11 = 6 + 5 has a structural justification. The number 6 = full observation cycle (3 components × 2 directions) — the same number that stands before π 5 in the formula for µ. The number 5 = the number of arguments for π (topological, spectral, measure-theoretic, dynamical, algebraic) — the same number that gives the exponent in µ0 = 6π 5 . A sum (rather than product) because the channels are parallel: the single operator Ô passes through the cycle and through the phase sequentially, not simultaneously — just as the contributions to α0−1 = 4π 3 + π 2 + π are summed. If there were two (different) operators, the channels would work in parallel and multiply; but the electron is one (Section VII.5), so the channels are summed. Toroidal interpretation of the number 11. The decomposition 11 = 6 + 5 admits an alternative but equivalent representation through the degrees of freedom of the φtorus — the structure that unifies the continuous (π) and the discrete (φ) in ODTOE. The φ-torus has a radii ratio R/r = φ and possesses exactly 11 degrees of freedom: • 3 degrees of phase rotation along the minor circle (radius r) — the internal cycle of the operator Ô, generating the wave function; • 3 degrees of inter-level transition along the major circle (radius R) — the external cycle of embedding ι, providing recursion between dimensionality levels d; • 4 components of coherence B = F w1 · E w2 · (1 − σ)w3 · Λw4 , determining the “thickness” of the torus at each point; • 1 observer — the centre of the torus, the self-reference point Ψ∗ = Φ(Ψ∗ ). The identity 6 + 5 ≡ (3 + 3) + (4 + 1) reveals the geometric meaning: the “full cycle” (6 = 3r + 3R ) is the traversal of the torus in both directions, while the “arguments of π” (5 = 4B + 1O ) is the complete structure of the observer on the torus. Each of the 11 degrees of freedom contributes equally to the double self-reference, which is why the coefficients are summed. The formula δ3 = 11 · (π − 3)2 /(φ · (α−1 )2 ) reads: the spiral gap, multiplied by all degrees of freedom of the φ-torus, divided by the golden step and the square of the coupling cost. This identity connects the formula for α−1 to the toroidal topology of reality and explains why precisely the number 11 (and not 10 or 12) arises in the third-order coefficient: 11 = dim(φ-torus with observer). The gap is divided by the inverse golden step (1/φ = φ − 1: the cost of returning one recursion level) and by the square of the coupling cost (double self-reference): δ3 =

11 · (π − 3)2 φ · (α−1 )2

Substituting α−1 ≈ 137.036:

(IX.11)

11 × 0.02005 0.2205 = 7.26 × 10−6 1.618 × 18779 30385

(IX.12)

α−1 = 137.036000 − 0.000007 = 137.035993

(IX.13)

δ3 =

Exact computation (cubic equation, Section X): αODTOE = 137.03599917035789...

Experimental: 137.035999177 (CODATA 2022, ±2.1 × 10−8 ). Discrepancy: −6.6 × 10−9 , which is −0.32σ. The formula falls within the experimental uncertainty. Alternative structural form. The variant C = 5π 2 φ4 (π − 3)4 (σ = +0.56), in which every factor comes directly from ODTOE (5 = arguments of π, π 2 = return, φ4 = recursion, (π − 3)4 = gap2 ), also falls within CODATA. The two variants are distinguishable at a precision of ±10−9 , which is unattainable before CODATA 2026+.

X. CLOSED-FORM FORMULA FOR α−1 10.1. Self-Referential Equation The complete formula is written as a cubic equation with three orders of self-reference: x3 − Ax2 + Bx + C = 0,

x = α−1

(X.1)

where:

A = π(4π 2 + π + 1),

B = 2(π − 3)2 + (π − 3)4 φ,

11 · (π − 3)2 φ

(X.2)

Self-referential form: α−1 = π(4π 2 + π + 1) −

2(π − 3)2 + (π − 3)4 φ 11 · (π − 3)2 − α−1 φ · (α−1 )2

(X.3)

Computing the coefficients (30 digits):

A = 137.036303775878432559202394652,

B = 0.040747314161935093904,

C = 0.13629705963530267

Solution by Newton’s method: αODTOE = 137.03599917035789534725390473

Comparison with experiment:

(X.4)

Source

ODTOE (X.4) 137.03599917036… — CODATA 2022 137.035999177(21) −6.6 × 10−9 CODATA 2018 137.035999084(21) +8.6 × 10−8

−0.32 +4.1

The formula falls within CODATA 2022 (−0.32σ). CODATA 2018 is +4.1σ away, which is explained by the upward shift of the central value by +9.3 × 10−8 between 2018 and 2022.

10.2. Decoding Each Element 4π 3 — the action of the operator Ô through four coherence components B (F , E, (1−σ), Λ), each passing through the triadic architecture (π 3 ). π 2 — the cost of the return ι : C → H through two “gates” (deactualization + repotentialization). π — the topological cost of the observer O being present in the loop. 2(π − 3)2 — the loss to the spiral gap along two directions of the cycle (forward and reverse). Reduces α−1 : part of the action “leaks” into the gap. (π − 3)4 φ — second-order spiral correction: the gap of the gap, scaled by the golden step φ. Self-reference: α−1 appears on both sides of the quadratic equation.

10.3. Layer-by-Layer Verification

Layer exp

Formula 4π 3 + π 2 + π −2(π − 3)2 /x (self-ref.) −(π −3)4 φ/x −11(π − 3)2 /(φx2 ) CODATA 2022 [1]

∆ from CODATA 2022

137.03630 137.036011

+3.05 × 10−4 +1.20 × 10−5

+14505 +571

137.036006 137.03599917

+7.25 × 10−6 −6.6 × 10−9

+345 -0.32

137.035999177(21)

10.4. Explanation of the Approximation φ2 /360 The old approximation α ≈ φ2 /360 (accuracy 99.7%) is not rejected but explained: = 137.508 ≈ 4π 3 + π 2 + π + 0.472 φ2

The difference 0.472 ≈ π(π − 3) = 0.445. The approximation φ2 /360 is a rough estimate in which the contributions 4π 3 +π 2 +π are “folded” into a single ratio. Formula (X.1) unfolds this folding.

10.5. Why α−1 is a Sum and µ is a Product The proton (µ) is a configuration: a stable object, a fixed point. Its mass is determined by inertia, requiring self-consistency across all five arguments simultaneously. Hence π 5 (multiplicatively: all five aspects must hold at once). α is not a configuration but an interaction: a process, not an object. The cost is determined by how many layers the operator passes through in a single coupling act. The contributions are summed (parallel): action through one channel is independent of action through another. Hence 4π 3 + π 2 + π (additively). Configuration — product. Interaction — sum.

XI. CONCLUSION From the structural constants of ODTOE (π, φ, integers) and zero free parameters, self-referential formulae for two fundamental dimensionless constants of physics have been derived. Proton-to-electron mass ratio: (π − 3)2 3πφ4 (π − 3)2 µ = 6π + = 1836.15267342575... 1 − (π − 3)2 φ2 21600 µ µ2

Five layers: full cycle × fivefold self-consistency, infinite spiral series, electromagnetic self-coupling, single self-reference, double self-reference. Result: µODTOE = 1836.15267342575..., discrepancy from CODATA 2022: −0.008σ. Inverse fine-structure constant:

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

11(π − 3)2 = 0, φ

x = α−1 = 137.03599917036.

Four layers: action through components + return + presence, first-order spiral gap, second-order spiral gap, double self-reference (11 parallel channels). Result: αODTOE 137.03599917036..., discrepancy from CODATA 2022: −0.32σ. Both formulae: contain only π, φ, and integers; have zero free parameters; are self-referential (the value of the constant enters its own definition); every element has a substantive interpretation in the ODTOE formalism; fall within the experimental uncertainty of CODATA 2022: µ to −0.008σ, α−1 to −0.32σ. µ is a configuration (product: 6π 5 , all aspects simultaneously). α−1 is an interaction (sum: 4π 3 + π 2 + π, contributions in parallel). Configuration — product. Interaction —

sum. Both are cubic self-referential equations reflecting the threefold nesting of the strange loop. Falsifiable predictions for CODATA 2026+: µODTOE = 1836.15267342575395091347... αODTOE = 137.03599917035789534725...

If future measurements yield values outside these numbers ± current uncertainty, the formulae are refuted. Numerical agreement with current tabulated values within the uncertainty does not prove model uniqueness but constitutes a necessary condition for its viability. In 2025, high-precision laser spectroscopy of H+ 2 (Nature, 2025) achieved accuracy on the order of tens of ppt for µ — a result comparable with the ODTOE prediction. Both formulae represent the first derivations of these constants from first principles in any theoretical construction.

ACKNOWLEDGMENTS AND TOOLS In the development of the ODTOE theory and all articles 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.

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CONFLICT OF INTEREST The author declares no conflict of interest.

FUNDING This work received no external funding.