# Cosmological Fractions from Toroidal Architecture: Deriving Dark Energy, Dark Matter and Baryonic Matter from π and φ

> Within the toroidal ODTOE model, the cosmological fractions of dark energy, dark matter and baryonic matter are derived from two structural invariants: π and φ. The φ-torus possesses three topological sectors: inter-level (R², gravitational inertia), intra-level (r²=1), and gap sector (Z=(π−3)/[1−(π−3)φ]). Normalized fractions: ΩΛ:ΩDM:Ωb = φ²:1:Z = 68.86%:26.30%:4.83%. Planck 2018 comparison: dark energy 0.54σ, dark matter 0.32σ, baryonic 1.64σ. Zero adjustable parameters.

Source: https://odtoe.org/en/articles/cosmological-fractions
Author: Anton Pankratov · Observer-Dependent Theory of Everything (ODTOE) · CC BY 4.0

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COSMOLOGICAL FRACTIONS FROM TOROIDAL ARCHITECTURE: DERIVING THE CONTENT OF DARK ENERGY, DARK MATTER AND BARYONIC MATTER FROM π AND φ Pankratov Anton Sergeevich Independent researcher, Kazan, Russia E-mail: anton.s.pankratov@gmail.com ORCID: 0009-0002-4870-2995

## UDC 524.8 + 530.12 + 514.7 + 167.7

ABSTRACT Within the toroidal model of ODTOE, the cosmological fractions of dark energy, dark matter and baryonic (visible) matter content of the Universe are derived from two structural invariants: π and φ. The φ-torus (a torus with radii ratio R/r = φ, maximally stable by the KAM theorem) possesses three topological sectors: the inter-level sector (major radius R, gravitational inertia ∝ R2 = φ2 ), the intra-level sector (minor radius r, inertia ∝ r2 = 1), and the gap sector (accumulated spiral gaps, full series Z = (π − 3)/[1 − (π − 3)φ]). Normalized fractions: ΩΛ : ΩDM : Ωb = φ2 : 1 : Z = 68.86% : 26.30% : 4.83%. Comparison with Planck 2018 data (TT,TE,EE+lowE+lensing) [1]: 68.47±0.73% (dark energy), 26.07±0.73% (cold dark matter Ωc ), 4.93±0.06% (baryonic). Dark energy and dark matter fall within the 1σ Planck confidence interval (0.54σ and 0.32σ respectively). Baryonic matter deviates by 1.64σ (within 2σ). A self-referential correction (by analogy with the formulas for µ and α−1 [10]) improves the agreement to 1.24σ. The formula contains zero adjustable parameters. All three components are expressed through π and φ, which arise from the Banach theorem [17] as the continuous and discrete invariants of convergence to the fixed point. Keywords: dark energy, dark matter, baryonic matter, cosmological fractions, ODTOE, φ-torus, KAM theorem, spiral gap, number π, golden ratio.

I. INTRODUCTION I.1. The Problem The observable Universe consists of three main components by energy density: dark energy (ΩΛ ≈ 68.5%), dark matter (ΩDM ≈ 26.5%), baryonic matter (Ωb ≈ 5%) [1]. The standard model of cosmology (ΛCDM) accepts these fractions as empirical parameters

fitted to data from the cosmic microwave background, Type Ia supernovae and baryon acoustic oscillations [18, 19]. The question of why these particular fractions — remains open. The cosmological constant Λ is not derived from first principles; the discrepancy between the prediction of quantum field theory and observation amounts to ∼ 10120 (the cosmological constant problem [2]).

I.2. The Approach ODTOE [3] models reality as a hierarchy of nested φ-tori [4]: each level of dimensionality d is represented by a torus with radii ratio R/r = φ, maximally stable by the KAM theorem [5, 6, 7]. The three topological sectors of the torus give rise to three components of cosmological content. Below it is shown that the normalized sector fractions coincide with Planck 2018 data [1] within 1σ–2σ. This work uses results from the ODTOE series of papers: the toroidal topology of reality [4], the structure of the number π as an invariant of observation [9], the derivation of the fundamental constants µ and α−1 [10], the model of the atom as a strange loop [11], the architecture of the quantum [14], the dimensionality of the observer [15] and Planck’s constant from the architecture of observation [16].

II. THE φ-TORUS: THREE TOPOLOGICAL SECTORS II.1. Definition √ A torus with major radius R and minor radius r, R/r = φ = (1 + 5)/2. The trajectory on the torus is described by two angular coordinates: θ (rotation around the minor radius, fast) and ϕ (rotation around the major radius, slow). By the KAM theorem [5, 6, 7]: when the frequency ratio ωθ /ωϕ = R/r = φ (the most irrational number [8]) the torus is maximally stable against perturbations. The trajectory is quasi-periodic: it never closes, densely filling the surface.

II.2. Three Sectors Sector I: inter-level (R-dynamics). Rotation along the major radius = transition between dimensionality levels d. Associated with macrostructure: expansion of the Universe, cosmological constant. Via ODTOE: pressure of the field H (infinite) on the finite configuration C. Sector II: intra-level (r-dynamics). Rotation along the minor radius = phase dynamics within a single level d. Associated with structure formation: gravitational binding, halo formation. Via ODTOE: coherent configurations at levels d > dour , invisible by D-Prot but gravitating by P5 [3]. Sector III: gap ((π−3)-dynamics). Accumulated spiral gaps: each revolution along θ does not close (length = π > 3, gap = π − 3), generating a remainder. The sum of

remainders from all windings = visible matter. Via ODTOE: everything that is born in the gap of the observation loop — photons, atoms, stars, observers [9, 14].

II.3. Gravitational Inertia of the Sectors Each sector contributes to the total gravitational inertia of the Universe. Here gravitational inertia denotes the contribution of the corresponding mode to the T 00 component of the energy-momentum tensor [20]. The fraction Ωi is determined by the gravitational weight of the corresponding degree of freedom. For rotational motion the gravitational weight is proportional to the moment of inertia: I = m · reff

## (II.1)

For R-rotation: IR = mR2 . For r-rotation: Ir = mr2 . The ratio: R2 IR = 2 = φ2 Ir r

## (II.2)

Justification: in general relativity, the contribution of a component to the total energy density is determined by the energy-momentum tensor Tµν [20]. For a perfect fluid: T 00 = ρc2 (energy density). For rotational motion the kinetic energy density ∝ Iω 2 /V . But by the KAM condition ωθ /ωϕ = R/r = φ, whence IR ωϕ2 /(Ir ωθ2 ) = (R2 /r2 )× (ωϕ /ωθ )2 = φ2 /φ2 = 1. The kinetic energies are equal (the virial theorem for a KAM torus). However, gravitational inertia is determined not by kinetic energy, but by the total energy (kinetic + potential + pressure). For the cosmological constant: p = −ρc2 (negative pressure), the contribution to the effective gravitational mass ∝ ρ + 3p/c2 = −2ρ [2, 20]. It is precisely this anomalous contribution (through pressure) that scales as R2 , not R2 ω 2 . The total effective mass of the sector: Meff, R ∝ R2 ,

Meff, r ∝ r2

## (II.3)

The ratio of gravitational weights = φ2 : 1 — is determined by the geometry of the torus, not by the dynamics.

III. CONTRIBUTION OF THE GAP SECTOR III.1. One Winding Each revolution along the minor radius (θ) does not close: the path length = π, the minimum closed path = 3 (ternary architecture [9]). The first-order gap: δ1 = π − 3 = 0.14159265358979...

## (III.1)

III.2. Full Spiral Series Each winding generates a gap scaled by φ (the step between windings on the torus). The k-th order gap: (π − 3)k · φk−1 . The full series: Z=

(π − 3)k · φk−1 =

k=1

(π − 3) 1 − (π − 3)φ

## (III.2)

Convergence: the ratio (π − 3)φ = 0.22910... < 1. The series is geometric. Numerical value (50 digits): Z = 0.18367229293062031020024539841572564569480...

## (III.3)

Decomposition by contributions: Order k

Contribution (π − 3)k φk−1

Fraction of Z

0.14159 0.03244 0.00743 0.00170 0.00051

77.1% 17.7% 4.0% 0.9% 0.3%

Visible matter consists of 77% from the “first winding” (k = 1) and 23% from higher orders. The higher orders are critical for agreement with Planck: without them Ωb = 3.77% (first order), with them Ωb = 4.83% (full series).

## IV. NORMALIZED FRACTIONS IV.1. Formula ΩΛ : ΩDM : Ωb = φ2 : 1 : Z

## (IV.1)

Σ = φ2 + 1 + Z

## (IV.2)

Normalization:

ΩΛ =

, Σ

## ΩDM =

, Σ

Ωb =

Z Σ

## (IV.3)

IV.2. Numerical Values (50 digits for reproducibility) Note. 50 digits are provided for exact reproducibility of computations; the precision of Planck data is ±0.73% (∼3 significant figures).

φ2 = 2.61803398874989484820458683436563811772031... Z = 0.18367229293062031020024539841572564569480... Σ = 3.80170628168051515840483223278136376341511... ΩΛ = 0.68864709548066742427504562258101833038578... = 68.865% ΩDM = 0.26303978421972085001664645325056078691342... = 26.304% Ωb = 0.04831312029961172570830792416842088270080... = 4.831% Check: ΩΛ + ΩDM + Ωb = 1.00000000000000000000000000000000000000000.

V. COMPARISON WITH PLANCK 2018 DATA V.1. Experimental Values Planck mission data [1] (TT,TE,EE+lowE+lensing, 68% CL): ΩΛ = 0.6847 ± 0.0073 Ωc = 0.2607 ± 0.0073

(cold dark matter)

## (V.1) (V.2)

Ωb = 0.0493 ± 0.0006 (baryonic matter) (V.3) Additionally: Ων ≈ 0.0014 (neutrinos, at mν = 0.06 eV), Ωr ≈ 0.0001 (cosmic microwave background radiation).

V.2. Identification of Components ODTOE sector

Cosmological component

φ2 (inter-level) ΩΛ (dark energy) 1 (intra-level) Ωc (cold dark matter) Z (gap) Ωb (baryonic)

Mechanism (ODTOE) Pressure of H on C via R Coherent structures at d > dour Matter from the spiral gap

Neutrinos (Ων ) and radiation (Ωr ) are not separated into distinct sectors: their contributions ∼ 0.15% are absorbed by the main components. In the four-component model (Section VII) neutrinos are separated out.

V.3. Comparison Table Component

ODTOE, % Planck 2018, %

Dark energy (ΩΛ ) Dark matter (ΩDM ) Baryonic (Ωb )

68.47 ± 0.73 26.07 ± 0.73 4.93 ± 0.06

68.86 26.30 4.83

Dev., %

+0.39 +0.23 −0.10

0.54 0.32 1.64

Dark energy and dark matter: within the 1σ confidence interval. Baryonic: within 2σ, deviation 1.64σ.

VI. SELF-REFERENTIAL CORRECTION VI.1. Justification The formulas for µ = mp /me and α−1 [10] contain self-referential terms: the proton mass enters its own definition ((π − 3)2 /µ), the fine-structure constant enters its own equation. For cosmological fractions: the baryonic matter fraction affects the total gravitational dynamics, which determines the conditions for the existence of baryons. The loop Ωb ↔ conditions for baryon creation [11].

VI.2. Quadratic Equation Denote x = Ωb , ε = (π − 3)2 , K = φ2 + 1: x=

Z + εx K + Z + εx

## (VI.1)

Expanding: εx2 + x(K + Z − ε) − Z = 0

## (VI.2)

Solution (positive root): x=

## −(K + Z − ε) +

## (K + Z − ε)2 + 4εZ 2ε

Numerical values: a = ε = 0.02004847955... b = K + Z − ε = 3.78165780213... c = −Z = −0.18367229293...

## (VI.3)

D = b2 + 4ac = 14.31566513325... √

x = Ωb

D = 3.78360478027...

= 0.04855675290... = 4.856%

VI.3. Recalculation of All Fractions

ΩΛ =

## ΩDM =

= 4.856%

φ2 ( (sr) ) 1 − Ωb = 68.847% φ +1

φ2 + 1

(sr) )

1 − Ωb

= 26.297%

VI.4. Comparison (with Self-Reference) Component

Without self-ref. With self-ref. Planck 2018

## ΩΛ ΩDM Ωb

68.86% 26.30% 4.83%

68.85% 26.30% 4.86%

σ (self-ref.)

68.47 ± 0.73 26.07 ± 0.73 4.93 ± 0.06

0.52 0.31 1.24

The self-referential correction improves the agreement for baryons: 1.64σ → 1.24σ.

VII. FOUR-COMPONENT MODEL (WITH NEUTRINOS) VII.1. Neutrinos as the Second-Order Gap Via ODTOE: neutrinos = spiral remainder of the observation loop [11, Section IV.3]. Their contribution = (π − 3)2 = 0.02005 (squared gap, second order): Σ4 = φ2 + 1 + Z + (π − 3)2 = 3.82175... (4)

ΩΛ = 68.50%,

(4)

ΩDM = 26.17%,

(4)

Ωb = 4.81%,

Ω(4) ν = 0.52%

## (VII.1) (VII.2)

VII.2. Comparison Planck gives Ων ≈ 0.14% (at the minimum mass sum mν = 0.06 eV) [1]. The ODTOE model: 0.52%. Discrepancy: ×3.7. Significant. However:∑the Planck upper limit on mν is < 0.12 eV (95% CL), which gives Ων < 0.27%. At mν ≈ 0.15 eV (admissible in extended models [21]): Ων ≈ 0.34%, closer to 0.52%. Status: the four-component model predicts a neutrino mass sum higher than the minimum. This is a falsifiable prediction in tension with the current Planck upper limit (Ων < 0.27% at 95% CL). If future data (KATRIN, DESI, CMB-S4) confirm Ων < 0.3%, the four-component model will be refuted. The three-component model (Sections IV–VI) retains its validity.

VIII. BINARY AND TERNARY φ-PROPORTIONS VIII.1. Binary (Work/Rest) With two components (without the gap): φ/(1 + φ) : 1/(1 + φ) = 61.8% : 38.2%. Observed in optimal work/rest regimes (62:38), inhalation/exhalation, systole/diastole [12].

VIII.2. Ternary (Universe) With three components (with the gap): φ2 : 1 : Z = 68.9% : 26.3% : 4.8%. Observed in the composition of the Universe [1].

VIII.3. Connection Dark energy in the ternary model (68.9%) is greater than in the binary model (61.8%), because the gap sector (4.8%) is a small third contribution, and its “share” is mostly compensated by the major radius. The binary φ-proportion is the limit of the ternary one as Z → 0 (the gap tends to zero, π → 3): φ = = = 61.8% π→3 φ2 + 1 + Z φ +1 1+φ lim

## (VIII.1)

The ternary proportion reduces to the binary one in the limit of zero gap. π > 3 is the reason why the cosmological fractions differ from the “pure” φ-proportion.

## IX. DEMARCATION

Statement

Status

KAM theorem: the φ-torus is maximally stable φ is the most irrational number Three sectors of the torus → three components Gravitational inertia ∝ R2 : r2 = φ2 : 1 Full series Z = (π − 3)/(1 − (π − 3)φ) ΩΛ = 68.86% (within 1σ of Planck) ΩDM = 26.30% (within 1σ of Planck) Ωb = 4.83% (1.64σ from Planck) Self-ref. correction: Ωb = 4.86% (1.24σ) Neutrinos = (π − 3)2 , Ων ≈ 0.52% Binary → ternary at π > 3

Proven [5, 6, 7] Proven [8] Interpretation via ODTOE Follows from Tµν for torus modes Follows from the geom. series of gaps Numerical result, zero fitting Numerical result, zero fitting Numerical result, refinement Follows by analogy [10] Falsifiable prediction Mathematical fact (limit)

X. CONCLUSION X.1. Result From two numbers (π and φ) and one geometric construction (the φ-torus), three cosmological fractions are derived without adjustable parameters: Z ΩΛ : ΩDM : Ωb = 2 : 2 : 2 φ +1+Z φ +1+Z φ +1+Z

(X.1)

(π − 3) 1 − (π − 3)φ

(X.2)

= 68.86% : 26.30% : 4.83%

(X.3)

where Z =

Agreement with Planck 2018: 0.54σ, 0.32σ, 1.64σ (all within 2σ, two out of three within 1σ).

X.2. Structure of the Formula φ2 = gravitational weight of inter-level dynamics (dark energy = pressure of H on C) [3, 4]. 1 = gravitational weight of intra-level dynamics (dark matter = coherent structures at d > 3) [3, 15]. Z = (π − 3)/(1 − (π − 3)φ) = accumulated gaps of all windings (visible matter = everything born in the gap) [9, 14].

X.3. What This Means The Universe consists of ∼ 95% “torus” (φ2 +1: two rotations, invisible to us) and ∼ 5% “gap” (Z: what is born each time the loop fails to close). We are the gap. We are (π −3), multiplied by φ and summed over all windings of the spiral. A small fraction, but the only visible one. The remaining 95% is the torus on which we live but which we do not see, as a fish does not see the water.

We =

(π − 3) = 4.83% of the Universe = sum of all spiral gaps 1 − (π − 3)φ

ACKNOWLEDGEMENTS AND TOOLS During the development of the ODTOE theory 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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