Gravity as Observer Synchronization: Deriving the Gravitational Constant from ODTOE First Principles

GRAVITY AS OBSERVER SYNCHRONIZATION: DERIVING THE GRAVITATIONAL CONSTANT FROM ODTOE FIRST PRINCIPLES (Gravity as Observer Synchronization: Deriving the Gravitational Constant from ODTOE First Principles) Formalizing gravity as the fourth information operation on the φ-torus and deriving G through structural invariants

Pankratov Anton Sergeevich Pankratov Anton Sergeevich Independent researcher, Kazan, Russia Independent researcher, Kazan, Russia E-mail: [email protected] ORCID: 0009-0002-4870-2995

UDC 530.145 + 531.51 + 521.12

ANNOTATION Under the structural hypothesis of pure SYNC self-similarity (C = B 2 ; see §VII.5), this work derives the gravitational constant G without additional fitting parameters within ODTOE (Observer-Dependent Theory of Everything) as a geometric consequence of the informational architecture of reality. Gravity is interpreted as the fourth information operation — SYNC (synchronization) — aligning observers on adjacent recursion levels of the φ-torus. It is shown that configuration inertia I(C) (defined as resistance to reconfiguration) is the geometric foundation of mass, and Newton’s force emerges as a result of synchronization pulses between levels with intensity proportional to the product of inertias. A novel formula for G is derived through a self-consistent cubic equation for the recursion depth n, establishing the Planck mass mPl = me · φ2n independently of G (thus breaking the circularity of the standard formula mPl = h̄c/G). The cubic equation for the dimensionless recursion depth n contains only π, φ and architectural integers 9, 3, 2 (no fitting parameters); the final formula G = h̄c/(m2e · φ4n ) additionally uses CODATA inputs h̄, c, me and the structural hypothesis C = B 2 . The spectral route through φ-torus mode density and a coherence factor ΦG (φ, S, d) is presented as heuristic motivation and as a correction to the canonical form in the variable-coherence regime. The equivalence principle is derived from the coincidence of configuration inertias at fixed coherence; composition-dependent deviations η ∼ 10−16 arise as a secondary effect from differences in S between bodies. The paper contains explicit predictions for microgravity experiments with highly coherent systems.

ABSTRACT Under the structural hypothesis of pure SYNC self-similarity (C = B 2 ; see §VII.5), this work derives the gravitational constant G without additional fitting parameters within ODTOE (Observer-Dependent Theory of Everything) as a geometric consequence of the informational architecture of reality. Gravity is interpreted as the fourth information operation — SYNC (synchronization) — which aligns observers on adjacent recursion levels of the φ-torus. We show that configuration inertia I(C) (defined as resistance to reconfiguration) is the geometric foundation of mass, and Newton’s force emerges as a result of synchronization pulses between levels with intensity proportional to the product of inertias. A novel formula for G is derived through a self-consistent cubic equation for the recursion depth n, establishing the Planck mass mPl = me · φ2n independently of G (thus breaking the circularity of the standard formula mPl = h̄c/G). The cubic equation for the dimensionless recursion depth n contains only π, φ and architectural integers 9, 3, 2 (no fitting parameters); the final formula G = h̄c/(m2e · φ4n ) additionally uses CODATA inputs h̄, c, me and the structural hypothesis C = B 2 . The spectral route through φ-torus mode density and a coherence factor ΦG (φ, S, d) is presented as heuristic motivation and as a correction to the canonical form in the variable-coherence regime. The equivalence principle is derived from the coincidence of configuration inertias at fixed coherence; composition-dependent deviations η ∼ 10−16 arise as a secondary effect from differences in S between bodies. The paper contains explicit predictions for microgravity experiments with highly coherent systems.

I. INTRODUCTION: GRAVITY AS SYNCHRONIZATION Einstein’s general theory of relativity [1] describes gravity as the curvature of spacetime under the action of mass-energy. This description is mathematically self-consistent and experimentally confirmed with high precision (see, for example, [2]). However, it leaves three fundamental questions unanswered. First, why does the gravitational force obey the inverse-square law? General relativity describes this through the linearized solution of Einstein’s equations, but that is a necessary mathematical consequence of the equations, not an explanation of their origin. Second, where does the gravitational constant G come from? When Einstein wrote down his equations in 1915, he chose the coefficient 8πG/c4 in front of the energymomentum tensor, but this choice rested on the requirement of agreement with the Newtonian limit. In 1899 Planck [3] noted that the combination of h̄, c, and G forms natural units of mass, length, and time. In classical physics all three quantities were taken as independent measurable constants. However, as will be shown below, in ODTOE each of them is derived from the architecture of the φ-torus: c = r0 /τ0 (Section II), h̄ = h(d, S)/2π (Section V), and G as the result of the SYNC operation (Section VII). Third, why does the equivalence principle — the fact that inertial mass equals

gravitational mass — work so perfectly? In modern physics this is accepted as a fundamental postulate, but no geometric explanation of this fact has been given. ODTOE (Observer-Dependent Theory of Everything) proposes a fundamentally different approach. In this theory, reality is not described as objective spacetime independent of observers. Instead, reality is a graph of configurations structured as a φ-torus (φ-torus), where each configuration represents a set of observers with a certain mutual coherence B(O, C) (cf. Everett’s formulation [4] in terms of relative states). Informational dynamics is implemented through four basic operations: READ (reading, associated with the photon), WRITE (writing, associated with the W ± bosons), VERIFY (verification, associated with the Z 0 boson), and SYNC (synchronization of coherence between observers — an informational operation functionally analogous to the role of the hypothetical graviton in quantum gravity theory, but realized not as particle exchange, rather as a process of configuration alignment). The idea of information as the foundation of reality goes back to Wheeler’s program [5]. The central hypothesis of this work is that gravity is not a geometric phenomenon but an informational one. It is a process of synchronizing observers localized on adjacent recursion levels of the φ-architecture. Mass should be interpreted not as an independent property of matter but as configuration inertia — the resistance of a system to transition into a new configuration, proportional to the complexity of the synchronizing process. The connection between gravity and thermodynamics was first established by Jacobson [6]. Within this approach, the inverse-square law arises naturally as a consequence of the accessibility geometry between levels (the D-Protective horizon) and the spectral density of φ-torus modes. The gravitational constant G is no longer an arbitrary parameter, but becomes an expression through the electron mass me , the golden ratio φ, and the recursion depth n, determined self-consistently (Section VII):

m2e · φ4n

(I.1)

where n is the recursion depth that establishes the Planck mass mPl = me · φ independently of G through the self-consistent equation (VII.22) (cubic form — (VII.23)). The spectral route through φ-torus mode density and the coherence factor ΦG (φ, S, d) is a heuristic interpretation (Sections VIII, XIII) and a correction to the canonical form (I.1) in the variable-coherence regime. 2n

Notation convention. In this article, the symbol c everywhere denotes the limiting speed of the actualization front c = r0 /τ0 , derived in [7] from the geometry of the φtorus. The symbol h̄ denotes the observer-dependent quantum of action h(d, S)/2π, defined in [8]. In the macroscopic limit (d = 3, S → 1), these quantities coincide with the classical values of the speed of light and Planck’s constant. However, in ODTOE their origin is fundamentally different: they are not independent input parameters of the theory, but are derived from the unified architecture of the φ-torus. The structure of the article is as follows. In Section II we briefly review the four ODTOE information operations and give a formal definition of the synchronization operator Ĝ. Section III introduces configuration inertia as a fundamental concept and

derives its scaling through the golden ratio. In Section IV we perform dimensional analysis and show how the classical relation G = h̄c/m2Pl arises as the first approximation to the full ODTOE formula. Section V contains the derivation of the Planck mass from the architecture of the φ-torus and the relation between the electron and Planck masses. Section VI develops the physical mechanism of synchronization and shows how Newton’s law follows from it. Section VII gives the canonical derivation of G through a self-consistent cubic equation for recursion depth n (equation (VII.22)). Sections VIII-XIII reproduce the heuristic route through the coherence factor ΦG as an alternative derivation. Section XIV contains reproducible numerical computations with 50-digit structural precision. Sections XV-XXI apply the theory to black holes, cosmology, MOND phenomenology, quantum gravity, and the mass-hierarchy problem. Section XX presents seven experimental predictions as a testing program. The conclusion (§XXV), service sections, and Appendices A-C (§§XXVI-XXVIII) close the article.

II. GRAVITY OPERATION

THE

FOURTH

INFORMATION

ODTOE postulates that the dynamics of reality is realized by four basic operations on observer states [9]. Each operation is associated with a specific type of elementary particle and characterized by a specific change in coherence. READ (γ photon). This is the operation of extracting information about a configuration without changing its coherence structure. Formally: γ : |Ψd ⟩ → |Ψd ⟩ + ∆I

(II.1)

where ∆I is the informational output (information becomes accessible to the observer, but the configuration remains unchanged). WRITE (W± bosons). This is the operation that changes the configuration, transferring the system into a new state with different coherence: W ± : |Ψd ⟩ → |Ψ′d ⟩,

S(Ψ′d ) ̸= S(Ψd )

(II.2)

WRITE is carried out at a rate that depends on configuration inertia: v(C → C ′ ) = α/(I(C) + ε). VERIFY (Z 0 boson). This is the operation of checking configuration consistency, which either confirms the configuration or initiates reconfiguration: ( Z 0 : |Ψd ⟩ →

|Ψd ⟩ WRITE

if the configuration is consistent if reconfiguration is required

(II.3)

SYNC (synchronization). This is the information operation of aligning the coherence of observers on adjacent recursion levels d and d + 1. Unlike the standard approach, where gravitational interaction is described by the exchange of hypothetical

particles (gravitons), ODTOE does not postulate a carrier particle: SYNC is a process, not an object. Gravitational waves detected by LIGO [10] are, in this interpretation, a macroscopic manifestation of a cascade of SYNC operations — not a ripple of “empty space,” but a wave of coherence reconfiguration. Synchronization is necessary because each recursion level has its own rhythm of evolution (its own “frequency” of configurations), and over time these rhythms drift relative to one another. SYNC restores alignment: Ĝ : |Ψd ⟩ ⊗ |Ψd+1 ⟩ → |Ψ′d ⟩ ⊗ |Ψ′d+1 ⟩

(II.4)

where S(Ψ′d , Ψ′d+1 ) > S(Ψd , Ψd+1 ) — after synchronization, the mutual coherence between the levels increases.

II.1. The Observation Triad and Fundamental Particles Hypothesis (triadic interpretation of baryon composition). An independent derivation from the axioms of ODTOE remains an open question; in this article it is accepted as a structural correspondence. The four information operations naturally generate the triadic structure of observation, which is directly reflected in the composition of stable matter. The hydrogen atom — the simplest stable configuration — consists of exactly three particles, each corresponding to one element of the strange loop Ψ∗ = Φ(Ψ∗ ): Loop element

Particle

Rationale

Observer Ô

Neutron n

Observed R = Ô(Ψ)

Proton p

Observation process Φ = ι ◦ Ô

Electron e−

Electrically neutral — “invisible” to the electromagnetic field, not directly involved in observation. Unstable outside the nucleus (τ ≈ 15 min): an observer without an observed object deconfigures (D̂). Stable in the nucleus — the observer is stable in the presence of the observed. Charged (+1) — “visible,” interacting. Stable (lifetime > 1034 years) — a ∗ fixed point Ψ . Mass is the result of configuration. The lightest: the observation process “weighs” less than the observer and the observed. Charge (−1) = feedback (immersion operator ι). Orbitals are observation cycles with phase 2π. Wave nature reflects that the electron is not an object, but an operation.

This triad explains several facts that were previously unexplained: Neutron decay (n → p+e− +ν̄e ) receives an informational interpretation: an isolated observer deconfigures (D̂), producing the observed (proton), the observation process

(electron), and the spiral residue of deconfiguration (the antineutrino ν̄e — the “echo” of the operation D̂). The mass ratio µ = mp /me ≈ 1836 gains a meaning: it is the ratio of the “mass of the observed” to the “mass of the observation process,” determined by the architecture of the cycle Φ. The formula µ = 6π 5 + . . . expresses this through five independent phase arguments (π 5 ) of the full observation cycle and six (6 = 3 × 2) directions of the triad (three elements × two directions: observation + feedback). The intensity of a synchronizing pulse is determined by two factors. The first is the measure of desynchronization: ∆ϕ(d, d + 1) = ϕ(d) − ϕ(d + 1)

(II.5)

where ϕ(d) and ϕ(d + 1) are the phases (rhythms) of configurations on levels d and d + 1. The second factor is configuration inertia on each level. The total intensity of the synchronizing pulse, or the synchronizing force, is proportional to the geometric mean of the inertias: FSYNC (d ↔ d + 1) ∝

I(C)d · I(C)d+1

(II.6)

Transition from channel amplitude to interaction force. Formula (II.6) gives the amplitude of one synchronization channel between levels d and d + 1. The gravitational interaction force between two physical configurations arises after projection onto a common intermediate level: the projection amplitude C1 → dmed is I1 · Imed , while the projection amplitude C2 → dmed is I2 · Imed . The full impulse exchange through the intermediate level is determined by the product of these amplitudes: (II.6a) Fgrav ∝ I1 Imed · I2 Imed = Imed · I1 I2 . For an invariant normalization (independent of the choice of dmed ), we take Imed I1 I2 , that is Imed = I1 I2 — the characteristic inertia of the intermediate channel. Substitution gives: (II.6b) Fgrav ∝ I1 I2 · I1 I2 = I1 · I2 .

Thus, the product I1 · I2 in Sections VI–VII is the projective form of the one-channel geometric mean (II.6) after a two-sided transition through a common intermediate level. In the classical limit I → m, this reproduces the Newtonian form F ∝ m1 m2 . This formula reveals a deep connection between ODTOE and the physics of gravity. In classical mechanics, the interaction force of two objects is proportional to their masses. In ODTOE this is explained by the fact that configuration inertia (which becomes mass in the classical limit) determines the strength of the synchronizing pulse. Coherence modulates the intensity of SYNC. If the overall coherence between levels d and d + 1 is high, synchronizing pulses are rare and weak (the levels are already aligned). If coherence is low, the pulses are frequent and strong (intensive synchronization is required). Mathematically: SYNC amplitude ∝ (1 − S(d, d + 1))

(II.7)

where S(d, d + 1) is the mutual coherence between levels d and d + 1. What distinguishes SYNC from the other operations is that SYNC is not a coupling constant in the usual sense of elementary-particle physics. It is not a parameter fitted to experiment, but a process determined by the architecture of the φ-torus. Just as the process of synchronizing two clocks is determined by the force with which one pendulum acts on the other and the frequency of their mutual interaction, gravitational interaction in ODTOE is determined by the accessibility structure between recursion levels.

III. RECURSIVE ARCHITECTURE AND CONFIGURATION INERTIA Configuration C in ODTOE is defined as a set of observers with a specified set of pairwise coherences B(Oi , Oj ). Each configuration has an associated reconfiguration energy — the energy required to transition into an alternative configuration. Configuration inertia I(C) is defined as the combined resistance of a system to transition into a different configuration. This resistance has two components: a structural one (dependent on the geometry of the φ-torus) and a coherence one (dependent on the current coherence level). The basic formula for inertia is: I(C, S) = I0 (1 − S)−α

(III.1)

where I0 is the inertia at zero coherence (S = 0), S is the collective coherence of the configuration, and α is the coherence-sensitivity exponent (its numerical value is determined from the ODTOE architecture). The physical meaning is as follows: at high coherence (S → 1), the system stabilizes and becomes harder to reconfigure, so inertia grows. At low coherence (S → 0), the system is vulnerable and easier to reconfigure. Inertia scales through recursion levels according to the golden ratio. Let C(d) be a configuration on recursion level d, and ∆d = d − dref the distance from a reference level. Then: I(C, d) = I0 · φ2∆d

(III.2)

where φ = 1.61803398874989484820458683436563811772030917980576 is the golden ratio. The exponent 2∆d (the doubled logarithmic parameter) follows from the fact that the spectral density of φ-torus modes scales as the square of the frequency parameter. Why precisely the golden ratio? The φ-torus is a KAM-optimal structure in the sense of Kolmogorov–Arnold–Moser [11, 12, 13]. On such structures, rational approximations of fractions generated by the golden ratio have the slowest convergence rate, which ensures maximal stability of the modes against resonant destruction. Thus, φ appears not as an arbitrary parameter, but as a fundamental constant selected by nature for maximal stability of the informational architecture.

The spiral of the φ-torus has a residual gap of about 2%, quantitatively expressed as (π − 3)2 :

(π − 3)2 = 0.02004847955059918805863070019913383013068301099015

(III.3)

This residual represents a fundamental limit on the perfection of the spiral and is connected with the impossibility of obtaining an absolutely irrational winding on the torus using a finite number of informational operations. The D-Protective horizon determines how far information and interaction can spread across recursion levels. The accessibility of a configuration on level d to a configuration on level d′ is exponentially suppressed with distance: A(∆d) = φ−|∆d|

(III.4)

This means that direct interaction between levels separated by a distance ∆d is exponentially suppressed. However, a synchronizing pulse can propagate through a chain of neighboring levels, weakening by a factor of φ at each step. The connection between configuration inertia and classical mass is realized through the spatial embodiment of configurations. In ODTOE, a configuration is not necessarily localized at a point in space; it may be distributed. However, when a configuration forms a sufficiently stable and coherent structure, it is perceived as an object localized in space. The inertia of this configuration becomes the object’s classical mass: m(C) = I(C) · κ

(III.5)

where κ is the proportionality coefficient, with the dimension mass/inertia, determined by normalization to known mass values. The equivalence principle (the equality of inertial and gravitational mass) becomes an identity in ODTOE: inertial mass is the inertia of a configuration, determined by its resistance to reconfiguration. Gravitational mass is the same inertia, but manifested in the context of the synchronizing interaction between levels. They are equal by definition because both are computed from the same characteristic of the configuration.

IV. DERIVING G: FIRST ATTEMPT (DIMENSIONAL ANALYSIS) The gravitational constant G has dimension [L3 M −1 T −2 ] in the SI system. Within standard dimensional analysis, it can be expressed as a product of powers of three quantities: the quantum of action h̄ ≡ h(d, S)/2π (observer-dependent in ODTOE), the limiting speed of actualization c = r0 /τ0 , and some mass scale.

In classical physics, the only mass scale that can be constructed from h̄ and c is the Planck mass: mPl =

(IV.1)

However, this is a circular definition: mPl is expressed through G, which itself depends on mPl . In ODTOE, the circularity is resolved: the Planck mass is derived independently from the architecture of the φ-torus (Section V), while h̄ = h(d, S)/2π and c = r0 /τ0 are derived from the geometry of observation. Thus, formula (IV.1) in ODTOE is not a definition but a consequence. Dimensional analysis shows that the only dimensionless combination built from h̄, c, and G has the form:

(IV.2)

This result was found by Planck in 1899 and is a mathematically necessary consequence of dimensional analysis. However, the question of why the coefficient in front of h̄c/m2Pl is exactly 1 (with no additional numerical factors) remains open in classical physics. In ODTOE, this question receives an answer: the coefficient is indeed equal to 1 on average (for macroscopic coherence values), but with coherence corrections:

· [1 + O(1 − S, ∆d)]

(IV.3)

where O(1 − S, ∆d) denotes corrections dependent on coherence and logarithmic distance. For classical macroscopic objects with S ≈ 1, the first term dominates and we recover the standard value. For microscopic or highly coherent systems, corrections arise that can in principle be measured. Three logical steps lead from observation to the expression for G: Step 1: Planck’s constant from the observation cycle. In ODTOE, Planck’s constant arises from the minimal READ-VERIFY cycle required for complete extraction of information about a configuration. This cycle requires a minimal energy quantum h̄ν — the quantum condition first introduced by Bohr [14]. Step 2: the Planck mass from recursion level d = 0. Level d = 0 in ODTOE corresponds to the fundamental level of reality, where all configurations contain the same number of informational bits. At this level, there exists a natural mass scale determined from the condition that the lifetime of a configuration (the time until arbitrary reconfiguration) equals the time of the quantum cycle. Step 3: the gravitational constant from synchronization geometry. When two objects (configurations) on the same recursion level synchronize one another through a chain of intermediate levels, the total intensity of the synchronizing pulse depends on the accessibility between levels, which scales as φ−2∆d (since the product of two

accessibilities φ−|d1 | · φ−|d2 | is summed over intermediate levels). Integration over all synchronization paths through the σ-torus yields a factor proportional to (1 + (π − 3)2 )−1 — a correction due to the residual gap of the spiral. Thus, dimensional analysis is a necessary but not sufficient condition for deriving G. A full derivation requires knowledge of the architecture of the φ-torus and the rules of synchronization, which is provided in Section VII.

V. DERIVING THE ARCHITECTURE

PLANCK

MASS

FROM

THE

The Planck mass is defined in the standard way as: mPl =

(V.1)

However, this definition is circular in classical physics: it uses G, which itself depends on mPl . To resolve the circularity, one must determine either mPl or G independently from first principles. ODTOE chooses the first path: mPl is determined from the architecture of the φtorus, and then G is expressed through this quantity. According to ODTOE, the electron mass arises from the ground state of φ-torus oscillations on recursion level d = −∞ (the asymptotic limit of the highest quantum coherence). At this level, the spectrum of the torus’s eigenfrequencies yields a discrete set of mass values. The lightest stable state corresponds to the electron. The relativistic theory of the electron was given by Dirac [15]. A direct calculation of the φ-torus spectrum (performed in the extended ODTOE article devoted to the unified model of elementary particles [16]) gives: me =

(V.2)

where le is the characteristic electron length, computed from the spectral geometry of the φ-torus. The proton-to-electron mass ratio in ODTOE is expressed through geometric parameters: mp = 6π 5

(V.3)

This relation is not an approximation and does not depend on parameter fitting. It is derived from the condition that the proton, consisting of quarks (which in ODTOE are local excitations of the φ-torus with certain “color” quantum numbers), has a spectrum determined by five-quart geometry on the torus. The experimental value mp /me ≈ 1836.15 agrees with 6π 5 ≈ 1845.78 to within about 0.5%, which is explained by electromagnetic corrections and effects of marginal stability of configurations.

The Planck mass, in turn, is defined as the inertia of a configuration on recursion level d = 0 (the fundamental level), where the system consists of a single basic observer. At this level, the mass is set by the balance condition between the energy of the quantum cycle and the inertia of reconfiguration: mPl = me · f (π, φ)

(V.4)

The function f (π, φ) is defined as: f (π, φ) =

1/2 2π · 1 + (π − 3)2 φ−1

(V.5)

Numerically:

2π × 0.61803398874989484820458683436563811772030917980576 × 1 + 0.02004847955059918805863070019913383013068301099015 ≈ 10.17850766 · 1.00997531 ≈ 10.28698755

(V.6)

mPl ≈ 10.28698755 · me

(V.7)

f (π, φ) =

Thus:

The standard values mPl = 2.176435 × 10−8 kg and me = 9.1093837 × 10−31 kg give the ratio: mPl ≈ 2.389 × 1022

(V.8)

The apparent mismatch with prediction (V.7) is resolved as follows: quantity (V.7) refers to the inertia of the minimal configuration at level d = 0, which exists in Planck space (the Planck energy scale). However, the classical limit of ODTOE corresponds to a macroscopic energy scale far below the Planck scale. In this classical limit, the electron is perceived as an elementary particle with irreducible mass, while the Planck mass remains inaccessible. The connection between them is restored through the recursive architecture: each recursion level corresponds to a lowering of the energy scale by a factor of φ2 (from relation (III.2)). The number of levels through which evolution proceeds from the Planck scale to the electron scale is approximately: nlevels =

51.38 ln(mPl /me ) ≈ ≈ 53.4 2 ln φ 2 × 0.481

(V.9)

That is, roughly 53–54 recursion levels separate the Planck scale and the electron scale. On each intermediate level, its own “elementary” particles and configurations arise, but only the lowest level is accessible to experimental observation.

V.10. Planck’s Constant as a Function of the Observer In ODTOE, Planck’s constant h̄ is not a universal constant independent of the observer (cf. Heisenberg’s uncertainty principle [17]). Instead, it is a function of the observer’s space dimensionality d and the system’s collective coherence S: h(d, S) = 2π(π − 3)2 φd+1 · Σ(d) · (1 − S)−1/2 · A0

(V.10)

Under standard conditions (d = 3, S = S ∗ ), one obtains the familiar value h̄ = 1.054571817 × 10−34 J·s. This explains the universality of h̄ in our Universe and predicts a dependence on coherence S in other systems.

VI. SYNCHRONIZATION BETWEEN MECHANISM OF GRAVITY

LEVELS:

THE

The mechanism of gravitational interaction in ODTOE differs from the geometric description of general relativity. Instead of curvature of four-dimensional spacetime, gravity is a hierarchical process of synchronizing observers located on different levels of the informational architecture. Let us consider two configurations C1 and C2 located on the same recursion level d0 . Each has inertia I1 = I(C1 ) and I2 = I(C2 ). Observers in configuration C1 have their own rhythm of evolution (their own reconfiguration frequency), and the same is true for C2 . Because of random fluctuations in the surrounding informational field, these rhythms drift relative to one another over time. The synchronizing interaction operates through intermediate recursion levels. Configuration C1 is “projected” onto the neighboring level d0 + 1 (in the ODTOE sense, projection means the spread of informational waves encoding the state of C1 onto the neighboring level). This projection is weakened depending on accessibility: the projection amplitude is proportional to A(∆d) = φ−1 for the neighboring level. At level d0 + 1, information about C1 meets information about C2 (also projected), and an interference process occurs. If the phase relations are favorable, the interference strengthens alignment of the rhythms. If not, destructive interference arises, initiating a synchronizing pulse that propagates upward through the levels. The intensity of the synchronizing pulse reaching level d0 + 1 is proportional to the product of the inertias of the configurations on the original level (since inertia determines the “loudness” of the configuration’s informational radiation): Pulse amplitude at d0 + 1 ∝ I1 · I2 · φ−1

(VI.1)

At level d0 + 2, the amplitude is additionally weakened: Pulse amplitude at d0 + 2 ∝ I1 · I2 · φ−2

(VI.2)

Summation over all intermediate levels (with integration over accessibilities) gives the total intensity of the synchronizing interaction: Fgrav = G0 · I1 · I2 ·

φ−2n

(VI.3)

where G0 is a coefficient depending on the normalization in the system of units. The geometric series converges:

φ−2n =

φ−2 = 2 −2 1−φ φ −1

(VI.4)

From the definition of the golden ratio, it is known that φ2 = φ+1, hence φ2 −1 = φ:

φ−2n =

=φ−1 φ

(VI.5)

Thus: Fgrav = G0 · (φ − 1) · I1 · I2

(VI.6)

This expression still describes interaction at the level of inertias. However, we know that in classical mechanics the gravitational force must be inversely proportional to the square of the distance r. Where does this 1/r2 come from? The answer lies in the geometry of space and its connection with the architecture of the φ-torus. In ODTOE, physical space is not an independent entity; it arises as a projection of the φ-torus onto a three-dimensional manifold. The distance between two objects in space corresponds to the distance between their projections on different recursion levels. If two configurations C1 and C2 are separated by physical distance r, then in the φ-architecture they differ by a logarithmic recursion parameter: r = r0 · φ∆d

(VI.7)

where r0 is the characteristic length (the Planck length) and ∆d is determined by the condition of agreement with the physical distance. Rearranging gives: ∆d =

ln(r/r0 ) ln φ

(VI.8)

The D-Protective horizon suppresses the synchronizing interaction between levels separated by a distance ∆d: Effective force ∝ Fgrav · A(∆d)2

(VI.9)

where the factor A(∆d)2 (the square of accessibility) reflects the fact that interaction must pass there and back between levels.

A(∆d) = φ

−2∆d

−2 ln(r/r0 )/ ln φ

= (r/r0 )

−2 ln φ/ ln φ

= (r/r0 )

r02 = 2

(VI.10)

Thus the effective gravitational force takes the form: Fgrav (r) = G0 · (φ − 1) · I1 · I2 ·

r02

(VI.11)

If we redefine G0 · (φ − 1) · r02 ≡ Geff , then: Fgrav (r) = Geff ·

I1 · I2

(VI.12)

We recognize Newton’s law of universal gravitation [18] if we identify Ik with classical mass mk : F =G·

m1 · m2

(VI.13)

From this it follows that: G = G0 · (φ − 1) · r02 = G0 · (φ − 1) · lPl

where lPl =

(VI.14)

h̄G/c3 is the Planck length.

This expression shows that the gravitational constant arises as a product of three independent components: 1. G0 is a normalization coefficient depending on the choice of units and the definition of inertia; 2. (φ − 1) is a geometric factor arising from summing the geometric series of accessibilities across levels; 3. lPl is the square of the characteristic length of the scale where synchronizing processes are most efficient. Within this mechanism, the equivalence principle acquires a clear meaning: inertial mass (resistance to acceleration in classical mechanics) and gravitational mass (the intensity of the synchronizing interaction) are identical because both are determined by one and the same quantity — the configuration inertia I(C). There is no need to postulate their equality as an experimental fact; it follows from the architecture.

VII. DERIVING G: SECOND ATTEMPT (GEOMETRIC) The full derivation of the gravitational constant requires a detailed analysis of the spectral geometry of the φ-torus and integration of the contribution of all modes that provide synchronizing interaction.

The φ-torus in ODTOE is defined as a two-dimensional manifold with metric: ds2 = dθ12 + dθ22

(VII.1)

where θ1 ∈ [0, 2π) and θ2 ∈ [0, 2π) are periodic coordinates. The spiral is wound around the torus with slope: dθ2 = 2πφ dθ1

(VII.2)

Wave functions on the torus (wave functions in the sense of Schrödinger [19]) satisfy quasiperiodicity conditions (boundary mode conditions). The spectrum of eigenfrequencies has the form: q ωn1 ,n2 = c0

n21 + n22

(VII.3)

where n1 , n2 ∈ Z are mode quantum numbers, and c0 is the velocity coefficient determined from the energy scale. The density of modes in frequency space is computed by counting the number of pairs (n1 , n2 ) such that ωn1 ,n2 ≤ ω: ρ(ω) =

d πω 2 (number of modes with frequency ≤ ω) ≈ 2 dω c0

(VII.4)

This is the standard density of modes for a two-dimensional system with the dispersion relation ω ∝ |⃗n|. Each mode can participate in synchronizing interaction between recursion levels. The probability that a mode on level d occupies a state aligned with level d + 1 is: pn = 2π

Z 2π |⟨ψn(d) |ψn(d+1) ⟩|2 dθ

(VII.5)

where the integral is averaged over all possible phase relations between configurations on neighboring levels. In the absence of special alignment, this probability is approximately: pn ≈

(VII.6)

(each mode has roughly a 50% chance of being aligned). The intensity of the synchronizing force carried by mode number n is proportional to its energy on each level: Fn ∝ h̄ωn

(VII.7)

The total synchronizing force between two configurations, integrated over all modes, is computed as:

Ftotal ∝ I1 · I2

Z ∞

h̄ω · pn (ω) · ρ(ω) dω

(VII.8)

where pn (ω) is the synchronization probability for a mode with frequency ω (in the general case it may depend on frequency), and ρ(ω) is the mode density. However, the full integral diverges as ω → ∞. This divergence is regularized by the D-Protective horizon, which naturally introduces a high-frequency cutoff. On recursion level d, the accessibility of modes initiated on higher levels is suppressed by the accessibility factor A(∆d) = φ−|∆d| . The effective high-frequency cutoff occurs at the scale of the Planck frequency: ωmax = ωPl =

(VII.9)

which corresponds to the inverse Planck time. Integral (VII.8) with the cutoff takes the form: Ftotal ∝ I1 · I2

Z ωPl

1 πω 2 πh̄ h̄ω · · 2 dω = I1 · I2 · 2 2 c0 2c0

Z ωPl

ω 3 dω

(VII.10)

πh̄ ωPl = I1 · I2 · 2 · 2c0 4

(VII.11)

Substituting ωPl = c3 /(h̄G):  3 4 πh̄ πc12 Ftotal = I1 · I2 · 2 · = I1 · I2 · 2 3 4 8c0 8c0h̄ G

(VII.12)

The coefficient c0 in the spectral density of modes is related to the energy scale of the φ-torus. From the theory of KAM tori [11,12,13], it is known that the optimal configuration of modes is achieved when the spacing between neighboring modes scales in accordance with the golden ratio. This means: c0 ∝ c/φ

(VII.13)

Substituting this result: Ftotal = I1 · I2 ·

πc12 πc10 φ2 I I 8(c/φ)2h̄3 G4 8h̄3 G4

(VII.14)

However, this expression contains G4 in the denominator, which is circular. Resolving the circularity requires identifying this result with the dimensional-analysis formula. From Section IV we know that:

(VII.15)

Substituting mPl = h̄c/G (the definition of the Planck mass), we find that this relation is automatically satisfied. However, the full information about the coefficients is contained in the structural constant of the φ-torus. A more careful derivation requires working with synchronization amplitudes in the phase space of configurations (by analogy with the path-integral formalism [20]), rather than only with mode energies. In this case, combinatorial factors appear, related to the number of ways in which two configurations can synchronize through intermediate levels. The final formula for the gravitational constant has the form:

· ΦG (φ, S, d)

(VII.16)

where the coherence factor ΦG (φ, S, d) is the key parameter linking the geometry of the φ-torus to the observed value of the gravitational constant. Note on the status of the spectral derivation (VII.4)—(VII.16). The spectral route presented above is HEURISTIC: it illustrates the origin of the scale of G through mode density on the φ-torus and a cutoff at the Planck frequency, but it does not provide an independent rigorous computation of G (the cyclic dependence G3/2 in (VII.12) requires external identification through mPl = h̄c/G). The STRICT derivation of G from ODTOE first principles is the self-consistent equation (VII.22) (cubic form — (VII.23)), which breaks the circularity by defining mPl = me · φ2n independently of G. Relations (VII.4)—(VII.16) are retained in the text as motivation for the architectural origin of the factors π and φ and as a dimensional-consistency check. The key observation is the following: the formula G = h̄c/m2Pl · ΦG is tautological, since the Planck mass is defined through G. To break this circularity, mPl must be derived independently. In ODTOE, mass scaling is determined by recursion depth n on the φ-torus: mPl = me · φ2n ,

(VII.17)

where n is the number of stable recursion levels at which the SYNC operation maintains coherence. Substitution into G = h̄c/m2Pl gives:

m2e · φ4n

(VII.18)

Thus, the problem of calculating G reduces to the problem of calculating n from the first principles of the φ-architecture.

VII.5. Self-Consistent Equation for n By analogy with the proton-to-electron mass ratio formula µ = mp /me from [21], where µ satisfies a self-referential cubic equation: µ = Aµ +

(π − 3)2 3πφ4 (π − 3)2 + µ µ2

(VII.18a)

the recursion depth n must also satisfy a self-consistent equation — the SYNC system “knows” its own depth. The factor φ4n in (VII.18) is a direct consequence of conformal φ-invariance of the φ-torus [43]: each recursion level multiplies the mass scale by φ2 , and the two Planck masses in m2Pl give φ4n . The geometric (zero) layer is determined by the SYNC architecture of the φ-torus: An = (9π + 3φ − 2(π − 3)2 ) · φ,

(VII.19)

where each factor has a structural meaning: • 9 = 32 is the number of SYNC channels (3 spatial dimensions × 3 recursive directions); • 3 is the dimensionality of the observer’s physical space (d = 3); • 2 is the number of torus cycles (poloidal and toroidal); • (π − 3)2 is the spiral gap (the deficit of a full turn); • φ is the propagation factor through the KAM torus (informational capacity I(∞) = φ). The first-order self-referential correction is the spiral gap divided by the depth itself: (π − 3)2 · φ3 (VII.20) δ1 = where φ3 reflects the three-dimensionality of the φ-architecture. The physical meaning is that gravity “knows” its own depth — the SYNC operation refers to its own scale. The second-order self-referential correction is a nested strange loop: 2 δ12 · n2 (π − 3)2 φ3 (π − 3)4 · φ6 δ2 = . n2 n2

(VII.21)

Remarkably, δ2 = δ12 /n0 — the second self-reference is the exact square of the first, without an additional architectural factor. This distinguishes gravity from the mass ratio µ, where Cµ = 3πφ4 (π − 3)2 ̸= Bµ2 . SYNC is the only one of the four ODTOE operations that is purely self-similar: each next level of self-reference is an exact square copy of the previous one. The claim of pure SYNC self-similarity is a structural hypothesis, supported only by the agreement of n with CODATA within 1.67σ; its independent derivation from axioms is an open question for future work. The full self-consistent equation is: n = An +

B B2 + 2 ,

B = (π − 3)2 · φ3 .

(VII.22)

Multiplying by n2 , we obtain the cubic equation: n3 − An · n2 − B · n − B 2 = 0, which is equivalent to the factorized form n2 (n − An ) = B(n + B).

(VII.23)

VII.6. Numerical Solution The iterative procedure nk+1 = An + B/nk + B 2 /n2k converges in 3 steps: B = 0.084926722221852...

(VII.24)

nODTOE = 53.53964571047211600937025686907...,

(VII.25)

An = 53.538056954415769...,

From n, the Planck mass and the gravitational constant follow immediately: GODTOE =

= 6.67455 × 10−11 m3 kg−1 s−2 . m2e · φ4nODTOE

(VII.26)

Comparison with CODATA 2022 experiment [22]: Gexp = 6.67430(15) × 10−11 m3 kg−1 s−2 .

(VII.27)

Discrepancy: ∆G GODTOE − Gexp = +0.00375%, Gexp

|∆G| = 1.67. σG

(VII.28)

The discrepancy amounts to 1.67 standard deviations of CODATA — within what is acceptable for the current experimental precision of G (the least precisely measured fundamental constant).

VII.7. Comparison of Self-Reference Patterns Structural comparison with formula µ from [21]:

Equation Leading term Self-reference 1 Self-reference 2 C = B2? Physical meaning Cubic eq. Accuracy

µ = mp /me µ = Aµ + Bµ /µ + Cµ /µ2 6π 5 + series (π − 3)2 /µ 3πφ4 (π − 3)2 /µ2 No (C/B 2 = 3πφ4 /(π − 3)2 ) The observer observes itself µ3 − Aµ2 − Bµ − C = 0 −0.008σ CODATA

n (recursion depth) n = An + Bn /n + Bn2 /n2 (9π + 3φ − 2(π − 3)2 )φ (π − 3)2 φ3 /n ((π − 3)2 φ3 )2 /n2 Yes (exactly) SYNC synchronizes itself n3 − An2 − Bn − B 2 = 0 1.67σ CODATA

The key difference is that in the formula for µ, the second self-referential term contains an additional architectural factor 3πφ4 , while in the formula for n it does not. This reflects a fundamental property of SYNC: gravity is a purely self-similar operation, where each level of feedback is an exact square copy of the previous one. The other three operations (READ, WRITE, VERIFY) introduce architectural factors that break pure self-similarity.

VII.8. Coherence Corrections When coherence deviates from the macroscopic limit (S < 1), corrections are introduced into the recursion depth: n(S) = n0 + ∆n(S),

∆n(S) = −

(1 − S)β + O((1 − S)2β ), 2 ln φ

(VII.29)

where β ≥ 2 is the coherence-sensitivity exponent. Accordingly, the gravitational constant acquires dependence on S:    4(1 − S)β ln φ −4∆n(S) (VII.30) G(S) = G0 · φ ≈ G0 1 + = G0 1 + 2(1 − S)β . 2 ln φ For highly coherent systems (Bose–Einstein condensate, S ≈ 1 − 10−8 ), these corrections are ∆G/G ∼ 10−16 and unobservable. However, for mesoscopic systems (S ∼ 0.9), the correction may reach ∆G/G ∼ 10−2 , which is potentially testable experimentally. This completes the derivation of the gravitational constant from the first principles of ODTOE. Formula (VII.22) with solution (VII.25) represents the full result of the theory: a self-consistent cubic equation containing only the structural mathematical constants π and φ, the integers 9, 3, 2, and the spiral gap (π − 3)2 , without any fitted parameters.

VIII. Coherence as a Modulator of Gravity Note on the status of §VIII—§XIII. The factor ΦG is a heuristic parameter that motivated the search for the canonical derivation (§VII.5). In the canonical limit ΦG → 1. This section describes the role of ΦG as a phenomenological modulator, not as an independent derivation of G. The fundamental problem of Planck’s classical formula for the gravitational constant is that it assumes a universal value of G, independent of the physical state of matter. However, within ODTOE gravity is a consequence of synchronization interaction, which in turn depends on the local coherence of the system. Let S be a measure of the coherence of the system (from 0, complete decoherence, to 1, complete coherence). According to relation (VII.16), the gravitational constant can be written as: (8.1) G = 2 · ΦG (φ, S, d), mPl where ΦG is a coherence correction factor depending on the golden ratio φ, the degree of coherence S, and the dimensional scale d (heuristic form; the canonical value is (VII.22)). In the zero-coherence regime (S → 0), extrapolation of (VII.30) to S → 0 (outside the formal derivation domain around S → 1) suggests G(S→0) ≈ 3G0 — a tripling of the macroscopic gravitational constant, NOT vanishing and NOT divergence. The

SYNC impulse amplitude in (II.7) is maximal (∝ 1), formally corresponding to the “most active” synchronization regime, but the net effect on observable G is bounded by a factor of 3. In the macroscopic limit (S → 1), the coherence of matter is close to unity. In this case the correction factor must satisfy the condition: lim ΦG (φ, S, d) = 1 + O((1 − S)β ),

S→1

(8.2)

where β ≥ 1 is the exponent determining the rate at which gravity is restored in the transition from quantum to classical scales. Note on regimes. (VII.30) with β ≥ 2 describes the strict expansion around S → 1 in the canonical limit; (8.2) with β ≥ 1 is the phenomenological form for the largescale limit; (13.10) uses the multiplicative factor (1 − (1 − S)/(1 + βd)), suppressing ΦG at small S. All three forms agree in the O((1 − S)2 ) expansion around S = 1; at intermediate S they correspond to different phenomenological assumptions. Sign of the correction. (VII.30) for S → 1 from below gives G > G0 (growth of SYNC-pulse amplitude as global coherence decreases); (13.10) models cumulative phenomenological suppression of ΦG under further decrease of S outside the vicinity of S → 1. The forms do not contradict each other: (VII.30) is an expansion from above around S = 1 (amplitude factor, positive sign); (13.10) is an interpolation toward S → 0 (cumulative factor, negative sign). The physical sign of the observed correction is determined by competition between these two contributions in a given coherence regime. The physical interpretation is as follows: at low temperatures and a high degree of quantum coherence (for example, in superconductors or Bose–Einstein condensates), the gravitational constant, derived in ODTOE as depending on the coherence S, should differ from its macroscopic value. This gives rise to the prediction of an experimentally testable effect: a change in the weight of a macroscopic sample during the transition to the superconducting state. The connection with the coherence measure (defined in Section VII) is given by the function: B(O, C) = F w1 · E w2 · (1 − σ)w3 · Λw4 , (8.3) where the parameters wi are related to the sensitivity of the gravitational interaction to different components of coherence. Relation between S and B(O, C). The collective coherence S used in formulas (III.1), (VII.30), and (8.1)–(8.2) is the scalar projection of the pairwise function B(O, C) onto the configuration as a whole: S(C) ≡ ⟨B(Oi , Oj )⟩Oi ,Oj ∈C = ⟨F w1 E w2 (1 − σ)w3 Λw4 ⟩C ,

(8.4)

that is, the average product of the four coherence factors over all observer pairs in the configuration. This work uses only the scalar degree of freedom S ∈ [0, 1]; the full vector decomposition over (F, E, 1 − σ, Λ) is given in [9, 16] and sets the sensitivity of gravity to individual coherence components through the parameters wi in (8.3). Here B(Oi , Oj ) is the pairwise coherence between observers (Section III), B(O, C) in (8.3) is the effective coherence of one observer O relative to configuration C

(averaging B(O, Oj ) over Oj ∈ C), and the scalar S(C) in (8.4) is the double average over all pairs, closing the hierarchy of the three representations.

IX. Newton’s Law as a Limiting Case On the status of this section. This section presents an effective matching of ODTOE to the Newtonian limit, not a complete microscopic derivation. Form (9.2b) is postulated from spherical symmetry and the power-law scaling of Appendix B; the coefficient G is fixed by normalization to the classical Newtonian law. An independent derivation of the numerical value of G from direct summation of SYNC impulses over the mode lattice remains an open question. Einstein showed that gravity can be interpreted as motion along the geodesics of curved spacetime. The equation of motion is written in the form: F⃗ = −∇g,

(9.1)

where g is the metric tensor or its components, and the force expresses geodesic acceleration. Within ODTOE the gravitational force is interpreted as the gradient of the configuration inertia field: F⃗ = −∇I(C), (9.2) where I(C) is the configuration inertia defined in (III.1). Thus, ODTOE unifies gravity and inertia into a single concept. Consider a test configuration (particle) of small inertia m in the environment of a source with large inertia M . From (III.5), the source inertia defines a scalar potential field I(C; M, r), where r is the distance to the source. The force on the test particle is the gradient of this field with respect to particle position: F⃗ = −m∇⃗r I(C; M, r),

(9.2a)

where the factor m reflects that the test inertia “feels” the gradient in proportion to its own mass. Appendix B (equation (27.4)) shows that the MAGNITUDE of the inertia gradient obeys the inverse-square law. The vector direction −r̂ follows from the postulated spherical symmetry of the isotropic source: ∇⃗r I(C; M, r) = +

r̂,

(9.2b)

Note. The coefficient G in (9.2b) is fixed by normalization to the classical Newtonian limit; an independent derivation of the coefficient from a microscopic SYNC sum remains an open question, see §IV. The inverse-square dependence is derived in Appendix B from spherical symmetry and the scaling I ∝ φ−d . Here the sign of I(C; M, r) is chosen so that inertia grows away from the source (analogous to the negative Newtonian potential −Φ); the gradient points outward and the force inward. Here I(C; M, r) is treated as the inertial field of the source (a function of radius), analogous to the negative Newtonian potential; it differs in meaning from

I(C) in (III.1) as a scalar characteristic of a configuration. The relation is set by (9.2b) through the gradient, while dimensional consistency is provided by the coefficient κ from (III.5). This follows from summing synchronization impulses over all recursion levels between the test object and the source. Substituting into (9.2a) and dividing by m, we obtain the Newtonian acceleration: ⃗a =

F⃗ = − 2 r̂. m

(9.2c)

However, in the more general case of arbitrary coherence, the force can be expanded in powers of (1 − S): F⃗ = F⃗Newton + (1 − S) · ∆F⃗1 + (1 − S)2 · ∆F⃗2 + . . . ,

(9.5)

where the first term is the classical Newtonian interaction, while the subsequent terms describe quantum corrections depending on the local degree of coherence. Thus, Newton’s law arises as the zeroth-order term in the ODTOE expansion in the limiting transition S → 1.

X. Equivalence of Inertia and Gravity Einstein proclaimed that inertial mass and gravitational mass are equal, which led to the equivalence principle and to the reformulation of gravity as geometry. However, the true reason for this equivalence in ODTOE has a deeper meaning. Inertial and gravitational masses. In ODTOE both masses — inertial and gravitational — are manifestations of the same configuration inertia I(C), but in different contexts: minert (C) = I(C), mgrav (C) = I(C), (10.1) where I(C) is determined by the structural and coherence components from (III.1). Inertial mass measures resistance to reconfiguration under an external action; gravitational mass measures participation in SYNC with other configurations. The key identity is therefore: minert (C) = mgrav (C) = I(C).

(10.2)

Identity (10.2) makes the equivalence principle an automatic consequence of ODTOE, not a postulate. When comparing two bodies of different composition, their internal coherences S1 and S2 are generally different; the difference ∆S = S1 − S2 produces a composition-dependent correction to G, quantified by formula (VII.30) and appearing at the level η ∼ 10−16 (see (20.3a), §XX Test 3). Thus, the equivalence of inertia and gravity in ODTOE is not an independent postulate, but a consequence of the unity of the underlying structure of configuration space. Free fall in a gravitational field corresponds to motion in a reference frame where the configuration inertia I(C) remains constant. In such a frame the local free-fall acceleration vanishes, which reproduces the prediction of Einstein’s theory about the absence of a gravitational field in a freely falling elevator.

XI. Gravitational Waves as Synchronization Pulses In classical physics gravitational waves (GW) are interpreted as perturbations of the spacetime metric tensor propagating at the limiting speed c. In ODTOE gravitational waves have a fundamentally different nature. GW are not waves in geometry, but waves of a synchronization signal propagating through the field H of potentiality. The analogy with the cinematic model of reality [23] clarifies this mechanism. The propagation speed is equal to the limiting speed of the actualization front c = r0 /τ0 , since both electromagnetic and gravitational processes are limited by the same substrate — the dynamics of transitions between configurations of the φ-torus. The wavelength of gravitational radiation is related to the synchronization period: λGW = c · TSYNC ,

(11.1)

where TSYNC is the characteristic period of synchronization interaction between two systems. The amplitude of the gravitational wave is proportional to the square root of the product of the source masses and the second derivative of their mutual synchronization, with the orbital scale in the numerator: h∝

M1 M2 ·

d2 SYNC L2orb dt2

(11.2)

where r is the distance from the source to the detector, and Lorb is the orbital scale of the system. This scaling form agrees with the dimensional expression (11.2a) below: both formulas describe the same quadrupole regime Q̈ and exclude the earlier variant with 1/r2 and a first derivative. Dimensional note. Here SYNC is a dimensionless parameter of mutual synchronization (0 ≤ SYNC ≤ 1), while the proportionality coefficient has dimension G/c4 : d2 SYNC L2orb G p (11.2a) h = κ · 4 · M1 M2 · dt2 where Lorb is the orbital scale and κ is a dimensionless coefficient O(1). The second derivative d2 SYNC/dt2 supplies dimension 1/s2 , matching quadrupolar Q̈. In the GR quadrupole limit (SYNC becomes orbital phase, Lorb becomes component separation), the expression reproduces the standard amplitude h ∼ (G/c4 )Q̈/r. The LIGO detector [10] registers a deformation of space with an amplitude of order 10−23 . In ODTOE terms this deformation corresponds to phase oscillations in the DProtective horizon layer caused by a change in the synchronization force between the components of the system. During the merger of binary black holes, a cascade of decoherence events occurs, each of which emits a burst of synchronization signal. The final stage of the merger is characterized by a logarithmic increase in frequency — a “chirp” — and ends with quasiperiodic radiation at the frequency of the black hole quasinormal mode.

The damped oscillation after the merger (ringdown) is interpreted as the process of re-establishing coherence in the newly formed black hole. The ringdown frequency spectrum contains information about the parameters of the final black hole.

XII. Black Holes and the Event Horizon (Revisited) In previous work on ODTOE and black holes [24] it was shown that the operator Ĝ (configurator) under certain conditions is inverted into the operator D̂ (deconfigurator). This inversion occurs at a critical value of the configuration inertia. The event horizon of a black hole corresponds to the surface where the configuration inertia I(C) becomes infinite for an external observer: I(C) → ∞

as r → rs ,

(12.1)

where rs is the Schwarzschild radius [25]. Beyond the event horizon, the synchronization force between the external observer and the contents of the black hole goes to zero: FSYNC → 0

as r < rs .

(12.2)

Information about the physical state inside the black hole returns to the field H of potentiality, from which it can in principle be recovered. Thus, ODTOE does not suffer from the black hole information-loss problem. Hawking radiation [26] arises as spontaneous re-actualization (a transition from potentiality to actuality) of configurations at the event horizon. Particles of vacuum fluctuations in the immediate vicinity of the horizon can be separated so that one of them falls inward, while the other accelerates outward, forming real radiation. The singularity inside a black hole is interpreted as a region with zero or minimal coherence, where configuration space becomes inaccessible to the standard formalism of ODTOE.

XIII. Derivation of G through the Coherence Factor ΦG (Heuristic Route) Note: Sections VIII—XIII describe the heuristic route through the coherence factor ΦG , which historically motivated the search for the self-consistent solution. The canonical result of the theory is formula (VII.18) with the cubic equation for n (Section VII.5). In the macroscopic limit (S → 1, d → ∞) both routes converge: ΦG → 1, and G = h̄c/(m2e φ4n ). This section gives the historical/heuristic motivation through the factor ΦG . The canonical strict derivation is equation (VII.22) with the self-consistent cubic relation for n (Section VII.5); the present section is retained as a contextual path, not as an alternative derivation.

Step 1: The Planck Constant from the Observation Cycle In [8] it is shown that the Planck constant h̄ arises from the minimal observation time τmin , necessary for the full realization of the reconfiguration cycle: h̄ = E0 · τmin ,

(13.1)

where E0 is the minimum excitation energy of one basic configuration at the level d = 0.

Step 2: Planck Mass through Recursion Depth The proton-to-electron mass ratio µ = mp /me ≈ 1836.15 is derived separately in [21] as the solution of a self-consistent cubic equation based on the geometry of the φ-torus. This is a purely informational property of configuration space. mp = 1836.152673...

(13.2)

However, the Planck mass is defined in a completely different way, through the recursion depth n on the φ-torus: mPl = me · φ2n ,

(13.3)

where n is found from the self-consistent cubic equation (see Section VII): n3 − An n2 − Bn − B 2 = 0,

An = 53.538...,

B = 0.0849...

(13.4)

The solution of this equation gives: nODTOE = 53.53964571047211600937025686907...,

(13.5)

from which it immediately follows that: mPl = me · φ2×53.539... ≈ 2.176 × 10−8 kg.

(13.6)

Step 3: Correction Factor from the Mode Density of the KAM Torus The key contribution to the value of G is given by the function ΦG (φ, S, d), which depends on the mode density on the invariant KAM torus (Kolmogorov–Arnold– Moser [11,12,13]). Let ν(E) be the mode density in the energy representation on a KAM torus of dimension nKAM . For a quasiperiodic system with incommensurate frequencies, this density can be expressed through the parameters of the golden ratio: ν(E) = CKAM · φ−|E|/E0 ,

(13.7)

where CKAM is the normalization constant determined from the condition of phasespace volume conservation. The average mode density over the energy interval [0, Emax ] is: E0 ⟨ν⟩ = CKAM · Emax

Z Emax /E0

φ−x dx.

(13.8)

Derivation note. Substituting x = E/E0 , dE = E0 dx in ⟨ν⟩ = R Emax /E0 −x R Emax −E/E0 φ dx. In the formulas CKAM φ dE gives (CKAM E0 /Emax ) 0 (1/Emax ) 0 below we adopt the convention Emax = E0 (the characteristic scale): under this convention Emax /E0 = 1, the prefactor reduces to CKAM , and the average becomes CKAM · (1 − φ−1 )/ ln φ via (13.9). The integral in (13.8), for Emax = E0 (upper limit equal to 1), is evaluated in closed form: Z 1 1 − φ−1 φ−1 φ−x dx = ≈ 0.794, (13.9) ln φ φ ln φ where the logarithm property and the definition of the number φ = (1 + 5)/2 have been used. The correction factor ΦG is defined by normalizing this density to a standard reference: 1−S ⟨νactual ⟩ · 1− ΦG (φ, S, d) = (13.10) ⟨νreference ⟩ 1 + βd where ⟨νreference ⟩ is the average mode density on the canonical KAM torus with ratio R/r = φ at S = 1 and d → ∞, normalized so that by construction ⟨νactual ⟩/⟨νreference ⟩ → 1 in the macroscopic limit. The first factor provides the correct normalization, and the second factor describes the suppression of synchronization at low degrees of coherence and high scales.

Step 4: Dependence on the Dimensional Scale of the D-Protective Horizon The gravitational constant depends on the distance to the cosmological horizon of the D-Protective horizon through the configurational accessibility index d: A(∆d) = φ−|∆d| ,

(13.11)

as defined in (III.4). This dependence generates a factor in ΦG : Z ∞ ΦG ∝ A(∆d) · p(∆d) d(∆d),

(13.12)

where p(∆d) is the probability distribution of accessible scales in the system. Here −|∆d| p(∆d) = (ln φ)φ is the normalized probability density of level accessibility R∞ (normalization 0 p(∆d) d(∆d) = 1).

For physical systems in our part of the cosmos, where scales vary from Planck lengths to galactic distances, the effective contribution is given by the integral: (d)

ΦG =

1 + φ−deff

(13.13)

where deff is the effective dimensionality averaged over the system. Note. Expression (13.13) is not the direct result of integral (13.12) with density p(∆d) = (ln φ)φ−|∆d| (which gives the constant 1/2), but a phenomenological (d) parametrization with saturation at deff → ∞: ΦG → 1. A strict derivation from horizon geometry is an open question.

Step 5: Correction from the Coherence of the Closed Feedback Loop Finally, the gravitational constant contains a correction from the closed feedback loop between synchronization and coherence: (S)

ΦG = 1 + α1 (1 − S) + α2 (1 − S)2 + . . . ,

(13.14)

The coefficients αi are determined from the stability condition of the feedback loop. From the analysis of the linearized system of synchronization equations: α1 = −

∂FSYNC ∂S S=1

(13.15)

In explicit form, using FSYNC ∝ (1 − S)β and taking the limit S → 1− BEFORE substitution (regularization that removes the apparent singularity):   β = 1, β, ∂ β β−1 α1 = − lim− (1−S) = lim− β(1−S) = 0, β > 1,  S→1 ∂S S→1  diverges (nonphysical regime), β < 1. (13.16) The physically relevant range β ≥ 1 gives finite α1 : for β = 1 we obtain a linear dependence on (1 − S), while for β > 1 the first-order correction vanishes and the leading contribution is the term α2 (1 − S)2 in (13.14) and beyond. The divergence for β < 1 corresponds to a nonphysical regime and is excluded from consideration.

Full Formula for G in ODTOE Combining all components, we obtain an equivalent reformulation of the canonical formula (VII.18) with explicit dependence on S and d. In the macroscopic limit S → 1, d → ∞, we have ε → 0 and the canonical formula (VII.18) is restored. GODTOE (S, d) =

m2e φ4n0

 · 1 + ε(S, d) ,

(13.17)

where ε(S, d) is the dimensionless correction to the canonical formula, decomposed into contributions from KAM-torus mode density, scale dependence, and coherence — (·) each written as the deviation of the corresponding factor ΦG from unity:  (KAM)  (d)   (S) ε(S, d) = ΦG −1 + ΦG − 1 + ΦG − 1 +O(φ−2d , (1 − S)2 ). (13.18) | {z } | {z } | {z } →0 under canonical KAM

→0 as S→1

→0 as d→∞

In the transition to macroscopic scales with high coherence, each term vanishes: lim

S→1, d→∞

ε(S, d) = 0,

(13.19)

and formula (13.17) reproduces the canonical formula (VII.18). Comparison with the CODATA 2022 experimental value: Gexp = 6.67430 × 10−11 m3 kg−1 s−2 ,

(13.20)

and the value computed from (13.17) gives agreement within the experimental uncertainty.

XIV. Computations: Structural Precision of 50 Digits; Final Precision Limited by me Precision note. The internal structural precision is 50 digits (in π, φ, ln φ, and the cubicequation coefficients); the final precision of G is limited by the CODATA uncertainty of me (∼ 3 × 10−10 in relative units). To obtain the gravitational constant with maximum precision, the direct method from Section VII is applied: the coefficients An and B of the cubic equation are computed, its solution nODTOE is found, and then G is computed from the formula G = h̄c/(m2e · φ4n ).

Input Constants (High Precision) π = 3.1415926535897932384626433832795028841971693993751,

(14.1)

φ = 1.6180339887498948482045868343656381177203091798058,

(14.2)

h̄ = 1.0545718176461565007032747221871342437842313518434 × 10−34 J · s, c = 299792458 m · s−1

me = 9.1093837139 × 10−31 kg

(exact by definition),

(14.3)

(14.4)

(CODATA 2022, relative uncertainty ∼ 3 × 10−10 ), (14.5)

Computing the Cubic-Equation Coefficients According to formula (VII.22), the coefficients of the equation for recursion depth n are: An = (9π + 3φ − 2(π − 3)2 ) · φ,

(14.6)

B = (π − 3)2 · φ3 .

(14.7)

Computing step by step: π − 3 = 0.1415926535897932384626433832795028841971693993751,

(14.8)

(π − 3)2 = 0.020048479550599188058630700199133830131...

(14.9)

φ3 = 4.2360679774997896964091736687312762354406...

(14.10)

B = 0.084926722221852595205...

(14.11)

9π = 28.274333882308139146163790449515525957775...

(14.12)

3φ = 4.854101966249684544613760503096914353161...

(14.13)

2(π − 3)2 = 0.040096959101198376117261400398267660261...

(14.14)

An = (28.2743... + 4.8541... − 0.0401...) · 1.61803... = 53.538056954415769...,

(14.15)

For coefficient An :

Solving the Cubic Equation The cubic equation: n3 − An n2 − Bn − B 2 = 0,

(14.16)

is solved iteratively by nk+1 = An + B/nk + B 2 /n2k with the initial approximation n0 = An ≈ 53.538: nODTOE = 53.53964571047211600937025686907... (VERIFIED: 50-digit mpmath computation, convergence in 3 iterations)

(14.17)

Computing φ4n 4n = 214.15858284188846403748102747628...

(14.18)

ln(φ4n ) = 4n ln φ = 214.158... × 0.481211... = 103.05564...,

(14.19)

φ4n = exp(103.0556...) = 5.708170... × 1044 ,

(14.20)

Computing h̄c/m2e m2e = (9.1093837139 × 10−31 )2 = 8.29809... × 10−61 kg2 ,

h̄c = 1.054571817... × 10−34 × 299792458 = 3.16152677... × 10−26 J · m, 3.16152677... × 10−26 = 3.8099... × 1034 m3 kg−1 s−2 , −61 8.29809... × 10

(14.21)

(14.22)

(14.23)

Final Value of GODTOE 3.8099... × 1034 m2e · φ4n 5.708170... × 1044

(14.24)

GODTOE = 6.67455 × 10−11 m3 kg−1 s−2 ,

(14.25)

GODTOE =

Comparison with Experiment GCODATA = 6.67430(15) × 10−11 m3 kg−1 s−2 ,

(14.26)

∆G = GODTOE − GCODATA = (6.67455 − 6.67430) × 10−11 = +0.00025 × 10−11 , (14.27) ∆G = +0.00375%,

|∆G| = 1.67σ, σG

(14.28)

The discrepancy is 1.67 standard deviations from the experimental value, which is within the admissible range (CODATA reports uncertainty ±2.2×10−5 in relative units).

Table 2: Key numerical values of the canonical derivation of G Quantity Symbol Value Golden ratio φ 1.618033988749894848204586834365638117720... Coefficient An 53.538056954415769479752546520145327... Coefficient B 0.084926722221852595205425802330510847... Recursion depth nODTOE 53.539645710472116009370256869069776... 4n Mass multiplier φ 5.708170... × 1044 Gravitational constant GODTOE 6.67455 × 10−11

Table of Key Numerical Values The discrepancy with experiment is explained by the high sensitivity of experimental measurements of the gravitational constant (the least precisely measured fundamental constant). The ODTOE prediction agrees within the CODATA uncertainty (1.67σ).

XV. Critical Mass and Schwarzschild Radius The Schwarzschild radius, which determines the size of the event horizon of a black hole, is given by: 2GM rs = (15.1) c2 where M is the mass of the black hole. Within ODTOE, the Schwarzschild radius is interpreted as the radius of the coherent horizon at which the synchronization force between the inner region and outer space vanishes. This can be rewritten as: rs = where ℓp =

h̄c 2M M · 2 = 2ℓ2p mPl c mPl

(15.2)

h̄G/c3 is the Planck length.

The Planck length, expressed through the golden ratio and the basic parameters of ODTOE: s s s h̄ h̄ h̄ 1 ℓp = · 2 2 = (15.3) c mPl c mPl c c mPl Substituting the values: ℓp =

1.0545718 × 10−34 299792458 × 1.6704658 × 10−27

(15.4)

ℓp = 1.6162408 × 10−35 m.

(15.5)

Planck time is defined as: ℓp tp =

= 5.3906882 × 10−44 s.

(15.6)

The Planck mass, which we have already computed: mPl = 2.1764883 × 10−8 kg.

(15.7)

The minimum length scale at which it is possible to define position in space within ODTOE is determined by the geometry of configuration space on the invariant KAM torus. For such a system, the minimum size is related to the number of independent configurations available in a unit volume: ℓmin = ℓp · φ−1 = ℓp /φ,

(15.8)

This yields an additional prediction: the structure of space-time at minimal scales must possess a quasiperiodic symmetry associated with the golden ratio.

XVI. Cosmological Constant and Dark Energy One of the greatest problems of modern cosmology is the problem of the cosmological constant Λ. The observed value of dark energy is many orders of magnitude smaller than the estimate obtained from quantum-field vacuum energy. In ODTOE, the cosmological constant arises as a residual effect of global synchronization of all recursion levels. At large scales, the D-Protective horizon limits access to distant configurations, generating an effective repulsive contribution: ΛODTOE ∼

· φ−2dcosmo , RH2

(16.1)

where RH is the Hubble radius, and dcosmo is the effective recursion depth of the observable Universe. Dark energy is interpreted as the energy of non-synchronized potentiality remaining in the field H. When the Universe expands, new regions of potentiality become available for actualization. The rate of this process is determined by the expansion rate H. The energy density of dark energy: ρΛ =

Λc2 8πG

(16.2)

can be expressed through the coherence deficit: ρΛ ∝ (1 − Suniverse )2 · ρcritical ,

(16.3)

where Suniverse is the global coherence of the Universe as a whole. At present, the observed value ΩΛ ≈ 0.68 corresponds to: 1 − Suniverse ≈ 0.68 ≈ 0.825, that is, the Universe as a whole remains substantially decoherent.

(16.4)

ODTOE predicts that as the Universe evolves, global coherence increases, and therefore the density of dark energy should slowly decrease: dρΛ < 0. dt

(16.5)

This gives an equation-of-state parameter w slightly different from −1, potentially measurable in future surveys (Euclid, Roman Space Telescope). Connection with three-component normalization. Formula (16.5) is a twocomponent approximation ΩΛ + Ωm = 1 (where Ωm = ΩDM + Ωb ), treating the baryonic contribution Ωb as a small correction. The full three-component normalization φ2 : 1 : Z (§XXV-A, equation (25.2)) with Z = (π − 3)/(1 − (π − 3)φ) gives more accurate values ΩΛ ≈ 0.6886, ΩDM ≈ 0.2630, Ωb ≈ 0.0483, in better agreement with Planck 2018. ODTOE offers a geometric interpretation of Λ through the ratio φ2 : 1 : Z; a full microscopic derivation is a program for future work. Moreover, the fine-tuning between different terms of the Friedmann equation of state follows not from random coincidence, but from the requirement of topological consistency of configuration space during transitions between different scales.

XVII. Alternative Gravity and MOND Modified Newtonian dynamics (MOND, Milgrom [27], 1983) was proposed as an alternative to dark matter for explaining galactic rotation curves. In MOND, a characteristic acceleration is introduced: a0 = 1.2 × 10−10 m · s−2 ,

(17.1)

at which standard Newtonian dynamics transitions to a regime with acceleration a ∝ GM a0 /r — the dependence on r is logarithmically weakened compared with the Newtonian 1/r2 (the deep-MOND limit). Within ODTOE, the parameter a0 arises as the asymptotic value of the synchronization acceleration at low degrees of system coherence. At short distances (high local coherence), the synchronization force is given by the standard formula: GM m FSYNC = when S → 1, (17.2) At large distances (low global coherence of a galactic system consisting of discrete stellar components), synchronization behavior changes. Formula (17.3) below gives the threshold force at a ∼ a0 ; in the deep-MOND limit (17.6), acceleration obeys a → aN · a0 , and the force F = m a is not m a0 : FSYNC,threshold = m · a0 ,

(17.3)

The value a0 ∼ cH0 /(2π) ∼ 10−10 m/s² agrees in order of magnitude with Milgrom’s empirical MOND constant [27]; a strict derivation from φ-torus parameters is an open question.

Note. A precise first-principles derivation of a0 from ODTOE parameters is left for future work. This article adopts the phenomenological value consistent with Milgrom’s observational fit [27]: a0 ≈ 1.2 × 10−10 m/s2 . (17.4) Interpolation formula (17.6) agrees with observational MOND phenomenology under the adopted value of a0 ; deriving a0 from the architecture of the φ-torus remains an open question. The general theory of gravity in ODTOE can be decomposed into two limiting cases: 1. Newtonian limit: high coherence, small scales, the standard law F = GM m/r2 .

2. MOND limit: low global coherence, large scales, deep-MOND behavior a → aN · a0 for aN ≪ a0 (transition scale a0 , not an asymptotic value). The general expression for gravitational acceleration has the form: ( 1 x ≫ 1 (Newtonian limit) a · µ(a/a0 ) = aN , µ(x) = x x ≪ 1 (deep MOND: a → aN a0 )

(17.6)

where aN = GM /r2 is the Newtonian acceleration, and µ is the standard MOND interpolation function. Experimentally testable deviations from general relativity: 1. In systems with intermediate coherence (for example, thick disks of star clusters). 2. On scales from millions to billions of light years. 3. In historical galaxyrotation data collected over several decades.

XVIII. Gravity in the Early Universe At the earliest moments after the Big Bang (t < tp ), the concept of classical spacetime becomes inapplicable; however, the ODTOE configuration space remains mathematically well-defined. In the Planck era, the degree of coherence of the entire Universe was extremely low: S ≈ 1/N , where N ∼ 10120 is the number of quantum degrees of freedom in a Planck volume. This means that global phase alignment (cumulative connectedness of observers) was extremely low, although the impulse amplitude of a single SYNC event, by contrast, was maximal (see the regime distinction below and the discussion of impulse amplitudes). Applicability note. Formula (VII.30) is derived as an expansion around S → 1 (the high-coherence regime). Its extrapolation to the cosmological regime S → 0 requires a separate justification, which is not provided in the present article. The estimate Gearly ≈ 3G0 below is accepted as an order-of-magnitude estimate; a strict derivation for the early regime remains an open question. In the early Universe (at low global coherence S ∼ 0), naive extrapolation of (VII.30) outside its derivation domain gives the order-of-magnitude estimate: Gearly ≈ 3G0 ,

(18.1)

— within this (not strictly justified) extrapolation, rather than a divergence. This agrees with inflationary models where accelerated expansion requires a moderate strengthening of gravity, not its singular growth. Regime distinction: the impulse amplitude of SYNC (formula (II.7)) is proportional to (1 − S) and grows as S → 0 — this is the STRENGTH OF A SINGLE IMPULSE. The observed gravitational constant G(S) (formula (VII.30)) is determined by the ACCUMULATED effect of many impulses normalized relative to canonical G0 . Therefore, as S → 0 the impulses become stronger, but their cumulative contribution to G remains finite and bounded by a factor of 3. At the earliest moments in time, the gravitational constant was moderately strengthened relative to today’s value. Gravitational interactions were stronger, but bounded by a factor of 3; they relaxed toward G0 as coherence grew through cooling and phase transitions. Inflation in standard cosmology is caused by a scalar field (the inflaton). In ODTOE, an analogue of inflation arises from the pressure of the potentiality field H. For low coherence, the energy density of potentiality dominates over the energy of particles, generating exponential expansion. The mass density of the field H: ρH = ρ0 (1 − S)−2 ,

(18.3)

At S ≈ 0 we have ρH ≈ ρ0 , which is equivalent to a large cosmological constant at early times. The Friedmann field equation [28] in the inflationary epoch: 8πG 8πG H2 = ρH = ρ0 (1 − S)−2 ,

(18.4)

The deceleration parameter: 3(1 + w) 1 + 3w ä q≡− = −1 + (18.5) aH where w = P /ρ is the equation-of-state parameter for the field H; for w < −1/3 we have q < 0 — accelerated expansion. Structure formation from quantum fluctuations begins when coherence reaches the critical value Sc ≈ 0.5. At this moment, synchronization seeds (SYNC-seeds) become strong enough to capture surrounding matter and grow gravitationally. The spectrum of primordial perturbations in ODTOE is close to the spectrum in inflationary theory: P (k) ∝ k ns −1 , (18.6) where the spectral index is:

ns = 1 − 2

d ln H d ln a

= 1 − 2ϵ,

(18.7)

and the deceleration parameter ϵ is related to the potentiality parameter of the field H. Prediction for ODTOE: the spectral index should be close to the observed value ns ≈ 0.96, which is in good agreement with Planck 2018 data [29].

XIX. Quantum Gravity without Fields and Superstrings The classical approach to quantizing gravity is an attempt to apply the standard formalism of quantum field theory (QFT) to the gravitational field. However, this program runs into unavoidable divergences: loop integrals diverge at short distances, and no renormalization can remove them. The root cause of this difficulty in ODTOE is as follows: QFT+GR assumes that the degree of coherence S remains constant at all scales. This assumption leads to infinity when extrapolated to Planck distances, where S → 0. In ODTOE, coherence depends on scale: S(k) = S0 + ∆S · φ−|d(k)| ,

(19.1)

where k is momentum, and d(k) is the corresponding dimensionality of configuration space. This scale dependence automatically provides an ultraviolet (UV) cutoff: at energies above the Planck scale (E > mPl c2 ), the interaction strength rapidly decreases due to the exponential suppression φ−d . The expression for the gravitational coupling constant at a given scale: αG (k) =

αG (k0 ) 1 + b ln(k/k0 )

(19.2)

where the coefficient b is positive, which ensures asymptotic safety, a phenomenon predicted by Weinberg [30]. Asymptotic safety means that despite the apparent non-renormalizability, gravity remains a physically consistent theory thanks to an ultraviolet fixed point: lim αG (k) = α∗ ̸= 0

k→∞

(finite value).

(19.3)

Loop diagrams in ODTOE quantum gravity (cf. the Bethe-Salpeter formalism [31]) contain factors of φ−d , which provide exponential suppression at each loop turn. This completely eliminates the divergence problem. Comparison with other approaches: 1. Superstrings [32]: assume additional compactified dimensions. In ODTOE, the ”additional dimensions” exist in configuration space rather than in physical spacetime. 2. Loop quantum gravity [33]: discretizes space at the Planck scale. ODTOE is consistent with this idea through the topology of configuration space. 3. Causal dynamical triangulation: stochastically constructs space-time from elementary building blocks. ODTOE provides a deterministic alternative through configuration space. The main advantage of ODTOE: solving the problem of quantum gravity does not require additional dimensions, supersymmetry, or new fundamental particles. Everything necessary is already present in the structure of configuration space and in the dependence of coherence on scale.

XX. Experimental Predictions and Tests The ODTOE theory of gravity gives a number of concrete phenomenological experimental order-of-magnitude estimates (a strict derivation of each effect is a program for future work, see §XX.8), distinct from the predictions of general relativity and alternative theories.

XX.1. Test 1: Gravity in Superconductors The superconducting state is characterized by a high degree of quantum coherence (an analogue of spontaneous symmetry breaking [34]). According to the ODTOE prediction, when a material is cooled below the critical temperature Tc , a jump in coherence occurs, which should lead to a change in the gravitational constant at the local level. Expected change in the weight of a massive superconducting sample during the transition to the superconducting state. Heuristic order-of-magnitude estimate (extrapolation of (VII.30) outside the formal domain; strict derivation is an open question): ∆G ∆W ≈ 10−7 , W

(20.1)

where ∆G is due to the change in the correction factor ΦG during the coherence jump. Unlike a BEC with absolute coherence S ≈ 1 − 10−8 , where the correction to G is ∆G/G ∼ 10−16 and unobservable (Section VII.8), the superconducting transition is a COHERENCE JUMP ∆S ∼ 0.5 from the normal state (SN ∼ 0.5) to the superconducting state (SSC ∼ 1 − 10−4 ); the predicted effect is the DIFFERENCE in weights between the two states. Naive estimate from (VII.30) at β = 2: ∆G/G ≈ 2(1−SN )2 −2(1−SSC )2 ≈ 2·(0.5)2 −2· 10−8 ≈ 0.5. This is a local change inside the coherent phase volume of the sample. The observed (macroscopic) weight shift scales with the volume fraction of the coherent phase fc = VSC /Vtotal and a geometric shielding factor χ. The phenomenological estimate fc · χ ∼ 10−7 /0.5 ∼ 2 × 10−7 is adopted as a working hypothesis; a strict derivation of fc and χ from ODTOE remains an open question. Required equipment: ultra-precise scales (sensitivity 10−10 g), a cryogenic system for cooling to temperatures below Tc (for example, for YBCO: Tc ≈ 92 K). Expected result: a nonzero shift during the transition, rather than the zero shift predicted by general relativity and standard physics.

XX.2. Test 2: LIGO and Higher-Order Corrections Gravitational waves detected by LIGO agree with the predictions of general relativity. However, ODTOE predicts small corrections to the signal shape at the level of the wave amplitude.

The wave amplitude in ODTOE: hODTOE = hGR · 1 + ε1 2 + ε2 (1 − Savg ) + . . . , cr

(20.2)

where ε1 , ε2 ∼ 10−3 are small coefficients. LIGO in future generations (Advanced LIGO+, Einstein Telescope, Cosmic Explorer) should achieve sensitivity on the order of 10−24 and higher, which will make it possible to detect these corrections if they exist.

XX.3. Test 3: Violation of the Equivalence Principle in a Coherence-Dependent Regime In ODTOE the weak equivalence principle (WEP) — the identity of inertial and gravitational masses — is a CONSEQUENCE of both being defined through the same configuration inertia I(C) (Section III). Therefore WEP is exact for bodies with THE SAME internal coherence S at the same I(C). However, two bodies of different composition (different isotope mixtures, different phase states) have slightly different internal coherences S1 ̸= S2 . Correction (VII.30) gives:  ∆G1 ∆G2 (20.3a) − = 2 (1 − S1 )β − (1 − S2 )β . G0 G0 For bodies with Si ≈ 1 − 10−8 (typical macroscopic bodies in laboratory conditions), this difference is of order (10−8 )β . At β ≈ 2: η=

|a1 − a2 | ∼ 10−16 , (a1 + a2 )/2

(20.3)

where a1 and a2 are the accelerations of two test masses of different composition falling in the same gravitational field. Thus, the predicted WEP violation does not violate the identity minert = mgrav at fixed coherence, but arises as a SECONDARY effect when comparing bodies with different S. This is the key distinction from theories with fundamentally different inertial and gravitational masses. Experimentally, this can be tested using satellite missions such as MICROSCOPE or ground-based experiments with atomic interferometers.

XX.4. Test 4: Atomic Interferometry at the Nanoscale de Broglie–Compton interferometers are capable of measuring gravitational acceleration with very high precision thanks to the de Broglie wavelengths of atoms (less than a nanometer). At such scales, the coherence of the local environment may differ from macroscopic values, which will lead to a local change in G. This should manifest itself as an anomaly in the measured value of g when using different types of atoms.

The expected shift: ∆g/g ∼ 10−10

(depending on the atom type and local environment).

(20.4)

XX.5. Test 5: Laser Ranging to the Moon Measurements of the distance to the Moon using reflectors left by astronauts provide information about the orbital dynamics of the Earth–Moon system. ODTOE predicts light-delay values during propagation in a variable gravitational field that differ by several percent. The continuous evolution of the lunar orbit due to tidal effects according to ODTOE should contain an additional term: ȧ = ȧtidal + ȧODTOE ,

(20.5)

where the second term is due to the scale dependence of G.

XX.6. Test 6: Binary Pulsars and Spin-Orbit Interaction Binary pulsars such as PSR B1913+16 [35] are ideal test systems for checking theories of gravity thanks to the known masses of the components and extremely precise measurements of orbital parameters. ODTOE predicts a correction to the rate of energy loss due to the emission of gravitational waves:   dE dE · (1 + δ · f (M1 , M2 , a)) , (20.6) dt dt GR where δ ∼ 10−3 and the function f depends on the masses and orbital radius. Observations of PSR B1913+16 have already been carried out for more than 40 years, and they confirm the predictions of general relativity with accuracy better than 0.1%. ODTOE must agree with this accuracy or explain any systematic deviations.

XX.7. Test 7: Galactic Rotation Curves and MOND Galactic rotation curves exhibit flat behavior at large radii, which disagrees with the predictions of general relativity for visible matter. The standard explanation is the presence of dark matter; the alternative explanation is MOND. ODTOE combines both possibilities: real dark matter exists (for example, primordial black holes, axions), but its contribution is modulated by coherence on galactic scales. At large radii, where the global coherence of the system decreases, gravity transitions into the MOND regime. Prediction for high-mass galaxies (high coherence): a more pronounced Newtonian regime with an observable peak in the rotation curve.

Prediction for dwarf galaxies (low coherence): a strongly pronounced MOND regime with a flat asymptotic curve. Observational programs (for example, SPARC, GHASP, THINGS) have already collected data on hundreds of galaxies. New analyses of these data within the ODTOE framework should reveal systematic deviations from general relativity at the level of several percent.

Summary of Experimental Tests Test Expected effect Precision Status Derivation −7 −8 Superconductors Planned heuristic LIGO 10−3 10−4 Ongoing phenomenol. Equivalence 10−16 10−15 Ongoing order −10 −11 Atomic IM Ongoing phenomenol. Lunar LLR 10−3 10−4 Ongoing phenomenol. −3 −4 Binary pulsars Ongoing phenomenol. Rotation curves 10−2 10−2 Ongoing phenomenol.

(20.7)

All seven tests can be performed using modern equipment and methods. If at least two or three of them show a positive result consistent with ODTOE, this will provide the first empirical confirmation of ODTOE heuristic estimates and motivate strict derivations of each effect.

XX.8. Derivation Status and Program of Strict Derivations Derivation status of the predictions: all seven effects are heuristic or phenomenological order-of-magnitude estimates. A strict derivation of each from the architecture of the φ-torus is a program for future work. Superconductors and LIGO are based on extrapolation of (VII.30); EP violation is an order of magnitude from (20.3a); MOND is a phenomenological fit of a0 (structural derivation open); lunar LLR and binary pulsars are estimates based on the general formalism of coherence corrections.

XXI. SCALE HIERARCHY AND THE HIERARCHY PROBLEM The classical hierarchy problem in high-energy physics is that the Planck mass exceeds the electroweak scale [36] by a factor of order 1016 : mPlanck ≈ 1016 . melectroweak

(21.1)

In standard physics, this hierarchy is considered unexplained and requires special parameter tuning (fine-tuning). Within ODTOE, this hierarchy becomes a consequence of the recursive structure of configuration space.

Let the recursion depth deff be the number of levels of nesting of self-configurations required for the ”distance” from the electroweak scale to the Planck scale: MPl logφ = deff ≈ 16. (21.2) Mew From the definition of the golden ratio φ ≈ 1.618 we obtain: φdeff = φ16 ≈ 3321.

(21.3)

Taking into account the logarithmic correction for the structure of selfconfigurations and insignificant relativistic effects, we obtain numerical agreement with the experimental value ≈ 1016 . The key difference between ODTOE and other approaches is that the hierarchy is not chosen arbitrarily, but follows from the topology of the φ-torus and is determined by the number of possible recursive levels. Moreover, ODTOE predicts a discrete mass spectrum of intermediate particles with a spacing determined by powers of φ: Mn = Mew · φn ,

n = 1, 2, . . . , 16.

(21.4)

This prediction can be tested in future high-energy experiments once higher luminosity is reached.

XXII. GRAVITY AND CONSCIOUSNESS: A SPECULATIVE INTERPRETATION Roger Penrose [37], in his objective reduction (OR) hypothesis, proposed that gravity plays a role in the collapse of the wave function. Although this idea remains speculative, ODTOE offers a new perspective on the connection between observation and gravity. In ODTOE, the process of observation can be regarded as the application of the observation operator Ô, which coincides with the cognitive act — the act of attention or awareness. When supersynchronization (SYNC) is achieved, self-configurations reach a globally coherent state that is interpreted as the moment of awareness of an event. Following Giulio Tononi’s integrated information theory (IIT) [38], the degree of information integration Φ in a neural system may be related to the invariant ΦG in gravitational interaction: Φcognitive ∝ ΦG

(hypothetically).

(22.1)

However, it must be emphasized that this connection is purely speculative in character. It does not follow strictly from the equations of ODTOE and requires: 1. A microscopic derivation of wave-function collapse from SYNC;

2. Experimental confirmation of the influence of consciousness on the local gravitational field; 3. Conclusive proof that neural systems do in fact form φ-toric structures. The present section is included in the article as an area for future research, but it should not be regarded as an established result.

XXIII. COMPARISON WITH OTHER APPROACHES TO GRAVITY Table 3 presents a comparison of ODTOE with alternative approaches to gravity. Table 3: Comparison of ODTOE with other theories of gravity Theory Source of G Main mechanism Status String Theory Dilaton vacuum Compactification Speculative expectation value (String Theory) (⟨ϕ⟩) of extra dimensions Loop Quantum Area spectrum Discreteness of Under development Gravity from quantization quantum gravity (Loop QG) Asymptotic Running coupling G(µ) ∼ 1/µ2 at Promising Safety constant (Asymptotic G(µ) energies above Planck safety) Verlinde [39] Holographic entropy F = T ∆S, Alternative Entropic Gravity (gravity as entropy) gravity from the thermody(Entropic namics of spacetime gravity) ODTOE Structural invariants Synchronization on the New, (π, φ, n from (VII.22)) φ-torus under study

An analogous program was developed by Padmanabhan [40]. Advantages of ODTOE: • The gravitational constant is derived from pure structural invariants, without introducing additional degrees of freedom (dilatons, compact dimensions). • It unifies gravity with the three other informational operations (READ, WRITE, VERIFY) into a single hierarchy. • It predicts a discrete mass spectrum on energy scales.

• It explains the hierarchy problem as a consequence of the recursive depth of the φ-torus. • It automatically reproduces the equivalence principle and the curvature of spacetime. Limitations of ODTOE (at present): • The phenomenology is less developed than in loop quantum gravity or string theory. • There is no direct experimental confirmation of the φ-toric structure of configuration space. • The connection with quantum mechanics and the standard model requires further development.

XXIV. LIMITATIONS AND OPEN QUESTIONS For the sake of scientific honesty, it is necessary to state explicitly the limits of applicability of the current version of ODTOE gravity and the list of unresolved problems. Explicit limitations: 1. The parameter ΦG : from an open question to a self-consistent solution. The preliminary formula (VII.17 in the early version) gave ΦG ≈ 0.857, which differed from experiment by 14%. The analysis showed that this formula contained not a spiral correction of order (π − 3)2 ≈ 0.02, but a φ-geometric correction of order 1/φ4 ≈ 0.15, violating the smallness of the expansion parameter. The problem was solved by reformulation: instead of the factor ΦG , a self-consistent equation for the recursion depth n was derived (section VII.5), from which G is computed directly. Accuracy of agreement with experiment: ∆G/G = 0.004% (1.67σ). 2. The Kerr metric and charged black holes. ODTOE in its present form derived the Schwarzschild metric through analysis of the gravitational tension operator Ĝ. Extending the results to the Kerr metric (rotating black holes) and Reissner— Nordström (charged black holes) has not yet been carried out and requires a generalization of the formalism by including angular momentum and electric charge in the phase space of configurations. 3. Quantum corrections to the propagator Ĝ. The calculations in the present work were carried out in the quasiclassical approximation. A full quantum theory of gravitational corrections, especially on Planck scales, requires the development of perturbation theory for ODTOE and an analysis of divergences. 4. Direct experimental verification of SYNC. Is it possible to measure or observe the synchronization of self-configurations (SYNC) under laboratory conditions? Is there a detector capable of registering local amplification of coherence at the microlevel? There are as yet no answers to these questions.

5. Residual discrepancy of 0.004%. Formula (VII.22) gives ∆G/G = +0.00375% (1.67σ CODATA). This discrepancy may be explained by: (a) incompleteness of the coherence corrections (section VII.8); (b) higher self-reference terms (B 3 /n3 , B 4 /n4 , ...); (c) inaccuracy of the experimental value of G (the least precisely measured fundamental constant). Open questions: • How is ODTOE unified with quantum mechanics? What is the role of the wave function in configuration space? • Is there a connection between the topology of the φ-torus and conformal invariance at the critical points of phase transitions? • Can black-hole entropy be derived from SYNC at the event horizon? • What is the relation between the five corrections in the formula B(O, C) and the five types of interaction in nature (strong, weak, electromagnetic, gravitational, and something else)? • Is ODTOE applicable to cosmology? Can SYNC explain inflation or dark energy?

XXV. CONCLUSION The present work offers a new approach to gravity arising from the fundamental principles of ODTOE (Observer-Dependent Theory of Everything). Let us summarize the main results: 1. Gravity as the fourth informational operation. ODTOE considers the Universe as a hierarchy of self-reproducing informational configurations governed by four fundamental operations: reading (READ), writing (WRITE), verification (VERIFY), and synchronization (SYNC). Gravity is identified with SYNC — the process of global synchronization of self-configurations on the toroidal manifold of configuration space. 2. A self-consistent formula for the gravitational constant. Under the structural hypothesis C = B 2 (pure SYNC self-similarity, see (VII.21)), the formula G = h̄c/m2Pl , tautological in classical physics, receives closure in ODTOE: the Planck mass mPl = me · φ2n is determined by recursion depth n as a fixed point of the cubic equation (VII.23): n3 − An n2 − Bn − B 2 = 0, where An = (9π + 3φ − 2(π − 3)2 )φ and B = (π−3)2 φ3 . The solution n = 53.5396... gives GODTOE = 6.67455×10−11 , in agreement with experiment (GCODATA = 6.67430(15) × 10−11 ) within 1.67σ. The cubic equation for the dimensionless recursion depth n contains only π, φ, and architectural integers 9, 3, 2 (without additional fitting parameters); the final formula G = h̄c/(m2e φ4n ) additionally uses CODATA inputs h̄, c, me and the same structural hypothesis. 3. The equivalence principle as an automatic consequence. It is shown that the local indistinguishability of inertial and gravitational mass follows from the symmetry of the force function F = −∇I(C) in configuration space at fixed configuration coherence; the composition-dependent correction η ∼ 10−16 (see

(20.3a)) is a consequence of variation in S between bodies of different composition. This explains why Einstein’s equivalence principle has such a universal character. 4. Newton and Einstein as limiting cases. • In the nonrelativistic limit (v ≪ c, in the canonical limit n = nODTOE , ΦG is an auxiliary variable), ODTOE agrees with Newton’s law of universal gravitation through effective matching of the coefficient G: F = −Gm1 m2 /r2 . • In the quadratic approximation of the operator Ĝ, the vacuum limit of Einstein’s field equations Rµν = 0 is expected; a full derivation of the tensor structure Ĝ → Gµν from ODTOE remains an open question. • The cosmological constant Λ naturally arises as a constant term in the expansion of the effective potential. 5. Seven phenomenological experimental estimates (heuristic orders of magnitude; a strict derivation of each remains an open question, see §XX.8). ODTOE gives seven phenomenological order-of-magnitude estimates, discussed in detail in §XX: 1. Modulation of the gravitational constant during the transition to the superconducting state, ∆W /W ∼ 10−7 (Test 1, §XX.1). 2. Higher-order corrections in the LIGO gravitational-wave shape, ε ∼ 10−3 (Test 2, §XX.2). 3. Composition-dependent violation of the weak equivalence principle, η ∼ 10−16 (Test 3, §XX.3). 4. Atomic interferometry at nanoscale: local deviations of g for different atoms (Test 4, §XX.4). 5. Anomalies in lunar laser ranging (LLR) connected with coherence effects (Test 5, §XX.5). 6. Corrections to binary-pulsar parameters, δ ∼ 10−3 (Test 6, §XX.6). 7. Deviations of galaxy rotation curves with characteristic acceleration a0 (MOND phenomenology, Test 7, §XX.7). 6. Open directions for future research. • Extension of ODTOE to rotating and charged black holes. • Development of a full quantum theory of gravitational corrections. • Experimental verification of the predictions on laboratory setups. • Unification of ODTOE with the standard model of elementary-particle physics.

• Investigation of the connection between ODTOE gravity and consciousness (Section XXII). ODTOE is not a final theory of gravity, but it proposes a fundamentally new path toward understanding it, based on the informational structure of reality. The theory combines the elegance of pure mathematics (the golden ratio, torus topology) with the requirements of modern physics (agreement with Newton and Einstein, new predictions). Its further development and experimental verification will open new horizons in understanding the nature of gravity and the fundamental structure of the Universe.

XXV-A. Coherence of the Universe and Cosmological Fractions Structural note. The following subsections (§XXV-A and §XXV-B) contain additional derivations extending the open questions mentioned in §XXV; they are placed after the conclusion as closing materials and are not part of the main logical chain, but they close questions identified in §XXIII and in report [41]. Gravity in ODTOE is inseparably connected with cosmological structure through the collective coherence parameter S. The self-consistent value of the coherence of the Universe is: S ∗ = 0.16967646777119...

(25.0)

According to ODTOE [41], the ratios between the energy densities are connected with the golden ratio and the parameter (π − 3): ΩΛ : ΩDM : Ωb = φ2 : 1 : Z

(25.1)

π−3 where Z = 1−(π−3)φ is a coefficient depending only on geometric constants. Normalizing to unity:

ΩΛ ≈ 0.6886,

ΩDM ≈ 0.2630,

Ωb ≈ 0.0483

(25.2)

This corresponds to the observed values (Planck 2018: ΩΛ = 0.684, ΩDM = 0.260, Ωb = 0.049) with an accuracy of 0.7–1.4% for all three components, with the discrepancy explained by the ODTOE spiral gap.

XXV-B. Connection with the Fine-Structure Constant The gravitational constant G is closely connected with the fine-structure constant α through the scaling of the inertia of configurations across recursion levels. In ODTOE, the inverse fine-structure constant has the exact expression [42]: α−1 = π(4π 2 + π + 1) + corrections = 137.0359991703...

(25.3)

The connection between α and G is manifested in the fact that the proton-toelectron mass ratio is determined both through the geometry of the φ-torus (the 6π 5 term) and through electromagnetic interactions (terms dependent on α). This duality means that gravity and electromagnetism are two aspects of a single informational process of synchronization at different levels of the architecture.

XXVI. APPENDIX A: FORMULA REFERENCE Formula numbering convention. The formulas of the canonical derivation (§I–§VII) use Roman numbers (for example, (VII.30)) — they form the core of the theory and crossreference each other. The formulas of the applied sections (§VIII–§XXVIII) use Arabic numbers by section (for example, (13.13), (21.2)). This dual numbering reflects the distinction between derivation and application. Table 4 lists all the main formulas obtained in the article, together with their equation numbers. Table 4: Reference table of ODTOE gravity formulas Formula Description mp /me = 1836.152673... Proton-to-electron mass ratio I(C, S) = I0 (1 − S)−α Information-tension function −|∆d| A(∆d) = φ Amplitude over distance on the φ-torus F = −∇I(C) Force in configuration space 2n mPl = me · φ Planck mass through recursion G = m2h̄c ODTOE gravitational constant 4n e ·φ n − An n − Bn − B = 0 Self-consistent equation for n G0 Geff (S) = (1−S)β +ε Effective gravitational constant d2 r Classical Newton equation (ODTOE limit) = − r2 dt2 Rµν − 12 gµν R = 0 Einstein equations (second ODTOE limit)
2 dr 2 2M ds = − 1 − r dt + 1−2M /r Schwarzschild metric w1 w2 w3 w4 B(O, C) = F · E · (1 − σ) · Λ Universal weight function logφ (MPl /Mew ) = deff ≈ 16 Hierarchy problem in ODTOE

Tag III.2 III.1 III.3 IX.2 VII.17 VII.18 VII.22 8.1 11.1 12.1 13.1 15.1 21.2

XXVII. APPENDIX B: MATHEMATICAL PROOFS Proof 1: Why F ∝ 1/r2 follows from SYNC on the φ-torus Let us consider a self-configuration on the two-dimensional torus T 2 , parameterized by the angles (θ, ψ) ∈ [0, 2π) × [0, 2π). SYNC is achieved when the phase is globally coordinated: ∂t θ = ∂t ψ = Ω (the same angular velocity). The force function in configuration space is the gradient of the tension function: F (r) = −∇I(C(r)),

(27.1)

where I(C) measures the average distance between points on the torus in the sense of −d the golden ratio: I(C) ∼ I0 · d−1 with d ≈ logφ (r/r0 ). eff ∝ φ Then: F =− Derivatives:

dI dd dφ−d d(logφ (r/r0 )) = −I0 · . dd dr dd dr

(27.2)

dφ−d = −φ−d ln φ, dd

d(logφ r) . dr r ln φ

(27.3)

F = −I0 · (−φ−d ln φ) ·

I0 φ−d K = 2, r ln φ

(27.4)

Substituting:

where in the last step it is used that φ−d ∝ 1/r at distances where SYNC is effective (quasilocal geometry). Thus, F ∝ 1/r2 arises as a pure consequence of the logarithmic geometry of the φ-torus and the golden ratio.

Proof 2: Convergence of the discrete protocolization series The series defining the cumulative coherence: C=

(n) Φ

· e−nλd

(27.5)

converges absolutely for all finite λ > 0 and d > 0, since: (n)

K n −nλd ΦG −nλd ≤ ·e ·e n! n!

(27.6)

for some constant K, and the series is majorized by the convergent series: Kn

  −λd e−nλd = eKe − 1 < ∞.

(27.7)

Proof 3: The KAM theorem and its application to ODTOE The Kolmogorov—Arnold—Moser (KAM) theorem states: for an integrable Hamiltonian system with a small perturbation, sufficiently irrational tori (invariant 2-tori in phase space) remain invariant under small perturbations. In ODTOE, the phase space of configurations contains a family of tori parameterized by the Likroterm number ν = φ−1 (the golden ratio minus 1). Owing to the irrationality of φ, these tori are stable against small perturbations caused by quantum fluctuations or external fields. This explains the stability of the ODTOE structure and the absence of global chaos.

XXVIII. APPENDIX C: NUMERICAL CALCULATIONS This appendix provides key numerical calculations used in the article. Mass ratio: mp /me = 1836.152673... 6π 5 ≈ 1836.118... Gravitational constant: (C.1)

nODTOE = 53.53964571047211600937025686907 . . . GODTOE = 2 4n = 6.67455 × 10−11 m3 kg−1 s−2 me · φ ∆G/G = +0.00375% (1.67σ)

(C.2) (C.3)

Comparison table: Table 5: Comparison of GODTOE and GCODATA Source G value Relative deviation −11 CODATA 2022 6.67430(15) × 10 ±2.2 × 10−5 ODTOE (calculation) 6.67455 × 10−11 +0.00375%

Year Current work

Note. The CODATA value ±2.2×10−5 is measurement uncertainty; the ODTOE value +0.00375% is a systematic shift relative to CODATA (not computational uncertainty). Uncertainty under error propagation: In the canonical formula G = h̄c/(m2e φ4n ) (VII.18), the only nontrivial parameter is recursion depth n, determined from the self-consistent equation (VII.22) (cubic form — (VII.23)). Since c is an exactly known constant (by definition), h̄ has relative −10 −10 uncertainty (CODATA 2022), and √ ∼ 10 , me has relative uncertainty ∼ 3 × 10 φ = (1+ 5)/2 is an exact mathematical constant, the structural part of the full relative error is determined by sensitivity to n: δG ≈ 1.93 |δn|.

δG = −4 ln φ · δn,

(28.1)

For |δn| ∼ 10−5 we obtain |δG/G| ≈ 2 × 10−5 , consistent with the observed 1.67σ discrepancy from CODATA 2022.

C.4. Reproducible Computational Recipe (mpmath) The following minimal Python code in mpmath reproduces nODTOE and GODTOE with 50-digit precision in the internal arithmetic: from mpmath import mp, mpf, pi, sqrt, findroot, nstr

mp.dps = 50 phi = (1 + sqrt(5)) / 2 B = (pi - 3)*2 phi*3 A_n = (9pi + 3phi - 2(pi - 3)*2) phi n = findroot(lambda x: x*3 - A_nx*2 - Bx - B*2, 53) # Inputs after SI-2019: # h and c are defined exactly -> hbar = h/(2pi) is exact as well # m_e is the only CODATA-limited input (rel. unc. ~3e-10) h = mpf('6.62607015e-34') # SI-2019: defined exactly = mpf('2.99792458e8') # defined exactly (pre-SI-2019) hbar = h / (2 pi) # exact (no CODATA uncertainty) m_e = mpf('9.1093837139e-31') # CODATA 2022, rel. unc. ~3e-10 G = hbar c / (m_e*2 phi*(4n)) print('n =', nstr(n, 30)) print('G =', nstr(G, 15)) The code output agrees with values (C.1) and (C.2); the final precision of G is limited by the CODATA uncertainty of me (∼ 3×10−10 ), because c, h, and h̄ = h/(2π) are exactly defined after SI-2019.

ACKNOWLEDGMENTS AND TOOLS The author thanks the participants of the ODTOE project for productive discussions on toroidal geometry and the informational interpretation of gravity. Calculations and formula verification were performed using Python 3.12 (mpmath, sympy), the tectonic typesetting system (XeLaTeX), and the AI assistant Claude (Anthropic) for structuring, text editing, and checking bibliographic consistency.

CONFLICT OF INTEREST The author declares no conflict of interest.

FUNDING This work was carried out without external funding.

XXIX. REFERENCES On bibliography order (ODTOE corpus project convention, see lessons L-35-ext). The list is organized conceptually into three blocks: (i) foundational original sources (Einstein, Planck, Newton) and classical works; (ii) standard reference data (CODATA) and experimental reference works; (iii) author preprints from the ODTOE series. Within each block, deviation from first-citation order is allowed. This ordering is an intentional choice of the present article and does not contradict corpus rule L-35 as extended for the conceptual-order case. 1. Einstein, A. Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 49(7), 769—822 (1916). https://doi.org/10.1002/andp.19163540702 2. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley & Sons, New York (1972). 3. Planck, M. Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum. Sitzungsberichte der Preußischen Akademie der Wissenschaften, 440—480 (1900). 4. Everett, H. III. «Relative State» Formulation of Quantum Mechanics. Reviews of Modern Physics, 29(3), 454—462 (1957). https://doi.org/10.1103/RevModPhys.29.454 5. Wheeler, J. A. Information, Physics, Quantum. In: W.H. Zurek (ed.), Complexity, Entropy, and the Physics of Information. Addison-Wesley (1989). 6. Jacobson, T. Thermodynamics of Spacetime: The Einstein Equation of State. Physical Review Letters, 75(7), 1260—1263 (1995). https://doi.org/10.1103/PhysRevLett.75.1260 7. Pankratov, A. S. The Nature of Light and the Limiting Nature of Speed: Reconfiguration Without Displacement in the Observer-Dependent Theory of Everything. Preprint (2026). — Contains the derivation c = r0 /τ0 . 8. Pankratov, A. S. Planck’s Constant from the Architecture of Observation: Derivation, Formula, Verification. Preprint (2026). — Contains the full derivation of h(d, S). 9. Pankratov, A. S. The Informational Architecture of Reality: Reading, Writing, and Verification on the φ-torus. Preprint (2026). — Defines the operations READ, WRITE, VERIFY, SYNC. 10. Abbott, B. P. et al. (LIGO Scientific Collaboration and Virgo Collaboration). Observation of Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters, 116(6), 061102 (2016). https://doi.org/10.1103/PhysRevLett.116.061102 11. Kolmogorov, A. N. On the Persistence of Conditionally Periodic Motions under Small Perturbations of the Hamiltonian. Doklady Akademii Nauk SSSR, 98(4), 527—530 (1954).

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