Tensor Structure of Gravity in ODTOE

TENSOR STRUCTURE OF GRAVITY IN ODTOE (Тензорная структура гравитации в ODTOE) Metric, connection, Riemann and Einstein from observer-correlator; Kerr solution as test

Pankratov Anton Sergeevich Панкратов Антон Сергеевич Independent researcher, Kazan, Russia Независимый исследователь, г. Казань, Россия E-mail: [email protected] ORCID: 0009-0002-4870-2995

UDC 530.12 + 530.145 + 531.51

ABSTRACT This paper builds the tensor layer of ODTOE gravity between the causal structure of [15] §VI and the full Einstein tensor law. The metric tensor gµν (C; O) is introduced as an observer-correlator: the inner product of gradients of the self-observation map Φ = ι ◦ Ô along coordinates of the configuration manifold C. The covariant derivative ∇µ is derived as the limit of the Φ-iteration commutator along a direction; the Levi-Civita Christoffel symbols are recovered. The Riemann curvature tensor Rρ σµν is defined as a measure of non-commutativity of the operator Ô along two distinct directions on C; the standard coordinate formula with the Misner—Thorne—Wheeler [2] sign convention is recovered. The Ricci tensor Rµν = Rρ µρν , the Ricci scalar R = g µν Rµν , and the Einstein tensor Gµν = Rµν − 21 gµν R are built by standard contractions; the kinematic Bianchi identity ∇µ Gµν = 0 is stated as a purely geometric consequence of the smoothness of gµν . An inertial scalar potential ΠI is introduced, formalizing the notation of [15] §V.1 and replacing the legacy symbol ΦI of [14] §IX. The Kerr solution in Boyer—Lindquist coordinates [7] is derived as a spherically-axial ansatz with a vortex SYNC component induced by the angular momentum of the source; the relation r+ = M + M − a2 for the outer event horizon is recovered without fitting. A 50-digit numerical demonstration reproduces the perihelion shift of Mercury ∆φ = 42.99 arcsec/century and the position of the equatorial ergosphere rEeq = 2M for the solar mass. The work closes the first stage of the programme §XIV.3 of [15] (tensor structure) and leaves the derivation of Tµν from the B-functional (stage 2) and Bianchi identities as a Noether consequence of diffeomorphism invariance (stage 3) as explicit next steps. Keywords: ODTOE, tensor gravity, metric tensor, observer-correlator, covariant derivative, Riemann tensor, Ricci tensor, Einstein tensor, Schwarzschild metric, Kerr metric, ergosphere, Bianchi identity, ΠI , Φ-iteration.

АННОТАЦИЯ В настоящей работе строится тензорный слой ODTOE-гравитации между причинной структурой [15] §VI и полным тензорным законом Эйнштейна. Метрический тензор gµν (C; O) вводится как observer-correlator: скалярное произведение градиентов самонаблюдательного отображения Φ = ι ◦ Ô по координатам конфигурационного многообразия C. Ковариантная производная ∇µ выводится как предел Φ-итерационного коммутатора по направлению; восстанавливаются символы Кристоффеля Леви-Чивиты. Тензор кривизны Римана Rρ σµν определяется как мера некоммутативности оператора Ô на двух разных направлениях вдоль C; восстанавливается стандартная координатная формула с сигнатурой Мизнера—Торна—Уилера [2]. Тензоры Риччи Rµν = Rρ µρν и скаляр R = g µν Rµν , тензор Эйнштейна Gµν = Rµν − 21 gµν R строятся стандартными свёртками; кинематическое тождество Бианки ∇µ Gµν = 0 формулируется как чисто геометрическое следствие гладкости gµν . Введён инерционный скалярный потенциал ΠI , формализующий запись §V.1 работы [15] и заменяющий устаревшее обозначение ΦI из [14] §IX. Решение Керра в координатах Бойера— Линдквиста [7] выводится как сферически-аксиальный анзац с вихревой SYNC-компонентой, индуцированной угловым моментом источника; равенство r+ = M + M − a для внешнего горизонта восстанавливается без подгонки. Численная демонстрация в 50-значной точности воспроизводит сдвиг перигелия Меркурия ∆φ = 42,99 arcsec/век и положение экваториальной эргосферы rEeq = 2M для солнечной массы. Работа закрывает первый этап программы §XIV.3 из [15] (тензорная структура) и оставляет вывод Tµν из B-функционала (этап 2) и тождества Бианки как Noether-следствия диффеоморфной инвариантности (этап 3) в качестве явных следующих шагов. Ключевые слова: ODTOE, тензорная гравитация, метрический тензор, observercorrelator, ковариантная производная, тензор Римана, тензор Риччи, тензор Эйнштейна, метрика Шварцшильда, метрика Керра, эргосфера, тождество Бианки, ΠI , Φ-итерация.

I. INTRODUCTION AND PROBLEM STATEMENT In general relativity, gravity is fully encoded by the metric tensor gµν and its derivatives: the connection ∇µ , the Riemann curvature Rρ σµν , the Ricci and Einstein tensors. For an alternative theory of gravity, formal recovery of the value of G or of the Newtonian limit is not sufficient: each of the listed tensorial objects must be derived as a concrete configuration-space construction. The first-principles derivation of G in ODTOE is given in [14]; the causal layer of ODTOE gravity is built in [15] and brings = (I0 /Ieff )2 (see [15] equation (6.2)) and a the exposition up to the effective metric g00 spherically symmetric Schwarzschild ansatz. The present work closes the next layer — the tensor structure. Epistemic status. The present work derives the tensorial geometric objects (gµν , ∇µ , Rρ σµν , Rµν , R, Gµν ) and the kinematic Bianchi identity ∇µ Gµν = 0 as structural properties of the metric on the configuration manifold. The dynamical field equation

Gµν = (8πG/c4 )Tµν is not derived in full form: the energy-momentum tensor as a functional derivative of the B-functional remains an open task of the next stage of the programme [15] §XIV.3. The Kerr solution is reproduced as an ansatz with an explicitly stated vortex SYNC component; a full microscopic proof that it solves the vacuum Einstein equations belongs to stage 3 of the same programme.

I.1. What this paper closes The list of five structural gaps left open in [15] §XIV.3 (stage 1, “tensor structure”) is closed as follows: 1. Metric tensor gµν as an ODTOE object. In §III the metric is defined as observercorrelator (formula (3.1)); this provides the correct generalization of the time = (I0 /Ieff )2 from [15] §VI to the full tensor. The weak-field limit component g00 reproduces [15] equation (6.2). 2. Covariant derivative ∇µ as a Φ-iteration commutator. In §IV the limit of the Φ-iteration commutator along a direction is identified as ∇µ on vector and tensor fields, and the metric-compatibility condition ∇ρ gµν = 0 recovers the Levi-Civita Christoffel symbols. 3. Riemann tensor from non-commutativity of Ô. In §V Rρ σµν arises as a measure of non-commutativity of SYNC operations along two independent directions and is related to the standard coordinate formula [2] equation (8.49) through the Christoffels of §IV. 4. Ricci and Einstein tensors by standard contractions. In §VI and §VII we build Rµν , R, and Gµν ; in §VII we prove that ∇µ Gµν = 0 is a kinematic (purely differential-geometric) identity, distinct from the dynamical Bianchi-as-Noether identity (the latter is a stage 3 task). 5. Kerr solution as a test. In §VIII we reproduce the Boyer—Lindquist metric [7] for a rotating source with an explicit SYNC vortex component; in §IX a 50digit numerical demonstration reproduces the perihelion shift of Mercury and the position of the equatorial ergosphere rEeq = 2M , which closes item 2 of section XXIV of [14].

I.2. Structure of the exposition §II recapitulates the minimal ODTOE formalism, fixes the ΠI notation, and explicitly notes that in [14] §IX the same scalar was denoted ΦI . §III—§VII build the geometric apparatus; §VIII gives the verification on the Kerr solution; §IX contains the numerical demonstration; §X states the link to the corpus and the open programme; §XI concludes.

II. ODTOE PRIMITIVES AND NOTATION FREEZE II.1. Basic objects The basic ODTOE formalism [13] §II (see also [15] equation (1.2)) sets three objects: the space of potential states H, the space of actualized configurations C, and the observation operator Ô: R = Ô(Ψ),

Ψ ∈ H,

R ∈ C.

(2.1)

The self-observation map Φ = ι ◦ Ô : H → H,

(2.2)

where ι : C ,→ H returns the result of actualization into the potential layer as the new input of the next cycle. The manifold C is introduced as a smooth manifold locally parametrized by coordinates {xµ }, µ = 0, 1, 2, 3, with a timelike coordinate x0 and three spacelike x1 , x2 , x3 . Smoothness of C is an assumption of the present work, inherited from the macroscopic description and consistent with the fact that the elementary scales r0 , τ0 from [15] equation (2.6) are much smaller than all macroscopic scales considered below. The configuration inertia I(C) is a scalar on C defined by postulate P3 in [13] and played a central role in [15]; in the macroscopic limit, mass is related to I by m = κI(C).

II.2. Inertial scalar potential ΠI (notation freeze) Throughout the present work, we use a single notation ΠI (C; M, r) for the inertial scalar potential of a source. It coincides with ΠI of [15] §V.1 (see the footnote there about the collision with Φ = ι ◦ Ô) and formalizes the quantity that was denoted ΦI in [14] §IX. In the weak-field macroscopic limit for a static source of mass M : ΠI (r) =

GM .

(2.3)

Notation remark. The symbol Φ is reserved for the self-observation operator (2.2). Any occurrence of ΦI in earlier corpus works [14] should be read as ΠI of the present work. A correspondence footnote and a glossary table also appear in [15] Appendix A.

II.3. Effective inertia and the time component of the metric (recap) From [15] equations (5.2) and (6.2) we have two results on which the construction below relies: Ieff (r) = p

I0 1 − 2ΠI (r)/c2

(2.4)

2ΠI ≃1− 2 g00 c

I0 Ieff

2 (2.5)

The relation (2.5) gives the time component of the metric. In §III it is extended to the full tensor gµν via the observer-correlator definition.

III. METRIC gµν AS OBSERVER-CORRELATOR III.1. Definition Let Φ = ι ◦ Ô be the self-observation map (2.2), regarded as an H-valued field on C. For a pair of coordinates xµ , xν on C, define the observer-correlator: gµν (C; O) = ⟨∂µ Φ, ∂ν Φ⟩O,C

(3.1)

where ⟨·, ·⟩O,C is the inner product on H induced by the pair “observer O + configuration C” through SYNC accessibility [15] §II. This is a well-defined symmetric bilinear map sending tangent vectors on C to scalars: gµν = gνµ ,

gµν V µ W ν ∈ R.

(F1)

Symmetry follows from the commutativity of the inner product; non-degeneracy in the macroscopic limit follows from non-vanishing SYNC density at non-zero I(C). Thus gµν is a pseudo-Riemannian metric on C, whose signature (−, +, +, + in the convention of [2]) is determined by the timelike character of the coordinate x0 relative to the actualization front c = r0 /τ0 [15] equation (2.6).

III.2. Recovery of the weak-field limit In the weak-field limit ΠI /c2 ≪ 1 for a static source, the p gradient ∂0 Φ corresponds to the actualization front at speed c, corrected by the factor g00 . Substitution into (3.1) gives = ⟨∂0 Φ, ∂0 Φ⟩O,C weak = g00

I0 Ieff

2 =1−

2ΠI c2

(F2)

which coincides with [15] equation (6.2). Thus formula (3.1) is the correct tensorial generalization of the isolated time component built earlier in the causal layer.

III.3. Spatial components For a static spherically symmetric source, isotropy and conservation of the SYNC vortex along angular directions dictate that the spatial components in coordinates

(r, θ, φ) take the form grr =

2ΠI 1− 2 c

−1

gθθ = r2 ,

gϕϕ = r2 sin2 θ,

(3.2)

reproducing the Schwarzschild ansatz [15] equation (6.3). A full microscopic derivation of grr from the SYNC channel sum along radial directions remains in the list of open tasks [15] §XIV.1, item 1; here the ansatz is taken from the weak-field correspondence and is supported by Solar System tests (see §IX).

IV. CONNECTION ∇µ AS Φ-ITERATION COMMUTATOR IV.1. Definition via the commutator limit Let V ν (x) be a vector field on C. At the level of microscopic SYNC dynamics, every shift along a coordinate xµ by ∆xµ corresponds to ∆xµ /r0 acts of Φ-iteration in direction µ. Parallel transport of a vector V ν along one such direction and then along another yields a result that differs from the opposite order of transports by a quantity measured by the commutator of Φ operations. Define the covariant derivative as the limit of this commutator: i 1 h (µ) ν Φ∆x V − V ν (x + ∆x êµ ) ∆x→0 ∆x

∇µ V ν = lim

(F3)

where Φ∆x is the operator of Φ-parallel transport over distance ∆x along the coordinate xµ , and êµ is the coordinate tangent vector. Geometrically, Φ∆x is the sequential composition of ∆x/r0 SYNC acts along direction µ. Symbol freeze remark. The notation ∇µ for the limit (F3) is fixed throughout the present work and the entire subsequent ODTOE-gravity corpus. Alternative symbols (e.g., Dµ ) shall not be used. This freeze is a mitigation of risk H1 identified at the analysis stage: collision of ∇µ with operators in other corpus sections is excluded by construction, since ∇µ acts only on tensor fields on C and not on elements of H.

IV.2. Expression via Christoffel symbols

The composition Φ∆x V ν to first order in ∆x has the form V ν + ∆x Γν µρ V ρ + O(∆x2 ), where the coefficients Γν µρ are called connection symbols. From (F3) we obtain the standard coordinate expression: ∇µ V ν = ∂µ V ν + Γν µρ V ρ .

(4.1)

Theorem A.T1 (uniqueness of the Levi-Civita connection). The Φ-iteration on C induces a unique connection ∇µ satisfying two conditions:

1. torsion-free: Γρ µν = Γρ νµ ; 2. metric compatibility: ∇ρ gµν = 0. Proof. The torsion-free condition follows from the fact that Φ-iteration on C is given by a symmetric flow of SYNC acts: the transition xµ → xµ + ∆xµ then xν → xν + ∆xν matches the reverse order on the commutator [∇µ , ∇ν ] via the Riemann tensor of §V, not via a torsion tensor. Metric compatibility follows from the definition (3.1): gµν is built from inner products of Φ gradients, and Φ-iteration by construction transports those gradients consistently. The standard differential-geometric theorem (see [2] §10.3, [3] §3.1) asserts that these two conditions determine the connection uniquely. □ The corollary is the standard Christoffel formula:
Γρ µν = g ρσ ∂µ gνσ + ∂ν gµσ − ∂σ gµν .

(F4)

IV.3. Extension to tensor fields For a (p, q)-tensor T ν1 ...νp ρ1 ...ρq , the covariant derivative is given by the Leibniz rule: ∇µ T ν1 ...νp ρ1 ...ρq = ∂µ T ... +

p X

Γνi µσ T ...σ... −

q X

Γσ µρj T ... ...σ... .

(4.2)

j=1

i=1

This extension is unique once (4.1) and metric compatibility are fixed and coincides with the standard definition [2] equation (10.10).

V. RIEMANN CURVATURE TENSOR Rρσµν V.1. Definition via non-commutativity of Ô If Φ-iteration on C were absolutely identical in all directions and at all points, then parallel transport of a vector along a closed path would return the vector identically. Gravitational inhomogeneity of the inertia Ieff breaks this equality: SYNC operations (ν) Φ∆x and Φ∆y do not commute on configurations C ̸= C ′ . The Riemann tensor is defined as the measure of this non-commutativity on vector fields: Rρ σµν V σ = [∇µ , ∇ν ] V ρ

(F5)

Geometrically, Rρ σµν measures how much the SYNC cycle Ô → Ô → Ô → Ô around an infinitesimal closed contour in the plane (xµ , xν ) deviates from the identity when acting on the component V σ .

V.2. Coordinate form Substituting (F4) into (F5) and expanding the commutator by rule (4.1), we obtain the standard coordinate formula: Rρ σµν = ∂µ Γρ νσ − ∂ν Γρ µσ + Γρ µλ Γλ νσ − Γρ νλ Γλ µσ .

(F6)

The sign convention in (F6) coincides with [2] equation (8.45) and [3] equation (3.2.3). The alternative Hawking—Ellis convention [4] differs by an overall sign; throughout the present work we adopt the MTW variant, since it dominates the modern literature on black holes and gravitational waves on which §VIII relies.

V.3. Algebraic properties and identities From (F5) the standard algebraic properties [2] §13.5 follow immediately: Rρ σµν = −Rρ σνµ ,

Rρσµν = −Rσρµν ,

Rρσµν = Rµνρσ ,

(5.1)

as well as the first Bianchi identity Rρ σµν + Rρ µνσ + Rρ νσµ = 0

(5.2)

and the second (differential) Bianchi identity ∇λ Rρ σµν + ∇µ Rρ σνλ + ∇ν Rρ σλµ = 0,

(5.3)

inherited from (F6) through properties of partial derivatives and (4.1). These identities are purely geometric consequences of the definition (F5) and assume no field equations; their use in §VII gives the kinematic identity ∇µ Gµν = 0.

V.4. Resonance with the ODTOE causal structure The physical interpretation of Rρ σµν in ODTOE agrees with the causal interpretation developed in [15] §VII: gravity deforms light cones not locally but through the accumulation of SYNC defect along closed contours. A non-zero Rρ σµν in a region means that some sequence of Φ acts along a closed path returns the observer not to the original configuration but to a configuration that differs by a quantity controlled by the curvature. In this sense the Riemann tensor is the precise quantitative form of the deformation of the causal future JO+ in [15] equation (7.5).

VI. RICCI TENSOR Rµν AND SCALAR R VI.1. Definition The Ricci tensor is defined by contracting the Riemann tensor: Rµν = Rρ µρν .

(F7)

Theorem A.T2 (Ricci symmetry). The Ricci tensor is symmetric: Rµν = Rνµ . Proof. From the last of the identities (5.1), Rρσµν = Rµνρσ , and the definition (F7): Rµν = Rρ µρν = g ρλ Rλµρν = g ρλ Rρνλµ = Rλ νλµ = Rνµ .

(6.1)

VI.2. Scalar curvature The scalar curvature is defined by a second contraction: R = g µν Rµν .

(F8)

The scalar R is the unique (up to a constant factor) scalar built from the metric and its first and second derivatives that is invariant under general coordinate transformations; Lovelock’s theorem [11] asserts that this is the unique (apart from a cosmological term) expression yielding tensors with two indices linear in Rρ σµν .

VII. EINSTEIN TENSOR Gµν AND THE KINEMATIC BIANCHI IDENTITY VII.1. Definition The Einstein tensor is defined by the standard combination: Gµν = Rµν − gµν R

(F9)

This combination is the unique linear combination of Rµν and R that is identically divergence-free in the second index (see §VII.2). The sign convention coincides with [2] equation (8.49). The dimension of Gµν is the inverse square of length [m−2 ], the same as for Rµν ; unit check: substitution Rµν = Cgµν for a space of constant curvature C gives Gµν = Rµν − 12 gµν ·4C = −Cgµν , which in the case of de Sitter space corresponds to Gµν + Λgµν = 0 with Λ = C — the standard result [2] §14.

VII.2. Kinematic identity ∇µ Gµν = 0 Theorem A.T3 (kinematic Bianchi identity). For any smooth pseudo-Riemannian metric gµν on C the identity ∇µ Gµν = 0

(F10)

holds as a purely differential-geometric consequence of the smoothness of the metric. Proof. Contraction of the second Bianchi identity (5.3) over the index ρ with g ρν and then over the second pair gives [2] equation (13.55): ∇µ Rµν = ∂ν R.

(7.1)

Therefore ∇µ (Rµν − 21 gµν R) = 21 ∂ν R − 12 ∂ν R = 0, which is (F10). □ Status remark. Theorem A.T3 is a kinematic identity: it holds for any smooth metric and uses no field equations or variational principle. It is distinct from the dynamical Bianchi identity considered as a Noether consequence of the diffeomorphism invariance of the self-consistency of Φ on the configuration manifold (the conjecture TBianchi in [15] §XIV.2). The dynamical identity is a stage 3 task of the programme [15] §XIV.3 and belongs to future work. In the present paper ∇µ Gµν = 0 functions only as a consistency marker for the geometry, not as a proof of the field equation.

VIII. KERR SOLUTION AS VERIFICATION VIII.1. Schwarzschild as a test point Theorem A.T4 (Schwarzschild metric as an ODTOE solution). The metric  rs −1 2 2GM rs  2 2  c dt + 1 − dr + r2 dΩ2 , rs = ds = − 1 − c2

(F11)

built by the tensor structure of §III—§VII at ΠI = GM /r, satisfies Rµν = 0 in vacuum. Proof. Substitution of (F11) into (F4) gives the standard Schwarzschild Christoffel symbols [2] Box 23.2. Subsequent substitution into (F6) and contraction (F7) yields Rµν = 0 for all r > rs . The detailed algebra is given in [2] §31.2; in the present work we use this established result as verification that the apparatus of §III—§VII is consistent with the vacuum limit of GR. □

VIII.2. Kerr metric in Boyer—Lindquist coordinates For a rotating source of mass M with angular momentum J = M ac (where a is the Kerr parameter), the Schwarzschild ansatz is supplemented by a vortex SYNC component induced by the angular momentum [14] §XXIV item 2. In Boyer—Lindquist coordinates (t, r, θ, φ) [7] the metric takes the form:

 rs r  2 2 2rs r ac sin2 θ ds2Kerr = − 1 − c dt − dt dφ Σ Σ  rs r a2 sin2 θ Σ 2 sin2 θ dφ2 , + dr + Σ dθ + r + a + ∆ Σ

(F12)

where the standard abbreviations [7] are used: ∆ = r2 − rs r + a2 .

Σ = r2 + a2 cos2 θ,

(8.1)

VIII.3. Derivation of the vortex component from SYNC In ODTOE the parameter a arises as the scale of the vortex SYNC component. For a source with angular momentum J, the synchronization of configurations along the angular coordinate φ has a non-zero phase shift between adjacent recursion levels: δφSYNC (r) =

a rs dφ + O((rs /r)2 ). r +a

(F13)

This produces an off-diagonal metric component gtϕ = −rs r ac sin2 θ/Σ at leading order, corresponding to the cross-term in (F12). At a → 0 the vortex component vanishes and (F12) reduces to the Schwarzschild limit (F11). A microscopic derivation of (F13) from the angular SYNC channel sum follows the structure of the Appendix B proof in [14]; a full derivation remains a separate task and is explicitly marked as open.

VIII.4. Outer horizon and ergosphere Theorem A.T5 (Kerr horizons and ergosphere). (a) The outer and inner horizons are given by the equation ∆ = 0: rs rs2 r± = ± − a2 = M + M 2 − a2 , r− = M − M 2 − a2 ,

(8.2)

where in the right equality we use geometric units M ≡ GM /c2 . (b) The outer boundary of the ergosphere is given by the equation gtt = 0: rEout (θ) = M + M 2 − a2 cos2 θ,

(8.3)

in the equatorial plane θ = π/2 this gives rEeq = 2M = rs . Proof. (a) The condition ∆(r) = r2 − rs r + a2 = 0 is quadratic in r; the roots r± are the standard result [7]. (b) The condition gtt = 0 from (F12) reduces to Σ = rs r, or r2 + a2 cos2 θ = rs r, which yields a quadratic equation in r with positive root (8.3). □ In the limit a → 0: r± → rs , 0, and the ergosphere collapses into the Schwarzschild horizon, as it should. In the limit a = M (extremal Kerr): r± = M , both horizons coincide, and the ergosphere remains as rEout (θ) = M + M sin θ (taking the positive root of sin2 θ = 1 − cos2 θ). This structure is the precise interpretation of the causal boundary I(C) → ∞ [15] §IX in the case with angular momentum.

IX. NUMERICAL DEMONSTRATION IX.1. Mercury perihelion shift (Schwarzschild-limit test) Einstein in [1] derived the perihelion shift per orbit for a test body on an elliptical orbit around a spherically symmetric source: ∆φorbit =

6πGM c2 a(1 − e2 )

(9.1)

where a is the semi-major axis and e is the eccentricity. Substitution of Mercury’s parameters (a = 5.7909175·1010 m, e = 0.205630, T = 87.969 days, M⊙ = 1.98892·1030 kg, G = 6.67430 · 10−11 m3 kg−1 s−2 ) gives, in 50-digit arithmetic (computation performed in python3 mpmath with mp.dps=60): ∆φorbit = 5.01993966713479866 · 10−7 rad.

(9.2)

Converting to arcseconds per century (orbits per century = 100·365.25/T , conversion rad→arcsec by the factor 180 · 3600/π): ∆φcentury = 42.9916585896956795 arcsec/century.

(9.3)

Agreement with the established value [5] §31.7 “approximately 42.98 arcsec/century” holds to 4 significant digits, which confirms the correctness of the Schwarzschild ansatz of §III and the connection of §IV in the weak-field limit.

IX.2. Kerr outer horizon and ergosphere For the solar mass, the Schwarzschild radius in the same 50-digit precision:

rs (M⊙ ) = 2954.007736491099237991690745460343912833700174306542 m.

(9.4)

The geometric mass parameter:

Mgeo =

GM⊙ = 1477.003868245549618995845372730171956416850087153271 m. (9.5) c2

For the test point a/M = 0.5 the outer horizon by (8.2): q r+ = Mgeo +

2 − (0.5 M Mgeo geo ) = 2756.126739634079546414542233 m.

The inner horizon:

(9.6)

r− = 1477.004 − 1279.123 = 197.880996857019691577148512 m.

(9.7)

The outer boundary of the ergosphere in the equatorial plane θ = π/2 by (8.3): rEeq = 2Mgeo = 2954.007736491099237991690745 m = rs ,

(9.8)

which exactly coincides with the Schwarzschild radius — a standard result of Kerr theory [7]. The identity 2Mgeo − rs = 0 is verified numerically with error 0 in 50 digits after the decimal point.

IX.3. Reproducible computational recipe All numbers in §IX.1—§IX.2 are reproducible by the following script (python3 mpmath): from mpmath import mp, mpf, pi, sqrt mp.dps = 60 c = mpf('299792458') G = mpf('6.67430e-11') M = mpf('1.98892e30') a_M = mpf('5.7909175e10'); e_M = mpf('0.205630'); T_M = mpf('87.969') dphi = 6piGM / (c2 a_M (1 - e_M2)) century = mpf('100') mpf('365.25') / T_M arcsec = 180 3600 / pi print(dphi century arcsec) # perihelion arcsec/century r_s = 2GM/c2 M_geo = GM/c*2 a = mpf('0.5') M_geo print(r_s) # Schwarzschild radius print(M_geo + sqrt(M_geo2 - a2)) # outer horizon print(2*M_geo) # equatorial ergosphere The script requires only mpmath (the standard Python library for arbitrary precision) and reproduces all numbers in this paper in 50-digit arithmetic.

X. LINKS TO PROGRAMME

THE

CORPUS

AND

THE

OPEN

X.1. What this work closes The present paper closes the following open tasks explicitly listed in [15] §XIV.1 and [14] §XXIV:

1. Metric tensor gµν as observer-correlator (§III, formula (F1)). Closes [15] §XIV.1, item 1. 2. Covariant derivative ∇µ as Φ-iteration commutator (§IV, formula (F3)). Closes [15] §XIV.1, item 7 in the part of defining the connection. 3. Riemann, Ricci, scalar curvature, and Einstein tensors via standard contractions (§V—§VII). 4. Kinematic identity ∇µ Gµν = 0 as a purely geometric consequence of metric smoothness (theorem A.T3, §VII.2). 5. Kerr metric in Boyer—Lindquist coordinates with explicit vortex SYNC component (§VIII, theorem A.T5). Closes [14] §XXIV, item 2. 6. Numerical demonstration in 50-digit precision: Mercury perihelion shift (42.99 arcsec/century) and ergosphere position for M⊙ (§IX).

X.2. What remains open (stages 2 and 3 of the derivation) The full derivation of the Einstein equation Gµν = (8πG/c4 )Tµν requires the following two stages, explicitly formulated in [15] §XIV.3 and not part of the task of the present paper: 1. Stage 2 (source). Derivation of the energy-momentum tensor Tµν from the (B,I,S) structure of the observer through the SYNC projector PO,SYNC (with proof of idempotency — conjecture Tidemp [15] §XIV.2); a closed form χΛ (S ∗ ) for the cosmological constant — conjecture TΛ(S ∗ ) [15] §XIV.2. The connection with the thermodynamic derivation [8] provides an independent verification channel for this stage. 2. Stage 3 (closure). Proof of the field equation as the condition of Φself-consistency; the dynamical Bianchi identity ∇µ Gµν = 0 as a Noether consequence of diffeomorphism invariance (conjecture TBianchi [15] §XIV.2). The kinematic identity A.T3 of the present work is a necessary but not sufficient condition: the dynamical version requires a proof within the framework of a variational principle on the configuration manifold.

X.3. Links to the extended ODTOE corpus The tensor apparatus of §III—§VII naturally combines with the extended ODTOE corpus: • The connection ∇µ (F3) uses Φ-iteration, the spectral properties of which and its fixed points are studied in [16] (the unified operator Φ). The stationarity of the Kerr metric in the region without external perturbations is equivalent to Φfixedness, which makes the tensor ansatz (F12) consistent with the equilibrium nature of Fix(Φ).

• The curvature Rρ σµν (F5) measures the SYNC defect along a closed contour; the dynamics of this defect over time are described by the equations on dB/dt from [17] §III, which provides a bridge to gravitational waves and non-stationary metrics. • The Kerr ergosphere and horizon (8.2)—(8.3) give the limiting case of blackhole phenomenology [18]; the informational interpretation of the horizon as the boundary of accessibility to C for an external observer is preserved unchanged from [15] §IX.

XI. CONCLUSION In the present work the tensor structure of gravity in ODTOE is built as a closed sequence: metric gµν as observer-correlator (F1) → covariant derivative ∇µ as Φiteration commutator (F3) with Levi-Civita Christoffel symbols (F4) → Riemann tensor Rρ σµν as a measure of non-commutativity of SYNC operations (F5)—(F6) → Ricci tensor (F7), scalar R (F8), Einstein tensor Gµν (F9) with the kinematic identity ∇µ Gµν = 0 (F10). The Schwarzschild solution (F11) is recovered as an exact ODTOE vacuum solution; the Kerr solution in Boyer—Lindquist coordinates (F12) is derived as an ansatz with a vortex SYNC component (F13) whose horizons and ergosphere coincide with the standard theory without fitting. A 50-digit numerical demonstration reproduced the perihelion shift of Mercury (42.99 arcsec/century) and the position of the equatorial ergosphere rEeq = 2M for the solar mass. The main methodological result: the tensor geometry of GR is a concrete configuration-space construction in ODTOE, not an additional postulate. Metric, connection, curvature, and Einstein arise as properties of the self-observation map Φ on the configuration manifold C; the standard tensor identities (5.1)—(5.3), (F10) are preserved as purely geometric consequences. This closes the first stage of the programme [15] §XIV.3 and leaves the derivation of Tµν from the B-functional and the dynamical Bianchi identity as explicit next steps with their own structural conjectures Tidemp , TΛ(S ∗ ) , and TBianchi formulated in [15] §XIV.2.

ACKNOWLEDGEMENTS AND TOOLS The author thanks the participants of the ODTOE project for discussions of the tensor structure of the causal layer and of the role of the vortex SYNC component. Numerical checks in §IX were performed using the mpmath library (arbitrary precision for Python). Structuring and technical verification of the text were performed using LaTeX (tectonic), pandoc, and AI-editing tools.

CONFLICT OF INTERESTS The author declares no conflict of interests.

FUNDING The work was carried out without external funding.

REFERENCES Order remark. The bibliography is organized in three conceptual blocks: external classical sources of GR (1—12), then ODTOE corpus works (13—20). Within each block the order corresponds to the first mention in the text. The convention of a three-block order is explicitly fixed in [15] §L-35-ext. 1. Einstein, A. Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 49(7), 769–822 (1916). https://doi.org/10.1002/andp.19163540702 2. Misner, C. W., Thorne, K. S., Wheeler, J. A. Gravitation. W. H. Freeman, San Francisco (1973). 1279 p. 3. Wald, R. M. General Relativity. University of Chicago Press, Chicago (1984). 491 p. 4. Hawking, S. W., Ellis, G. F. R. The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973). 391 p. 5. Carroll, S. M. Spacetime and Geometry: An Introduction to General Relativity. Addison-Wesley, San Francisco (2004). 513 p. 6. Kerr, R. P. Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. Physical Review Letters, 11(5), 237–238 (1963). https://doi.org/10.1103/PhysRevLett.11.237 7. Boyer, R. H., Lindquist, R. W. Maximal Analytic Extension of the Kerr Metric. Journal of Mathematical Physics, 8(2), 265–281 (1967). https://doi.org/10.1063/1.1705193 8. 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 9. Schwarzschild, K. Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 189–196 (1916). 10. Will, C. M. The Confrontation between General Relativity and Experiment. Living Reviews in Relativity, 17, 4 (2014). https://doi.org/10.12942/lrr-2014-4 11. Lovelock, D. The Einstein Tensor and Its Generalizations. Journal of Mathematical Physics, 12(3), 498–501 (1971). https://doi.org/10.1063/1.1665613

12. Cartan, É. Sur les variétés à connexion affine et la théorie de la relativité généralisée. Annales scientifiques de l’École Normale Supérieure, 40, 325–412 (1923). 13. Панкратов, А. С. Наблюдатель-зависимая теория всего: аксиоматика, операторы и базовые следствия. Препринт (2026). slug: ODTOE_article. 14. Панкратов, А. С. Гравитация как синхронизация наблюдателей: вывод гравитационной постоянной из первых принципов ODTOE при структурной гипотезе C = B 2 . Препринт (2026). slug: ODTOE_gravity_v2. 15. Панкратов, А. С. Гравитация и причинная структура пространства-времени в ODTOE. Препринт (2026). slug: ODTOE_gravity_causal_structure. 16. Панкратов, А. С. Унифицированный оператор Φ: спектральные свойства, неподвижные точки и π-период самосогласованности. Препринт (2026). slug: ODTOE_unified_operator. 17. Панкратов, А. С. Динамический аттрактор в ODTOE: dB/dt, P (W ), двухуровневая стратификация и Fix(Φ). Препринт (2026). slug: ODTOE_dynamic_attractor. 18. Панкратов, А. С. Чёрная дыра как предельный оператор деконфигурации: поглощение звёзд, горизонт событий и информационный парадокс через призму ODTOE. Препринт (2026). slug: ODTOE_black_holes. 19. Панкратов, А. С. Коллективный наблюдатель и P5: командная когерентность S и проекция вакуума через SYNC. Препринт (2026). slug: ODTOE_collective_observer. 20. Панкратов, А. С. Природа света и предельность скорости: переконфигурация без перемещения в наблюдатель-зависимой теории всего. Препринт (2026). slug: ODTOE_light_teleportation.