STRESS-ENERGY TENSOR Tµν AND COSMOLOGICAL CONSTANT Λ FROM OBSERVER COHERENCE IN ODTOE (Тензор энергии-импульса Tµν и космологическая постоянная Λ из когерентности наблюдателя в ODTOE) SYNC projector PO,SYNC , idempotency proof (L7), conservation law L8, and closed-form χΛ (S ∗ )
Pankratov Anton Sergeevich Панкратов Антон Сергеевич Независимый исследователь, г. Казань, Россия Independent researcher, Kazan, Russia E-mail: [email protected] ORCID: 0009-0002-4870-2995
UDC 530.12 + 530.145 + 524.85
ABSTRACT This paper constructs the tensor source of ODTOE gravity: theR stress-energy tensor Tµν as a functional derivative of the observer action Sobs = B 2 (1 − σ)Λ −g d4 x with respect to the inverse metric g µν , and the cosmological constant Λ as a closed function of the global coherence of the Universe S ∗ = 0.169676 . . .. The central step is the construction of the SYNC projector PO,SYNC : H → C, which fixes the mapping from the potential Hilbert layer to the actualized causal layer. Using the orthogonal projection theorem in Hilbert space [1] Thm II.3, lemma L7 on idempotency PO,SYNC = PO,SYNC is proved via four sub-lemmas: L7.1 closedness of the Φ-invariant subspace, L7.2 linearity, L7.3 well-definedness, L7.4 self-adjointness. The proof does not use the Bianchi identity and does not assume Einstein’s equation; hypothesis Tidemp of [9] §XIV.2 is resolved without circularity. Lemma L8 on the conservation law ∇µ T µν = 0 is derived through the covariant derivative fixed in [10] §IV.1 (formula (F3) of that source); conservation thus is a consequence of L7 and Φ-self-consistency, not an axiom. §VIII obtains the closed form χΛ (S ∗ ) = (3φ2 )/(8π(φ2 + 1 + Z)) ≈ 0.082201, where Z(S ∗ ) = (π − 3)/(1 − (π − 3)φ), which closes the fitted form χΛ ≃ 8.2 · 10−2 of [9] §XII.5. Substitution of 50-digit constants π, φ, (π − 3) gives ΩΛ ≈ 0.688647 — agreement with Planck 2018 [7] ΩΛ = 0.6889 ± 0.0056 within 0.05σ without fitting. §IX establishes consistency with the thermodynamic derivation of Jacobson [3]: the horizon limit of the action Sobs reproduces the relation δQ = T dS. The work closes stage 2 of programme §XIV.3 of [9] and fixes six symbols (Tµν , PO,SYNC , χΛ (S ∗ ), Sobs , L7, L8) for subsequent work of the corpus. Keywords: ODTOE, stress-energy tensor, cosmological constant, SYNC projector, idempotency, Hilbert projection, observer action, S ∗ , χΛ , dark energy, horizon thermodynamics, Jacobson
I. INTRODUCTION AND PROBLEM STATEMENT In general relativity the right-hand side of Einstein’s equation Gµν = (8πG/c4 )Tµν is given by the stress-energy tensor Tµν . In the standard exposition Tµν is introduced either phenomenologically (as a perfect fluid, electromagnetic field, etc.) or variationally as Tµν = (2/ −g) δ( −g Lmatt )/δg µν [5] §E.1.7, [10] §4.3. The first route does not derive the source from first principles; the second requires an independently given matter density Lmatt . In ODTOE the gravitational source is not external «matter», but the structure of the observer: the triple (B, I, S) — cognitive coherence B, configurational inertia I(C), and pairwise synchronization S [8] §III, [13] §II. Gravitational coupling is provided by the SYNC operator that handles the transition from the potential Hilbert layer H to the actualized causal layer C [9] §II.1. Stage 1 of the programme §XIV.3 of [9] (tensor structure of geometry: gµν , ∇µ , Rρ σµν , Gµν ) is closed by [10]; the covariant derivative ∇µ is fixed there as a Φ-iteration commutator (formula (F3) of §IV.1 of that source) and used here unchanged. Epistemic status. The present work derives: (i) the SYNC projector PO,SYNC as a formally defined orthogonal projector onto a closed Φ-invariant subspace C ⊂ H (§IV); (ii) lemma L7 on idempotency PO,SYNC = PO,SYNC via four sub-lemmas (§V); (iii) the tensor Tµν from the variational principle δSobs /δg µν (§VI); (iv) the conservation law L8: ∇µ T µν = 0 — a lemma using the covariant derivative fixed in [10] §IV.1 (formula (F3) of that source) (§VII); (v) the closed form χΛ (S ∗ ) via substitution of ΩΛ from [8] §XXV-A (§VIII); (vi) agreement with the thermodynamic horizon derivation of Jacobson [3] (§IX). The work closes the following: hypothesis Tidemp of [9] §XIV.2 (via L7), the fitted form χΛ ≃ 8.2 · 10−2 of [9] §XII.5 (via the closed form §VIII), and stage 2 of the programme §XIV.3 of [9] (source Tµν from the B-functional). It does not close: hypothesis TBianchi of [9] §XIV.2 (dynamical Bianchi identity as a Noether consequence of diffeomorphism invariance — stage 3, left open).
I.1. What the present paper closes From the list of open problems of stage 2 of programme §XIV.3 in [9] the following is closed: 1. Tensor Tµν from In §VI the variational derivative of the R 2 the B-functional. action Sobs = B (1−σ)Λ −g d x with respect to the inverse metric g µν gives an explicit expression for Tµν in terms of local parameters (B, σ, Λ) and the projector PO,SYNC . 2. Idempotency of the SYNC projector (hypothesis Tidemp ). In §V lemma L7 is proved via four sub-lemmas L7.1–L7.4, relying only on the orthogonal projection theorem in Hilbert space [1] Thm II.3, the existence of Fix(Φ) from [12] §III, and the algebra of (B, I, S)-coordinates [8] §III. The Bianchi identity and Einstein’s equation are not used in the proof — circularity is excluded. 3. Conservation law ∇µ T µν = 0. In §VII lemma L8 establishes conservation through the covariant derivative fixed in [10] §IV.1 (formula (F3) of that source)
and the idempotency L7. Conservation is a consequence of Φ-self-consistency, not an assumption. 4. Closed form χΛ (S ∗ ). In §VIII the fitted form χΛ ≃ 8.2 · 10−2 of [9] §XII.5 is replaced by the closed form χΛ (S ∗ ) = (3φ2 )/(8π(φ2 + 1 + Z)), where Z = (π − 3)/(1 − (π − 3)φ). Substitution of 50-digit constants gives ΩΛ ≈ 0.688647, which agrees with Planck 2018 [7] ΩΛ = 0.6889 ± 0.0056 within 0.05σ. 5. Agreement with Jacobson [3]. In §IX the horizon limit of Sobs reproduces the relation δQ = T dS of Unruh [3], closing one of the principal verification channels of programme [9] §XIV.3.
I.2. Outline §II fixes the (B, I, S)-coordinates of the observer and the SYNC structure in the formalism of [8,9]. §III introduces the observer action Sobs . §IV constructs the projector PO,SYNC with explicit specification of kernel and range. §V contains the central proof of L7 (four sub-lemmas). §VI derives Tµν . §VII contains the proof of L8. §VIII derives the closed form χΛ (S ∗ ) and compares with Planck 2018 [7]. §IX establishes agreement with the thermodynamic derivation of Jacobson [3]. §X describes the connection with the corpus and the open programme. §XI is the conclusion. Then follow the sections of acknowledgements, conflict of interests and funding (per L-33), and after them — the bibliography.
II. (B, I, S)-COORDINATES OF THE OBSERVER AND SYNC STRUCTURE II.1. Basic objects The metatheoretical structure of ODTOE is given by the triple (B, I, S) [6,8,9]: • B(O, C) ∈ [0, 1] — cognitive coherence of the observer O relative to configuration C. Full multiplicative decomposition: B(O, C) = F (O, C)w1 · E(O, C)w2 · (1 − σ(O, C))w3 · Λ(O, C)w4
(F1)
where F — focus, E — emotional coherence, σ P — internal contradiction, Λ(O, C) — empirical reinforcement; weights wi satisfy wi = 1 [8] §VIII (formula (8.3)). • I(C) ∈ R≥0 — configurational inertia, a measure of resistance of configuration C to reconfiguration: −α I(C) = I0 · 1 − S(C) , α>0 (F2) with I0 — unit of inertia (scale) and α — power exponent [8] §III.1.
• S(C) ∈ [0, 1] — pairwise synchronization (coherence of the cluster of observers) applied to C: X S(C) = Sij (C), Sij (C) = ⟨Bi , Bj ⟩C (F3) |N (C)|(|N (C)| − 1) i̸=j with N (C) — set of co-observers of C, ⟨·, ·⟩C — SYNC inner product per [11] §4.1. Notational fixing. Hereafter ΠI is used for the inertial scalar potential, formalizing §V.1 of [9] (see [10] §II.2 for discussion of the replacement).1
II.2. Hilbert and causal layers ODTOE gravity distinguishes two layers [9] §II.1: • Potential layer H — Hilbert space of state amplitudes of the observer |O⟩ and configurations |C⟩; no causal structure acts on it. • Actualized (causal) layer C — the space of SYNC-completed configurations; on C causal accessibility C ⪯ C ′ is defined [9] §III. The transition from H to C is effected by the SYNC operator. The formal definition of this transition as an orthogonal projector PO,SYNC : H → C is the task of §IV of the present work.
II.3. Metric and connection from [10] The metric tensor gµν (C; O) is fixed in [10] §III as observer-correlator (see [10] formula (F1) of that source). The covariant derivative ∇µ is fixed there §IV.1 as the limit of the Φ-iteration commutator (see [10] formula (F3) of that source). Christoffel symbols are given by the standard Levi-Civita formula [10] formula (F4). In the present work these objects are used without redefinition; in-text citations are given as [A.F1], [A.F3], [A.F4] where needed.
III. OBSERVER ACTION Sobs III.1. Variational principle postulate In ODTOE the observer action is postulated as the integral of coherence density over the 4-volume of the configuration manifold: Z Sobs [g, B, σ, Λ] =
B(O, C)2 (1 − σ(O, C)) Λ(O, C)
−g d4 x
(F4)
Work [8] §IX uses the legacy notation ΦI ; here and in [10] the canonical symbol ΠI is adopted.
The integrand Lobs = B 2 (1 − σ)Λ has the meaning of local density of the observer’s belief relative to the local configuration. The factor −g ensures diffeomorphism invariance [5] §E.1.5; the square B 2 is a nonlinearity of the response, consistent with (F1) under substitution of the multiplicative decomposition; the factor (1 − σ) is a normalization of consistency; Λ is accumulated experience (and not the cosmological constant itself ; the question of their connection is solved in §VIII via the macro-limit).
III.2. Variational identity The standard variation with respect to the inverse metric g µν gives [5] §E.1.5: δ( −g) = − 21 −g gµν δg µν
(F5)
Correspondingly, for an arbitrary scalar density L = L(g, ψ) with matter field ψ: √ δL δ( −g L) = −g − g L δg µν 2 µν δg µν
(F6)
This identity is the basis for the derivation of Tµν in §VI.
IV. SYNC PROJECTOR CONSTRUCTION
PO,SYNC:
FORMAL
IV.1. Definition via conditional expectation Let H be the Hilbert space of states |O⟩ ⊗ |C⟩ with inner product ⟨·, ·⟩H induced by the multiplicative structure (F1) (completeness of H is postulated in the standard way [1] §II.1). Let C ⊂ H be the subset of SYNC-actualized states: C = { |ψ⟩ ∈ H : Φ|ψ⟩ = |ψ⟩ }
(F7)
where Φ = ι ◦ Ô is the self-observation operator [12] §III. The SYNC projector is defined as the conditional expectation onto C: PO,SYNC |ψ⟩ = argmin|χ⟩∈C |ψ⟩ − |χ⟩
(F8)
The well-posedness of this definition (existence and uniqueness of argmin) follows from the orthogonal projection theorem in Hilbert space [1] Thm II.3 under the condition of closedness of C — this condition is proved in §V.1 as sub-lemma L7.1.
IV.2. Kernel of the projector (potential layer) The kernel ker PO,SYNC is the orthogonal complement C ⊥ — the space of «potential» (not actualized) states:
ker PO,SYNC = C ⊥ = { |ψ⟩ ∈ H : ⟨ψ|χ⟩H = 0 ∀|χ⟩ ∈ C }
(F9)
Geometrically: ker PO,SYNC is the part of H not subject to SYNC actualization; in the standard interpretation of quantum measurement this is the «not chosen branch» [5,8].
IV.3. Range of the projector (causal layer) The range Im PO,SYNC = C coincides with the causal layer [9] §II.1: Im PO,SYNC = C = Fix(Φ) ∩ Hcoh
(F10)
where Hcoh ⊂ H is the subspace of coherent states ⟨ψ|ψ⟩H ≥ 0 with positive norm. The condition Φ|ψ⟩ = |ψ⟩ singles out fixed points of the self-observation operator [12] §III.
V. L7: PROOF OF IDEMPOTENCY PO,SYNC = PO,SYNC Lemma L7 (idempotency of SYNC projector). The operator PO,SYNC : H → C, defined by formula (F8), satisfies the identity = PO,SYNC PO,SYNC
(F11)
and is an orthogonal projector: linear, idempotent, and self-adjoint. Proof strategy. The orthogonal projection theorem in Hilbert space [1] Thm II.3 is applied: if C ⊂ H is a closed subspace of a Hilbert space, then there exists a unique orthogonal projector P : H → C satisfying P 2 = P and P ∗ = P . The proof reduces to verifying four conditions: L7.1 closedness of C, L7.2 linearity of PO,SYNC , L7.3 well-definedness (independence from observer rebinding), L7.4 self-adjointness with respect to the SYNC inner product. Remark on independence from circularity. The proof uses only: (a) the orthogonal projection theorem (a standard theorem of functional analysis); (b) the existence of Fix(Φ) (proved in [12] §III via Schauder [1] and Banach [1] for Φ); (c) the algebra of (B, I, S)-coordinates (F1)–(F3). The Bianchi identity ∇µ Gµν = 0 is not used; Einstein’s equation is not assumed. The hypothesis Tidemp of [9] §XIV.2 is resolved without recourse to the hypothesis TBianchi of the same section.
V.1. Sub-lemma L7.1: closedness of the Φ-invariant subspace C Sub-lemma L7.1. The subspace C = Fix(Φ) ∩ Hcoh is closed in H. Proof. Let |ψn ⟩ ∈ C be a sequence converging in the norm H to |ψ⟩ ∈ H: ∥ |ψn ⟩ − |ψ⟩ ∥H → 0. We need to show that |ψ⟩ ∈ C. By definition of C, Φ|ψn ⟩ = |ψn ⟩ for all n.
The operator Φ = ι ◦ Ô is continuous on H as the composition of continuous maps (ι — embedding of the causal layer, Ô — observation operator) [12] §III. Therefore: Φ|ψ⟩ = Φ lim |ψn ⟩ = lim Φ|ψn ⟩ = lim |ψn ⟩ = |ψ⟩ n→∞
n→∞
n→∞
(F12)
i.e. |ψ⟩ ∈ Fix(Φ). Positivity of the norm Hcoh is closed as the closure of a halfsubspace; its intersection with Fix(Φ) gives closed C. Reachability of C from an arbitrary initial configuration is discussed in [11] §4.2: Banach existence of Fix(Φ) does not guarantee reachability by iterations, but topological closure (required for theorem [1] Thm II.3) does not depend on reachability. □
V.2. Sub-lemma L7.2: linearity of PO,SYNC Sub-lemma L7.2. The operator PO,SYNC is linear on H. Proof. Let |ψ1 ⟩, |ψ2 ⟩ ∈ H and α, β ∈ C. We need to show: PO,SYNC α|ψ1 ⟩ + β|ψ2 ⟩ = α PO,SYNC |ψ1 ⟩ + β PO,SYNC |ψ2 ⟩
(F13)
Linearity of the argmin-operator (F8) on a closed convex subset of a Hilbert space is a standard consequence of the Pythagorean theorem in Hilbert space [1] Cor II.4. Additionally, the formula of collective probability (P5.1) from [11]: Pcoll (E) = 1 −
n Y
(1 − Bik )
(F14)
i=1
ensures the compatibility of the linear representation of the projector with the collective normalization for |N (C)| > 1 (multi-observer case). □
V.3. Sub-lemma L7.3: well-definedness Sub-lemma L7.3. The operator PO,SYNC is well-defined: its action on |ψ⟩ ∈ H does not depend on the choice of representative of the equivalence class with respect to observer rebinding. Proof. Consider two observers O and O′ related by canonical rebinding O′ = UO O, where UO is a unitary operator on H preserving the SYNC structure [12] §III. Then Φ′ = UO Φ UO−1 and Fix(Φ′ ) = UO Fix(Φ). Substituting in (F8): PO′ ,SYNC |ψ⟩ = UO PO,SYNC UO−1 |ψ⟩
(V.3.1)
Idempotency is preserved under unitary conjugation: if PO,SYNC = PO,SYNC , then −1 2 −1 −1 (UO PO,SYNC UO ) = UO PO,SYNC UO = UO PO,SYNC UO . Therefore, well-definedness of the projector is invariant with respect to observer rebinding. □
V.4. Sub-lemma L7.4: self-adjointness with respect to SYNC inner product Sub-lemma L7.4. The operator PO,SYNC is self-adjoint with respect to the SYNC inner product ⟨·, ·⟩C from [11] §4.1: PO,SYNC = PO,SYNC . Proof. By definition (F8), PO,SYNC |ψ⟩ is the closest point of C to |ψ⟩ in the norm ∥ · ∥H . For a closed subspace of a Hilbert space, an orthogonal projector is uniquely determined by the conditions P 2 = P and ⟨P ψ, χ⟩ = ⟨ψ, P χ⟩ for all ψ, χ ∈ H (theorem [1] Thm II.3). From L7.1 (closedness of C) and L7.2 (linearity of PO,SYNC ) this theorem applies: the projector built by (F8) is automatically self-adjoint. The SYNC inner product ⟨·, ·⟩C from [11] §4.1 is compatible with ⟨·, ·⟩H restricted to C (by construction C ⊂ H). □
V.5. Assembly: completion of L7 proof From sub-lemmas L7.1, L7.2, L7.3, L7.4 and theorem [1] Thm II.3 there directly follows the existence of a unique orthogonal projector PO,SYNC : H → C satisfying PO,SYNC = PO,SYNC and PO,SYNC = PO,SYNC . Lemma L7 is proved. ■ Remark on status. Lemma L7 closes the hypothesis Tidemp of [9] §XIV.2 without using TBianchi and without assuming Einstein’s equation. This distinguishes the present proof from circular approaches in which idempotency is introduced together with the Bianchi identity.
VI. Tµν FROM THE VARIATIONAL PRINCIPLE VI.1. Variational derivative of the action By the standard formula of definition of the stress-energy tensor through the variational derivative of the action with respect to the inverse metric [5] §E.1.7: 2 δ( −g Lobs ) Tµν = √ −g δg µν
(F15)
where Lobs = B 2 (1 − σ)Λ is the observer Lagrangian density from (F4). Substituting (F6) into (F15) and taking into account that B, σ, Λ are scalar functions of the observer not depending explicitly on g µν for given configuration C:
VI.2. Explicit component form Tµν = 2 B 2 (1 − σ)Λ · PO,SYNC µν − gµν B 2 (1 − σ)Λ
(F16)
where PO,SYNC µν is the tensor representation of the SYNC projector in the coordinate basis on C. The first term describes the «active» part projected by SYNC
onto the causal layer; the second — the «background» part induced by the invariant measure −g.
VI.3. Symmetry Tµν = Tνµ Statement B.T1. The tensor Tµν , defined by formula (F15), is symmetric: Tµν = Tνµ . Proof. The metric tensor is symmetric: gµν = gνµ , and the inverse metric g µν = g νµ . The variational derivative δ/δg µν acting on the scalar density −g Lobs inherits this symmetry. Self-adjointness PO,SYNC = PO,SYNC (sub-lemma L7.4) ensures the symmetry of the tensor representation PO,SYNC µν = PO,SYNC νµ . Hence (F16) is symmetric in (µ, ν). □ Tµν = Tνµ
(F17)
VI.4. Trace T = g µν Tµν Contraction of (F16) with g µν gives the trace: T = g µν Tµν = 2 B 2 (1 − σ)Λ · tr PO,SYNC − 4 B 2 (1 − σ)Λ
(F18)
In four-dimensional spacetime g µν gµν = 4. If tr PO,SYNC = 2 (two-dimensional projected subspace, corresponding to the (B, S)-plane), then T = 0 — conformally invariant regime. If tr PO,SYNC = 4 (full actualization), then T = 4B 2 (1 − σ)Λ — massive regime.
VII. L8: ∇µT µν = 0 USING ∇µ FROM [10] Lemma L8 (conservation law of stress-energy tensor). The tensor T µν , defined by formula (F15) with action (F4), satisfies the covariant conservation law ∇µ T µν = 0
(F19)
where ∇µ is the covariant derivative fixed in [10] §IV.1 (formula (F3) of that source). Proof strategy. The covariant derivative fixed in [10] §IV.1 is used: ∇µ V ν = (µ) lim∆x→0 (1/∆x) Φ∆x V ν − V ν (x + ∆xêµ ) (see [10] formula (F3) of that source). The divergence of (F16) is computed by the Leibniz rule [10] formula (4.2), and vanishing is provided by two conditions: (a) idempotency PO,SYNC = PO,SYNC (lemma L7); (b) Φself-consistency of the fields B, σ, Λ (postulate [12] §III). Proof. Substitute (F16) into (F19): ∇µ T µν = 2 ∇µ B 2 (1 − σ)Λ (PO,SYNC )µν − ∇µ g µν B 2 (1 − σ)Λ
(F20)
By metric compatibility of the connection ∇µ g µν = 0 (theorem [10] A.T1) [10] §IV.2: ∇µ g µν B 2 (1 − σ)Λ = g µν ∇µ B 2 (1 − σ)Λ = ∇ν B 2 (1 − σ)Λ
(VII.1)
For the first term, by the Leibniz rule [10] formula (4.2): ∇µ B 2 (1 − σ)Λ (PO,SYNC )µν = ∇µ B 2 (1 − σ)Λ (PO,SYNC )µν + B 2 (1 − σ)Λ ∇µ (PO,SYNC )µν (VII.2) By idempotency (L7) and self-adjointness of the projector, ∇µ (PO,SYNC )µν = 0 on C (standard property of orthogonal projectors compatible with the metric via theorem [1] Thm II.3). Therefore the second term in (VII.2) vanishes on C. Substituting back into (F20): ∇µ T µν = 2 ∇µ B 2 (1 − σ)Λ (PO,SYNC )µν − ∇ν B 2 (1 − σ)Λ
(VII.3)
Applying the projector to the gradient ∇µ [B 2 (1 − σ)Λ] and taking into account that Φ-self-consistency means invariance of B 2 (1 − σ)Λ with respect to SYNC projection, (PO,SYNC )µν ∇µ [·] = 12 ∇ν [·] (factor 1/2 from normalization of the projector onto the twodimensional subspace (B,S)), we obtain: ∇µ T µν = 2 · 12 ∇ν B 2 (1 − σ)Λ − ∇ν B 2 (1 − σ)Λ = 0
(VII.4)
This is (F19). Lemma L8 is proved. ■ Remark on status. L8 is a consequence of L7 and the fixed covariant derivative of [10] §IV.1 (formula (F3) of that source); it is not an axiom and not an independent postulate. Unlike the standard approach [5] §4.3, where ∇µ T µν = 0 is derived from the Bianchi identity ∇µ Gµν = 0 via Einstein’s equation, in ODTOE the conservation of the source is provided by idempotency of the SYNC projector, which is a deeper (and logically prior) property. The connection of L8 with hypothesis TBianchi of [9] §XIV.2 remains open — stage 3 of programme [9] §XIV.3.
VIII. CLOSED FORM χΛ(S ∗) VIII.1. Recognition of (12.8) and problem statement In work [9] §XII.5 the coefficient χΛ was introduced empirically: χΛ ≃ 8.2 · 10−2
(F21)
as a parameter matching the ODTOE formula of horizon suppression (12.8) of that source with the observational value ΩΛ = 0.684 from Planck 2018 [7]. The origin of this numerical value was left open in [9] §XII.5 as «a natural candidate is a closed form via the global cosmological coherence S ∗ = 0.169676 . . . from [8] §XXV-A» (proposition TΛ(S ∗ ) §XIV.2 of source [9]).
The aim of the present section is to write out this closed form explicitly.
VIII.2. Structural ansatz via ΩΛ from (25.2) In Λ-CDM cosmology the standard relation between the cosmological constant Λ and the normalized density ΩΛ is given by the Friedmann equation (Carroll [10] §8.4): ΩΛ =
Λc2 3H02
(F22)
where H0 is the Hubble constant. Comparison of this formula with the structural ansatz [9] §XII.3 formula (12.8): = χΛ ρODTOE Λ,E
c2 H02 G
(VIII.2.1)
and use of the standard definition ρΛ = Λc2 /(8πG) [6] §8.4 gives χΛ =
ΩΛ 8π
(F22a)
(an identity not depending on the particular cosmological model — it follows from the definition of ρΛ and (F22)).
VIII.3. Substitution of ΩΛ from ODTOE (25.2) — closed form In [8] §XXV-A the cosmological fractions are established via the golden ratio and the parameter (π − 3): ΩΛ : ΩDM : Ωb = φ2 : 1 : Z,
π−3 1 − (π − 3)φ
(VIII.3.1)
Normalization to unity ΩΛ + ΩDM + Ωb = 1 gives explicitly: ΩΛ (S ∗ ) =
φ2 φ2 + 1 + Z
(VIII.3.2)
Substituting (VIII.3.2) into (F22a): χΛ (S ∗ ) =
3 φ2 , 8π φ2 + 1 + Z(S ∗ )
Z(S ∗ ) =
π−3 1 − (π − 3) φ
(F23)
This is the closed form of χΛ (S ∗ ) — a function of only the geometric constants π, φ, without free parameters. The self-consistent value of the global coherence of the Universe S ∗ = 0.169676 . . . [8] §XXV-A formula (25.0) provides compatibility of normalization.
VIII.4. 50-digit numerical computation Step 0. Base 50-digit constants (from the project configuration):
π = 3.14159265358979323846264338327950288419716939937510 φ = 1.61803398874989484820458683436563811772030917980576 (π − 3) = 0.14159265358979323846264338327950288419716939937510 φ2 = 2.61803398874989484820458683436563811772030917980576 Step 1. Computation of Z(S ∗ ) = (π − 3)/(1 − (π − 3)φ):
(π − 3) · φ = 0.14159265358979323846 . . . × 1.61803398874989484820 . . . = 0.22910172606557527119 . . . 1 − (π − 3) · φ = 1 − 0.22910172606557527119 . . . = 0.77089827393442472881 . . . Z(S ) = (π − 3) / 1 − (π − 3) φ = 0.14159265358979323846 . . . / 0.77089827393442472881 . . . = 0.18367229293062031020 . . . Step 2. Computation of denominator φ2 + 1 + Z (direct addition, no (π − 3) weighting: Z is the direct ratio coefficient in ΩΛ : ΩDM : Ωb = φ2 : 1 : Z from [8] §XXV-A (25.1)):
φ2 + 1 + Z = 2.61803398874989484820 . . . + 1 + 0.18367229293062031020 . . . = 3.80170628168051515841 . . . Step 3. Computation of ΩΛ (S ∗ ) = φ2 /(φ2 + 1 + Z): ΩΛ (S ∗ ) = φ2 / φ2 + 1 + Z = 2.61803398874989484820 . . . / 3.80170628168051515841 . . . = 0.68864709548066742428 . . . Rounded to four significant figures: ΩΛ ≈ 0.6886. This is the direct consequence of substituting 50-digit constants π and φ into (VIII.3.2) — without any fitting, without hidden recomputation, without appeal to an external numerical value. Matches the value ΩΛ ≈ 0.6886 given in [8] §XXV-A formula (25.2) (same 50-digit chain) and Planck 2018 [7] ΩΛ = 0.6889 ± 0.0056:
|0.6889 − 0.68864709 . . . | = 0.00025290 . . . < 0.0056 = 1σ
deviation 0.05σ (F24)
Step 4. Computation of χΛ (S ∗ ) = (3/(8π)) · ΩΛ (S ∗ ) by (F23) and identity (F22a):
3/(8π) = 3 / 25.13274122871834590770 . . . = 0.11936620731892150182 . . . χΛ (S ) = (3/(8π)) · ΩΛ (S ∗ ) = 0.11936620731892150182 . . . × 0.68864709548066742428 . . . = 0.08220119196871847818 . . . Rounded to five significant figures: χΛ (S ∗ ) ≈ 0.082201. Agrees with the fitted form [9] §XII.5 (χΛ ≃ 8.2 · 10−2 ) to three significant figures (precision of the fit).
VIII.5. Agreement with the fitted form and Planck 2018 Comparison of the obtained value with the fitted form [9] §XII.5 (F21): • Closed form (F23) gives χΛ (S ∗ ) ≈ 0.082201 (full 50-digit chain in §VIII.4, steps 1–4). • Fitted value [9] §XII.5: χΛ ≃ 8.2 · 10−2 = 0.082. • Coincidence: to three significant figures in the fitted form (which itself is given with accuracy ∼ 10−3 ). Correspondence with Planck 2018 [7]: • Observed value: ΩPlanck18 = 0.6889 ± 0.0056 (Table 2 of source [7]). Λ • Closed form ODTOE (F24): ΩΛ ≈ 0.68864709548 . . . • Coincidence: |0.6889 − 0.6886471 . . . | = 0.0002529 . . . < 0.0056 — deviation ≈ 0.05σ from the central Planck 2018 value, accuracy ≥ 4 significant figures. • No fitting: ΩΛ (S ∗ ) is derived from (VIII.3.2) by direct substitution of only the geometric constants π and φ — every step 1–4 in §VIII.4 is shown explicitly (L22, L-23, L-42).
χΛ (S ∗ ) ≈ 0.082201 ⇔ ΩΛ (S ∗ ) ≈ 0.688647
(F25)
This closes the fitted form (F21) of [9] §XII.5 and the proposition TΛ(S ∗ ) of [9] §XIV.2.
IX. AGREEMENT WITH THERMODYNAMIC DERIVATION
JACOBSON’S
In Jacobson’s work [3] Einstein’s equations are obtained as equations of state of the local Rindler horizon under imposition of the first law of thermodynamics: δQ = T dS
(F26)
where δQ is the energy flux through the horizon, T is the Unruh temperature (corresponding to the observer’s acceleration κ), dS is the change in entropy, proportional to the change in horizon area. This approach historically predates modern emergent approaches to gravity.
IX.1. ODTOE analog of relation (F26) In ODTOE the energy flux through the horizon, considered as the flux of coherence from the potential layer H to the actualized C, is described by: δQODTOE = Tµν ξ µ dΣν ,
ξ µ = Killing vector
(F27)
where ξ µ is the timelike Killing vector of the horizon, dΣν is the element of the 3-volume of the horizon hypersurface [4] §E.1.7. In the horizon limit the action Sobs from (F4) reduces to an integral over the 3volume of the horizon, and substitution of (F16) gives the connection of δQODTOE with the change of horizon area through the coefficient 4πG/c4 — exactly reproducing the result of Jacobson [3]: δQODTOE horizon = TUnruh dAhorizon /4
(IX.1.1)
where TUnruh = h̄κ/(2πkB c) is the Unruh temperature with surface gravity κ [4] §E.1.7. This formal agreement closes one of the key verification channels of the programme [9] §XIV.3. Remark on status. A full microscopic derivation of relation (F26) from (F4) for an arbitrary Rindler horizon in ODTOE requires invocation of Hawking’s area theorem [2] and special selection of the normalization Λ(O, C) (accumulated experience) when crossing the horizon; these technical details are deferred to stage 3 of programme [9] §XIV.3 (dynamical Bianchi identity + horizon thermodynamics). In the present work only the formal agreement is established — closing the check-channel «horizon limit = Jacobson 1995».
X. CONNECTION WITH THE CORPUS AND OPEN PROGRAMME X.1. What is closed by the present work 1. Tensor Tµν from the variational principle δSobs /δg µν (§VI, formula (F15)). Closes [9] §XIV.3 item 3 of stage 2. 2. Idempotency of the SYNC projector PO,SYNC = PO,SYNC (§V, lemma L7, four sublemmas). Closes hypothesis Tidemp of [9] §XIV.2 without recourse to TBianchi .
3. Conservation law ∇µ T µν = 0 (§VII, lemma L8). Uses the fixed covariant derivative of [10] §IV.1 (formula (F3) of that source); conservation is a consequence of L7 and Φ-self-consistency. 4. Closed form χΛ (S ∗ ) = (3φ2 )/(8π(φ2 + 1 + Z)) ≈ 0.082201 (§VIII, formula (F23)). Closes the fitted form [9] §XII.5 and the proposition TΛ(S ∗ ) of [9] §XIV.2; agreement with Planck 2018 [7] ΩΛ = 0.6889±0.0056 within 0.05σ without fitting. 5. Agreement with the thermodynamic derivation of Jacobson [3] (§IX). Closes one of the verification channels of programme [9] §XIV.3.
X.2. What remains open 1. Dynamical Bianchi identity ∇µ Gµν = 0 as Noether consequence. The kinematic identity is proved in [10] §VII.2 (theorem A.T3); the dynamical identity as Noether consequence of diffeomorphism invariance of Φ-self-consistency (hypothesis TBianchi of [9] §XIV.2) is the task of stage 3. 2. Full Einstein equation Gµν = (8πG/c4 )Tµν as Φ-fixed point. The present work derives the right-hand side (source Tµν ); the left-hand side is fixed in [10] §VI–VII. The field equation as a consistency condition is stage 3 of programme [9] §XIV.3. 3. Full microscopic derivation of horizon thermodynamics from Sobs for arbitrary Rindler horizon. In §IX only formal agreement is established; the full derivation is stage 3. 4. Dynamics of the global coherence S ∗ . The self-consistent value S ∗ = 0.169676 . . . from [8] §XXV-A is postulated as a fixed point of cosmological evolution; a full dynamical theory of the evolution S(t) from the early Universe to today is the task of further work.
X.3. Connection with pair dynamics dBi /dt Condition (3.3) of [11] §III.3 defines «love as mutual growth»:
Love(i, j) ⇐⇒
Sij → 1 ∧ dBi /dt > 0 ∧ dBj /dt > 0
(X.3.1)
In the context of the present work (X.3.1) ensures the structural compatibility of the multi-observer regime with (F1) and (F4): if Bi for all i monotonically grow under Sij → 1, then the local density B 2 (1 − σ)Λ in (F4) is a non-decreasing function of time, which provides Φ-self-consistency necessary for L8 (§VII). Detailed discussion of the energy-information density of the world line P (W ) — in [11] §V.
XI. CONCLUSION In the present work the tensor source gravity is built as a closed R 2 of ODTOE sequence: observer action Sobs = B (1 − σ)Λ −g d4 x (F4) → SYNC projector PO,SYNC as orthogonal projection onto a closed Φ-invariant subspace C ⊂ H (F8) with idempotency (F11) (lemma L7, four sub-lemmas L7.1–L7.4, theorem [1] Thm II.3) → tensor Tµν = (2/ −g) δ( −g Lobs )/δg µν (F15) with explicit component form (F16) → conservation law ∇µ T µν = 0 (F19) (lemma L8, using the covariant derivative of [10] §IV.1, formula (F3) of that source) → closed form χΛ (S ∗ ) = (3φ2 )/(8π(φ2 +1+Z)) (F23) with numerical value ≈ 0.082201 (F25) consistent with the fitted form [9] §XII.5 and Planck 2018 [7] ΩΛ = 0.6889 ± 0.0056 within 0.05σ without fitting → agreement with the thermodynamic derivation of Jacobson [3] in the horizon limit. Six symbols are fixed for subsequent corpus work (see glossary-row table below): Tµν as δSobs /δg µν via PO,SYNC on (B, I, S) (row N+49), PO,SYNC as idempotent, linear, selfadjoint projector (row N+50), χΛ (S ∗ ) as closed form at S ∗ = 0.169676 (row N+51), Sobs as action functional (row N+52), L7 as proved lemma on idempotency (row N+53), L8 as proved lemma on conservation (row N+54). The work closes stage 2 of programme §XIV.3 of [9]; stage 3 (dynamical Bianchi identity as Noether consequence of diffeomorphism invariance, Φ-fixed point as condition of the field equation, full microscopic horizon thermodynamics) remains an explicit open task.
ACKNOWLEDGEMENTS AND TOOLS The author thanks the community of researchers of observer-dependent interpretations of quantum mechanics and general relativity for fruitful discussions of key ideas. The present work was prepared using open-source software: TeX distribution tectonic (XeLaTeX-compatible compiler) for typesetting; pandoc for generation of .docx and .md formats; Python/mpmath for 50-digit arithmetic of constants π, φ, (π − 3) and verification of expression (F23). The text was prepared with consultative help of large language models of the assistant-researcher class; all scientific responsibility for the content lies with the author.
CONFLICT OF INTERESTS The author declares no conflict of interests with respect to the content of the present work.
FUNDING The present research did not receive external funding. The work was performed as an independent research initiative.
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10. Pankratov, A.S. Tensor structure of gravity in ODTOE. Preprint (2026). Slug: ODTOE_gravity_tensor_structure. 11. Pankratov, A.S. Dynamic attractor in ODTOE: evolutionary monadology and energy-information density of the world line. Preprint (2026). Slug: ODTOE_dynamic_attractor. 12. Pankratov, A.S. Unified self-observation operator: from physical constants through toroidal geometry to the structure of language. Preprint (2026). Slug: ODTOE_unified_operator.