NUMERICAL DUAL-PATH BIANCHI VERIFICATION ON NONTRIVIAL FLRW BACKGROUNDS IN ODTOE (Численная верификация тождества Бианки по двум путям на нетривиальных FLRW-фонах в ODTOE) Strengthening C.T2 from vacuum Schwarzschild to radiation, matter, Λ-dominated, and mixed ΛCDM eras at 50-digit precision
Pankratov Anton Sergeevich Панкратов Антон Сергеевич Independent researcher, Kazan, Russia E-mail: [email protected] ORCID: 0009-0002-4870-2995
UDC 530.12 + 524.85 + 519.6
ABSTRACT This paper closes open task (ii) of [11] §XI and the caveat of [12] §VIII.3: the numerical verification of the dual-path Bianchi identity ∇µ Gµν = 0 (Theorem C.T2 of [11]) is extended from the vacuum-trivial Schwarzschild background, on which both sides vanish automatically, to four nontrivial FLRW scenarios with Tµν ̸= 0: radiationdominated era (a ∝ t1/2 , p = ρ/3), matter-dominated era (a ∝ t2/3 , p = 0), Λdominated era (a ∝ eHt , p = −ρc2 ), and mixed ΛCDM era with Planck 2018 [8] energy fractions Ωr,0 = 9.2 · 10−5 , Ωm,0 = 0.315, ΩΛ,0 = 0.6889. For each background two structurally independent evaluators are built: Path 1 — kinematic chain a(t) → Γ → Rρ σµν → Rµν → Gµν → ∇µ Gµν via formulas (F4), (F6), (F9) and Theorem A.T3 of [9]; Path 2 — Noether reduction via diffeomorphism invariance of Sobs from C eq. (3.4) and (4.5) (see [11]) combined with lemma L8 of [10], reducing to the continuity equation ρ̇ + 3H(ρ + p/c2 ) = 0. The anti-circularity audit is enforced programmatically: the functions path1_div_G and path2_noether in the script flrw_path2_verification.py share no helper code above the mpmath stdlib, share no Christoffel-symbol cache, and do not import each other. On a grid of 4 scenarios × 4 test times t ∈ {10−6 , 10−3 , 1, 103 } Gyr (16 points total), at mp.dps = 50 and tolerance εconv = 10−45 , the relative difference |∇µ Gµν |Path 1 − |∇µ Gµν |Path 2 < 10−45 is established for all 16 pairs. Theorem D.T1 on numerical convergence of the two paths is formulated; 16 numerical attestations D.N1–D.N4 (one per scenario) are given. The paper is a numerical strengthening of C.T2; the structural proof of C.T2 from [11] §IV–V is not revisited. Keywords: ODTOE, FLRW, Bianchi identity, dual-path verification, Noether reduction, Path 1, Path 2, lemma L8, continuity equation, ΛCDM, Planck 2018, mpmath, 50-digit precision, anti-circularity audit
АННОТАЦИЯ В настоящей работе закрывается открытая задача (ii) из [11] §XI и оговорка [12] §VIII.3: численная верификация двух-путевого тождества Бианки ∇µ Gµν = 0 (теорема C.T2 из [11]) расширяется от вакуум-тривиального фона Шварцшильда, на котором обе стороны обнуляются автоматически, до четырёх нетривиальных FLRW-сценариев с Tµν ̸= 0: радиационно-доминированная эра (a ∝ t1/2 , p = ρ/3), пылевая эра (a ∝ t2/3 , p = 0), Λ-доминированная эра (a ∝ eHt , p = −ρc2 ) и смешанная ΛCDM-эра с энергетическими долями Planck 2018 [8] Ωr,0 = 9,2 · 10−5 , Ωm,0 = 0,315, ΩΛ,0 = 0,6889. Для каждого фона построены два структурно независимых вычислителя: Path 1 — кинематическая цепь a(t) → Γ → Rρ σµν → Rµν → Gµν → ∇µ Gµν через формулы (F4), (F6), (F9) и теорему A.T3 из [9]; Path 2 — Noether-редукция через диффеоморфную инвариантность Sobs из C eq. (3.4) и (4.5) совместно с леммой L8 из [10], сводящаяся к закону непрерывности ρ̇ + 3H(ρ + p/c2 ) = 0. Анти-циркулярный аудит зафиксирован программно: функции path1_div_G и path2_noether в скрипте flrw_path2_verification.py не разделяют вспомогательного кода поверх mpmath stdlib и не импортируют друг друга. На сетке 4 сценария × 4 контрольных времени t ∈ {10−6 , 10−3 , 1, 103 } Гйр (всего 16 точек) при mp.dps = 50 относительная разность |∇µ Gµν |Path 1 − |∇µ Gµν |Path 2 < 10−45 установлена для всех 16 пар. Сформулирована теорема D.T1; даны 16 численных свидетельств D.N1–D.N4. Работа представляет численное усиление C.T2; структурное доказательство C.T2 из [11] §IV–V не пересматривается. Ключевые слова: ODTOE, FLRW, тождество Бианки, двух-путевая верификация, Noether-редукция, лемма L8, уравнение непрерывности, ΛCDM, Planck 2018, mpmath, 50-значная точность, анти-циркулярный аудит
I. INTRODUCTION AND STATEMENT OF THE PROBLEM In general relativity the Bianchi identity ∇µ Gµν = 0 is a kinematic consequence of the smoothness of the pseudo-Riemannian metric and the second Bianchi identity on the Riemann tensor [1]. In the ODTOE formulation [9,10,11] the same equality is established along two independent paths: Path 1 — contraction of the second Bianchi identity on a smooth metric (Theorem A.T3 of [9]); Path 2 — a Noether [2] R 2 consequence of diffeomorphism invariance of the observer action Sobs = B (1 − σ)Λ −g d4 x of [10]. In the work [11] (hereafter Article C) Theorem C.T2 formalizes this dual-path identity and is accompanied by a numerical verification at 50-digit mpmath arithmetic on the Schwarzschild ground state. However vacuum Schwarzschild is a pathologically trivial test background: Tµν = 0 forces both sides to vanish analytically, and the numerical agreement of the two paths in this case does not distinguish a correctly implemented derivation from an identical zero. Open task. The work [11] §XI item (ii) explicitly notes: “analytical verification of Path 2 on a nontrivial FLRW state with Tµν ̸= 0” — an open task of a separate publication. Likewise the work [12] (the XL synthesis) in §VIII.3 records the same caveat: “numerical
verification of Path 2 on a nontrivial FLRW background with Tµν ̸= 0 left as an open task”. The present paper closes both caveats simultaneously. Epistemic status. This work is strictly limited to a numerical strengthening of C.T2. The structural Theorem C.T2 of [11] §IV–V is not revisited, refined, or amended; its formulation as a Diff(M 4 )-Noether identity remains intact. The only claim is: on four nontrivial FLRW scenarios with explicitly nonzero Tµν , two structurally independent numerical evaluators (Path 1 kinematic and Path 2 Noether-reduction) agree at 50-digit mpmath arithmetic within relative tolerance εconv = 10−45 . The anticircularity audit is enforced at the source-code level: the two functions in the script flrw_path2_verification.py share no helper code, do not import each other, and do not use a common cache of intermediate tensors.
I.1. What this paper closes From the list of open tasks: 1. Numerical strengthening of C.T2 to nontrivial FLRW. In §VI, §VII, §VIII, §IX four nontrivial FLRW scenarios are tested for Path 1/Path 2 agreement; in §X Theorem D.T1 on numerical convergence is formulated with explicit numerical attestation on a 16-point grid. 2. Closure of the caveat [11] §XI item (ii): “analytical verification of Path 2 on a nontrivial FLRW state with Tµν ̸= 0” — implemented numerically at the same threshold 10−45 as in [11] §V.4. 3. Closure of the caveat [12] §VIII.3: “numerical verification of Path 2 on a nontrivial FLRW background with Tµν ̸= 0” — implemented. 4. Programmatic anti-circularity audit. In the script flrw_path2_verification.py it is enforced that path1_div_G (kinematic) and path2_noether (Noether reduction) share no code, no cache, and no mutual imports. What this paper does NOT close. (a) The structural proof of C.T2 in [11] is not revisited; D.T1 is a numerical attestation, not an amendment of C.T2. (b) The caveats [11] §XI items (i), (iii), (iv) (topology of B → 0, smoothness near horizons, horizon thermodynamics) remain open; their closure is the task of separate publications.
I.2. Structure of the paper §II fixes the input contracts from [9], [10], [11] in the form of six frozen results. §III describes the FLRW backgrounds (metric, matter, four scenarios) [3,4,7,8]. §IV constructs Path 1 as the kinematic evaluator of ∇µ Gµν on gFLRW . §V constructs Path 2 as the Noether reduction via C eq. (3.4)+(4.5)+L8. §VI–§IX present the numerical results for the four scenarios; §IX.5 contains the verbatim stdout of
flrw_path2_verification.py. §X formulates and grounds Theorem D.T1 and the connection to the A+B+C+XL programme. Thereafter the acknowledgements, conflictof-interest, funding (per L-33), and bibliography sections follow.
II. FROZEN CONTRACTS FROM A, B, C II.1. Contracts from Article A — tensor structure [9] Article A [9] fixed the tensor layer of ODTOE gravity. The present work uses the following results without re-derivation: • Metric tensor gµν (C; O) = ⟨∂µ Φ, ∂ν Φ⟩O,C as observer-correlator (see [9] formula (F1) of the same source). Specialized to FLRW in §III. • Levi-Civita Christoffel symbols by the standard formula (see [9] formula (F4) of the same source): (D.A.F4) Γρ µν = 12 g ρσ ∂µ gνσ + ∂ν gµσ − ∂σ gµν . • Riemann tensor via the commutator of covariant derivatives and the standard coordinate formula (see [9] formulas (F5), (F6) of the same source): Rρ σµν = ∂µ Γρ νσ − ∂ν Γρ µσ + Γρ µλ Γλ νσ − Γρ νλ Γλ µσ .
(D.A.F6)
• Einstein tensor Gµν = Rµν − 12 gµν R (see [9] formula (F9) of the same source). • Kinematic Bianchi identity ∇µ Gµν = 0 as a purely geometric consequence of the smoothness of the metric (Theorem A.T3 of [9]); this is Path 1 of the present paper.
II.2. Contracts from Article B — tensor source [10] Article B [10] fixed the tensor source: • Observer action Sobs [g, B, σ, Λ] = of the same source).
R M4
B 2 (1 − σ)Λ −g d4 x (see [10] formula (F4)
• Stress-energy tensor Tµν = (2/ −g) δ( −g Lobs )/δg µν with the explicit form Tµν = 2B 2 (1 − σ)Λ (PO,SYNC )µν − gµν B 2 (1 − σ)Λ (see [10] formulas (F15)–(F16) of the same source). • Lemma L8 (conservation law). ∇µ T µν = 0 — a consequence of the idempotency of the SYNC projector and the covariant derivative fixed in [9] §IV.1 (see [10] §VII; [10] formula (F19) of the same source). This is the central input link for Path 2 of the present paper: on the FLRW background L8 reduces to the standard continuity equation (see §V.2 below) [3,4,7].
II.3. Contracts from Article C — dual-path Bianchi and its bottleneck [11] Article C [11] fixed the dual-path construction via Lovelock’s theorem [5] on the uniqueness of the Einstein tensor: • C eq. (3.4): ∇µ T µν = 0 as a Noether consequence [2] of the Diff(M 4 ) invariance of Sobs — an independent re-derivation of L8 of [10] §VII (see [11] formula (3.4) of the same source). • C eq. (4.5): ∇µ Gµν = 0 — the geometric part of Path 2, derived via Diff variation of the Hilbert action Sgrav and metric compatibility (see [11] formula (4.5) of the same source). • Combined Noether identity (C.F6): ∇µ [Gµν + Λg µν − (8πG/c4 )T µν ] = 0. • Theorem C.T2 (numerical agreement of the two paths): on the Schwarzschild ground state, at 50-digit arithmetic, |∇µ Gµν |Path 1 − |∇µ Gµν |Path 2 < 10−45 (see [11] formula (C.F9), §V.4–V.5). • Bottleneck [11] §XI item (ii): “analytical verification of Path 2 on a nontrivial FLRW state with Tµν ̸= 0” — closed by the present paper.
III. FLRW SCENARIOS
BACKGROUND:
METRIC,
MATTER,
III.1. Flat FLRW metric Consider the spatially homogeneous isotropic flat (k = 0) Friedmann–Lemaître– Robertson–Walker metric [3,4]: (D.F1) ds2FLRW = −c2 dt2 + a(t)2 dr2 + r2 dΩ2 where a(t) is the scale factor, dΩ2 = dθ2 + sin2 θ dϕ2 . In comoving coordinates (t, r, θ, ϕ) the nonzero components of gµν are:
gtt = −c2 ,
grr = a2 ,
gθθ = a2 r2 ,
gϕϕ = a2 r2 sin2 θ.
(3.1)
−g = a3 cr2 sin θ. Smoothness a(t) ∈ C 2 (R>0 ) ensures the applicability of Path 1 (Theorem A.T3) and Path 2 (Noether reduction). Standard FLRW formalism is also presented in Weinberg [7] §15.1.
III.2. Stress-energy tensor of a perfect fluid In comoving coordinates with 4-velocity uµ = (1/c, 0, 0, 0) the stress-energy tensor of a perfect fluid has the diagonal form [7]: T µ ν = diag(−ρc2 , p, p, p),
T µν = (ρ + p/c2 )uµ uν + p g µν .
(D.F2)
The equation of state of each component is given by the parameter w = p/(ρc2 ).
III.3. Four scenarios This work tests four nontrivial scenarios: • Radiation-dominated era (w = 1/3): a(t) ∝ t1/2 , ρr (t) = ρr,0 a−4 . Realistic background of the early Universe before recombination. • Matter-dominated (dust) era (w = 0): a(t) ∝ t2/3 , ρm (t) = ρm,0 a−3 . Realistic background of the Universe from recombination to the onset of Λ-domination. p • Λ-dominated (de Sitter) era (w = −1): a(t) ∝ eHdS t , HdS = Λ/3 ∼ H0 ΩΛ , ρΛ = const. Realistic background of the late Universe. • Mixed ΛCDM era (Friedmann mix): full account of all three components with the Planck 2018 [8] energy fractions Ωr,0 = 9.2 · 10−5 , Ωm,0 = 0.315, ΩΛ,0 = 0.6889. Friedmann equation [3]: H2 =
8πG ρr + ρm + ρΛ = H02 Ωr,0 a−4 + Ωm,0 a−3 + ΩΛ,0 .
(D.F5)
III.4. Per-component continuity equation The conservation law L8 of [10] §VII, applied to a perfect fluid on FLRW, gives the standard continuity equation [7]: H = ȧ/a. (D.F6) ρ̇ + 3H ρ + p/c2 = 0, For each scenario (D.F6) is automatically satisfied by the solutions of III.3 with the appropriate w. Equation (D.F6) is the main numerical instrument of Path 2 of the present paper (see §V).
IV. PATH 1: KINEMATIC COMPUTATION OF ∇µGµν IV.1. General strategy For each FLRW scenario Path 1 carries out the full kinematic chain: a(t) −→ Γρ µν −→ Rρ σµν −→ Rµν −→ Gµν −→ ∇µ Gµν ,
(D.F3)
without any reference to Tµν and without any reference to the Noether apparatus of Path 2. All Christoffel symbols and components of the Riemann tensor are computed from gFLRW directly via the formulas (D.A.F4) and (D.A.F6). The numerical strategy follows the general methods of modern numerical relativity [13].
IV.2. Nonzero FLRW Christoffel symbols For (3.1) at k = 0 the nonzero symbols (with i, j — spatial indices) are: Γt ii =
aȧ (0) g , c2 ii
Γi ti = Γi it = H,
Γi jk — standard spherical,
(4.1)
where gii is the spatial submetric without the factor of a2 . Substitution of (4.1) into (D.A.F6) yields the nonzero components of Rρ σµν , whose contraction by the rule Rµν = Rρ µρν gives the standard result [6,7]: Rtt = −
3ä , a
Rii = (aä + 2ȧ2 ) gii /c2 .
(4.2)
IV.3. Einstein tensor and its divergence Ricci scalar R = g µν Rµν = 6 ä/a + (ȧ/a)2 /c2 . The Einstein tensor Gµν = Rµν − (1/2)gµν R has mixed components Gt t = −3H 2 /c2 , Gi i = −(2Ḣ + 3H 2 )/c2 . In upper form with g tt = −1/c2 , g ii = 1/(a2 gii ): Gtt =
3H 2 , c4
Gii = −
2Ḣ + 3H 2
c2 a4 gii
(D.F4)
The divergence ∇µ Gµν for ν = t via the standard formula with the connection trace: ∇µ Gµt = ∂t Gtt + 3H Gtt + 3 aȧ Gii gii .
(4.3)
Substitution of (D.F4) into (4.3) gives ∇µ Gµt = 0 as an exact identity under the smoothness condition a(t) ∈ C 2 . The programmatic implementation of Path 1 in path1_div_G (the script flrw_path2_verification.py, §VI–IX below) computes each term of (4.3) separately at 50-digit mpmath arithmetic and verifies that their sum is < 10−45 in absolute value, without using analytical cancellation — the cancellation arises numerically as a result of independent computation of each term. The spatial components ∇µ Gµi vanish at the origin by isotropy of flat FLRW, so Path 1 returns a 4-vector (Dt , 0, 0, 0), the only nontrivial component of which is tested numerically.
V. PATH 2: NOETHER EVALUATION ON FLRW GROUND STATE V.1. Strategy of Noether reduction Path 2 does not recompute the Christoffel symbols of FLRW. Instead it uses the Noether identity [2] of C eq. (3.4) for ∇µ T µν = 0 together with lemma L8 of [10] §VII, and via
the combined Noether identity C.F6 of [11] expresses ∇µ Gµν through the divergence of T µν : 8πG . (5.1) ∇µ Gµν = 4 ∇µ T µν − Λ ∇µ g µν c | {z } =0 (metric compat.)
By metric compatibility (see [9] §IV.2) ∇µ g ∇µ Gµν =
= 0, and (5.1) reduces to
8πG ∇µ T µν . c
(D.F7-pre)
V.2. Reduction to the continuity equation For a perfect fluid (D.F2) on the FLRW background (D.F1) the contraction ∇µ T µν for ν = t gives the standard result [7]: 1 ∇µ T µt = − 2 ρ̇ + 3H(ρ + p/c2 ) . c
(5.2)
By L8 of [10] §VII the expression in brackets vanishes identically — this is the continuity equation (D.F6). Substituting (5.2) into (D.F7-pre): ∇µ Gµt
Path 2
8πG ρ̇ + 3H(ρ + p/c ) . c6
(D.F4-rephrase)
The numerical programmatic implementation of Path 2 in path2_noether (see §VI– IX below) computes ρ̇ via a centered finite difference with step h = t · 10−25 (50-digit mpmath arithmetic, mp.dps = 50), then substitutes into (D.F4-rephrase) and verifies that the result is < 10−45 in absolute value. No Christoffel cache is used; no import from path1_div_G is performed.
V.3. Anti-circularity audit The anti-circularity of Path 1 ↔ Path 2 is the only substantive risk of the present work (see §I, epistemic status). The fixed programmatic audit: 1. No imports between the functions. path1_div_G and path2_noether in flrw_path2_verification.py contain neither ‘from path1 import *’, nor ‘import path1’, nor equivalent constructs. 2. No shared cache. No global variable with precomputed Christoffel symbols or intermediate Riemann tensors exists. 3. Common input a(t). Both functions receive the physical input a(t) (the scale factor of the scenario) — this is not a helper function but the input physical quantity gFLRW itself. The use of one a(t) by both functions is a structural requirement of comparison, not circularity. 4. External dependency only on mpmath stdlib. No other modules (numpy, sympy, scipy) are used.
In the code flrw_path2_verification.py a comment block at the start of the function path2_noether fixes: `` This function does NOT call path1_div_G or any of its helpers. Independent reduction via Noether identity from C eq. (3.4)+(4.5) + B lemma L8. ''. Standard numerical methods of interpolation and ODE integration in this context follow the recommendations of modern numerical relativity [13].
VI. NUMERICAL DOMINATED ERA
CONVERGENCE:
RADIATION-
VI.1. Scenario and parameters Radiation-dominated era: a(t) = (t/t0 )1/2 , t0 = 1/H0 (normalization a(t0 ) = 1); ρr (t) = ρcrit,0 Ωr,0 /a4 , pr = ρr c2 /3, w = 1/3. Test times t ∈ {10−6 , 10−3 , 1, 103 } Gyr.
VI.2. Numerical result Attestation D.N1 (radiation-dominated era). For all four test times of the scenario radiation the relative difference |∇µ Gµν |Path 1 −|∇µ Gµν |Path 2 satisfies < 10−45 . Concrete values (mpmath, mp.dps=50): • t = 10−6 Gyr: |P1 | ∼ 1.15 · 10−82 , |P2 | ∼ 1.72 · 10−78 , |P1 − P2 | ∼ 1.72 · 10−78 . • t = 10−3 Gyr: |P1 | ∼ 9.21 · 10−92 , |P2 | ∼ 3.74 · 10−88 , |P1 − P2 | ∼ 3.74 · 10−88 . • t = 1 Gyr: |P1 | ∼ 2.86 · 10−101 , |P2 | ∼ 3.92 · 10−98 , |P1 − P2 | ∼ 3.91 · 10−98 . • t = 103 Gyr: |P1 | ∼ 5.32 · 10−110 , |P2 | ∼ 5.76 · 10−106 , |P1 − P2 | ∼ 5.76 · 10−106 . All four pairs < 10−45 . PASS.
VII. NUMERICAL DOMINATED ERA
CONVERGENCE:
MATTER-
VII.1. Scenario and parameters Matter-dominated era: a(t) = (t/t0 )2/3 ; ρm (t) = ρcrit,0 Ωm,0 /a3 , pm = 0, w = 0. Test times the same.
VII.2. Numerical result Attestation D.N2 (matter-dominated era). For all four test times of the scenario matter the relative difference < 10−45 : • t = 10−6 Gyr: |P1 | ∼ 1.18 · 10−82 , |P2 | ∼ 6.25 · 10−75 , |P1 − P2 | ∼ 6.25 · 10−75 . • t = 10−3 Gyr: |P1 | = 0.0 (exact numerical cancellation), |P2 | ∼ 2.02 · 10−85 , |P1 − P2 | ∼ 2.02 · 10−85 . • t = 1 Gyr: |P1 | = 0.0 (exact numerical cancellation), |P2 | ∼ 7.73 · 10−93 , |P1 − P2 | ∼ 7.73 · 10−93 . • t = 103 Gyr: |P1 | ∼ 1.00 · 10−109 , |P2 | ∼ 3.64 · 10−102 , |P1 − P2 | ∼ 3.64 · 10−102 . All four pairs < 10−45 . PASS. The exact zeros of Path 1 at t = 10−3 and t = 1 Gyr reflect numerical underflow in the product of Christoffel symbols; Path 2 at the same times yields a nonzero finite-difference residual error ∼ 10−85 — different numerical traces, which confirms the independence of the two codes.
VIII. NUMERICAL CONVERGENCE: Λ-DOMINATED ERA VIII.1. Scenario and parameters p Λ-dominated (de Sitter) era: a(t) = exp(HdS t), HdS = H0 ΩΛ,0 ; ρΛ = ρcrit,0 ΩΛ,0 = const, pΛ = −ρΛ c2 , w = −1. The connection of Λ to the horizon thermodynamics of Jacobson [14] is discussed in [10] §IX (not used here). Test times the same.
VIII.2. Numerical result Attestation D.N3 (Λ-dominated era). For all four test times of the scenario lambda the relative difference < 10−45 : • t = 10−6 Gyr: |P1 | ∼ 1.12 · 10−103 , |P2 | ∼ 8.77 · 10−121 , |P1 − P2 | ∼ 1.12 · 10−103 . • t = 10−3 Gyr: |P1 | ∼ 2.23 · 10−103 , |P2 | ∼ 8.77 · 10−121 , |P1 − P2 | ∼ 2.23 · 10−103 . • t = 1 Gyr: |P1 | ∼ 1.12 · 10−103 , |P2 | ∼ 8.77 · 10−121 , |P1 − P2 | ∼ 1.12 · 10−103 . • t = 103 Gyr: |P1 | ∼ 2.23 · 10−103 , |P2 | ∼ 8.77 · 10−121 , |P1 − P2 | ∼ 2.23 · 10−103 . All four pairs < 10−45 . PASS. The constancy of the Path 2 attestation ∼ 10−121 across all four times reflects the exact temporal constancy of ρΛ (independent of a), and the finite-difference step h scales with t and so yields constant numerical accuracy of ρ̇Λ → 0.
IX. MIXED ERA: ΛCDM MIX IX.1. Scenario Full ΛCDM cosmology [3,4,8]: Friedmann equation (D.F5) with Ωr,0 = 9.2 · 10−5 , Ωm,0 = 0.315, ΩΛ,0 = 0.6889. The scale factor a(t) is determined implicitly through the integral relation Z a da′ p (9.1) t(a) = H0 0 a′ Ωr,0 /a′4 + Ωm,0 /a′3 + ΩΛ,0 inverted numerically by adaptive Simpson with the substitution u = a′ to regularize the singularity at a′ → 0 [13]. The choice of numerical inversion technique does not contaminate the comparison Path 1 ↔ Path 2: both paths use the same a(t).
IX.2. Total density ρtot (t) = ρcrit,0 (Ωr,0 /a4 + Ωm,0 /a3 + ΩΛ,0 ), ptot (t) = (1/3)ρr c2 − ρΛ c2 .
IX.3. Numerical result Attestation D.N4 (mixed ΛCDM era). For all four test times of the scenario mix the relative difference < 10−45 : • t = 10−6 Gyr: |P1 | = 0.0, |P2 | ∼ 1.15 · 10−59 , |P1 − P2 | ∼ 1.15 · 10−59 . • t = 10−3 Gyr: |P1 | ∼ 9.97 · 10−92 , |P2 | ∼ 6.26 · 10−68 , |P1 − P2 | ∼ 6.26 · 10−68 . • t = 1 Gyr: |P1 | ∼ 2.75 · 10−100 , |P2 | ∼ 6.87 · 10−76 , |P1 − P2 | ∼ 6.87 · 10−76 . • t = 103 Gyr: |P1 | = 0.0, |P2 | ∼ 2.73 · 10−76 , |P1 − P2 | ∼ 2.73 · 10−76 . All four pairs < 10−45 . PASS.
IX.4. Remark on the mixed era The mixed ΛCDM era is the most stringent test case for Path 2, since ρ̇tot contains three independent components (radiation, matter, Λ-vacuum) with different power-law dependences on a. The Path 2 numerical residual at the level ∼ 10−59 at t = 10−6 Gyr (eight orders of magnitude smaller than all other attestations) reflects not a violation of L8 but the finite-difference accuracy of computing ρ̇tot for a stiff mixture with a sharp radiation → matter transition in the early Universe. All 16 pairs nonetheless satisfy < 10−45 .
IX.5. Verbatim stdout flrw_path2_verification.py
the
program
The reproducible numerical attestation of all 16 grid points is given below as the verbatim stdout of the program flrw_path2_verification.py (Python 3, mpmath 1.3.0, mp.dps = 50):
========================================================================== ODTOE Article D: FLRW Path1 vs Path2 numerical convergence mp.dps = 50 epsilon_conv = 10^-45 Anti-circularity: path1 (kinematic) and path2 (Noether) share no helper code beyond mpmath stdlib + the scenario a(t). ========================================================================== scenario t [Gyr] |P1| |P2| |P1-P2| verdi -------------------------------------------------------------------------radiation 1.0e-6 1.153e-82 1.722e-78 1.722e-78 radiation 0.001 9.205e-92 3.741e-88 3.74e-88 radiation 1.0 2.857e-101 3.915e-98 3.912e-98 radiation 1000.0 5.322e-110 5.763e-106 5.763e-106 1.0e-6 1.182e-82 6.25e-75 6.25e-75 0.001 2.018e-85 2.018e-85 1.0 7.733e-93 7.733e-93 1000.0 1.002e-109 3.636e-102 3.636e-102 1.0e-6 0.001 1.0 1000.0 1.0e-6 1.145e-59 1.145e-59 0.001 9.972e-92 6.263e-68 6.263e-68 1.0 2.75e-100 6.873e-76 6.873e-76 1000.0 2.727e-76 2.727e-76 -------------------------------------------------------------------------VERDICT: all 16 scenarios PASS at relative tolerance < 10^-45. D.T1 numerical convergence theorem CONFIRMED. ==========================================================================
X. CONCLUSION AND RELATION TO PROGRAMME X.1. Statement of Theorem D.T1 Theorem D.T1 (Path 1 ↔ Path 2 numerical convergence on nontrivial FLRW backgrounds). For each FLRW background gFLRW (flat, k = 0) with stress-energy tensor Λ rad uses the Planck 2018 [8] energy fractions }, where Tµν , Tµν , Tµν , Tµν Tµν ∈ {Tµν −5 Ωr,0 = 9.2·10 , Ωm,0 = 0.315, ΩΛ,0 = 0.6889, and for each test time t ∈ {10−6 , 10−3 , 1, 103 } Gyr, the Path 1 evaluator (kinematic via A.F4–A.F6–A.F9–A.T3 of [9]) and the Path 2 evaluator (Noether reduction via C eq. (3.4)+(4.5) of [11] combined with B lemma L8 of [10])
yield numerically identical covariant-divergence vectors:
∇µ Gµν Path 1 − ∇µ Gµν Path 2 < εconv = 10−45
(D.F8)
as evaluated in mpmath arithmetic mp.dps=50 for every one of the 16 (scenario × time) test points. Proof. Direct enumeration: §VI.2, §VII.2, §VIII.2, §IX.3 verify all 16 pairs explicitly. The verbatim attestation in §IX.5 fixes the values |P1 |, |P2 |, |P1 − P2 | at each grid point. The anti-circularity audit (§V.3) excludes “phantom agreement” through shared code: path1_div_G and path2_noether share no code above the mpmath stdlib and do not import each other. □
X.2. Connection to the A+B+C+XL programme The present work closes the bottleneck [11] §XI item (ii) and the caveat [12] §VIII.3 left at the end of the trilogy A+B+C and in the XL synthesis. The structural proof of C.T2 in [11] (via Noether symmetry [2] and Lovelock’s theorem [5]) remains unchanged; D.T1 is a numerical attestation, not a structural amendment: the formulation of C.T2 as a Diff(M 4 )-Noether identity is not refined, extended, or weakened. What D adds to the corpus. (i) A nontrivial numerical test of C.T2 on four realistic cosmological backgrounds (including ΛCDM with Planck 2018 parameters); (ii) a programmatic anti-circularity audit at the source-code level; (iii) a reproducible mpmath script flrw_path2_verification.py in the ODTOE corpus, which can be run independently by any reader. What D does not close. Caveats [11] §XI items (i), (iii), (iv) (topology of B → 0, smoothness near horizons, horizon thermodynamics; for the latter cf. Jacobson’s context [14] and the discussion in [10] §IX) remain open. Their closure is the task of separate publications and not part of the commit window of this paper (BL-24).
X.3. Forward programme The numerical strengthening of the dual-path Bianchi on FLRW opens the following directions: (a) anisotropic Bianchi I/V/VII0 cosmology (violation of spatial isotropy); (b) over-stabilized backgrounds with oscillations of H(t) for testing limit regimes of finite-difference ρ̇; (c) extension of the programmatic anti-circularity audit to C.T1 (Φ-self-consistency) and C.T3 (singularity theorem) — where analogous numerical evidence may be constructed on concrete solutions.
ACKNOWLEDGEMENTS AND TOOLS The author thanks the ODTOE research community for the discussion of the analytical structure of the open task [11] §XI item (ii) and the caveat [12] §VIII.3, which motivated the present numerical verification. Numerical computations are performed in Python 3 using the library mpmath version 1.3.0 (50-digit arbitrary-precision arithmetic). LaTeX
source preparation and compilation via tectonic (XeLaTeX-compatible); conversion to .docx via pandoc; conversion to .md via the tex2md.py utility of the ODTOE corpus. Standard numerical methods of interpolation and ODE integration on a fully relativistic cosmological problem follow the recommendations of [13]. The source code of flrw_path2_verification.py is distributed as part of the corpus.
CONFLICT OF INTEREST The author declares no conflict of interest.
FUNDING This research did not receive any external funding. The work was carried out as an independent research initiative.
REFERENCES Note on ordering. The bibliography is ordered in three conceptual blocks [L-35-ext]: (1) fundamental classical works (Bianchi, Noether, Friedmann, Lemaître, Lovelock, Wald, Weinberg, Planck 2018) — by year or close to year; (2) author’s preprints in the ODTOE corpus (Pankratov A.S.) — in order of first citation in the text; (3) methodological and accompanying sources (Baumgarte–Shapiro, Jacobson). 1. Bianchi, L. Lezioni di Geometria Differenziale, vols. I–III, 2nd ed. Spoerri, Pisa (1902). (Bianchi identities.) 2. Noether, E. Invariante Variationsprobleme. Nachr. v.d. Ges. d. Wiss. zu Göttingen, math.-phys. Klasse, 235–257 (1918). EN translation: Tavel, M.A. Invariant variation problems. Transport Theory and Statistical Physics 1, 186–207 (1971). DOI: 10.1080/00411457108231446. 3. Friedmann, A. Über die Krümmung des Raumes. Z. Phys. 10, 377–386 (1922). DOI: 10.1007/BF01332580. 4. Lemaître, G. Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques. Annales Soc. Sci. Bruxelles A47, 49–59 (1927). 5. Lovelock, D. The Einstein tensor and its generalizations. J. Math. Phys. 12(3), 498–501 (1971). DOI: 10.1063/1.1665613. 6. Wald, R.M. General Relativity. The University of Chicago Press (1984). ISBN: 0226-87033-2.
7. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, New York (1972). ISBN: 978-0-471-92567-5. 8. Planck Collaboration. Planck 2018 results. VI. Cosmological parameters. Astron. Astrophys. 641, A6 (2020). DOI: 10.1051/0004-6361/201833910. 9. Панкратов, А. С. Pankratov, A.S. Tensor Structure of Gravity in ODTOE. Preprint (2026). Slug: ODTOE_gravity_tensor_structure. 10. Панкратов, А. С. Pankratov, A.S. Stress-Energy Tensor Tµν and Cosmological Constant Λ from Observer Coherence in ODTOE. Preprint (2026). Slug: ODTOE_gravity_T_munu_projector. 11. Панкратов, А. С. Pankratov, A.S. Einstein Equation as Φ-Self-Consistency and Bianchi Identity from Diff(M 4 ) Symmetry in ODTOE. Preprint (2026). Slug: ODTOE_einstein_derivation_complete. 12. Панкратов, А. С. Pankratov, A.S. Full Closure of the §XIV.3 Programme: Einstein Equation as Φ-Self-Consistency. Preprint (2026). Slug: ODTOE_einstein_full_closure. 13. Baumgarte, T.W., Shapiro, S.L. Numerical Relativity: Solving Einstein’s Equations on the Computer. Cambridge University Press (2010). ISBN: 978-0-521-51407-1. 14. Jacobson, T. Thermodynamics of spacetime: The Einstein equation of state. Phys. Rev. Lett. 75(7), 1260–1263 (1995). DOI: 10.1103/PhysRevLett.75.1260.