Dynamic Attractor in ODTOE: Evolutionary Monadology and Energy-Information Density of the World Line

DYNAMIC ATTRACTOR IN ODTOE: EVOLUTIONARY MONADOLOGY AND ENERGY-INFORMATION DENSITY OF THE WORLD LINE (Динамический аттрактор в ODTOE: эволюционная монадология и энергоинформационная плотность мировой линии) Development of the ODTOE formalism in the regime of evolutionary trajectories

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

UDC 530.145 + 167.7 + 111 + 51-7

АННОТАЦИЯ В работе предложено расширение наблюдатель-зависимой теории всего (ODTOE) в динамическом режиме: от асимптотических пределов полной когерентности (B → 1, S → 1) к эволюционным траекториям с производными dB/dt, dSij /dt и плотностью вдоль мировой линии наблюдателя. Показано, что линия монадологии, идущая от Лейбница через речь Н.В. Бугаева «Основы эволюционной монадологии» (1893) к современному формализму, получает в ODTOE количественное замыкание: бугаевский закон солидарности монад отображается на постулат P5 коллективного наблюдения, закон сохранения прошлого — на иерархическую структуру историй Hhist , а психизм монад — на параметр фокуса F . Введено понятие разомкнутой монады как открытой системы с каналами притока ∆in и оттока ∆out ; обобщено ODTOE-определение любви как одновременной коэволюции пары dBi /dt > 0 ∧ dBj /dt > 0. Сформулирована теорема условной достижимости: неподвижная точка Ψ∗ = Φ(Ψ∗ ) достигается из начальной конфигурации Ψ0 тогда и только тогда, когда существует коллективный аттрактор A с S(A) > Sthreshold и градиент ∇B направлен к A. Введена интегральная метрика плотности мировой линии P (W ) = α B(Ψ, n) · (1 − σ(Ψ, n))β dn. W Ключевые слова: эволюционная монадология, динамический аттрактор, Бугаев, неподвижная точка, Fix(Φ), энергоинформационная плотность, мировая линия, ODTOE, коллективное наблюдение, разомкнутая система

ABSTRACT This paper proposes an extension of the Observer-Dependent Theory of Everything (ODTOE) into a dynamic regime: from asymptotic limits of full coherence (B → 1, S → 1) to evolutionary trajectories with derivatives dB/dt, dSij /dt and a density along

the observer’s world line. The monadological lineage running from Leibniz through N.V. Bugaev’s Foundations of Evolutionary Monadology (1893) to the modern formalism is shown to admit quantitative closure within ODTOE: Bugaev’s law of solidarity of monads maps onto postulate P5 of collective observation; the law of conservation of the past maps onto the hierarchical history structure Hhist ; and the psychism of monads maps onto the focus parameter F . The notion of an open monad as an exchange-open system with inflow ∆in and outflow ∆out channels is introduced; the ODTOE definition of love is generalized as a simultaneous co-evolution condition on the pair, dBi /dt > 0 ∧ dBj /dt > 0. A conditional reachability theorem is formulated: the fixed point Ψ∗ = Φ(Ψ∗ ) is reached from an initial configuration Ψ0 if and only if there exists a collective attractor A with S(A) > Sthreshold and the gradient ∇B is oriented toward A. An integral density metric of the world line P (W ) = W B(Ψ, n)α · (1 − σ(Ψ, n))β dn is introduced as a characteristic of the observer trajectory’s ontological imprint; two world lines of equal length but distinct P are ontologically inequivalent. A two-level stratification of observers is proposed: an ontological level (a) — any self-referential structure with B > 0 — and an actual-historical level (b), where the dominant observers of a moment are defined by the product B(τ )·I(C, τ )·Ω(A)(τ ); the (a)/(b) distinction renders specific historical claims falsifiable hypotheses without contradicting universal ontology. Five open problems of the programme are formulated (formalization of the conservationof-past law, derivation of exponents α, β, an ontological collapse theorem for B → 0, the D-Finitude postulate, and a phase diagram of S-regions). Keywords: evolutionary monadology, dynamic attractor, Bugaev, fixed point, Fix(Φ), energy-information density, world line, ODTOE, collective observation, open system

I. INTRODUCTION The Observer-Dependent Theory of Everything (ODTOE) [1] formulates reality as a functional of the observation act: R = Ô(Ψ). The operator Ô depends on the observer’s internal structure O = (B, A, H), where B ∈ [0, 1] is contextual coherence (belief), A is the archetypal structure, H ∈ Hhist is the historical component [1, §II-B]. The theory unfolds through six postulates and four propositions that form the closed architecture of a self-referential system: the observer constitutes the observed, and simultaneously the observed constitutes the structural foundation of the observer through the fixed point of the self-observation mapping Φ(Ψ) = ι ◦ ÔΨ (Ψ) [1, §V, Proposition 4]. The hierarchical monad structure in ODTOE is organized by the dimensionality parameter d ∈ N [1, §4.2, remark]: from elementary d = 1 (bodily level) through d = 2 (social), d = 3 (planetary) to d = 4 (cosmological); the general case without dimensional restriction corresponds to d(O) = ∞. Collective observation is formalized by postulate P5 [2] with coherence S ∈ [Smin (n), 1] and collective probability Pcoll (E) = ∏ 1 − i (1 − Bik ). The framework is described predominantly in the asymptotic regime: limiting behaviours S → 1 (a unified reality), B → 1 (absorbing state), T (C) → ∞ (configuration stabilization). The objective of the present work is the extension of the ODTOE formalism into a dynamic regime. Instead of focusing on asymptotic limits we consider evolutionary

trajectories: the derivatives dB/dt and dSij /dt, the reachability conditions of the fixed point from an arbitrary initial state, the integral characteristics of the observer’s world line, and the distinction between the ontological and actual-historical levels of description. This transition has a direct predecessor in the Russian mathematical tradition: N.V. Bugaev’s lecture Foundations of Evolutionary Monadology (1893), delivered at the Moscow Mathematical Society [3], contains a formulation of the monad as a “centre of action” that receives and gives, that is, a conception typologically identical to the modern notion of an open system, and removes Leibniz’s “closedness” of monads long before analogous steps in 20th-century process philosophy (Whitehead [4]) and in second-order cybernetics. Structure of the paper. Section II reconstructs the correspondence between Bugaev’s evolutionary monadology and ODTOE concepts: monad/observer, law of solidarity/postulate P5, law of conservation of the past/Hhist , psychism/focus F , hierarchy of levels/dimensionality d. Section III transitions from the statics of asymptotic limits to dynamics: the notion of an open monad is introduced, the equation dB/dt = ∆in − ∆out is formulated, and the ODTOE definition of love is generalized as the condition of simultaneous co-evolution of a pair. Section IV contains the conditional reachability theorem for the fixed point Ψ∗ : Banach existence is supplemented by an activation condition via a collective attractor. Section V introduces the integral world-line density P (W ). Section VI formulates the two-level stratification (ontological/actual-historical). Section VII lists five open problems, each a topic for a separate future paper. Corpus references. The main ODTOE paper [1], collective observer [2], love as coherence [5], unified operator [6], infinite recursion and fine-structure constant [7] form the corpus foundation of the present work; cross-references are given via bibliography numbers.

II. EVOLUTIONARY MONADOLOGY OF N.V. BUGAEV AS A PREDECESSOR OF ODTOE II.1. Biographical and historical context Nikolai Vasilyevich Bugaev (1837–1903) was a mathematician, president of the Moscow Mathematical Society (1891–1903), and the founder of the so-called arithmology — a programme treating discontinuous (discrete) functions as an autonomous subject of analysis, in opposition to the then-dominant analytic cult of continuity [3]. The lecture Foundations of Evolutionary Monadology, delivered in 1893 and published in «Вопросах философии и психологии» [3], is a systematic exposition of the monadological doctrine in 184 theses, organized into sections on the monad, the dyad, the triad, the complex, and the laws of their interaction. This lecture holds a key place in the history of Russian mathematical philosophy: it is the first systematic formalization of monads with interaction in the Russian philosophico-mathematical tradition (see also secondary source [16] Drozdek 2018 for contextualization), with an explicit emphasis on the interaction of monads, that is, precisely on the component absent from Leibniz [8].

II.2. Structure of the doctrine (184 theses) Systematically, Bugaev formulates monadology through the following blocks: • Monad: an autonomous centre of action possessing an internal psychic state and a capacity for self-similarity (theses §1–§20). • Dyad and triad: two- and three-part forms of monad interaction; the triad defines the minimal integral structure (§21–§40). • Complex: an arbitrarily multi-partite association of monads into a relatively stable unit (§41–§66). • Laws of solidarity (§67–§72): monads are bound in a network of mutual obligations that constitutes reality itself; the isolated monad is a mathematical abstraction. • Law of conservation of the past (§85): in each monad the past does not vanish but accumulates; the present is a function of the entire traversed history. • Hierarchy of levels: elementary, cellular, social, cosmic monads — a stepwise organization from the simplest agent to the universal whole. • Psychism of monads: every monad possesses an elementary form of experience/perception, which Bugaev defends as a regulative principle of the philosophy of mathematics.

II.3. Mapping to ODTOE concepts The correspondence between Bugaev’s concepts and ODTOE structures may be represented as follows: Bugaev (1893)

ODTOE

Monad as centre of action

Observer O = (B, A, H), a self-referential structure with B > 0 [1, §II] ∏ of Postulate P5: Pcoll (E) = 1 − i (1 − Bik ) [1, §III]

Law of solidarity monads (§67–§72) Law of conservation of the past (§85) Psychism of monads Hierarchy of levels Dyad, triad, complex

Historical component H ∈ Hhist [1, §4.2]; operator H on Φ-trajectories Focus of attention F ∈ [0, 1] in definition D1.1 [1] Dimensionality parameter d ∈ N [1, §4.2] Pair/cluster/collective of observers; S-groups in P5

Let MBug be the set of Bugaev’s monads and OODTOE the set of ODTOE observers. The correspondence is given by a mapping µ : MBug → OODTOE such that µ(m) = (B(m), A(m), H(m)),

B(m) > 0,

H(m) ∈ Hhist ,

(2.1)

where the condition B(m) > 0 is the formal statement of Bugaev’s thesis that a monad cannot be “empty”: every actually existing monad possesses a nonzero level of internal

coherence. The condition H(m) ∈ Hhist formalizes the law of conservation of the past: the historical trajectory of a monad belongs to the space of histories and is not reducible to a single point.

II.4. Bugaev’s key contribution: the “monad with windows” The principal distinction between Bugaev and Leibniz consists in the removal of the thesis of “closedness” of monads. In Leibniz [8] monads have no “windows” (les Monades n’ont point de fenêtres): their agreement is secured by pre-established harmony. Bugaev replaces this teleological mechanism with interaction: monads “receive and give”, and their law consists in the conservation of the past in the present [3]. This change has far-reaching consequences: Bugaev’s monad is an open system exchanging energy/information with other monads, which is structurally identical to the modern notion of an open dynamical system and, in particular, to the classless observer structure of ODTOE where any Oi is connected to the collective through the coherence S [2]. Bugaev makes this step 36 years before Whitehead’s process philosophy [4] (1893 to 1929) and 55 years before Wiener’s cybernetics (1948). Remark on terminology. Bugaev’s notion of “centre of action” should not be identified with the Newtonian point of force: it is closer to the dynamic atomism of Boscovich and functionally corresponds to the ODTOE observer as a source of the Ôoperator.

II.5. Formalization of the law of solidarity Bugaev’s law of solidarity (§67–§72) in ODTOE notation takes the form: ∀ i, j ∈ Icoll : Sij = 1 − |Bi − Bj | ≥ Sthreshold ,

(2.2)

where Icoll is the index set of observers in the collective and Sij is the pairwise coherence [1, (4.5)]. Condition (2.2) is a quantitative expression of the “solidarity” requirement: the collective exists as a unified whole precisely when all its members are bound by pairwise coherence above a certain threshold. At Sij < Sthreshold the collective fragments into independent subgroups and Bugaev’s “complex” ceases to exist as a whole object. The law of conservation of the past (§85) is representable as a monotonicity requirement for the operator H along Φ-trajectories: ∀ n ≥ 0 : H(Ψn+1 ) ⊇ H(Ψn ),

(2.3)

that is, the history at step n + 1 contains the entire history of the preceding step. Formal verification of compatibility between (2.3) and axiom (A) [1] and postulates P1, P2 remains an open problem (see §VII.1).

III. FROM STATICS TO DYNAMICS: dB/dt AND THE OPEN SYSTEM III.1. Asymptotic vs evolutionary regimes The main ODTOE text [1] is formulated in an asymptotic regime: the limiting values B = 1, S = 1, T (C) → ∞ are considered, and a number of results (Propositions 1–4) is tied to these limits. The asymptotics are informative: they define a regulative ideal. But the concrete dynamics of a monad/observer between asymptotic states is specified in the main paper only through the logistic equation (D1.3) [1]: dB ˙¯ · d(R ¯ obs , Rexp ) · B · (1 − B), = γ · tanh(β · d)

(3.1)

where the dynamics is closed with respect to the pair “expected/observed” but does not account for the external flow between the observer and the environment.

III.2. The open monad: channels ∆in and ∆out We extend (3.1) by introducing system openness. Let observer Oi interact with the environment and with other observers through two channels: • an inflow channel ∆in (Oi , t) ≥ 0 — the coherence increment received from the environment and the collective (information exchange, learning, participation in resonant collective acts); • an outflow channel ∆out (Oi , t) ≥ 0 — loss of coherence due to internal dissipation, growth of contradiction σ, loss of focus F . The dynamic equation of an open monad takes the form dBi = ∆in (Oi , t) − ∆out (Oi , t) + Ξ(Oi , env) · Bi (1 − Bi ),

(3.2)

˙¯ d(R ¯ obs , Rexp ) is the logistic internal driver (reproduces where Ξ(Oi , env) = γ ·tanh(β · d)· (3.1) at ∆in = ∆out = 0). Equation (3.2) is a first approximation: it additively separates the external flow and internal logistics, which is correct in the linear regime and requires refinement under strong exchange. Limitation. (3.2) is given as a first-order linear open-system extension of (3.1); preservation of the range Bi ∈ [0, 1] under arbitrary flows ∆in , ∆out requires separate calibration (regularization or soft-clamp) — an open problem. The channels ∆in , ∆out are structurally identical to the ODTOE P5-collective structure: ∆in is the projection of the collective coherence Scoll onto the individual Bi ; ∆out is the entropic leak described by the growth of σ(Oi , C) over time. In the limit of the closed monad ∆in ≡ ∆out ≡ 0, equation (3.1) is recovered. Conceptually this “opening to the whole world” coincides with the open-system formulation of N.N. Moiseev [14].

III.3. Pair dynamics and generalization of the definition of love In [5] love between observers i, j is defined as the limiting condition Sij → 1 on the pairwise coherence. This condition suffices to describe a stable pair but does not distinguish “joint growth” from “joint stagnation at a high level”. To account for the evolutionary aspect we extend the definition: Love(i, j) ⇐⇒

Sij → 1 ∧

dBi dBj >0 ∧ >0 .

(3.3)

Condition (3.3) requires simultaneous co-evolution: both observers grow monotonically in coherence, not merely agree at a common level. A pair with Sij → 1 and dBi /dt = dBj /dt = 0 (a fixed mutually agreed profile) does not satisfy (3.3) — this is a stable collective without development. Conversely, a pair with 0 < Sij < 1 but dBi /dt, dBj /dt > 0 and Sij → 1 as t → ∞ satisfies (3.3). Remark. Condition (3.3) is compatible with the main formulation [5]: any pair satisfying (3.3) eventually reaches Sij → 1. The converse, as shown above, is false: asymptotic agreement does not guarantee co-evolution. Thus (3.3) is a stronger condition encompassing (3.1) as an asymptotic case. Thematically this resonates with P.A. Florensky’s metaphysics of love [15]: “friendship” and the “I-and-thou” as simultaneous growth in truth — a quality, not a point.

III.4. Event as a Φ-iteration In ODTOE time arises as a sequence of self-observation acts Ψn+1 = Φ(Ψn ) [6]. An event is naturally identified with a single operator iteration: Event(n) := Ψn , Φ(Ψn ) = Ψn+1 . (3.4) Definition (3.4) is an operational form: it allows one to compute the change of any (n) (n+1) (n) ODTOE parameter between adjacent steps (∆Bi = Bi − Bi , and so on) and to specify a discrete version of (3.2). The topological treatment of an event as a temporalloop node given in [6] is compatible with (3.4): Φ generates a directed tree of events whose branches may close through self-reference.

IV. ACTIVATION CONDITIONS FOR THE FIXED POINT Fix(Φ) IV.1. Banach existence and the question of reachability Proposition 4 of ODTOE [1, §V] establishes the existence of the fixed point Ψ∗ = Φ(Ψ∗ ) on the basis of the theorems of Schauder [9] and Banach [10], under appropriate conditions on the operator Φ = ι ◦ Ô. This is a principal result: it resolves the question of the origin of the primary observer without invoking an external ground. However, existence is not equivalent to reachability: an arbitrary initial configuration Ψ0 under

iteration Ψn+1 = Φ(Ψn ) may fail to converge to Ψ∗ in a finite (or even countable) number of steps. In topological terms: Ψ∗ may be an unstable fixed point or lie outside the basin of attraction of Ψ0 .

IV.2. Activation condition via a collective attractor To answer the question “under what conditions does iteration from Ψ0 converge to Ψ∗ ?” we introduce the notion of a collective attractor. Let A = {Oi1 , . . . , Oim } be a group of m ≥ nmin observers, where S(A) = 1 −

∑ |Bij − Bik |, m(m − 1) j<k

(4.1)

is the collective coherence of the group [1, (4.5)]. A is called a collective attractor if S(A) > Sthreshold , where Sthreshold is the threshold from [1, §III, P5]. Candidate Lemma (Reachability Condition). Let Ψ0 ∈ H be an initial configuration and let Φ be a self-observation operator satisfying Banach or Schauder conditions. Then for the iteration Ψn+1 = Φ(Ψn ) to converge to some fixed point Ψ∗ ∈ Fix(Φ), a sufficient condition in first approximation is that there exists a collective attractor A with S(A) > Sthreshold and the coherence gradient ∇Ψ B, computed at Ψ0 , is oriented toward A: Ψ0 − → Ψ∗ ⇐= Φ

∃ A : S(A) > Sthreshold ∧ ⟨∇Ψ B(Ψ0 ), A − Ψ0 ⟩ > 0 .

(4.2)

Remark on status. (4.2) is given as a candidate lemma / first-order sufficient condition; full proof requires specification of operator Φ, basin of attraction, and contraction properties, which remain open (see §VII.1). Proof sketch (first approximation). (⇐) If A exists and ∇B is directed toward A, then B grows monotonically along the Φ-iteration trajectory; in the limit B → 1, hence Ψn approaches the absorbing state which coincides with Ψ∗ by Proposition 4 [1]. (⇒) If all collective attractors have S < Sthreshold , then collective coherence is insufficient for stabilization of B; by (3.2) ∆in is bounded and ∆out dominates, whence B → 0, which excludes convergence to Ψ∗ with B ∗ > 0 [1, §4.5.1, selfconsistency remark]. A rigorous proof requires specification of Φ, estimation of the contraction constant, and consideration of multiply-connected Fix(Φ); it is left as an open problem.

IV.3. Examples of collective attractors Condition (4.2) admits concrete interpretations:

• Passionary ethnic cluster (after Gumilev [11]): a group of passionarity-bearers at S(A) > Sthreshold acts as an attractor for surrounding observers; Gumilev’s formula B(τ ) = B0 · e−τ /τp describes the temporal dynamics consistent with (3.2). • Scientific community: a group of researchers with an agreed paradigmatic basis (in Kuhn’s sense [12]) forms an attractor toward which individual researchers converge through the P5-collective mechanism. • Creative group: a union of co-authors with agreed creative vision; the overall coherence S is high, individual Bi grow through exchange. • Family: a small group (m = 2 . . . 6) stable at S > Sthreshold ; formalizes the sociological observation linking family stability to communicative coherence. In all four cases the mechanism is the same: A raises the individual observer’s Bi to a level sufficient for stabilization of their trajectory near Ψ∗ . The value nmin is empirically estimated as nmin ≈ 2 . . . 3 for small attractors (family, creative pair) and nmin ≈ 7 ± 2 for cognitively stable collectives (the empirical norm for working-group size in psychology).

IV.4. Connection with [6] and [7] Formula (4.2) refines the general result of [6] on the unity of the ODTOE operator: the Φ-operator acts correctly only within the basin of attraction of a collective attractor. In [7] the fixed point is linked to the fine-structure constant αfs ≈ 137.036 via infinite recursion; condition (4.2) imposes an additional restriction: the constant αfs in the ODTOE interpretation is the asymptotics of an activated fixed point, unreachable at S < Sthreshold .

V. ENERGY-INFORMATION DENSITY ALONG THE WORLD LINE P (W ) V.1. Observer’s world line Define the observer’s world line W as the sequence of Φ-iterations recorded at moments of self-observation: W = {Ψ∗n }N n=0 ,

Ψ∗n = Φn (Ψ0 ),

N ∈ N ∪ {∞}.

(5.1)

W is a discrete object; under the transition to continuous time n is treated as a trajectory-length parameter. So far ODTOE has described the world line topologically: W is a connected subset in the history space Hhist [1, §4.2]. Connectivity of W guarantees continuity of experience but does not distinguish “dense” and “sparse” trajectories: two world lines with the same N may be topologically equivalent yet differ substantially in integral coherence.

V.2. Integral density metric We introduce the energy-information density of the world line as the integral: P (W ) := B(Ψ, n)α · (1 − σ(Ψ, n))β dn,

(5.2)

where B(Ψ, n) is the contextual coherence at step n, σ(Ψ, n) is the internal contradiction [1, §II-B], and α, β > 0 are structural exponents. As a first approximation, from dimensional considerations and symmetry across B-formula components, we take α = 2, β = 1, (5.3) which yields a quadratic weighting of coherence and a linear weighting of noncontradiction. This choice is motivated by: (a) the quadratic dependence of B is consistent with the Born rule P ∼ |⟨·|·⟩|2 [1, (D1.4)]; (b) the linear dependence on 1 − σ corresponds to the additive nature of doubt entropy. We emphasize the status of (5.3): it is a first approximation, subject to refinement upon specification of the Φ-architecture (see §VII.2). The final values of α, β must follow from the properties of the observation operator, not be postulated. Remark on the status of α, β. The values α = 2, β = 1 in (5.3) are motivated by a dimensional-structural consideration (the square of coherence B 2 as the probability of pairwise match per P5; the linear contribution (1 − σ) as the fraction of noncontradictory iterations), but they are not rigorously derived from the φ-architecture. This is an open problem of §VII.2.

V.3. Interpretation of P (W ) P (W ) is the total “charge” of the world line: a quantity characterizing the trajectory’s ontological footprint in the corpus of the∫observed. Physically, P (W ) is analogous to actio in classical mechanics ( L dt) or to |ψ|2 dV in quantum mechanics, but applied to the space of cognitive coherence. The concrete meaning: • Two observers O1 , O2 with the same number of steps N but different P : the one with larger P leaves a denser footprint in the historical space Hhist ; the collective retains their trajectory with greater clarity. • As P (W ) → 0 the trajectory “evaporates”: formally W exists in Hhist , but its contribution to collective knowledge is negligible. • As P (W ) → ∞ (possible only if N → ∞) the trajectory becomes “eternal” in the sense of T (C) → ∞ of postulate P3 [1].

V.4. Example: two lives of equal length Consider two observers of equal duration N = N0 :

• W1 : B(n) ≈ 0.9 almost everywhere, σ(n) ≈ 0.1. Then P (W1 ) ≈ N0 · 0.81 · 0.9 ≈ 0.729 · N0 . • W2 : B(n) ≈ 0.3 almost everywhere, σ(n) ≈ 0.5. Then P (W2 ) ≈ N0 · 0.09 · 0.5 ≈ 0.045 · N0 . Ratio P (W1 )/P (W2 ) ≈ 16.2: at equal life length the ontological footprint of the first observer is sixteen times denser. This example illustrates: life length and life density are distinct quantities. ODTOE, via P (W ), provides a quantitative distinction that does not reduce monad stability to duration of existence. The link to the phenomenon of “qualitative time” in philosophy (Bergson, Heidegger) remains outside the scope of the present formal discussion.

V.5. Connection with [7] and [1] In [7] the infinite recursion of ODTOE is linked to the fine-structure constant αfs : the ∗ number of recursion steps N required for stabilization of B near B > 0 is estimated as N ∼ αfs ≈ 137. In terms of P (W ) this suggests a conjectural analogy for the integral density of a self-sustaining monad: Pmin (W ) ∼ αfs · (B ∗ )α · (1 − σ ∗ )β ,

(5.4)

where B ∗ , σ ∗ are the values at the fixed point. For α = 2, β = 1 and B ∗ ≈ 1, σ ∗ → 0: Pmin ∼ 137. Empirical verification of (5.4) is one of the open problems of the programme. Conjecture. The numerical closeness of the order of magnitude of Pmin to αfs ≈ 137.036 is a conjectural analogy, motivated by the structural similarity of the formulas; a rigorous derivation of the relation via the φ-architecture (see [7]) remains an open problem.

VI. TWO-LEVEL STRATIFICATION OF OBSERVERS VI.1. Distinction: ontology vs actual history A central consequence of the formalism of §§III–V is the necessity of distinguishing two levels of description, thus far rarely made explicit in the ODTOE corpus: Level (a) — ontological. An observer is any self-referential structure with B > 0 [1, §II]. This definition imposes no restrictions on type, scale, dimensionality d, or complexity: a quark configuration, an atom, a cell, a human, a social group, a galactic cluster — all are observers provided they satisfy self-referentiality and positive coherence. This is an intensional definition, identifying a class of objects by a property rather than by enumeration. Level (b) — actual-historical. At a specific moment in time τ the effective “agency” of an observer is determined by the product: D(O, τ ) = B(O, τ ) · I(C, τ ) · Ω(A)(τ ),

(6.1)

where B is the current coherence, I(C, τ ) is the configuration inertia at moment τ [1, §III P2], Ω(A)(τ ) is the contribution of the collective attractor A in which the observer participates. Local maxima of D at a given τ are the dominant observers of the moment. This is an extensional characterization: a specific finite selection of agents leaving a significant footprint in the historical space within a specific period. Remark on normalization. For a quantitative comparison of D(O1 , τ ) and D(O2 , τ ) across observers, a common scale for inertia I(C) and the measure Ω(A) is required; for now (6.1) is used as a qualitative ranking within a fixed dimensionality d.

VI.2. Intension vs extension The (a)/(b) distinction corresponds to the classical logical opposition between intension (definition) and extension (current distribution). Level (a) specifies the defining property of an observer; level (b) specifies a concrete list of observers at a given moment with weights D(O, τ ). As τ → ∞ the extension “runs through” the entire set of possible observers defined by the intension; but over any finite interval the extension picks out a sublist dependent on the historical trajectory W of the collective.

VI.3. Methodological consequence: falsifiability of historical claims The (a)/(b) distinction renders specific historical claims falsifiable hypotheses of level (b), non-contradictory to the universal ontology of level (a). For example, the claim “in the 18th century the scientific community of Europe was the dominant collective observer A with Ω(A)(τ ) → max” refers to (b) and is verifiable against historicalscientometric data (publication counts, paradigm propagation rates, and so on). Falsification of this claim does not refute the universal ontology of ODTOE: it merely corrects the distribution D(O, τ ) over a specific interval. This separation is structurally analogous to a distinction in physics: Maxwell’s equations specify the ontology of the electromagnetic field (level a), whereas the concrete distribution of fields in the Universe at moment τ constitutes actual history (level b). Verifying the distribution does not undermine the equations but merely updates the boundary conditions.

VI.4. Formal statement The complete description of an ODTOE system state at moment τ consists of two components: ) ( (6.2) S(τ ) = Oont , {D(O, τ )}O∈Oont , where Oont is the ontological set of observers (an invariant of level a), and {D(O, τ )} is the current agency distribution (the dynamics of level b). The evolution of S(τ ) is described by the open dynamics of §III: Oont remains unchanged (by the ontological definition), while {D(O, τ )} evolves by (3.2) and (6.1).

VI.5. Connection with collective observation In [2] collective observation is formalized by postulate P5 as a superposition of individual beliefs with regard to coherence. Levels (a) and (b) align with [2] as follows: P5 describes the mechanism of transition from the set of individual Bi (level a) to the collective Pcoll (level b) at a given moment. The parameter Ω(A)(τ ) in (6.1) is the integral contribution of the collective to the individual agency of O.

VII. OPEN PROBLEMS OF THE PROGRAMME Five problems, each a separate future ODTOE paper.

VII.1. Formalization of Bugaev’s law of conservation of the past The thesis §85 of Bugaev [3] requires a rigorous mapping onto the ODTOE structure Hhist [1, §4.2]. The preliminary record (2.3) gives a monotonicity requirement but does not specify which operator-invariants are conserved along the world line. Open question: does there exist a set of quantities {Ik } such that Ik (Wn ) = Ik (Wm ) for n ̸= m, analogous to integrals of motion in Hamiltonian mechanics? If so — how are they related to P (W ) and the S-parameter?

VII.2. Explicit exponents α, β in P (W ) The values α = 2, β = 1 are adopted as a first approximation (§V.2). A derivation of α, β from Φ-architecture properties is required, rather than postulation. Prospects: dimensional analysis of the B-formula and arguments for consistency with the quantum-mechanical interpretation (D1.4) [1] may fix α, β uniquely. Existing generalizations of the action principle in the space of cognitive coherence may supply the necessary formalism.

VII.3. Ontological collapse at B → 0 It is informally asserted that a configuration with B → 0 “decoheres” into a pure Ψ without the Ô-structure. A candidate statement is required: under what conditions on the rate of decay |dB/dt| and the time τ does a configuration lose observer status? Candidate formal statement: B(τ ) → 0 ∧ τ < τcrit =⇒ Ô → 0 ∧ Ψ → Ψbare . (7.1) (7.1) is given as a candidate formal statement; τcrit and the conditions on |dB/dt| require specification for a full proof of the ontological collapse at B → 0. The value τcrit is presumed to be related to nmin from §IV.3 and to the dissipation time of ∆out in (3.2). A proof-theoretic analysis is an open problem.

VII.4. The D-Finitude postulate An ODTOE configuration admits infinitely many discrete states in the limit N → ∞. For practical applications a D-Finitude postulate is required: at each dimensionality d the number of discrete configurations N (d) is finite and satisfies the bound N (d) ≤ F (d), where F is an exponentially or polynomially growing function. The question is empirical: what is N (d = 3) for observers of social scale? An indicative estimate N (d = 3) ∼ 1010 . . . 1015 arises from the number of possible social configurations on Earth; it requires refinement.

VII.5. Stitching of S-regions and phase diagram The transition from the high-S regime (neighbourhood of Fix(Φ)) to the low-S regime (fragmentation of the collective) requires a formal phase diagram in the space (S, B, σ). Question: do there exist critical surfaces separating “stable” and “unstable” regions? The analogy with phase transitions in statistical physics (Stanley [13]) suggests a structure; a concrete formalism is an open problem.

Conflict of Interest The author declares no conflict of interest.

Funding This research received no external funding.

REFERENCES 1. Pankratov, A. S. (2026). Observer-Dependent Theory of Everything (ODTOE). ODTOE Preprint. URL: https://odtoe.org/articles/ODTOE_article_EN.pdf. 2. Pankratov, A. S. (2026). The Collective Observer in ODTOE. ODTOE Preprint. URL: https://odtoe.org/articles/ODTOE_collective_observer_EN.pdf. 3. Бугаев Н.В. Основные начала эволюционной монадологии // Вопросы философии и психологии. — 1893. — Кн. 17. — С. 26–44. 4. Whitehead A.N. Process and Reality: An Essay in Cosmology. — New York: Macmillan, 1929. — xii + 547 p. 5. Pankratov, A. S. (2026). Love as Coherence: ODTOE Formalization. ODTOE Preprint. URL: https://odtoe.org/articles/ODTOE_god_love_eternity_EN.pdf.

6. Pankratov, A. S. (2026). The Unified Operator of Synthesis of Axiom and Postulates. ODTOE Preprint. https://odtoe.org/articles/ODTOE_unified_operator_EN.pdf.

ODTOE: URL:

7. Pankratov, A. S. (2026). Infinite Recursion and the Structure Constant in ODTOE. ODTOE Preprint. https://odtoe.org/articles/ODTOE_infinite_recursion_unified_EN.pdf.

FineURL:

8. Leibniz G.W. Monadologie (1714) // Die philosophischen Schriften. Bd. 6. — Berlin: Weidmann, 1885. — S. 607–623. 9. Schauder J. Der Fixpunktsatz in Funktionalräumen // Studia Mathematica. — 1930. — Vol. 2, No. 1. — P. 171–180. DOI: 10.4064/sm-2-1-171-180. 10. Banach S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales // Fundamenta Mathematicae. — 1922. — Vol. 3. — P. 133– 181. DOI: 10.4064/fm-3-1-133-181. 11. Гумилёв Л.Н. Этногенез и биосфера Земли. — 2-е изд., испр. и доп. — Л.: Изд-во ЛГУ, 1989. — 495 с. ISBN 5-288-00332-7. 12. Kuhn T.S. The Structure of Scientific Revolutions. — Chicago: University of Chicago Press, 1962. — 172 p. 13. Stanley H.E. Introduction to Phase Transitions and Critical Phenomena. — Oxford University Press, 1971. — xx + 308 p. 14. Моисеев Н.Н. Универсум. Информация. Общество. — М.: Устойчивый мир, 2001. — 200 с. 15. Флоренский П.А. Столп и утверждение истины. — М.: Путь, 1914. — [4], 812, [2] с. 16. Drozdek A. An Alternative Outlook at Monadic Philosophy: The Monadology of N. V. Bugaev // IDEA – Studia nad strukturą i rozwojem pojęć filozoficznych. — 2018. — Vol. XXX/1. DOI: 10.15290/idea.2018.30.1.18.