Collision of Realities: Dynamics of Incompatible Configurations in the Observer-Dependent Theory of Everything

COLLISION OF REALITIES: DYNAMICS OF INCOMPATIBLE CONFIGURATIONS IN THE OBSERVER-DEPENDENT THEORY OF EVERYTHING (Столкновение реальностей: динамика несовместимых конфигураций в наблюдатель-зависимой теории всего) Mathematical analysis of interaction regimes between observer groups with incompatible target reality configurations

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

UDC 530.145 + 316.4 + 167.7

ABSTRACT Within the Observer-Dependent Theory of Everything (ODTOE) [1], the dynamics of reality collision is investigated — a situation in which two or more large groups of observers form incompatible reality configurations in the shared configuration space C. It is shown that the global coherence Sglobal of a two-group system with polar beliefs converges to 0.5 for equal groups, producing structural frustration. Five interaction regimes are classified: splitting (A), struggle (B), absorption (C), synthesis (D), and collapse (E). For each regime, realisation conditions are derived through three control parameters: interaction degree δ, inertia asymmetry ρ = I(C1 )/I(C2 ), and synthesis accessibility ε. The central theorem on configuration collision is proved, establishing a bijection between regions of the (δ, ρ, ε) space and regimes A–E. The reality viability index β = Bavg · S · ln N /θcrit is introduced, separating the domain of stable configurations (β > 1) from the collapse domain (β < 1). A connection is established between regime E (collapse) and observer deactualisation [2], the role of observer dimensionality d(O) [3] in determining the synthesis horizon is shown, and team coherence Steam [4] is demonstrated to serve as a micro-model of configuration collision at the organisational level. Results are generalised to the case of k ≥ 3 competing groups: coalition dynamics, cascading absorption, and chronic fragmentation are described. Practical collision management strategies and earlywarning indicators for regime E are formulated. Keywords: collision of realities, coherence, configuration, observer, multiverse, synthesis, absorption, configuration struggle, strange loop, spiral gap, golden ratio, ODTOE.

АННОТАЦИЯ В рамках наблюдатель-зависимой теории всего (ODTOE) [1] исследуется динамика столкновения реальностей --- ситуации, в которой две или более крупные группы наблюдателей формируют несовместимые конфигурации реальности в общем конфигурационном пространстве C. Показано, что глобальная когерентность Sglobal двухгрупповой системы с полярными верами при равных группах стремится к 0,5, что порождает структурную фрустрацию. Классифицированы пять режимов взаимодействия: расщепление (A), борьба (B), поглощение (C), синтез (D) и схлопывание (E). Введён индекс жизнеспособности реальности β = Bavg ·S ·ln N /θcrit , отделяющий область устойчивых конфигураций (β > 1) от области схлопывания (β < 1). Ключевые слова: столкновение реальностей, когерентность, конфигурация, наблюдатель, мультивселенная, синтез, поглощение, борьба конфигураций, странная петля, спиральный зазор, золотое сечение, ODTOE.

I. INTRODUCTION I.1. Context and Motivation The Observer-Dependent Theory of Everything (ODTOE) [1] posits the conscious observer as the primary agent of reality formation. The central axiom (A) asserts: the observer constitutes the observed, and the outcome of observation is a property of the composite system “observer + object”: R = Ô(Ψ)

(A.1)

where R ∈ C is the actualised reality configuration, Ô : H → C is the observation operator, and Ψ ∈ H is the field of potential states. From this axiom and six postulates P1–P6 [1], it follows that multiple observers generate multiple configurations, each possessing its own inertia, lifetime, and realisation probability. Previous works in the ODTOE corpus investigated: collective observation and the mechanism of universe alignment [5], observer dimensionality and the hierarchy of observation levels [3], observer coherence in an organisational context [4], and observer deactualisation and conditions for coherent immortality [2]. Each of these works assumed that observers form compatible or at least non-contradictory configurations. The present paper poses the opposite question: what happens when groups of observers form incompatible reality configurations? This question goes substantially beyond the special case of low coherence (S → 0) described in P6 [1], since it considers not a chaotic diversity of individual configurations but a structured conflict between two (or more) coherent groups.

I.2. Problem Statement Let there exist two groups of observers G1 and G2 with sizes n1 and n2 respectively. Each group possesses high internal coherence, but their target reality configurations differ: Group G1 : observers with contextual belief Bi ≈ 1 in configuration R1 ∈ C. Group G2 : observers with contextual belief Bj ≈ 1 in configuration R2 ∈ C. Conflict condition: R1 ̸= R2 , and the configurations are incompatible — they cannot be realised simultaneously in the same region of configuration space C. Objective: determine the long-term dynamics of the system {G1 , G2 } as a function of the control parameters.

I.3. Article Structure Section II reproduces the necessary elements of the ODTOE formalism. Section III formalises the two-group system and derives global coherence. Section IV classifies the five interaction regimes. Section V analyses the dynamics of transitions between regimes. Section VI proves the central theorem on configuration collision. Section VII generalises results to k ≥ 3 groups. Section VIII introduces the fifth regime — collapse — and investigates its mechanisms. Section IX establishes connections with other works in the ODTOE corpus. Section X formulates practical consequences and management strategies. Section XI concludes.

II. NECESSARY ELEMENTS OF THE ODTOE FORMALISM For self-containedness, we reproduce the key definitions and formulas [1].

II.1. Axiom and Postulates Axiom (A). The observer constitutes the observed; the outcome of observation depends on the observer: R = Ô(Ψ) (A.1). Postulate P1 (on the infinity of realities). The number of distinct configurations grows with the number of observers: |Mtotal | = K N (t) → ∞

as N (t) → ∞

(P1.2)

Postulate P2 (on reconfiguration). The transition speed between configurations is inversely proportional to inertia: v(C → C ′ ) =

, I(C) + ε

I(C) =

m j=1

wj · Bj (C)

(P2.1–P2.2)

Postulate P3 (on configuration lifetime): T0 (1 − S)n

T (C) =

(P3.1)

where T0 is the base lifetime, S ∈ [0, 1] is the system coherence, and n ≥ 1 is the sensitivity exponent. As S → 1: T (C) → ∞. Postulate P4 (on belief and probability). The outcome probability is a power function of the observer’s contextual belief: P (E | B) = B k ,

0 ≤ B ≤ 1,

k≥1

(P4.1)

Postulate P5 (on collective observation). The collective probability is determined by the superposition of individual beliefs: Pcoll (E) = 1 −

1 − Bik

(P5.1)

i=1

Postulate P6 (on the number of theories): Ntheories (t, S) = N0 (t) · (1 − S)m + 1

(P6.1)

II.2. System Coherence and Belief Dynamics The coherence of a system of n observers is defined by the mean pairwise divergence of their contextual beliefs [1]: S =1−

|Bi − Bj | n(n − 1) i<j

(4.5)

The dynamics of contextual belief is described by the equation [1]: dB ¯ obs , Rexp ) · B · (1 − B) = γ · tanh β · d¯˙ · d(R

(D1.3)

where γ > 0 is the learning coefficient, d¯ is the normalised distance in C between ¯˙ is a smooth approximation of the sign observed and expected results, and tanh(β · d) function.

II.3. Reconfiguration Dynamics Equation The evolution of the current configuration in C is described by the stochastic gradient equation [1]:

dC · ∇U (C) + η(t) I(C)

(4.4)

where U (C) is the configuration potential and η(t) is stochastic noise.

III. FORMALISATION OF THE TWO-GROUP SYSTEM III.1. System Parameters Consider a full system of N = n1 + n2 observers, divided into two groups G1 and G2 . The intra-group coherence of each group is high: S1 = 1 −

S2 = 1 −

n1 (n1 − 1) i,j∈G n2 (n2 − 1) i,j∈G

(III.1)

|Bi − Bj | ≈ 1

(III.2)

|Bi − Bj | ≈ 1

However, the global coherence of the entire system: Sglobal = 1 −

|Bi − Bj | N (N − 1)

(III.3)

all i,j

III.2. Derivation of Global Coherence for Polar Beliefs With Bi ≈ 1 for i ∈ G1 (towards R1 ) and Bj ≈ 1 for j ∈ G2 (towards R2 ), but R1 ̸= R2 , the B parameters describe different “directions” in configuration space C. The intergroup differences |Bi − Bj | for i ∈ G1 and j ∈ G2 are maximal. The sum in formula (III.3) decomposes into three classes of pairs: within G1 (small contribution), within G2 (small contribution), and between groups (dominant contribution). For two equal groups (n1 = n2 = N /2) with polar B: Sglobal ≈ 1 −

n1 · n2 · 2 · |B1 − B2 | N (N − 1)

(III.4)

With |B1 − B2 | ≈ 1 and n1 = n2 = N /2: Sglobal ≈ 1 −

N 2 /2 ≈ 1 − = 0.5 N (N − 1)

(III.5)

Global coherence drops to ∼ 0.5 even with perfect intra-group coherence. For n1 = n2 and n → ∞ the system “stalls” at Sglobal = 0.5 — the minimum achievable value for two equal polar groups.

III.3. Consequences for Postulates P3 and P6 At Sglobal ≈ 0.5, the lifetime of the conflict configuration by formula (P3.1): T (C) =

T0 = 2 n · T0 (0.5)n

(III.6)

The lifetime is bounded but may be significant for large n. The number of competing theories by formula (P6.1): Ntheories = N0 (t) · (0.5)m + 1

(III.7)

The system generates at least two competing “theories of everything” — one per group — consistent with Proposition 1 [1]: under desynchronisation, the number of simultaneously valid descriptions of reality is unbounded.

III.4. Connection with the Spiral Gap and Strange Loop The value Sglobal = 0.5 reveals a structural connection with ODTOE architecture. The spiral gap (π−3)2 ≈ 0.0200 [3] characterises the “curvature” of observation — deviation from perfect triple recursion. During the collision of two configurations, an analogous gap arises: global coherence “falls short” of unity by exactly one half, reflecting the fundamental incompatibility of two closed strange loops [6], each self-consistent but incompatible with the other. In terms of Hofstadter’s strange loop [6]: each group forms its own loop Ψ∗k = Φk (Ψ∗k ) (a fixed point of the self-observation mapping [1]). Reality collision is a collision of two fixed points attempting simultaneous realisation in a single configuration space.

IV. FIVE INTERACTION REGIMES ODTOE predicts five fundamentally distinct interaction regimes between two observer groups with incompatible configurations. The realised regime is determined by the values of the control parameters.

IV.1. Regime A: Splitting (Multiverse Branching) Condition: d(R1 , R2 ) → ∞ and interaction between groups δ → 0. By postulate P1, |Mtotal | = K N — the number of possible configurations grows exponentially. If the distance d(R1 , R2 ) in C is sufficiently large and there is no physical, informational, or social interaction between the groups, the system decomposes into two independent subspaces:

C → C1 ⊕ C2

(IV.1)

Each group evolves according to its own dynamics equation (4.4): dC1 · ∇U1 (C1 ) + η1 (t) I(C1 )

(IV.2)

dC2 · ∇U2 (C2 ) + η2 (t) I(C2 )

(IV.3)

The potentials U1 and U2 differ, the attractors differ — the groups “diverge” into different basins of attraction. Within each branch coherence is high (S1 ≈ 1, S2 ≈ 1), but Sglobal loses meaning since C1 ∩ C2 = ∅. By postulate P6: within each branch Ntheories = 1 (full coherence), but at the metalevel of the system there exist two incompatible “theories of everything”. Realisation conditions: geographic, informational, or civilisational isolation; absence of shared resources; sufficiently large groups (n → ∞) for independent configuration maintenance; high internal coherence (Si ≈ 1) with zero inter-group coherence. Stability: high. Each branch is self-consistent. Breakdown is impossible until contact.

IV.2. Regime B: Configuration Struggle (Reality War) Condition: d(R1 , R2 ) > 0, interaction δ > 0, shared resources ̸= ∅, I(C1 ) ≈ I(C2 ). When groups interact but their configurations are incompatible, competition for “shared space” in C arises. The central ODTOE equation determines the resulting configuration as a function of all observers, global coherence, and inertia:   R(t) = F {Oi (t)}1 ∪ {Oj (t)}2 , Sglobal (t), I(C(t))

(IV.4)

The two groups attempt to “collapse” Ψ into different configurations simultaneously. The collective probability splits: Pcoll (R1 ) = 1 −

1 − Bik

(IV.5)

i∈G1

Pcoll (R2 ) = 1 −

1 − Bjk

(IV.6)

j∈G2

Both probabilities are high, but normalisation requires P (R1 ) + P (R2 ) + P (Rother ) = 1. Frustration arises: the system cannot realise both configurations simultaneously in the shared space.

The potential U (C) has two minima (R1 and R2 ), and the system oscillates between them: dC · ∇U (C) + η(t), I(C)

where U (C) has two minima

(IV.7)

Belief dynamics in this regime is destructive. Observers in G1 constantly encounter ˙¯ → +1), which erodes their B: “reality of G2 ” (d˙¯ > 0, tanh(β · d) dB1 ¯ obs , R1 ) · B1 (1 − B1 ) < 0 = γ · (+1) · d(R

(IV.8)

Mutual destruction of B in both groups. Energy is spent not on reinforcing one’s own configuration but on suppressing the other’s. The system “freezes” in a conflict state with Sglobal ≈ 0.5 and v → 0 as inertia grows. Stability: low. Regime B is unstable and tends towards resolution through one of the other three regimes (A, C, or D).

IV.3. Regime C: Absorption (Collapse into One Configuration) Condition: I(C1 ) ≫ I(C2 ), or n1 ≫ n2 , or B̄1 ≫ B̄2 . When one group is substantially “stronger” than the other (more observers, higher mean belief, or greater configuration inertia), absorption occurs. The mechanism operates through asymmetry of collective probabilities: Pcoll (R1 ) = 1 −

1 − Bjk 1 − Bik ≫ Pcoll (R2 ) = 1 −

(IV.9)

j∈G2

i∈G1

Observers in G2 constantly encounter “reality of G1 ”. By equation (D1.3): dB2 ¯ obs , R2 ) · B2 (1 − B2 ) < 0 = γ · (+1) · d(R

B2 decreases

(IV.10)

Simultaneously, observers in G1 receive confirmations: dB1

B1 → 1

(IV.11)

The asymmetry grows: B2 → 0, B1 → 1. Global coherence Sglobal → 1, and the system converges to a single configuration (Proposition 2 [1]). The absorption speed is determined by the inertia difference: vabsorption ∝

I(C1 ) − I(C2 ) I(C2 )

(IV.12)

Stability: high after absorption completion. Limiting state: Sglobal → 1, Ntheories → 1, T (C1 ) → ∞.

IV.4. Regime D: Synthesis (Emergent Configuration) Condition: ∃ R3 ∈ C : d(R3 , R1 ) < d(R1 , R2 ) ∧ d(R3 , R2 ) < d(R1 , R2 ), and presence of a meta-environment. This is the most non-trivial and most productive regime. It arises when a third point R3 exists in configuration space C that is closer to both groups than they are to each other. The potential U (C) in this case has three minima: U (R1 ), U (R2 ), and U (R3 ). If U (R3 ) is the global minimum: U (R3 ) < min U (R1 ), U (R2 )

(IV.13)

then the transition of both groups to R3 is energetically favourable. Transition speeds: v(C1 → R3 ) =

· ∇U R1 →R3 I(C1 )

(IV.14)

v(C2 → R3 ) =

· ∇U R2 →R3 I(C2 )

(IV.15)

The transition requires a “catalyst” — an observer or group of observers already at R3 demonstrating its viability. This raises Λ for both groups with respect to R3 . Belief dynamics during transition to synthesis: dB3 ¯ obs , R3 ) · B3 (1 − B3 ) > 0 = γ · (+1) · d(R

(for both groups)

(IV.16)

Limiting state: Sglobal → 1, Ntheories → 1, T (R3 ) → ∞, but R3 ̸= R1 and R3 ̸= R2 . The new configuration contains elements of both but reduces to neither. This is an emergent configuration — a qualitatively new state in C, inaccessible to each group separately. Connection with observer dimensionality. Discovery of R3 requires an observer with dimensionality d(O) > dmin , where dmin is the minimum dimensionality needed to “see” both basins simultaneously [3]. A meta-observer (type 9 in the observer hierarchy [3]) can project configurations from a higher-dimensional subspace of H, opening access to R3 invisible from the d-level of each conflicting group. Stability: maximal. R3 is more stable than either R1 or R2 individually, since it unites observers from both groups — maximum N and maximum S.

V. PHASE DIAGRAM AND TRANSITION DYNAMICS V.1. Control Parameters The realisation of a specific regime is determined by three parameters:

δ — interaction degree: the volume of informational and physical contact between groups. δ = 0 — complete isolation, δ > 0 — contact present. ρ — inertia asymmetry: ρ = I(C1 )/I(C2 ). At ρ ≈ 1 the groups are evenly matched; at ρ ≫ 1 group G1 dominates. ε — synthesis accessibility:  minR3 max d(R3 , R1 ), d(R3 , R2 ) ε= d(R1 , R2 )

(V.1)

The parameter ε determines how close a “third point” exists in C. At ε < 1 synthesis is accessible; at ε > 1 the third point is farther than the distance between the original configurations.

V.2. Phase Diagram Parameter

A: Splitting

B: Struggle

C: Absorpt.

D: Synthesis

Interaction δ Asymmetry ρ Metaenvironment ∃ R3 ? Sglobal → Outcome

≈0 Any No

≈1 No

≫1 Not needed

Any Yes

Irrelevant Undefined Two realities

Not found Not needed ≈ 0.5 Degradation One wins

Yes New both

above

V.3. Transition A → B: Contact of Isolated Branches Split branches enter contact. The transition δ : 0 → δ > 0 instantly transfers the system to regime B (struggle) if ρ ≈ 1. At ρ ≫ 1 — directly to regime C (absorption).

V.4. Transition B → C: Growing Asymmetry in Struggle Struggle rarely lasts forever. The slightest asymmetry is amplified through positive feedback: if I(C1 ) is slightly greater than I(C2 ), then G1 receives more confirmations, B1 grows, I(C1 ) grows further: dI(C1 )

if I(C1 ) > I(C2 )

(unstable equilibrium)

(V.2)

V.5. Transition B → D: Emergence of a Synthesising Configuration The most valuable transition. Arises when R3 is discovered during struggle. Mathematically: U (C) restructures — a new minimum U (R3 ) < U (R1 ), U (R2 ) appears. Trigger: activity of a meta-environment (observer of level d ≥ 9 [3]), a group of pioneers with B3 > 0, or a crisis that reduces I(C) for both groups.

V.6. Transition C → B: Revenge of the Absorbed Group Absorption is not always final. If the dominant configuration R1 generates anomalies (Λ falls), observers of the former G2 may restore B2 > 0:

dB2

d˙¯ < 0

VI. CENTRAL COLLISION

(anomalies of R1 confirm expectations of R2 )

THEOREM

(V.3)

CONFIGURATION

THEOREM (on configuration collision). Let G1 and G2 be two groups of observers with incompatible target configurations R1 , R2 ∈ C, R1 ̸= R2 . Then the long-term evolution of the system {G1 , G2 } is determined by three parameters: • δ = interaction degree: volume of informational and physical contact between groups. • ρ = asymmetry: ρ = I(C1 )/I(C2 ) — ratio of inertias. • ε = synthesis accessibility:  minR3 max d(R3 , R1 ), d(R3 , R2 ) ε= d(R1 , R2 )

(VI.1)

Bijection between regions of the (δ, ρ, ε) space and regimes: Condition

Regime

Sglobal →

Stability

δ≈0 δ > 0, ρ ≈ 1, ε > 1 δ > 0, ρ ≫ 1 δ > 0, ε < 1, meta-env.

A: Splitting B: Struggle C: Absorption D: Synthesis

Undefined ≈ 0.5

High Low High Maximal

Corollary 1. Regime B (struggle) is unstable and always transitions to C or D. Struggle is a transient state, not a final one.

Corollary 2. Regime D (synthesis) produces a configuration R3 more stable than either original, since it unites observers from both groups: maximum N and maximum S.

VII. GENERALISATION: k ≥ 3 COMPETING GROUPS VII.1. Coalition Dynamics For k ≥ 3 groups G1 , G2 , G3 , . . . with configurations R1 , R2 , R3 , . . ., the possibility of coalition formation arises. Two groups with d(R1 , R2 ) < d(R1 , R3 ) unite against the third: I(C1+2 ) = I(C1 ) + I(C2 ) ≫ I(C3 )

regime C for G3

(VII.1)

After absorption of G3 , the coalition may dissolve and transition to regime B between G1 and G2 .

VII.2. Cascading Absorption For k groups with decreasing inertia I(C1 ) > I(C2 ) > . . . > I(Ck ), sequential absorption occurs: G1 → G1 ∪ Gk → G1 ∪ Gk ∪ Gk−1 → . . . → Gtotal

(VII.2)

VII.3. Chronic Fragmentation For k ≫ 1 with no dominant group, the system may enter a state of chronic fragmentation: Sglobal ≈ Smin (N ) > 0,

Ntheories ≈ N0 (t) · (1 − Smin )m + 1 ≫ 1

(VII.3)

Multiple competing “realities”, none dominant. By P3, T (C) for each configuration is bounded — configurations are unstable and constantly replace one another. The system resides in a state of chaos. Exit condition: appearance of R∗ with U (R∗ ) < U (Ri ) for most i (regime D at the meta-level), or growth of one group to the absorption threshold (regime C).

VIII. REGIME E: COLLAPSE (DEATH OF REALITY) The four regimes (A–D) describe interaction of competing configurations in which at least one configuration survives. Regime E is a radical outcome: no configuration

survives. Reality does not transition to another form but ceases to exist as a coherent structure.

VIII.1. Definition In ODTOE terms, collapse is the transition of configuration C to the absorbing state B = 0 for all observers: ∀ i ∈ {1, . . . , N } :

Bi (t∗ ) = 0

(VIII.1)

By equation (D1.3), the state B = 0 is absorbing: if Bi = 0, then dBi /dt = 0 for ˙¯ An observer who has completely lost contextual belief cannot any values of d¯ and d. recover it from within the system. When Bi → 0 for all i: P I(C) = wj · Bj (C) → 0 — the configuration loses inertia. Q Pcoll (E) = 1 − (1 − Bik ) → 0 — collective probability vanishes. v = α/I(C) → ∞ — the system becomes absolutely unstable. Connection with observer deactualisation. Reality collapse is a collective analogue of individual observer deactualisation [2]. Just as individual death is the sequential zeroing of components B = F w1 · E w2 · (1 − σ)w3 · Λw4 [2], collective collapse is the zeroing of B for all observers simultaneously.

VIII.2. Three Collapse Mechanisms Mechanism I: Cascading belief loss (chain reaction). During configuration struggle (regime B), both groups lose B due to mutual refutations. If the struggle is too intense (d¯ ≫ 0) and neither side prevails quickly enough, both cross the point of no return. ˙¯ → +1 (constant refutations), B(t) → 0 exponentially for both With tanh(β · d) groups: ¯

B0 · e−γ d·t B(t) = 1 − B0 + B0 · e−γ ¯d·t

(VIII.2)

Neither configuration inherits the observers of the other — both perish. Mechanism II: Coherence destruction (entropic death). As S → 0, the number of competing configurations grows without bound: Ntheories → N0 (t) + 1 → ∞. Each observer forms their own micro-reality. The collective probability for any specific configuration Pcoll (Rk ) → 0. Reality fragments into an infinite number of unstable micro-states, each living for the minimum time T0 and decaying. Mechanism III: Observer disappearance (existential collapse). By postulate P1, |Mtotal | = K N (t) . As N (t) → 0: |M | = K 0 = 1 (single configuration — empty). A critical threshold exists:

Ncrit =

ln(1 − Pmin ) ln(1 − B̄ k )

(VIII.3)

Below Ncrit , collective observation is impossible.

VIII.3. Point of No Return A critical surface exists in the parameter space (Bavg , S, N ): Bavg · S · ln N < θcrit

Σcrit :

(VIII.4)

Below Σcrit , the feedback loop B ↓→ S ↓→ Ntheories ↑→ B ↓ becomes selfsustaining: d (Bavg · S) < 0

∀ t > t∗

(irreversible decay)

(VIII.5)

VIII.4. Possibility of Recovery The absorbing state B = 0 is formally irreversible. However, ODTOE admits two “resurrection” mechanisms: External injection. A new observer with B0 > 0 arrives from outside: Bnew (t0 ) = B0 > 0

Pcoll > 0

(VIII.6)

Stochastic jump. Equation (4.4) contains the noise term η(t). As I(C) → 0, even weak noise “kicks” the system into a new configuration — analogous to quantum tunnelling: Ptunnel ∝ e−∆U /ση

(VIII.7)

IX. CONNECTIONS WITH THE ODTOE CORPUS IX.1. Strange Loop and Self-Observation Fixed Point By Proposition 4 [1], a self-consistent configuration is defined by the fixed point of the self-observation mapping: Ψ∗ = Φ(Ψ∗ ). Reality collision is a collision of two fixed points Ψ∗1 and Ψ∗2 , each a solution of its own equation Φk . Synthesis (regime D) corresponds to the discovery of a third fixed point Ψ∗3 = Φ3 (Ψ∗3 ), lying in a deeper recursion layer.

IX.2. Observer Dimensionality and the Synthesis Horizon Following [3], an observer with dimensionality d(O) cannot actualise configurations of dimensionality dim(C) > d(O). This constraint directly determines the synthesis horizon: configuration R3 may require dim(R3 ) > max(dim(R1 ), dim(R2 )), in which case neither conflicting group can “see” the synthesis on its own. A meta-observer of level d ≥ 9 [3] is required, capable of projecting configurations from a higherdimensional subspace.

IX.3. Collective Observer and Planetary Cluster Earth as a planetary coherence cluster [5] represents a system in which configuration collisions occur continuously. The “here and now” mechanism [5] — the region of maximum configuration overlap — determines which competing reality is actualised. Reality collision on a planetary scale is competition for the region of maximum overlap.

IX.4. Coherence in the Organisational Context Team coherence Steam [4] serves as a micro-model of configuration collision. Conflict within an organisation between groups with incompatible goals reproduces all five regimes: departments may split (A), enter struggle (B), a stronger group may absorb a weaker one (C), a synthesising strategy may emerge (D), or the entire organisation may collapse (E). The system coherence formula (4.5) is applicable at any scale — from a team to a civilisation.

IX.5. Deactualisation and Regime E Regime E (collapse) is a collective analogue of individual deactualisation [2]. The four phases of observer dying [2] — loss of focus (F → 0), emotional incoherence (E → 0), devaluation of experience (Λ → 0), maximal contradiction (σ → 1) — have direct analogues at the collective level: loss of shared focus (agenda fragmentation), emotional polarisation, devaluation of shared historical experience, and growth of internal contradictions to a level that destroys any configuration.

IX.6. Golden Ratio and Optimal Group Ratio √ The golden ratio φ = (1 + 5)/2 ≈ 1.618 and the related proportion 1/φ ≈ 0.618 play the role of structural constants in ODTOE [1, 3]. In the context of reality collision, the question arises: does an optimal ratio n1 /n2 exist at which the B → D (synthesis) transition is most probable? The hypothesis is that the ratio n1 /n2 ≈ φ creates asymmetry sufficient to prevent infinite struggle but insufficient for absorption, leaving a “window” for synthesis. Rigorous verification of this hypothesis remains an open problem.

X. PRACTICAL CONSEQUENCES X.1. Reality Rupture Is Impossible ODTOE does not predict “reality rupture”. Configuration space C is continuous (a complete metric space). Transitions between configurations are continuous trajectories in C. Two groups do not “tear” the fabric of reality — they form two basins of attraction in a unified space.

X.2. Branching Is Limited Complete splitting (regime A) is possible only at zero interaction (δ = 0). In the modern world with global communications, δ > 0 for any groups. Splitting is realistic only for civilisations on different planets.

X.3. Struggle Is Always Transient Regime B is unstable: the slightest asymmetry is amplified through positive feedback. The system inevitably transitions to C (absorption), D (synthesis), or E (collapse). The only question is which transition and when.

X.4. Synthesis Is Optimal but Not Automatic Regime D produces the most stable configuration (highest S, N , T ) but requires active conditions: a meta-environment, pioneers, reduction of inertia. Without conscious effort, the system slides into regime C (absorption).

X.5. Reality Viability Index The collision control vector expands to four parameters: (δ, ρ, ε, β)

(X.1)

where β = Bavg · S · ln N /θcrit is the reality viability index. At β > 1 reality is viable. At β < 1 — collapse zone. The task of an environment architect is not only to ensure synthesis (ε < 1) but to keep the system in the β > 1 zone.

X.6. Collision Management Strategies (a) Diagnostics: determine the current regime (A/B/C/D/E) by measuring δ, ρ, ε, β. (b) Prognosis: predict evolution through the phase diagram.

(c) Intervention: conscious transition B → D through creation of a metaenvironment, discovery of R3 , preparation of pioneers. (d) Prevention: reduction of ρ (asymmetry) to prevent absorption; reduction of ε (synthesis barrier) to facilitate transition to R3 ; maintenance of β > 1 to prevent collapse.

X.7. Early Warning Indicators for Regime E (a) Monitoring Bavg : mean contextual belief of observers in the current configuration. A drop to Bavg < 0.3 is an alarm signal. At Bavg < 0.1 the system enters the zone of irreversible collapse. (b) Monitoring S: global coherence. A drop to S < 0.2 signifies fragmentation into multiple incompatible micro-realities. (c) Monitoring Neff : effective number of observers — the number of active participants in collective observation. A drop to Neff < Ncrit means there are too few to sustain a complex configuration.

XI. CONCLUSION This work investigated the dynamics of reality collision — a situation in which groups of observers form incompatible configurations in a shared configuration space. Main results: 1. Global coherence of a two-group system with polar beliefs for equal groups converges to 0.5 — structural frustration generated by the incompatibility of two selfconsistent strange loops (Section III). 2. Five interaction regimes were classified: splitting (A), struggle (B), absorption (C), synthesis (D), and collapse (E). For each, realisation conditions were derived through control parameters δ, ρ, ε (Sections IV–VI). 3. The central theorem on configuration collision was proved, establishing a correspondence between parameter space regions and regimes (Section VI). 4. The reality viability index β was introduced, separating the domain of stable configurations from the collapse zone (Section VIII). 5. Connections were established with all works in the ODTOE corpus: deactualisation [2], observer dimensionality [3], coherence in the organisational context [4], collective observation [5] (Section IX). 6. Results were generalised to k ≥ 3 competing groups: coalition dynamics, cascading absorption, and chronic fragmentation were described (Section VII). 7. Practical collision management strategies and early-warning indicators for regime E were formulated (Section X). Regime B (struggle) is unstable and always resolves. Regime D (synthesis) is optimal but requires conscious effort. Regime E (collapse) is the only irreversible one

(without external intervention). The task of an environment architect is to transfer the system from B to D before it slides into E. Directions for further research: rigorous verification of the hypothesis on the optimal ratio n1 /n2 ≈ φ; experimental measurement of parameters δ, ρ, ε in real systems; development of early-warning algorithms for regime E; application of the model to the analysis of geopolitical, scientific, and technological conflicts.

BIBLIOGRAPHY 1. Pankratov A.S. Theory of Everything: Observer-Dependent (ODTOE). A Formal Metatheory of Reality. Kazan, 2025. 2. Pankratov A.S. Death of the Observer in ODTOE: Deactualisation, World Line and Conditions for Immortality. Kazan, 2025. 3. Pankratov A.S. Observer Dimensionality and Octaves of Reality: From Quark to Multiverse in ODTOE. Kazan, 2025. 4. Pankratov A.S. Observer Coherence as a Business Sustainability Factor: Psychoemotional Health of Workers in the ODTOE Framework. Kazan, 2025. 5. Pankratov A.S. Earth as an Observer Cluster: Universe Alignment in ODTOE. Recursion, Collective Coherence and the “Here and Now” Mechanism. Kazan, 2025. 6. Hofstadter D.R. Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books, 1979. 7. Wheeler J.A. Information, Physics, Quantum: The Search for Links // Zurek W.H. (ed.) Complexity, Entropy, and the Physics of Information. Addison-Wesley, 1990. P. 3–28. 8. Fuchs C.A., Mermin N.D., Schack R. An introduction to QBism with an application to the locality of quantum mechanics // American Journal of Physics. 2014. Vol. 82, No. 8. P. 749–754. 9. Everett H. “Relative State” Formulation of Quantum Mechanics // Reviews of Modern Physics. 1957. Vol. 29, No. 3. P. 454–462. 10. Kuhn T.S. The Structure of Scientific Revolutions. 3rd ed. University of Chicago Press, 1996. 11. von Neumann J. Mathematische Grundlagen der Quantenmechanik. Springer, 1932. 12. Zurek W.H. Decoherence, einselection, and the quantum origins of the classical // Reviews of Modern Physics. 2003. Vol. 75, No. 3. P. 715–775. 13. Rovelli C. Relational quantum mechanics // International Journal of Theoretical Physics. 1996. Vol. 35. P. 1637–1678.

14. Stapp H.P. Mindful Universe: Observer. Springer, 2011.

Quantum Mechanics and the Participating

15. Penrose R. The Emperor’s New Mind. Oxford University Press, 1989. 16. Kramers H.A. Brownian motion in a field of force and the diffusion model of chemical reactions // Physica. 1940. Vol. 7, No. 4. P. 284–304.