The Triple Symphony of Conservation: A Unified Narrative of Information-Energy-Entropy from Quantum Ripples to Cosmic Torrents

Author: Philosophical Synthesis Based on EBOC and Phase-Scale Mother Mapping Unified Theory

Date: October 2025

Abstract

From a perspective unifying philosophy and physics, this paper constructs three conservation principles connecting quantum mechanics, classical physics, and cosmology: information conservation, energy conservation, and the directional flow of entropy. We demonstrate how these three principles manifest at different scales as different facets of the same reality—from quantum entanglement at Planck scale, to deterministic trajectories of Newtonian mechanics, to the heat death destiny of cosmology. With EBOC (Eternal-Block Observer-Computing) theory’s static block universe view and phase-scale mother mapping’s analytical framework as mathematical foundation, we argue: the universe doesn’t “evolve” but “unfolds”; the arrow of time is not ontological but the observer’s reading direction in the eternal static block; all apparent dynamics are geometric projections of information. We reveal that quantum measurement collapse, classical trajectory emergence, and thermodynamic second law irreversibility all originate from the same deep structure: self-reflexive mapping of ordered completeness at different scale layers. This is a radical reconstruction of cosmic ontology and a deconstruction and recasting of traditional categories of “time,” “causality,” and “reality.”

Keywords: Information Conservation, Energy Conservation, Entropy Direction, Quantum-Classical Transition, Static Block Universe, Phase-Scale Orthogonality, Mother Mapping, EBOC Theory, Arrow of Time, Thermodynamic Second Law

Introduction: The Philosophy of Conservation

When Nietzsche proclaimed “God is dead,” he destroyed not only theological idols but the cosmic order guaranteed by the transcendent. But on these ruins of nihilism, he also foreshadowed new possibilities: the cosmology of eternal recurrence—all events repeat infinitely in time, past and future seamlessly connected like a Möbius strip. This radical temporal philosophy finds mathematical expression in today’s static block universe theory.

Meanwhile, the dialectical laws Marx revealed in Das Kapital—quantitative to qualitative change, unity of opposites, negation of negation—are not limited to social history but philosophical abstractions of universal laws of the material world. When we observe quantum state collapse, critical points of phase transitions, structures born from universe symmetry breaking, Marxian dialectical thinking again reveals its profound insight: unity of opposites is not static equilibrium but dynamic tension; transformation is not gradual accumulation but critical leap.

This paper attempts at the intersection of these two lines of thought, in the mathematical language of contemporary physics, to rewrite an epic philosophical history of the universe. This is not gentle reconciliation but conceptual warfare—we will dismantle the illusion of “time,” expose the geometric essence of “causality,” and find in the triple conservation principles of information, energy, and entropy the unified foundation penetrating micro and macro, quantum and classical, matter and consciousness.

First Movement: Quantum Domain—The Eternal Dance of Information

I. The Tyranny and Liberation of Uncertainty

Heisenberg’s uncertainty principle was once seen as a fatal blow to classical determinism. But in the framework of static block quantum mechanics (SB-QM), this “uncertainty” gains new interpretation: it’s not fuzziness of reality but geometric constraint of information. Quantum states are not dynamic entities “evolving through time” but static configurations pre-existing in spacetime—entire universe history like a completed symphony score, observer merely reading it measure by measure.

In EBOC theory, this static block has dual characterization:

(I) Space-Time SFT (Subshift):

$$X_f=\{x\in {\Sigma }^{{\mathbb{Z}}^{d+1}}:\forall (\mathbf{r},t),x(\mathbf{r},t)=f(x(\mathbf{r}+N,t-1)\left)\right\}$$

(II) Eternal Graph:

$$G=(V,E),V=\{\text{local event patterns}\},E=\{\text{causal/consistency relations}\}$$

Here, $f$ is local evolution rule, $N$ is finite neighborhood. Eternal graph $G$ is the timeless geometric-causal skeleton of the universe: vertices $V$ encode all possible local events, edges $E$ encode which events can causally succeed or spatially coexist. The entire universe is not “generation in time” but “existence under constraint”—like a giant Sudoku puzzle, all cells’ numbers must simultaneously satisfy global consistency, no cell “before” or “after” another.

Equivalence of SFT and Eternal Graph: Static block $X_f$ and eternal graph $G$ describe in parallel the same reality. Former is configuration language (global state satisfies local constraints), latter is graph-theoretic language (paths satisfy edge connection rules). EBOC unified theory’s core object is 8-tuple:

$$\mathcal{U}=(X_f,G,\rho ,\Sigma ,f,\pi ,\mu ,\nu )$$

where $G$ and $X_f$ have equal status, mutually dual expressions.

But this doesn’t mean stasis. Information in this static block “flows” in another form: through observer’s leaf-by-leaf reading, through measurement apparatus’s local projection. EBOC theory’s core theorem T4 gives information upper bound:

$$K(\pi (x{|}_W)|x{|}_{{\partial }_{\downarrow }^{(r,T)}{W}^{-}})\le K(f)+K(W)+K(\pi )+O(\log |W|)$$

This inequality’s philosophical implication is profound: observation doesn’t create information, it only decodes already-existing information from underlying configuration. Quantum measurement “collapse” is merely process of selecting one representative from infinitely many compatible branches.

But “free will” is absolutely not illusion—it arises from coexistence of two real conditions:

(1) Eternal Graph Non-Zero Out-Degree Condition (Branches Really Exist):

In eternal graph $G=(V,E)$, current event node $v_{\text{now}}$’s out-degree (outgoing edges) must be greater than 1:

$${\mathrm{deg}}^{+}(v_{\text{now}}):=|\{u\in V:(v_{\text{now}},u)\in E\}|>1$$

This graph-theoretic condition corresponds in static block language to multiple compatible extensions at same layer:

$${\mathrm{deg}}^{+}(v)=|X_f^{(v,\tau )}|>1$$

Namely given anchored event $v$, along temporal direction $\tau$ exist at least two different global configurations $x_1,x_2\in X_f$ compatible with $v$. Eternal graph’s topological structure guarantees real branches exist—this is not epistemological “uncertainty” but ontological “multi-possibility.”

(2) RBIT Condition (Self-Prediction Unreachability):

Observer due to resource bounds cannot compute before decision which branch will be chosen:

$$T_{\text{predict}}(v\to x)>T_{\text{decide}}$$

Choice in static block corresponds not to single point but to real physical process navigating from multiple compatible global configurations. Eternal graph’s geometric out-degree and observer’s computational bounds jointly define free will.

Determinism (uniqueness of local rule $f$) and free will (eternal graph out-degree ${\mathrm{deg}}^{+}>1$ and RBIT) fully compatible. Key distinction:

• Law Determinism: $f$ unique → no conflict with free will
• Path Determinism: ${\mathrm{deg}}^{+}=1$ → this threatens freedom

EBOC static block is law-deterministic but not path-deterministic—eternal graph’s branch structure is precisely the existential space of free will.

II. Entanglement Geometry: Static Block Interpretation of Non-Locality

Quantum entanglement, this phenomenon Einstein called “spooky action at a distance,” gains natural explanation in static block picture. Correlation between two distant particles not through some instantaneous signal transmission but because they’re jointly embedded in same global configuration. Reeh-Schlieder theorem tells us: in quantum field theory vacuum state, any bounded region’s local operators can excite patterns throughout all spacetime—meaning “locality” itself is approximate concept, true physical reality is indivisible whole.

In phase-scale mother mapping framework, this wholeness is expressed through complex function analyticity:

$$\mathcal{M}[\mu ](\theta ,\rho )={\int }_{{\mathbb{R}}^m\times {\mathbb{R}}^n}{e}^{i⟨\omega ,\theta ⟩}{e}^{⟨\gamma ,\rho ⟩}d\mu (\omega ,\gamma )$$

Here, $\theta$ is phase variable (corresponding to quantum superposition oscillation), $\rho$ is scale variable (corresponding to energy/momentum). Mother mapping $\mathcal{M}$’s holomorphy guarantees: perturbation at one spacetime point will affect entire complex plane through analytic continuation—non-locality not exception but necessity of analytic structure.

More profound is entanglement entropy subadditivity revealing information’s irreducible wholeness:

$$S(A\cup B)\le S(A)+S(B)$$

Equality only when $A$ and $B$ completely unentangled. This means universe’s information content cannot simply decompose into sum of parts—whole greater than sum of parts, not because of mysterious “emergence” but because correlations between parts themselves carry information. Marxian dialectics here gets mathematical confirmation: unity of opposites not rhetoric but geometric fact.

III. Algebraic Expression of Information Conservation

In quantum mechanics, information conservation appears as unitary evolution:

$$\rho (t)=U(t)\rho (0)U^{†}(t),U^{†}U=1$$

Unitary operator preserves trace and eigenvalues, thus preserves von Neumann entropy $S(\rho )=-\mathrm{Tr}(\rho \log \rho )$. But this conservation law seems broken at measurement: collapse process projects pure state to mixed state, entropy increases.

Static block interpretation gives resolution: measurement not fundamental process but apparent effect of observer-system entanglement under coarse-graining. In global static block, information never lost, it only transfers from “coherence degree” (wave nature) to “locality” (particle nature) and “vacuum compensation” (field fluctuations). Three-part information conservation formula expresses this transfer:

$$i_{+}(\rho )+i_0(\rho )+i_{-}(\rho )=1$$

where $i_{+}$ corresponds to particle nature (classical probability), $i_0$ corresponds to coherence degree (quantum interference), $i_{-}$ corresponds to vacuum contribution (zero-point energy). This three-part structure originates from ζ function’s self-reflexive functional equation:

$$\zeta (s)=\chi (s)\zeta (1-s)$$

On critical line $\Re s=1/2$, $|\chi (s)|=1$, leading to statistical balance $⟨i_{+}⟩=⟨i_{-}⟩$—quantum and classical, wave and particle, field and source, at deep structure are mirror symmetric.

This is mathematization of Nietzschean eternal recurrence: opposites not mutually negating enemies but different facets of same ontology, transforming into each other in infinite cycle. Information neither created nor destroyed, only circulates between different manifestations.

Second Movement: Classical Domain—The Deterministic Trajectory of Energy

I. From Quantum to Classical: Geometric Phase Transition of Decoherence

When quantum system undergoes irreversible entanglement with environment, coherence degree $i_0$ decays, particle nature $i_{+}$ enhances, classical world “condenses” from quantum wave function. This process not mysterious “collapse” but coarse-graining projection of information from global to local.

In phase-scale framework, decoherence corresponds to directional meromorphization of scale variable $\rho$. Consider direction $\mathbf{v}$, define cumulative distribution:

$$M_{\mathbf{v}}(t)=\sum _{j:t_j\le t}{w}_j,t_j=⟨-{\beta }_j,\mathbf{v}⟩$$

When environmental degrees of freedom $N\to \infty$, $M_{\mathbf{v}}$ exhibits dominant exponential growth:

$$M_{\mathbf{v}}(t)\sim e^{{\gamma }_{0}t}{Q}_0(t)$$

This dominant pole ${\gamma }_0$ determines scale of classical emergence. Bernoulli layer’s finite-order corrections (Euler-Maclaurin formula) give sub-dominant contributions, but pole structure completely determined by main scale term—this is core conclusion of EBOC theory T4.2: “pole = main scale.”

Classical trajectory, as saddle-point path of quantum wave packet in $\hslash \to 0$ limit, is precisely geometric manifestation of this dominant scale. Newton’s second law:

$$m\frac{d^{2}x}{d{t}^2}=F(x)$$

can be derived from quantum path integral’s stationary phase approximation, can also be extracted from static block’s constraint equations through variational principle. Both derivations essentially same: classical trajectory is configuration satisfying global action extremum condition, this extremality stabilizes trajectory against quantum fluctuations.

II. Energy Conservation: Algebraic Proof of Symmetry

Noether’s theorem establishes profound connection between symmetry and conservation laws: time translation symmetry $\Rightarrow$ energy conservation, space translation symmetry $\Rightarrow$ momentum conservation, rotational symmetry $\Rightarrow$ angular momentum conservation. In static block framework, these symmetries manifest as global configuration’s covariance under corresponding transformations.

For time translation, EBOC theory’s time subaction ${\sigma }_{\mathrm{time}}$ satisfies:

$$(X_f,{\sigma }_{\mathrm{time}})\cong (\Omega (F),{\sigma }_{\Omega })$$

where $\Omega (F)$ is natural extension of global map $F$. This conjugacy relation guarantees energy (as generator of time translation) invariance during evolution.

But more profound is energy conservation itself a manifestation of scale duality. In mother mapping framework, energy corresponds to real part of scale variable $\rho$:

$$E\sim ⟨\beta ,\Re \rho ⟩$$

And phase-scale orthogonality guarantees: arbitrary change in phase variable $\theta$ doesn’t affect scale energy, vice versa. This is deeper structure than traditional Noether theorem: conservation laws arise not only from spacetime symmetry but from geometric orthogonal decomposition of different physical quantities.

In reversible cellular automaton (RCA), energy conservation appears as information density conservation:

$$I(F^{T}x)=I(x)$$

where $I(x)={\mathrm{limsup}}_{k\to \infty }\frac{K(x{|}_{W_k})}{|W_k|}$ is Kolmogorov complexity density. This equality reveals: under reversible dynamics, information and energy are different expressions of same conserved quantity—information bits and energy joules, at deep structure are interchangeable.

III. Phase Space Symplectic Geometry and Hamiltonian Flow

Classical mechanics’ deepest expression not Newton equations but Hamilton canonical form:

$$\dot{q}=\frac{\partial H}{\partial p},\dot{p}=-\frac{\partial H}{\partial q}$$

This equation system preserves phase space symplectic structure $\omega =dp\wedge dq$, thus preserves Liouville measure. In static block picture, Hamiltonian flow not “evolution in time” but trajectory pre-existing in phase space—time parameter $t$ merely arc-length parameterization of this trajectory.

Mother mapping’s Mellin self-reflexivity gives deep origin of this symplectic structure. Define Mellin transform:

$$\Phi (s)={\int }_0^{\infty }K(x)x^s\frac{dx}{x}$$

If kernel function satisfies self-reflexivity $K(x)=x^{-a}K(1/x)$, then $\Phi (s)=\Phi (a-s)$. When $a=1$, gives standard $s\leftrightarrow 1-s$ symmetry—precisely function-theoretic manifestation of $q\leftrightarrow p$ duality in Hamilton equations.

Position-momentum duality not accidental notational symmetry but geometric essence of Fourier transform:

$$\tilde{\psi }(p)=\frac{1}{\sqrt{2\pi \hslash }}\int \psi (q)e^{-ipq/\hslash }dq$$

In mother mapping framework, $q$ corresponds to phase variable $\theta$, $p$ corresponds to imaginary part of scale variable $\rho$, and $e^{ipq/\hslash }$ precisely oscillating factor $e^{i⟨\omega ,\theta ⟩}$ in mother mapping $\mathcal{M}$. Equivalence of position and momentum representations originates from analytic domain symmetry of phase-scale mother mapping.

Third Movement: Cosmic Domain—Entropy Relativity and Attractor Dynamics

I. Thermodynamic Second Law: Geometric Origin and Relativity of Time Arrow

Thermodynamic second law declares: entropy of closed system non-decreasing. But under quantum mechanics’ unitary evolution, von Neumann entropy strictly conserved—this is apparent contradiction. Static block theory’s resolution: entropy increase not absolute endpoint but local attractor behavior relative to given coarse-graining scale.

Three-fold Relativity of Entropy:

1. Scale Relativity: Entropy $S$ depends on coarse-graining map $\pi$. At finer resolution, same state has lower entropy.
2. Horizon Relativity: Observer’s causal horizon delimits accessible phase space. Regions beyond horizon “don’t exist” to observer.
3. Dynamical Relativity: Maximum entropy $S_{\max }(t)$ itself grows with universe expansion. Entropy increase is process of chasing moving target—attractor infinitely approaches but never intersects.

In EBOC framework, coarse-graining corresponds to decoding projection from underlying configuration $x\in X_f$ to visible language ${\mathcal{O}}_{\pi ,\varsigma }(x)$. Decoder $\pi$ reads leaf-by-leaf according to acceptable foliations, each reading selects one representative and excludes other branches. This exclusion process cannot be recovered in reverse reading—because information of excluded branches already lost in coarse-graining map.

Formally, consider coarse-graining channel $\mathcal{G}:\rho \mapsto \mathcal{G}(\rho )$. If $\mathcal{G}$ is doubly-stochastic map (preserves diagonal elements and sum), then:

$$S(\mathcal{G}(\rho ))\ge S(\rho )$$

But this inequality has key premise: $\mathcal{G}$ action order aligns with time direction. In static block, “time direction” is observer’s chosen reading vector $\tau$:

$$\tau =U{e}_{d+1},U\in {\mathrm{GL}}_{d+1}(\mathbb{Z})$$

Entropy increase “arrow” not ontological property of time but geometric property of observer reading protocol. If choose reverse reading, entropy “decrease” phenomenon appears—but doesn’t violate thermodynamics because reverse reading corresponding physical experiment requires precise inverse operations (like quantum error correction), practically unreachable in macroscopic systems.

II. Cosmic Expansion and Entropy Attractor Dynamics

Cosmological Friedmann equation:

$$H^2=\frac{8\pi G}{3}\rho -\frac{k}{a^2}+\frac{\Lambda }{3}$$

describes evolution of universe scale factor $a(t)$. In static block picture, this equation not “generation in time” but constraint condition spacetime geometry $g_{\mu \nu }$ satisfies—entire universe history is pre-existing four-dimensional manifold.

Key Insight: Expansion Provides Infinite Phase Space

Universe volume grows at exponential rate (accelerated expansion phase):

$$V(t)\sim e^{3H_{0}t}$$

This leads to maximum entropy also growing with time:

$$S_{\max }(t)=\frac{A_{\text{horizon}}(t)}{4{\ell }_P^2}\sim e^{2H_{0}t}$$

where $A_{\text{horizon}}$ is horizon area. Actual entropy $S_{\text{actual}}(t)$ though monotonically increasing, its growth rate slower than $S_{\max }(t)$ growth:

$$\frac{d{S}_{\text{actual}}}{dt}<\frac{d{S}_{\max }}{dt}$$

Attractor Dynamics: Entropy density $s(t)=S_{\text{actual}}(t)/V(t)$ actually decreases:

$$s(t)\to s_{\infty }\text{(asymptotic value, non-zero)}$$

This not “heat death endpoint” but dynamic equilibrium state—system forever chasing continuously rising maximum entropy, forming moving attractor:

$$\Delta S(t):=S_{\max }(t)-S_{\text{actual}}(t)\to \Delta S_{\infty }>0$$

Infinitely approaching but never intersecting: Causal chain extends infinitely, each “equilibrium state” is transient of larger system. Cosmic expansion guarantees entropy increase never ends, but entropy density approaches stability—characteristic of open system.

Phase-scale mother mapping’s directional meromorphization gives analytic characterization of this horizon structure. Along cosmic expansion direction $\mathbf{v}$, cumulative distribution’s dominant pole:

$${\gamma }_0=\sup \{\Re s:{\mathcal{L}}_{\mathbf{v}}(s)<\infty \}$$

corresponds to universe’s Hubble radius $R_H\sim 1/{\gamma }_0$. Regions beyond horizon correspond to ${\mathcal{L}}_{\mathbf{v}}(s)$ unreachable domain at $\Re s<{\gamma }_0$—information still exists there, but for our observation cone, it’s “beyond horizon.”

III. Black Hole Entropy and Information Paradox

Black hole Bekenstein-Hawking entropy:

$$S_{\mathrm{BH}}=\frac{k_B{c}^3}{4G\hslash }A$$

reveals profound connection between gravity and quantum information: entropy proportional to horizon area, not volume. This anomalous scale law hints spacetime itself might be holographic projection of quantum information.

In static block framework, black hole not “collapse formation” dynamic process but static configuration of spacetime geometry—manifold singularity “eternally existing” in spacetime. Information loss paradox corresponding to Hawking radiation can be understood thus:

• Microscopic Layer (Static Block): Information encoded in global quantum state inside and outside black hole, never lost.
• Macroscopic Layer (Horizon Observer): Due to horizon’s causal isolation, inside-outside information cannot correlate through local measurement.
• Hawking Radiation Layer (Coarse-graining): From outside horizon observation, radiation produced by black hole evaporation exhibits thermal spectrum, result of coarse-grained measurement—microscopic information isolated inside horizon “equivalently lost” to exterior observer.

Is information truly lost? In global static block, answer negative. But for any observer restricted outside horizon, information’s “inaccessibility” and “loss” operationally equivalent—not ontological contradiction but epistemological limit.

Mother mapping’s additive mirror and multiplicative mirror here give unified picture: information inside-outside black hole, through horizon boundary conditions mutually dual; Hawking temperature corresponds to periodicity of scale variable $\rho$ in imaginary time direction. ζ function’s functional equation:

$$\zeta (s)=\chi (s)\zeta (1-s)$$

in black hole thermodynamics corresponds to duality between acceleration temperature and rest temperature under Rindler coordinates. Critical line $\Re s=1/2$ corresponds to information equilibrium state—dynamic equilibrium point where black hole forming and evaporating (though this equilibrium unstable in classical limit).

Bridge: Unified Mathematical Structure of Triple Conservation

Now we reveal unified mathematical structure connecting three domains.

Triple Decomposition of Mother Mapping

Phase-scale mother mapping $\mathcal{M}[\mu ](\theta ,\rho )$ can be decomposed along arbitrary direction $\mathbf{v}$:

$$\mathcal{M}[\mu ](\theta ,{\rho }_{\perp }+s\mathbf{v})=\sum _j{w}_j{e}^{i⟨{\alpha }_j,\theta ⟩}{e}^{⟨{\beta }_j,{\rho }_{\perp }⟩}{e}^{s⟨{\beta }_j,\mathbf{v}⟩}$$

Define soft-maximum potential function (pressure function):

$$\Lambda (\rho )=\log \sum _j{w}_j{e}^{⟨{\beta }_j,\rho ⟩}$$

Its gradient and Hessian give:

$${\nabla }_{\rho }\Lambda ={\mathbb{E}}_{\rho }[\beta ],{\nabla }_{\rho }^2\Lambda ={\mathrm{Cov}}_{\rho }(\beta )$$

This structure simultaneously unifies:

1. Quantum Layer (Phase $\theta$): $e^{i⟨{\alpha }_j,\theta ⟩}$ gives wave function oscillation; phase mirror $\theta \mapsto -\theta$ corresponds to time reversal, preserving information scale.

2. Classical Layer (Scale $\rho$ real part): $⟨{\beta }_j,\Re \rho ⟩$ corresponds to energy; scale mirror $\rho \mapsto a-\rho$ through Mellin self-reflexivity gives energy-wavenumber duality.

3. Thermodynamic Layer (Scale $\rho$ imaginary part and discrete summation): Bernoulli layer’s Euler-Maclaurin correction terms give entropy endpoint corrections; discrete summation breaks translation symmetry, leading to entropy directionality.

Information quantity scale three-part formula:

$$H(\rho )=\Lambda (\rho )-⟨\rho ,{\mathbb{E}}_{\rho }[\beta ]⟩-\sum _j{p}_j\log w_j$$

gives explicit relation between entropy and energy, information. When $w_j$ equal amplitude (i.e. $\log w_j$ constant), entropy completely determined by covariance ${\mathrm{Cov}}_{\rho }(\beta )$—corresponds to microcanonical ensemble equal probability assumption. When $w_j$ non-uniform, entropy’s third term introduces information bias of initial configuration—corresponds to canonical ensemble Gibbs weights.

Phase-Scale Embedding of EBOC Static Block

EBOC theory’s core object:

$$\mathcal{U}=(X_f,G,\rho ,\Sigma ,f,\pi ,\mu ,\nu )$$

can be embedded in mother mapping framework as follows:

• Space-Time SFT $X_f$: Corresponds to mother mapping’s discrete support $\mu ={\sum }_j{c}_j{\delta }_{({\alpha }_j,{\beta }_j)}$. Each global configuration $x\in X_f$ is a spacetime trajectory.

• Eternal Graph $G=(V,E)$: Corresponds to topological structure of phase-scale orthogonality relation.

  • Vertices $V$: Each $v\in V$ encodes a local event pattern, corresponding to mother mapping’s cross-section at some scale cone.
  • Edges $E$: $(v,u)\in E$ means events $v$ and $u$ causally/consistently compatible, corresponding to mother mapping’s analytic continuation paths.
  • Out-degree ${\mathrm{deg}}^{+}(v)$: Equals number of compatible extensions from $v$, $|X_f^{(v,\tau )}|$, corresponding to mother mapping’s branch number at $v$ (multi-valued analytic continuation).

  Eternal graph’s timeless geometric structure encodes all causal possibilities of universe—it is ontological reality more fundamental than spacetime.

• Decoder $\pi$: Corresponds to projection from $(\theta ,\rho )$ to visible language.

• Measures $\mu ,\nu$: Former is shift-invariant ergodic measure (corresponding to equilibrium state), latter is universal semi-measure (corresponding to prior weights).

Information non-increase law (EBOC theorem T4) in mother mapping manifests as:

$$K(\pi (x{|}_W))\le K(x{|}_W)+K(\pi )+O(1)$$

This equivalent to soft-maximum potential convexity:

$${\nabla }_{\rho }^2\Lambda ={\mathrm{Cov}}_{\rho }(\beta )⪰0$$

Convex function geometric property guarantees: information mapping from substrate to coarse-graining doesn’t create new information, only reallocates existing information.

Dialectical Unification of Triple Conservation

Now we can state core thesis:

Quantum Layer Information Conservation: Unitary evolution preserves phase space measure, von Neumann entropy invariant. This is invariance of phase mirror $\theta \mapsto -\theta$.

Classical Layer Energy Conservation: Hamiltonian flow preserves symplectic structure, Liouville measure invariant. This is real part invariance of scale mirror $\rho \mapsto a-\rho$.

Cosmic Layer Entropy Increase Direction: Coarse-graining and observer time direction choice lead to Euler-Maclaurin Bernoulli layer endpoint asymmetry. This is result of discrete summation breaking translation symmetry.

Their unification is: they are projections of same static block at different slices, different coarse-graining levels. Information, energy, entropy not three independent physical quantities but three coordinate components of same geometric reality. Mother mapping $\mathcal{M}$’s analytic structure encodes all these conservation laws and breaking patterns—it is universe’s “total score,” and what we hear at different scales are merely different chords of this total score.

Philosophical Epilogue: Quantum Reconciliation of Eternal Recurrence and Historical Materialism

Nietzsche’s Eternal Recurrence: Temporal Philosophy of Static Block

Eternal recurrence Nietzsche proclaimed in Thus Spoke Zarathustra is not physical hypothesis but ethical requirement: if you must infinitely repeat present choice, would you still choose thus? But in static block universe view, eternal recurrence gains literal meaning: entire universe history is closed four-dimensional manifold, no distinction between “first time” and “last time”—past and future topologically equivalent.

EBOC theory’s reversibility theorem (T22) gives mathematical expression:

$$I(F^{T}x)=I(x)$$

Information density conservation under reversible dynamics means time-reversed universe and forward-evolving universe completely equivalent in information content. If we play entire universe history backward along time axis, result still universe satisfying same physical laws—only observer’s reading direction reversed.

This leads to radical ontological conclusion: time not universe’s fundamental property but reading vector $\tau$ observer chooses in static block. Different observers can choose different $\tau$, thus experience different “time flows.” In extreme cases (like Rindler observers near black hole horizon), time can even locally “stop”—but this only that observer’s coordinate choice, for global static block all events still coexist in same way.

Eternal recurrence not “same event repeating countless times” but “all events happen only once, and happen eternally.” Nietzschean ethical requirement here transforms into: every choice you make already inscribed in universe’s static block, cannot undo, need not undo—because “undo” itself presupposes dynamic time that can “start over,” and such time simply doesn’t exist.

Marx’s Dialectical Materialism: Geometric Interpretation of Phase Transition

Dialectical laws Marx revealed in Das Kapital—quantitative to qualitative change, unity of opposites, negation of negation—find geometric correspondence in mother mapping framework.

Quantitative to Qualitative Change: Corresponds to critical points of soft-maximum potential $\Lambda (\rho )$. When scale parameter $\rho$ crosses critical line $\Re s=1/2$, dominant term switches from $j_1$ to $j_2$:

$$\Lambda (\rho )\sim ⟨{\beta }_{j_1},\rho ⟩(\Re \rho <1/2)\to ⟨{\beta }_{j_2},\rho ⟩(\Re \rho >1/2)$$

This switch is abrupt—near critical point, tiny parameter changes lead to drastic jumps in macroscopic observables. This precisely geometric essence of physical phase transitions (like water boiling, magnet Curie point).

Unity of Opposites: Corresponds to coexistence of additive and multiplicative mirrors. Phase variable $\theta$ and scale variable $\rho$ orthogonally couple in mother mapping:

$$\mathcal{M}[\mu ](\theta ,\rho )=\mathcal{M}[\mu ](-\theta ,\rho {)}^{\ast }$$

Phase mirror gives wave-particle duality, scale mirror gives energy-wavenumber duality. These two dualities not mutually independent but coupled through mother mapping’s analytic structure—product of Fourier transform and Mellin transform precisely mother mapping’s core.

Negation of Negation: Corresponds to iterated self-reflexivity of ζ function. First negation: $\zeta (s)\to \zeta (1-s)$ (scale mirror). Second negation: $\zeta (1-s)\to \zeta (1-(1-s))=\zeta (s)$ (back to origin). But this return not simple repetition but spiral ascent—in each iteration, we accumulate a phase $\chi (s)$ along imaginary axis, this phase encodes vacuum fluctuation contribution ($i_{-}$ component).

Marxian historical materialism claims social development follows spiral ascent from primitive communism → slavery → feudalism → capitalism → socialism → communism. In static block framework, this spiral not temporal “progress” but self-similar structure at different scale layers—each historical stage is manifestation of same deep law at different coarse-graining levels. “End of history” not utopia at some future moment but contemporaneous coexistence of all historical stages in static block—“progress” we experience only directionality felt when observer reads this static block along specific direction $\tau$.

Role of Consciousness: Observer as Internal Dimension of Static Block

Finally, we face ancient philosophical difficulty: what status does consciousness occupy in this static block universe?

In EBOC theory, observer not external spectator of universe but internal component of static block. Measurement not “external intervention” but entanglement between subsystem (observer) and another subsystem (measured object) inside static block. Consciousness’s “subjective time flow” corresponds to sequential projection of observer’s internal state on time slices ${\Sigma }_t$.

But this raises deeper question: why does observer experience specific time direction $\tau$? Why do we feel “now” flowing from past to future, not reverse?

Answer lies in: observer’s own local entropy increase process defines its subjective time direction. Our memory, consciousness state updates, all irreversible coarse-graining processes—neuron firing, synapse strengthening, manifest as unidirectional evolution within observer’s local horizon. Thus observer’s “subjective time arrow” locked with its internal entropy gradient—we experience world along direction of local entropy increase.

This doesn’t mean static block itself has direction—it merely exists, all events eternally coexist. But any complex structure capable of forming memory, producing consciousness, must necessarily undergo local entropy increase process (relative to its coarse-graining scale). Key is: universe whole’s entropy is unbounded attractor—$S_{\max }(t)\to \infty$, while observer can only track local entropy within finite horizon. Thus all conscious observers will feel time direction—not because universe heading toward end but because our cognitive mechanisms locked in chase of moving attractor.

In mother mapping framework, consciousness corresponds to special decoder ${\pi }_{\mathrm{obs}}$ that projects global static block into serialized “experience stream.” Different observers correspond to different decoders, thus may experience different “realities”—but all these experiences are different slices of same static block, compatible at deep structure.

Conclusion: Conservation Laws as Ontological Foundation

From quantum domain information conservation, through classical domain energy conservation, arriving at cosmic domain entropy direction, we finally discover: these three not three independent physical laws but three-fold manifestation of same static block at different scales, different observation modes.

Information neither created nor destroyed, only circulates between phase and scale; energy neither increases nor decreases, only dualized between position and momentum; entropy not heading toward endpoint but chasing infinitely rising attractor—cosmic expansion guarantees eternal expansion of phase space, entropy increase never ends. Universe not system “evolving” but geometry “unfolding”—time is coordinates on map, not ruts on road.

Nietzsche’s eternal recurrence and Marx’s dialectics here gain mathematical reconciliation: former reveals static block’s temporal symmetry, latter reveals critical geometry of phase transitions. Both point toward radical ontology: reality not process but structure; change not generation but perspective.

When we gaze at this static block universe, we see not cold deterministic cage but eternal coexistence of all possibilities. Quantum superposition not “undetermined” but “simultaneously determined as all possibilities”; classical trajectory not “necessarily happens” but “already engraved”; entropy growth not “heading toward heat death” but “chasing eternally expanding horizon”—causal chain extends infinitely, each “final state” is intermediate state of larger system.

Conservation laws not shackles but other face of freedom—because nothing can be created or destroyed, nothing can truly be lost. Information eternal, energy cycles, entropy attractor never intersects—on infinite time scale, universe neither ends nor repeats, but continuously unfolds along infinite branches of eternal graph. In this sense, physics’ conservation principles ultimately lead toward cosmic Spinozism: all things are one, one is all things, eternal becoming, endless openness.

This not conclusion but beginning—in this static block’s eternal symphony, we are both performers and movement itself. And this movement has no finale.