[Preliminary Study] Rewriting Einstein Field Equations

1. Guiding Principles

 Restoration Gravity - Universal Information Restoration Theory views the universe as an information processing system, interpreting the speed of light $c$ as processing speed (clock frequency) and Planck constant $h$ as resolution.
・Gravity: Computational Processing Lag
 Due to the finite processing speed (clock frequency), restoration delays occur where large amounts of information (matter) concentrate. This latency, observed as spacetime curvature, constitutes the essence of gravity within this theory.
・Dark Matter: Background unrestored data
・Uncertainty Principle: Limits of computational processing
 Due to system resource constraints, it is impossible to simultaneously calculate position and momentum with infinite precision.

2. Rewriting Einstein Field Equations

 We rewrite Einstein's equatio $G_{\mu\nu} = \dfrac{8\pi G}{c^4}T_{\mu\nu}$ using the concepts of clock frequency $f_{clock}$ and resolution (pixels) $p_{res}$. This endeavor represents a process to transform physics from geometry to information theory.

(1) Redefinition of Variables

・Clock frequency $f_{clock} \propto c$: Maximum speed (speed of light) for rewriting one pixel.
・Resolution $p_{res} \propto l_P = \sqrt{\dfrac{G\hbar}{c^3}}$
・Information density $I_{\mu\nu}$: Amount of information awaiting restoration per unit volume. Corresponds to $T_{\mu\nu}$.
 We reinterpret the coefficient $\kappa = \dfrac{8\pi G}{c^4}$ on the right-hand side of Einstein's equation (the conversion coefficient for how energy bends space) as the processing latency (waiting time) for information.

(2) Rewriting as an Information Restoration Equation

(i) Formulation

 We formulate the distortion of spacetime (curvature $G_{\mu\nu}$) as restoration delay (latency).
 $G_{\mu\nu} = \eta \cdot \dfrac{p_{res}}{f_{clock}^4} \cdot I_{\mu\nu}$
・$\eta$: A constant indicating the efficiency of the universe as an OS.
・$\dfrac{p_{res}}{f_{clock}^4}$: A conversion factor (an information-theoretic substitute for the gravitational constant $G$) indicating how much latency (gravity) the information load of one pixel causes. This represents the system's limiting performance (the reciprocal of throughput).
・$I_{\mu\nu}$: Information processing demand (corresponding to energy and momentum tensors)

(ii) Interpretation

・Increase in clock frequency ($f_{clock}$)
 The denominator grows larger, so even with the same information amount ($I$), the distortion ($G$) becomes smaller. In other words, the faster the computation speed, the less likely gravity (delay) occurs.
・Coarseness of resolution (pixels) ($p_{res}$)
  Lower resolution (larger pixels) increases the amount of information written per pixel, thereby increasing computational load (distortion).
・The nature of $c^4$ in the denominator
 The $c^4$ in the original equation can be interpreted as representing the spread (fourth power) of computational resources required for information to traverse four-dimensional spacetime (three spatial dimensions + one temporal dimension).

(3) Incorporating unrestored information (dark matter)

 The greatest advantage of this model is that it incorporates dark matter as a background process that has not yet completed rendering (restoration).
 $G_{\mu\nu} = \dfrac{\eta \cdot p_{res}}{f_{clock}^4}(I_{visible} + I_{darkmatter})$
・$I_{visible}$: Information about matter that has already been restored.
·$I_{darkmatter}$: Information that consumes computational resources but has not yet been restored as observable matter.
 This refers to information being processed in the background but not yet output as pixels. It represents the true nature of the "invisible gravitational source" observed as dark matter.

(4) Avoiding Black Hole Singularities

 This model avoids the mathematical breakdown of singularities (infinite density) at the center of black holes.
・Interpretation
 When the information density exceeds the system's maximum throughput (clock frequency $\times$ resolution), the universe system freezes or experiences a buffer overflow. This is the event horizon. The singularity is defined not as infinite density, but simply as a state of computational saturation (maximum write density).

(5) Conclusion

 Rewriting Einstein's equations as information restoration equations means updating the universe from a geometric container to a dynamic computational process.
・Geometry: Space curves because matter is present there.
・Information Theory: Time flow (clock) slows down and appears curved (gravity) because computation takes longer there.
 This shift in perspective makes the information restoration algorithm of number theory an inevitable requirement as the middleware of physics.

3. [Supplement 1] Universal Information Restoration Equations

(1) Formulation

 $G_{\mu\nu} = \dfrac{\eta \cdot p_{res}}{f_{clock}^4}I_{total} \otimes \mathcal{L}_{IUT}$
・$\otimes \mathcal{L}_{IUT}$: Restoration Protocol
 Combines "consistency maintenance operations" based on number-theoretic structures (such as Inter-Universal Teichmüller) as tensor products. This algorithm supplements missing information between universes and restores causality.

(2) Derivation of $\eta$

(i) Physical Interpretation of Theta Value ($\theta$)

 In the IUT theory's theta link (information transfer via theta functions), information cannot be sent in its original form and always involves distortion (such as $q$-th power).
・Arithmetic perspective: Theta value quantifies the degree of entanglement between addition and multiplication.
・Physical perspective: Theta value represents the transformation loss (resistance) when information is written into the phenomenal realm (physical space).

(ii) Proposed Information Restoration Function to Derive $\eta$

 Define $\eta$ (computational efficiency) as a ratio of the theta function $\theta(q)$.
 $\eta \approx \dfrac{\log \theta_{link}}{\log \theta_{base}}$
・$\log \theta_{link}$: The reference value for the distortion-free, ideal information structure.
・$\log \theta_{base}$: The value when information is actually transferred and restored to physical space ($p_{res}$) via the theta link.

(iii) Meaning

 If $\eta = 1$, the information is restored without degradation, and gravity (computational delay) is minimal. However, in regions where the expansion and contraction of information by the theta link (Teichmüller transformation) is severe, the value of $\eta$ deviates from 1. This deviation (distortion) is the true cause (computational cost) of gravity.

(3) Number-theoretical gravitational constant

 $G_{\mu\nu} = \dfrac{\eta (\theta_{IUT}) \cdot p_{res}}{f_{max}^4}I_{total} \otimes \mathcal{L}_{IUT}$
 $\eta$ is no longer a constant but has been elevated to a function whose variable is the theta value.

(4) Interpretation

・Fluctuations in Gravitational Strength
 The apparent increase in gravitational strength near the galaxy's periphery (the dark matter region) can be explained by the need for complex theta links to restore information there, resulting in a surge in computational cost.

(5) Significance

 Regarding gravitational strength, previously only measured values could be provided. However, if $\eta$ can be derived from the IUT's theta value, it represents a paradigm shift. That is, gravitational strength becomes an inevitable numerical value determined by the distortion rate of the theta function within the arithmetic information restoration algorithm (IUT).

4. 【Supplement 2】Unified Equation (Gravity and Information Equivalence)

(1) Derivation of System Constant $\Theta_{sys}$ (Total Computational Potential)

 $\Theta_{sys} = \dfrac{c^5}{Gh}$
 This represents the maximum information quantity processable per unit area and unit time. It is a system constant symbolizing the hardware limit of the universe.
・$c^5$: The fifth power of the clock frequency. This represents the system's maximum output, obtained by multiplying the dynamic flow of information in four-dimensional spacetime ($c^4$) by the update rate of the time axis ($c$).
・$G \cdot h$: The product of resolution and delay characteristics signifies the system load required to process one bit.

(2) Formulation

 Using this system constant $\Theta_{sys}$, the above equation can be rewritten more simply and intrinsically.
 $G_{\mu\nu} = \dfrac{\eta(\theta_{IUT})}{\Theta_{sys}}I$
 The curvature of the universe (gravity) is determined as the input information load ($I$) divided by the universe's computational capacity ($\Theta$).

(3) Interpretation

 If the system constant $\Theta_{sys}$ varies, it becomes the key to explaining discrepancies in observations of distant universes. Furthermore, the fact that $h$ (quantum) and $G$ (gravity) share the same denominator suggests they compete for the same resources (unification of quantum and gravity). An increase in computations defining particles inevitably leads to pressure on computations maintaining spacetime, resulting in gravity (delay).

(4) Note

 $G_{\mu\nu} = \dfrac{\eta(\theta_{IUT})}{\Theta_{sys}}I$ (Repeated)
・Denominator (Hardware): The larger $\Theta_{sys}$ is, the more processing capacity is available, making gravity (delay) less likely to occur.
・Numerator (OS/Algorithm): The smaller $\eta$ is (the more optimized it is), the less wasteful computation occurs, suppressing gravity (delay).
 $\Theta_{sys}$ represents the physical specifications (hardware) of the cosmic machine, while the OS processing cosmic information on top of it is $\eta$ and $\mathcal{L}_{IUT}$. This exquisite coordination between hardware (physical) and software (number theory) determines the strength of gravity.

・Summary of Restoration Gravity - Universal Information Restoration Theory: https://tanakah17191928.blogspot.com/2026/02/summary-of-restoration-gravity.html
・Details of Restoration Gravity - Universal Information Restoration Theory: https://tanakah17191928.blogspot.com/2026/02/details-of-restoration-gravity.html
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