Adaptive-Resolution Screens as a Toy Model for Emergent Curvature

Status: exploratory / M-arena candidate mechanism Related: Emergence of GR (C_s / M split), 3-sphere topology selection

1. Setup

Consider the monitor industry's ultimate ideal: a display featuring non-uniform 2D pixel or 3D voxel sampling density governed by local information content—concentrating high density in detailed regions and low density in uniform ones.

An internal observer, made of those pixels, can only measure distance by counting pixels along a path effectively measures:

$$ L_{obs}[\gamma] = \int_\gamma n(x(s)) \left|\frac{dx}{ds}\right| ds $$

where $n(x)$ is the local linear pixel (sampling) density. This is exactly the arc length of the conformally flat metric

$$ ds^2 = n(x)^2 , dx^2 \qquad \text{(isotropic density assumed)} $$

i.e. the density field is a conformal factor. This is the same construction as the optical/refractive-index metric from analogue gravity (Gordon 1923): a medium with variable refractive index bends geodesics exactly as a curved space would.

2. Curvature from density gradients

Write $n(x) = e^{\varphi(x)}$. For a conformally deformed flat metric in $D$ spatial dimensions, linearizing in $\varphi$:

$$ R \approx -2(D-1), \nabla^2 \varphi $$

(In 2D this reduces to Gaussian curvature $K \approx -\nabla^2 \varphi$.)

Key point: curvature is sourced by the Laplacian of log-density, not density itself. A uniform high-resolution plateau is flat; curvature concentrates at the edges of detail-rich regions.

3. Dimension-dependent falloff (the actual result)

Treating an excess-detail region as a source term for $\nabla^2 \varphi$, the resulting potential is the Green's function of the Laplacian in $D$ dimensions:

$$ \varphi(r) \sim \begin{cases} \ln(r) & D = 2 \ 1/r & D = 3 \ 1/r^{D-2} & \text{general } D \end{cases} $$

This falloff law is not a free choice of the mechanism — it is forced by the dimensionality of the space being resolved.

Numerical check

Built a localized Gaussian "excess detail density" blob and solved $\nabla^2 \varphi = -4\pi\rho$ via FFT, once as a 2D pixel field and once as a 3D voxel field.

• 3D: $\varphi \cdot r$ stays roughly constant ($\approx 95$–$108$) over $r \approx 5$–$15$ grid units from the blob — the signature of $\varphi \propto 1/r$. Log-log slope in the clean window: $-1.24$ (pure $1/r$ gives exactly $-1$; residual from source finite-size + periodic-box artifacts).
• 2D: fit against $\ln(r)$ gives $R^2 = 0.9994$; fit of the same data against $1/r$ gives only $R^2 = 0.909$ — decisively logarithmic.

Same mechanism, same code, two different laws, because the ambient dimensionality does the work.

4. Why this matters for the framework?

• The framework's Solomonoff-induction argument independently selects a 3-sphere topology ($D=3$ space). This gives a second, independent reason $D=3$ is singled out: it is the unique dimension in which density-induced curvature reproduces genuine inverse-square (Newtonian) falloff.
• This is compatible with, not contradicted by, the existing negative result that $C_s$ (translation-invariant spectral complexity) cannot produce $1/r$ on its own — that result lives in the spectral ($C_s$) arena; this mechanism lives in the counting/geometric ($M$) arena. The $C_s / M$ split predicts exactly this kind of division of labor.
• Natural temporal partner: variable frame rate (temporal sampling density) as the time-dilation analog of variable spatial density — higher local frame density in high-detail/high-motion regions, same way slow-motion capture increases frame rate for fast action. This maps onto the "two numbers — spatial resolution and temporal resolution" already identified as the model's current output.

5. MPEG-codec analogy for fermions/bosons

• Fermionic ("pixel") data: sample position/occupation — a grid slot is filled or not, no two samples occupy the same slot in a frame. Hard $0/1$ occupation number, i.e. exclusion.
• Bosonic ("residual") data: motion-compensated residual / DCT coefficients — unbounded, additive amplitudes, no exclusion, multiple components can pile onto the same mode.

This reproduces the fermion/boson norm split already established

$$ |F|^2 + |B|^2 = 1 $$

from an independent (compression-codec) starting point.

6. Caveats

• What's shown is the correct conformal-curvature Poisson equation and its dimension-dependent Green's function — a legitimate, checkable result.
• This is not yet a derivation of the full linearized Einstein equations: no tensor structure, no fixed coupling to $G$, and the time-dilation half of the metric has not been consistently coupled to the same source term. That coupling is the natural next step.

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