Exhibit #5

Implementation

Ada Source Code — Toy Model

Emergent Spacetime from Quantum Entanglement Networks

Toy Model: 2D Lattice Demonstrating Gravity as Emergent Phenomenon

Mathematical Basis:

  1. Entanglement entropy E(i,j) on lattice edges
  2. Emergent distance ds = α / E (Ryu-Takayanagi inspired)
  3. Conformal factor φ = ln(ds)
  4. Gaussian curvature K = -e^(-2φ) · ∇²φ
  5. "Mass" = localized entanglement disruption
  6. Result: mass creates curvature → gravity emerges
with Ada.Text_IO;                    use Ada.Text_IO;
with Ada.Float_Text_IO;              use Ada.Float_Text_IO;
with Ada.Integer_Text_IO;            use Ada.Integer_Text_IO;
with Ada.Numerics.Elementary_Functions;
use Ada.Numerics.Elementary_Functions;

procedure Emergent_Spacetime is

   -- Grid parameters (odd N so mass sits at exact center)
   N : constant Positive := 21;
   subtype Grid_Range is Integer range 1 .. N;

   type Scalar_Field is array (Grid_Range, Grid_Range) of Float;

   -- ============================================================
   -- PHYSICAL PARAMETERS
   -- ============================================================

   -- Background entanglement (uniform => flat space)
   E0 : constant Float := 1.0;

   -- Distance-entanglement coupling: ds = Alpha / E
   Alpha : constant Float := 1.0;

   -- Mass influence width (Gaussian sigma)
   Sigma : constant Float := 3.0;

   -- Minimum entanglement floor (prevents division by zero)
   E_Min : constant Float := 0.01;

   -- Mass position: center of lattice
   Cx : constant Integer := (N + 1) / 2;  -- = 11
   Cy : constant Integer := (N + 1) / 2;

   -- ============================================================
   -- FIELD VARIABLES
   -- ============================================================

   Entanglement : Scalar_Field := (others => (others => E0));
   Distance     : Scalar_Field := (others => (others => 0.0));
   Phi          : Scalar_Field := (others => (others => 0.0));
   Curvature    : Scalar_Field := (others => (others => 0.0));

   -- ============================================================
   -- HELPER FUNCTIONS
   -- ============================================================

   function Lattice_Radius (I, J : Integer) return Float is
   begin
      return Sqrt (Float ((I - Cx) ** 2 + (J - Cy) ** 2));
   end Lattice_Radius;

   function Max_F (A, B : Float) return Float is
   begin
      if A > B then return A; else return B; end if;
   end Max_F;

   function Abs_F (A : Float) return Float is
   begin
      if A < 0.0 then return -A; else return A; end if;
   end Abs_F;

   -- ============================================================
   -- CORE COMPUTATION: Build fields for a given mass parameter M
   -- ============================================================

   procedure Compute_Fields (M : Float) is
      R           : Float;
      Perturbation : Float;
      Laplacian    : Float;
      Scale_Factor : Float;
   begin
      -- Step 1: Entanglement field with mass perturbation
      -- E(i,j) = E0 - M * exp(-r^2 / 2*sigma^2)
      -- Mass disrupts local entanglement proportional to M
      for I in Grid_Range loop
         for J in Grid_Range loop
            R := Lattice_Radius (I, J);
            Perturbation := M * Exp (-(R ** 2) / (2.0 * Sigma ** 2));
            Entanglement (I, J) := Max_F (E0 - Perturbation, E_Min);
         end loop;
      end loop;

      -- Step 2: Emergent proper distance  ds = alpha / E
      -- More entanglement => shorter distance (Ryu-Takayanagi)
      for I in Grid_Range loop
         for J in Grid_Range loop
            Distance (I, J) := Alpha / Entanglement (I, J);
         end loop;
      end loop;

      -- Step 3: Conformal factor  phi = ln(ds)
      -- For metric g = e^(2*phi) * delta
      -- On unit lattice, ds = e^phi => phi = ln(ds)
      for I in Grid_Range loop
         for J in Grid_Range loop
            Phi (I, J) := Log (Distance (I, J));
         end loop;
      end loop;

      -- Step 4: Gaussian curvature  K = -e^(-2*phi) * Laplacian(phi)
      -- Using discrete Laplacian on the lattice
      -- Boundary nodes get zero curvature (not enough neighbors)
      Curvature := (others => (others => 0.0));
      for I in 2 .. N - 1 loop
         for J in 2 .. N - 1 loop
            Laplacian := Phi (I + 1, J) + Phi (I - 1, J)
                       + Phi (I, J + 1) + Phi (I, J - 1)
                       - 4.0 * Phi (I, J);
            Scale_Factor := Exp (-2.0 * Phi (I, J));
            Curvature (I, J) := -Scale_Factor * Laplacian;
         end loop;
      end loop;
   end Compute_Fields;

Algorithm Walkthrough

Step 1: Build Entanglement Field

E(i,j) = E₀ - M · exp(-r² / 2σ²)

Mass disrupts local entanglement proportional to its strength M. The Gaussian falloff ensures the disruption is localized.

Step 2: Compute Emergent Distances

ds = α / E

Higher entanglement means shorter emergent distance — the Ryu-Takayanagi intuition inverted. This is the bridge from quantum information to geometry.

Step 3: Calculate Conformal Factor

φ = ln(ds)

For a conformally flat metric g = e^(2φ) · δ, the conformal factor captures how the metric deviates from flat space.

Step 4: Compute Gaussian Curvature

K = -e^(-2φ) · ∇²φ

Using the discrete Laplacian on the lattice, we calculate how curved the emergent space is at each point.

Four Experiments

01

Single Mass Analysis

Full field computation showing entanglement cross-section and radial curvature profile compared to analytical predictions.

02

Linearity Test

Verifies K ∝ M — the equivalence principle. Peak curvature should scale linearly with mass in the weak field limit.

03

2D Curvature Map

ASCII visualization showing the gravitational well pattern: positive curvature at center, negative stretching ring, asymptotically flat edges.

04

Flat Space Verification

With M = 0, curvature should be exactly zero everywhere. A sanity check that the math works.

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