Configuration Space is Five-Dimensional

1. Setting

Configuration Space is Five-Dimensional is anchored in Constants.CurvatureSpaceDerivation. The page is not a loose explainer: it is a public map from the Recognition Science forcing chain into one Lean-checked declaration bundle. The primary anchor determines what is proved, and the surrounding declarations show how the result is used.

2. Equations

(E1)

$$ 2^D=8,\qquad D=3,\qquad \Omega_\Lambda^{\mathrm{RS}}=\frac{11}{16}\ \text{where applicable} $$

Shared cosmological forcing frame.

3. Prediction or structural target

• Structural target: Constants.CurvatureSpaceDerivation must keep resolving in the Lean canon, and all downstream pages that cite this anchor must continue to type-check.

This page is currently a structural derivation. Where the claim has direct empirical content, the prediction table gives the measurable target; otherwise the claim is a formal bridge inside the Lean canon.

4. Formal anchor

The primary anchor is Constants.CurvatureSpaceDerivation..config_space_is_5D.

/-- The dimension is exactly 5. -/
theorem config_space_is_5D : configSpaceDim = 5 := rfl

/-! ## Part 2: Each Dimension is Forced -/

/-- **Spatial Dimensions (D=3)**: Forced by the linking requirement.

In D dimensions, linking of closed curves is only non-trivial when D ≥ 3.
For D > 3, the link penalty vanishes (curves can pass through each other).
Therefore D = 3 is the unique stable dimension. -/

5. What is inside the Lean module

Key theorems:

• config_space_is_5D
• spatial_dims_eq_3
• eight_tick_forces_temporal
• balance_from_conservation
• total_angular_is_pi5
• seam_ratio_from_topology
• curvature_correction_eq_formula
• curvature_matches_alpha_derivation
• pi3_incomplete
• pi4_incomplete
• pi6_excess
• pi_power_eq_pi5_iff

Key definitions:

• configSpaceDim
• ConfigSpaceDecomposition
• canonicalDecomposition
• spatial_dims_forced
• temporal_dim_forced
• balance_dim_forced
• angular_contribution_per_dim
• total_angular_factor

6. Derivation chain

• config_space_is_5D - Config space is 5D
• spatial_dims_eq_3 - Spatial = 3
• eight_tick_forces_temporal - 8-tick forces temporal
• balance_from_conservation - Balance from conservation
• config_space_complete - Config space complete

7. Falsifier

A complete physical theory consistent with all RS primitives but using a different config space dimension refutes config_space_is_5D.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

11. Why this belongs in the derivations corpus

The corpus is organized around load-bearing consequences, not around file names. This entry is included because Constants.CurvatureSpaceDerivation contributes a reusable theorem or definitional bridge that other pages can cite. Keeping the page public gives readers a stable URL, a JSON record, and a direct path into the Lean theorem page. If the entry becomes redundant with a stronger derivation later, the current slug should be retired rather than silently rewritten; the replacement should absorb its anchors and preserve the audit history.