The pi^5 Curvature Tuple is Forced

1. Setting

The pi^5 Curvature Tuple is Forced 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)

$$ J(x)=\frac12(x+x^{-1})-1,\qquad \varphi^2=\varphi+1 $$

Shared constant-forcing backbone.

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..total_angular_is_pi5.

theorem total_angular_is_pi5 : total_angular_factor = Real.pi ^ 5 := by
  unfold total_angular_factor configSpaceDim
  rfl

/-! ## Part 4: The Curvature Integral Structure -/

/-- The curvature integrand is the seam mismatch 103/102.

The seam count arises from:

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

• total_angular_is_pi5 - Total angular = pi^5
• pi5_uniquely_forced - pi^5 uniquely forced
• curvature_term_complete_derivation - Curvature term complete derivation
• seam_ratio_from_topology - Seam ratio from topology

7. Falsifier

Any consistent RS scheme that produces an angular factor other than pi^5 refutes pi5_uniquely_forced.

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.

10. Audit path

To audit pi5-from-curvature, start with the primary Lean anchor Constants.CurvatureSpaceDerivation.total_angular_is_pi5. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.