Foundation Circle Winding Chain Closed Singular One Cycle Boundary Generate Of Z

Winding numbers on the circle

The winding number is a classical invariant: for a continuous loop on the circle, it counts the net number of times the loop travels around the circle, with counterclockwise positive. A loop that goes around once has winding number 1; a loop that goes around twice has 2; a loop that goes back and forth has 0. The winding number is unchanged by continuous deformation of the loop, which makes it a powerful tool for distinguishing loops that cannot be continuously reshaped into each other.

The formal library of the Recognition Science framework proves a consistency statement about winding numbers at the level of singular chains. A singular 1-simplex is a continuous map from the standard interval into the circle, and a singular 2-simplex is a continuous map from a triangle into the circle. The framework assigns a real displacement to each singular 1-simplex by reparameterizing the interval and taking the path displacement, which equals 2π times the winding number. The theorem simplexDisplacement_boundary states that for every singular 2-simplex, the alternating sum of the displacements of its three boundary edges is zero: disp(δ₀F) − disp(δ₁F) + disp(δ₂F) = 0.

This identity is the chain-level statement that winding numbers kill boundaries. Its proof is a telescoping argument: the boundary walk along two edges of the triangle is homotopic, rel endpoints, to the direct edge, because the standard 2-simplex is convex and hence simply connected. Pushing that homotopy through the map F and combining additivity of path displacement with its homotopy invariance gives the vanishing of the alternating sum. Together with the fact that the invariant sends the once-around generator to 1, this yields a winding homomorphism on 1-cycles that is a left inverse to the fundamental class: the split-injective half of H₁(S¹; ℤ) ≅ ℤ.

In Recognition Science, this result is part of the framework's foundational chain. The framework models physical structure from a discrete ledger of recognition events, and the winding invariant on the circle is a step toward understanding how continuous topology emerges from discrete data. The theorem is proved in the machine-checked library of formal theorems with no axioms, no sorry, and no project-local replacements for the circle.

What the declaration does not claim is the converse direction. The full statement that every 1-cycle is homologous to an integer multiple of the fundamental cycle, which is the surjectivity half of H₁(S¹; ℤ) ≅ ℤ, requires a simplicial prism or subdivision operator that the library's singular homology does not yet provide. That generation half remains open in the framework.

/-- **The displacement of a singular `1`-simplex**: the lift-independent angular
travel `2π × (winding number)`, obtained from the path-level displacement. -/
def simplexDisplacement (f : OneSimplex) : ℝ := pathDisplacement (oneSimplexPath f)

/-- **The displacement of a singular `1`-simplex**: the lift-independent angular
travel `2π × (winding number)`, obtained from the path-level displacement. -/
def simplexDisplacement (f : OneSimplex) : ℝ := pathDisplacement (oneSimplexPath f)

What this page does not claim

The declaration does not prove the full isomorphism H₁(S¹; ℤ) ≅ ℤ, only the split-injective half. The declaration does not establish the generation half, that every 1-cycle is homologous to an integer multiple of the fundamental cycle. The declaration does not claim that the framework's derivation of three spatial dimensions follows from this winding result.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/CircleWindingChain.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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