Foundation Circle Winding Chain Cycle Winding Integral Of Free Boundary Kernel D

The winding invariant

The winding number is a classical invariant of a closed curve in the plane: it counts how many times the curve loops around a chosen point, with direction taken into account. For a circle, the once-around loop has winding number 1, the twice-around loop has 2, and a loop that goes out and back without encircling the center has 0. The winding number is a homotopy invariant: continuously deforming the curve without passing through the point leaves the count unchanged.

The framework's machine-checked library of formal theorems studies this invariant at the level of singular simplices, the standard building blocks of singular homology. A singular 1-simplex on the circle is a continuous map from the unit interval into the circle, and its displacement is the net change in angle along the path, measured in units of 2π. The key theorem, simplexDisplacement_boundary, states that for any singular 2-simplex, the alternating sum of the displacements of its three boundary edges is zero. This is the chain-level statement that winding kills boundaries: the boundary of any 2-simplex has total winding zero.

This single fact, together with the fundamental loop having winding 1, gives a winding homomorphism on 1-cycles that is a left inverse to the fundamental class. The library proves this provides the split-injective half of the isomorphism H₁(S¹; ℤ) ≅ ℤ, meaning every integer multiple of the fundamental cycle is detected by the winding number. The converse, that every 1-cycle is homologous to an integer multiple of the fundamental cycle, requires a subdivision operator that the library's underlying homology theory does not yet provide, so that direction remains open.

In Recognition Science, this winding invariant is what the framework's ledger of recognition events uses to distinguish a genuine cycle from a boundary. The declaration cycleWinding_integral_of_freeBoundaryKernel_decomposesIntoCyclicEdgeLists expresses this decomposition: a 1-chain whose winding is zero can be written as a sum of boundaries of 2-simplices, provided the chain is a cycle and the subdivision operator is available. The framework proves the winding homomorphism exists and is split-injective; it does not prove the full homology isomorphism without the subdivision step.

/-- **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 exact remaining generation statement, in homology-level form.  It says
that every degree-`1` homology class is an integer multiple of the fundamental
circle class.  This is deliberately stated as surjectivity of the already-built
comparison morphism `fundamentalHomologyClass`; proving this is the remaining
surjective half of `H₁(S¹;ℤ) ≅ ℤ`.

Note: this is **not** needed for the strict T-1-to-T8 frontier closure, which only
requires nonvanishing (`circleH1ZNonzero_unconditional`).  Surjectivity is the
stronger statement that upgrades nonvanishing to the full isomorphism. -/
def fundamentalHomologyClass_surjective : Prop :=
  Function.Surjective (ModuleCat.Hom.hom fundamentalHomologyClass)

What this page does not claim

The full isomorphism H₁(S¹; ℤ) ≅ ℤ is not proved, only the split-injective half. The winding number is not defined for arbitrary continuous maps into higher-dimensional spaces. The declaration does not establish the physical recognition-to-linking bridge.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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