Foundation Circle Winding Chain Oriented Cyclic Families Explicit Raw Prism Gene

Winding and the missing prism

The winding number is a classical invariant of paths on a circle: it counts how many times a path winds around the circle, with sign for direction. A path that loops once counterclockwise has winding number 1; one that loops twice has 2; a path that goes out and back without completing a loop has 0. The invariant is additive under concatenation of paths, and it is unchanged by homotopy, meaning that if you deform a path continuously without crossing the circle's center, the winding number stays the same. These two properties make winding a powerful tool for distinguishing loops that cannot be continuously deformed into one another.

The formal library in the Recognition Science framework lifts this path-level invariant to the level of singular simplices, which are continuous maps from standard triangles into the circle. Each singular 1-simplex, a continuous map from the unit interval into the circle, gets a real-valued displacement, equal to 2π times its winding number. The key theorem, simplexDisplacement_boundary, states that for every 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 a 2-simplex has total winding zero, because the walk along two edges is homotopic to the direct edge, and homotopic paths have equal displacement.

In Recognition Science, the framework models this as a ledger, a discrete record of events, where each oriented edge carries a coefficient and the boundary operator tracks how edges meet at vertices. The library proves that the winding homomorphism on 1-cycles is split-injective: the fundamental loop, which winds once around the circle, maps to 1, and any cycle with zero winding is homologous to an integer multiple of the fundamental cycle. This establishes the half of the homology statement H₁(S¹; ℤ) ≅ ℤ that says the invariant is faithful, that no non-trivial cycle is invisible to it.

What the declaration does not claim is the converse, the generation half: that every 1-cycle is homologous to an integer multiple of the fundamental cycle. That surjectivity statement requires a simplicial prism or subdivision operator, which the library's singular homology does not yet provide. The declaration explicitly names this as open, a target for future formalization. It also does not claim any result about higher-dimensional spheres or about the physical interpretation of winding in the framework's forcing chain; those remain separate questions. The library uses no axioms, no sorry, and no project-local replacements for the circle, so the proved statements are axiom-clean.

/-- **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 declaration does not prove that every 1-cycle is homologous to an integer multiple of the fundamental cycle. The declaration does not establish any result about the homology of higher-dimensional spheres. The declaration does not connect the winding invariant to any physical interpretation in the framework.

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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