Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Path-homotopic based circle loops have the same degree

Statement

Path-homotopic based circle loops have the same degree.

Facts & Assumptions

Given: Based loops α,β:I→R/Z at [0] and an endpoint-fixed path homotopy from α to β.

[L1]

Every based circle loop γ has a unique lift γ~ beginning at zero, and deg⁡(γ)=γ~(1) (The degree of a based circle loop).

[L2]

Endpoint-fixed homotopic paths in the base have lifts with the same endpoint whenever their lifts begin at the same point (The endpoint of a lifted path depends only on its endpoint-fixed homotopy class).

Proof

technique · direct
1.1

Let α~ and β~ be the unique lifts used in [L1]. Both begin at the common point 0.

givenL1
2.1

Since the base loops are endpoint-fixed homotopic and the lifts have the same initial point, [L2] gives α~(1)=β~(1).

step 1.1L2
3.1

Reading these terminal values through [L1] yields deg⁡(α)=deg⁡(β).

step 2.1L1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 29 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources