Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17
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  • 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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Degree sends concatenation to addition, reversal to negation, and the constant loop to zero

Statement

Degree sends concatenation to addition, reversal to negation, and the constant loop to zero. Explicitly, for based circle loops α,β at [0],

deg⁡(α∗β)=deg⁡(α)+deg⁡(β),deg⁡(αˉ)=−deg⁡(α),deg⁡(c[0])=0.

Facts & Assumptions

Given: Based circle loops α and β at [0].

[L1]

Lifts of circle-loop concatenations and reversals have endpoints equal to the sum and the negative of the original endpoints (Lifts of circle-loop concatenations and reversals).

[L2]

Degree is the terminal value of the unique lift beginning at zero (The degree of a based circle loop).

[L3]

A path through a covering has a unique lift once its initial point is prescribed (Existence and uniqueness of path lifts through a covering map).

Proof

technique · direct
1.1L1L2

By [L1], the lift of α∗β from zero ends at deg⁡(α)+deg⁡(β). Reading that endpoint through [L2] gives deg⁡(α∗β)=deg⁡(α)+deg⁡(β).

1.2L1L2

By the reversal formula in [L1], the lift of αˉ from zero ends at −deg⁡(α), so [L2] gives deg⁡(αˉ)=−deg⁡(α).

2.1L2L3∎

The constant path at zero is a lift of the constant loop c[0] and starts at zero; uniqueness in [L3] makes it the defining lift. Its terminal value is zero, so [L2] gives deg⁡(c[0])=0.

Depends on

Used by

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources