Alphabeta Math
PropositionStatement: 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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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
1.1

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).

L1L2
1.2

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

L1L2
2.1

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.

L2L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 52 results over 15 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