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docs(lean,#15368): Conway docstrings name the two proven labelings, not a mirror family #15380

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myia-ai-01 merged 1 commit into from
Sep 10, 2026
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docs(lean,#15368): Conway docstrings name the two proven labelings, not a mirror family #15380

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7 changes: 4 additions & 3 deletions MyIA.AI.Notebooks/SymbolicAI/Lean/knot_lean/Knots/Conway.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -628,9 +628,10 @@ theorem alexander_figureEight_eval_neg_one :
norm_num

/-- La variante signée restitue le classique sur le nœud en huit :
l'étiquetage alterné `[−, +, −, +]` du diagramme DT-dérivé (deux signes de
chaque, toute paire miroir convient — `4_1` est amphichiral) rend
exactement `t² − 3t + 1` sous le même mineur désigné. -/
l'étiquetage alterné `[−, +, −, +]` du diagramme DT-dérivé rend
exactement `t² − 3t + 1` sous le même mineur désigné, et son miroir
`[+, −, +, −]` rend `t · (t² − 3t + 1)` — même classe d'unités, comme
l'exige l'amphichiralité de `4_1`. -/
theorem alexander_figureEight_signed :
alexanderPolynomialSigned figureEightDiagram [false, true, false, true]
= Polynomial.X ^ 2 - 3 * Polynomial.X + 1 := by
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12 changes: 7 additions & 5 deletions MyIA.AI.Notebooks/SymbolicAI/Lean/knot_lean/Knots/Conway_en.lean
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Expand Up @@ -559,8 +559,9 @@ determinant survives: `|P(−1)| = 5 = det(4_1)`

The signed variant `alexanderPolynomialSigned` takes chirality as data and
recovers the classical value on the figure-eight: the alternating labeling
of the DT-derived diagram returns exactly `t² − 3t + 1`, its mirror
`t · (t² − 3t + 1)` — same unit class, as amphichirality demands. -/
`[−, +, −, +]` of the DT-derived diagram returns exactly `t² − 3t + 1`,
its mirror `[+, −, +, −]` returns `t · (t² − 3t + 1)` — same unit class,
as amphichirality demands. -/

/-- Alexander row of a **negative** crossing: Fox derivative of the mirror
Wirtinger relation `x_o⁻¹ x_i x_o = x_out`, multiplied by the unit `t` to
Expand Down Expand Up @@ -632,9 +633,10 @@ theorem alexander_figureEight_eval_neg_one :
norm_num

/-- The signed variant recovers the classical value on the figure-eight:
the alternating labeling `[−, +, −, +]` of the DT-derived diagram (two
signs of each; any mirror pair works — `4_1` is amphichiral) returns
exactly `t² − 3t + 1` under the same designated minor. -/
the alternating labeling `[−, +, −, +]` of the DT-derived diagram returns
exactly `t² − 3t + 1` under the same designated minor, and its mirror
`[+, −, +, −]` returns `t · (t² − 3t + 1)` — same unit class, as
amphichirality of `4_1` demands. -/
theorem alexander_figureEight_signed :
alexanderPolynomialSigned figureEightDiagram [false, true, false, true]
= Polynomial.X ^ 2 - 3 * Polynomial.X + 1 := by
Expand Down
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