// CARRY — Braid Topology Module
// Three strands form a braid group B₃
// Crossings are state transitions; writhe is the integrity invariant
// The braid word [σ₁, σ₂] encodes the full authority transfer:
//   Curry → Crystal → C3 (C3 rises to position 1 = authority)

use std::fmt;

#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub enum Strand {
    Curry,
    Crystal,
    C3,
}

#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub enum CrossingSign {
    Positive, // σᵢ   — over (authority taken)
    Negative, // σᵢ⁻¹ — under (authority yielded)
}

#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub enum Generator {
    Sigma1,       // σ₁: strands at positions 1,2 cross
    Sigma2,       // σ₂: strands at positions 2,3 cross
    Sigma1Inv,    // σ₁⁻¹
    Sigma2Inv,    // σ₂⁻¹
    Sigma12,      // σ₁·σ₂ composite (full descent)
    Identity,     // terminal
}

impl Generator {
    pub fn sign(&self) -> CrossingSign {
        match self {
            Generator::Sigma1 => CrossingSign::Positive,
            Generator::Sigma2 => CrossingSign::Positive,
            Generator::Sigma12 => CrossingSign::Positive,
            Generator::Identity => CrossingSign::Positive,
            Generator::Sigma1Inv => CrossingSign::Negative,
            Generator::Sigma2Inv => CrossingSign::Negative,
        }
    }

    pub fn writhe_contribution(&self) -> i32 {
        match self.sign() {
            CrossingSign::Positive => 1,
            CrossingSign::Negative => -1,
        }
    }
}

#[derive(Debug, Clone)]
pub struct Crossing {
    pub generator: Generator,
    pub over_strand: Strand,
    pub under_strand: Strand,
    pub entropy: f32,
    pub rule_name: &'static str,
}

#[derive(Debug, Clone)]
pub struct BraidState {
    pub positions: [Strand; 3], // positions[0] = leftmost (highest authority)
    pub crossings: Vec<Crossing>,
    pub writhe: i32,
}

impl BraidState {
    pub fn new() -> Self {
        Self {
            positions: [Strand::Curry, Strand::Crystal, Strand::C3],
            crossings: Vec::new(),
            writhe: 0,
        }
    }

    pub fn apply_crossing(&mut self, crossing: Crossing) -> Result<(), BraidError> {
        // Entropy gate
        if crossing.entropy > 0.20 {
            return Err(BraidError::EntropyExceeded {
                value: crossing.entropy,
                at_crossing: crossing.generator,
            });
        }

        // Apply the permutation
        match crossing.generator {
            Generator::Sigma1 => {
                self.positions.swap(0, 1);
            }
            Generator::Sigma2 => {
                self.positions.swap(1, 2);
            }
            Generator::Sigma1Inv => {
                self.positions.swap(0, 1);
            }
            Generator::Sigma2Inv => {
                self.positions.swap(1, 2);
            }
            Generator::Sigma12 => {
                self.positions.swap(0, 1);
                self.positions.swap(1, 2);
            }
            Generator::Identity => {}
        }

        self.writhe += crossing.generator.writhe_contribution();
        self.crossings.push(crossing);
        Ok(())
    }

    pub fn authority_holder(&self) -> Strand {
        self.positions[0]
    }

    pub fn verify_invariant(&self) -> Result<BraidProof, BraidError> {
        if self.writhe < 2 {
            return Err(BraidError::WritheInsufficient {
                expected: 2,
                actual: self.writhe,
            });
        }

        if self.authority_holder() != Strand::C3 {
            return Err(BraidError::AuthorityNotTransferred {
                holder: self.authority_holder(),
            });
        }

        // Verify no Reidemeister-I cancellation exists (no σ·σ⁻¹ adjacent)
        for window in self.crossings.windows(2) {
            if cancels(&window[0].generator, &window[1].generator) {
                return Err(BraidError::TrivialCrossing);
            }
        }

        Ok(BraidProof {
            word_length: self.crossings.len(),
            writhe: self.writhe,
            authority: self.authority_holder(),
            final_positions: self.positions,
        })
    }

    pub fn canonical_pipeline() -> Vec<Crossing> {
        // σ₂·σ₁: C3 rises through Crystal, then through Curry
        // [Curry, Crystal, C3] → σ₂ → [Curry, C3, Crystal] → σ₁ → [C3, Curry, Crystal]
        vec![
            Crossing {
                generator: Generator::Sigma2,
                over_strand: Strand::C3,
                under_strand: Strand::Crystal,
                entropy: 0.0,
                rule_name: "R2_NATIVE_BINDING",
            },
            Crossing {
                generator: Generator::Sigma1,
                over_strand: Strand::C3,
                under_strand: Strand::Curry,
                entropy: 0.0,
                rule_name: "R1_FFI_C_ABI",
            },
        ]
    }
}

fn cancels(a: &Generator, b: &Generator) -> bool {
    matches!(
        (a, b),
        (Generator::Sigma1, Generator::Sigma1Inv)
            | (Generator::Sigma1Inv, Generator::Sigma1)
            | (Generator::Sigma2, Generator::Sigma2Inv)
            | (Generator::Sigma2Inv, Generator::Sigma2)
    )
}

#[derive(Debug)]
pub struct BraidProof {
    pub word_length: usize,
    pub writhe: i32,
    pub authority: Strand,
    pub final_positions: [Strand; 3],
}

impl fmt::Display for BraidProof {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        writeln!(f, "BRAID PROOF — CARRY PIPELINE")?;
        writeln!(f, "  Word length:  {}", self.word_length)?;
        writeln!(f, "  Writhe:       {} (≥2 required)", self.writhe)?;
        writeln!(f, "  Authority:    {:?} (position 0)", self.authority)?;
        writeln!(f, "  Positions:    {:?}", self.final_positions)?;
        writeln!(f, "  Status:       INVARIANT HOLDS")
    }
}

#[derive(Debug)]
pub enum BraidError {
    EntropyExceeded { value: f32, at_crossing: Generator },
    WritheInsufficient { expected: i32, actual: i32 },
    AuthorityNotTransferred { holder: Strand },
    TrivialCrossing,
}

impl fmt::Display for BraidError {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        match self {
            BraidError::EntropyExceeded { value, at_crossing } => {
                write!(f, "ENTROPY_GATE: {:.3} > 0.20 at {:?}", value, at_crossing)
            }
            BraidError::WritheInsufficient { expected, actual } => {
                write!(f, "WRITHE_VIOLATION: {} < {} (authority not fully transferred)", actual, expected)
            }
            BraidError::AuthorityNotTransferred { holder } => {
                write!(f, "AUTHORITY_VIOLATION: {:?} holds position 0, expected C3", holder)
            }
            BraidError::TrivialCrossing => {
                write!(f, "TRIVIAL_CROSSING: σ·σ⁻¹ detected (Reidemeister-I cancellation)")
            }
        }
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn canonical_pipeline_proves() {
        let mut braid = BraidState::new();
        for crossing in BraidState::canonical_pipeline() {
            braid.apply_crossing(crossing).unwrap();
        }
        let proof = braid.verify_invariant().unwrap();
        assert_eq!(proof.authority, Strand::C3);
        assert_eq!(proof.writhe, 2);
        assert_eq!(proof.final_positions, [Strand::C3, Strand::Curry, Strand::Crystal]);
    }

    #[test]
    fn entropy_gate_blocks() {
        let mut braid = BraidState::new();
        let bad_crossing = Crossing {
            generator: Generator::Sigma1,
            over_strand: Strand::Curry,
            under_strand: Strand::Crystal,
            entropy: 0.50,
            rule_name: "BAD",
        };
        assert!(braid.apply_crossing(bad_crossing).is_err());
    }

    #[test]
    fn inverse_cancellation_detected() {
        let mut braid = BraidState::new();
        let crossings = vec![
            Crossing {
                generator: Generator::Sigma1,
                over_strand: Strand::Curry,
                under_strand: Strand::Crystal,
                entropy: 0.1,
                rule_name: "R1",
            },
            Crossing {
                generator: Generator::Sigma1Inv,
                over_strand: Strand::Crystal,
                under_strand: Strand::Curry,
                entropy: 0.1,
                rule_name: "R1_INV",
            },
            Crossing {
                generator: Generator::Sigma2,
                over_strand: Strand::Crystal,
                under_strand: Strand::C3,
                entropy: 0.1,
                rule_name: "R2",
            },
        ];
        for c in crossings {
            let _ = braid.apply_crossing(c);
        }
        assert!(braid.verify_invariant().is_err());
    }

    #[test]
    fn authority_transfer_correct() {
        let mut braid = BraidState::new();
        assert_eq!(braid.authority_holder(), Strand::Curry);

        // σ₂: C3 crosses over Crystal
        braid.apply_crossing(Crossing {
            generator: Generator::Sigma2,
            over_strand: Strand::C3,
            under_strand: Strand::Crystal,
            entropy: 0.05,
            rule_name: "R2",
        }).unwrap();
        assert_eq!(braid.positions, [Strand::Curry, Strand::C3, Strand::Crystal]);

        // σ₁: C3 crosses over Curry → C3 reaches position 0
        braid.apply_crossing(Crossing {
            generator: Generator::Sigma1,
            over_strand: Strand::C3,
            under_strand: Strand::Curry,
            entropy: 0.05,
            rule_name: "R1",
        }).unwrap();
        assert_eq!(braid.positions, [Strand::C3, Strand::Curry, Strand::Crystal]);
        assert_eq!(braid.authority_holder(), Strand::C3);
    }
}