M468 Member Table

Abstract

The m468 side column is computed by the frozen Jacobi selector; unsupported m-side and 1729 orbit claims are omitted.

The phase classifier and the frozen selector column are separate definitions. The finite phase-member table records only prime labels and residue classes; it does not assume selector values.

Theorem 1.1 (Frozen selector side characterization).

$$\forall p\in \mathbb{N},\forall Psi\in \mathbb{Z},(phaseMember(p,Psi)\Rightarrow (sameSide(p,Psi)\iff Psi\mathrm{mod}24=0\wedge differentSide(p,Psi)\iff Psi\mathrm{mod}24=12)$$

Proof. Machine-checked in Lean as D5/S1/Phase/Interference/M468MemberTable.m468_split_prime_characterization (✓ std3). ∎

Source. Repository-derived.

Commentary.

The side is defined from the split-factor Jacobi value J(-384 | p) in the frozen selector factorization. The checked selector column is J(-384 | 7) = 1 and J(-384 | 67) = -1; the independent phase-member table then connects those computed values to the two residue classes.

Theorem 1.2 (Zero-only selector column fails at m468).

$$\neg zeroOnl{y}_{468}$$

Proof. Machine-checked in Lean as D5/S1/Phase/Interference/M468MemberTable.m468_zero_only_fails (✓ std3). ∎

Source. Repository-derived.

Commentary.

Zero-only means that every proper prime divisor has frozen selector value zero. The equivalence to successor primality is proved separately; 469 = 7 * 67 and the prime divisor 7 with selector value J(-384 | 7) = 1 provides the non-vacuity witness.

Disclosure: the frozen repository surface provides no m-side selector semantics and no three-prime orbit bridge at 1729, so neither claim is asserted here.

References

• Truth anchor: D5/S1/Phase/Interference/M468MemberTable.m468_split_prime_characterization
• Truth anchor: D5/S1/Phase/Interference/M468MemberTable.m468_zero_only_fails
• Dependency: D5/S1/Phase/Interference/ZolotarevSelector
• Dependency: D5/S1/Phase/SeatTowerArithmetic