Quaternion Orientation
on Z₂₅₆

Half-turn reversal, exact Q₈ embedding, CW/CCW orientation, and 720° state closure · September 2026

Abstract

Quaternion Sign as a Ring Half-Turn

This chapter studies Q₈ on the MPRC quarter-lattice {0,64,128,192} ⊂ Z₂₅₆. Hamilton multiplication is the declared external starting algebra. The embedding is exact: quaternion orientation reversal is represented by the MPRC half-turn R₁₂₈, preserving the resulting axis while shifting its ring position by exactly 128. The same construction gives I²=J²=K²=IJK=128 and separates 360° rotational closure from 720° full-state closure.

Claim boundary. The Q₈→Z₂₅₆ embedding and its half-turn/double-cover properties are proved. The stronger claim that MPRC CW/CCW geometry alone generates Hamilton anti-commutation remains an open conjecture.

01 · Starting Algebra

Hamilton Is the Input

i² = j² = k² = ijk = −1

ij=k, jk=i, ki=j

ji=−k, kj=−i, ik=−j

The cyclic products preserve orientation; reversing the traversal changes only the orientation of the resulting axis.

02 · Quarter-Lattice Embedding

Q₈ on {0,64,128,192}

χ(a,0)=1

χ(a,64)=a

χ(a,128)=−1

χ(a,192)=−a

No Euler formula, trigonometric evaluation, or floating-point phase is required.

03 · Identity

MPRC Half-Turn Orientation Identity

χ(a,s+128 mod 256)=−χ(a,s)

Verified for every a∈{I,J,K} and s∈{0,64,128,192}. Thus R₁₂₈:s↦s+128 is exactly quaternion orientation reversal inside this embedding.

04 · Theorem

MPRC Quaternion Orientation Theorem

CW	Site	Reverse	Site
IJ	K₆₄	JI	K₁₉₂
JK	I₆₄	KJ	I₁₉₂
KI	J₆₄	IK	J₁₉₂

JI=R₁₂₈(IJ), KJ=R₁₂₈(JK), IK=R₁₂₈(KI)

Reversal preserves the resulting axis and displaces the ring orientation by exactly 128.

05 · Half-Turn Theorem

Squares Land on 128

I²=J²=K²=IJK=128

64+64=128 (mod 256)

06 · Double Cover

360° Orientation, 720° State Closure

Ring	Element	Rotation
0	+1	0°
64	+I	180°
128	−1	360°
192	−I	540°
256≡0	+1	720°

T₃₆₀ ≠ I_state

T₃₆₀²=T₇₂₀=I

07 · U Dimension

Rotation and Observer Freedom

U is not a fourth quaternion basis unit. In MPRC, U is the fourth dimension and time is defined separately as t=⟨U|φ⟩. The observer walk remains:

z_U=7Θ mod 256

7⁻¹=183 mod 256

7·183≡1 mod 256

08 · Status Register

Proven vs Open

Statement	Status
χ(a,s+128)=−χ(a,s)	Exact embedding identity
Reverse basis product = same axis at +128	Proven under Q₈ embedding
I²=J²=K²=IJK=128	Proven under embedding
360° orientation / 720° full-state closure	Proven for embedded quaternion representation
MPRC CW/CCW alone generates Hamilton anti-commutation	Open conjecture

09 · Open Conjecture

MPRC Anti-Commutation Conjecture

R_CCW(A,B)=R₁₂₈(R_CW(B,A))

The next proof must remove Hamilton multiplication from the result side and derive this relation from MPRC traversal rules alone.