1 Technical Draft: The Grand Translation Symmetries 1.1 The Global Boundary Parameter and Sign Vector Let a physical system state be defined by the four-vector (x, y, z, t) over the three-element field F3 , evaluated in base-9B using the canonical digit set...
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1 Technical Draft: The Grand Translation Symmetries 1.1 The Global Boundary Parameter and Sign Vector Let a physical system state be defined by the four-vector (x, y, z, t) over the three-element field F3 , evaluated in base-9B using the canonical digit set where ¯1 = −1, ¯2 = −2, ¯3 = −3, and ¯4 = −4. We define the global coordinate tracker s as the linear scalar sum of the total state envelope: s=x+y+z+t (1) The parameter s acts as the sole determinant of the system’s global parity orientation. The sign alternator (−1)s establishes a binary phase switch (+1 or −1). 1.2 The Algebraic Squaring of the Invariant Metric When evaluating the self-coupling or scalar intensity of a state space, we square the invariant formula f (x, y, z, t)2 . Because (−1)2s ≡ +1 for all integer values of s, the binary phase alternator naturally clears from the quadratic product, leaving the expanded polynomial core modulo 9: f (x, y, z, t)2 ≡ (x − 2y + 4z + 3t)2 (mod 9) (2) Expanding this trinomial-time core
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