field members — yin and yang

proposal

name the two 32-bit halves of a Goldilocks field element yin (low) and yang (high), and retire Word (= a field proven in [0,2³²)) — which frees word for neural's lexical unit.

field = yang·2³² + yin
  yin  ∈ [0,2³²)   the low half     (was `Word` / surface u32)
  yang ∈ [0,2³²)   the high half

why two members, and why they interpenetrate

a Goldilocks field element is 64-bit (p = 2⁶⁴ − 2³² + 1). it splits at 32 bits into two equal-width members — but they are not symmetric, and that asymmetry is the whole point of Goldilocks. reducing higher powers:

2⁶⁴ ≡ 2³² − 1   (mod p)
2⁹⁶ ≡ −1

so a 128-bit product (two field elements multiplied) reduces with no division, by folding the high members down:

x = x₃·2⁹⁶ + x₂·2⁶⁴ + x₁·2³² + x₀
  ≡ (x₀ − x₂ − x₃) + (x₁ + x₂)·2³²

yin (low) stays; yang (high) folds back into the field, carrying a ∓1. low is the residue, high is what wraps — they pour into each other. exactly yin and yang.

why the name is structure, not decoration

p = Φ₆(2³²) = (2³²)² − 2³² + 1 — the sixth cyclotomic polynomial at 2³². 2³² is a primitive sixth root of unity (g³ ≡ −1, g⁶ ≡ 1). the +1 — the cyclotomic constant — does three jobs at once:

1. primality — 2⁶⁴ − 2³² is even; +1 makes p prime (a field needs a prime modulus)
2. NTT — p − 1 = 2³²(2³² − 1) is divisible by 2³², so there is a 2³²-th root of unity → fast transforms for STARKs
3. the fold — 2⁶⁴ ≡ 2³² − 1; the −1 you see in reduction is that +1 returning

that lone unit is the seed each half carries of the other — yin's dot of yang. the pair is the structure of the field, not a metaphor pasted on.

the rename (delta for the terms-map)

word (the noun) is hereby free. it goes to neural as the lexical unit — a typed particle. see neural.

scope

surface and naming only: the 32-bit value type, plus the names for field decomposition. it does NOT touch field, reduction, order, or particle. on acceptance, update the terms-map rows and propagate through nox / trident code and docs in the next sweep.