netSkew
plain-language theorem explainer
Net skew of an eight-tick complex signal is the sum of log-moduli over the cycle. Admissible (balanced-ledger) signals are those with vanishing skew. The complex-structure forcing argument uses this as the ledger-balance functional that unitarity must preserve. The body is a direct finite sum of real logarithms of norms.
Claim. For a signal $f:\{0,\ldots,7\}\to\mathbb{C}$, the net skew is $\sigma(f)=\sum_{k=0}^{7}\ln\|f(k)\|$. Admissibility of the ledger requires $\sigma(f)=0$.
background
The module Complex Structure Forcing shows that the eight-tick shift (T7) cannot be diagonalized over $\mathbb{R}$: its eigenvalues include $i$ and $-i$, which have no real square roots of $-1$. Complexification and the DFT-8 are therefore algebraically forced, not optional.
A signal on the cycle is a map $f:\mathrm{Fin},8\to\mathbb{C}$. The net skew $\sigma$ is the total log-charge of that signal. The module's admissibility criterion is precisely $\sigma=0$ (balanced ledger). Downstream in the same argument, a linear map on signals preserves admissibility if and only if it preserves the Euclidean norm, which forces unitarity of the recognition operator.
Related cost notions elsewhere in the monolith (J-cost on ratios, event costs, rung-coarsened costs) measure recognition expense; skew is the separate ledger-balance scalar that those costs sit on top of.
proof idea
Pure definition: evaluate $\sum_{k:\mathrm{Fin},8}\log|f(k)|$ with the standard real logarithm and complex modulus. No lemmas or tactics; the sum is the meaning of the symbol.
why it matters
This scalar is the concrete $\sigma$ in the module's unitarity bridge: a recognition map preserves admissibility ($\sigma=0$) if and only if it preserves norm, hence is unitary. That step closes the registry gap "complex Hilbert space from cost," linking T5 (J-uniqueness), T7 (eight-tick octave), and the forced complex structure.
Without a named net-skew functional, the Parseval and phase-invariance claims in the same file have nothing to conserve. Even with no recorded downstream edges yet, every later lemma about admissible mode coefficients or unitary diagonalization of the shift must quantify over $\sigma(f)=0$.
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