w_total
plain-language theorem explainer
The total kernel is the product of secular and resonant weight factors. Gravity and discrete cosmology researchers cite this when deriving resonance-induced weight reductions in eight-tick models. The definition is realized by direct multiplication of the two real inputs with no lemmas required.
Claim. Let $w_ {sec}$ be the secular kernel weight and $w_{res}$ the resonant factor. The total kernel is defined by $w_{total}(w_{sec}, w_{res}) = w_{sec} · w_{res}$.
background
In the EightTickResonance module the resonant factor is given by $w_{res}(r) = 1 + C_{lag} ·$ interpolation cost at rung $r$, attaining its minimum value of 1 at integer resonance points. Secular weights are supplied by upstream kernel definitions: constant, inverse-linear, or exponential forms from Cosmology.BITKernelFamilies.kernel and the ILG.Kernel construction $w(k,a) = 1 + C · (a/(k τ_0))^α$. The product separates baseline secular evolution from resonant modulation in discrete vorticity sums and error budgets.
proof idea
This is a direct definition realized by multiplication of the two arguments. No lemmas are invoked; downstream theorems simply unfold the definition to obtain equalities and inequalities such as $w_{total}(w_{sec},1) = w_{sec}$.
why it matters
The definition supplies the algebraic core for the resonance weight reduction ratio theorem and the comparisons $w_{total}$ at resonance versus off-resonance. These results feed the eight_tick_period theorem that recovers the T7 eight-tick octave ($8 = 2^3$) from the unified forcing chain. It bridges ILG power-law kernels to resonant gravity weights without introducing new hypotheses.
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