N_modes_saturated
plain-language theorem explainer
The integer 11 counts Q₃-saturated phase modes in the 8-tick DFT budget: the modes fixed by three spatial axes plus gauge structure under the [4,2,2] Gray-code asymmetry. Cosmologists deriving Ω_Λ from phase saturation cite it as the numerator of the raw fraction 11/16. It is a bare natural-number constant, not a proved identity.
Claim. The number of $Q_3$-saturated modes is the natural number $11$, arising from the $[4,2,2]$ Gray-code plus gauge structure (equivalently $4+4+2+1$).
background
The module derives the dark-energy fraction $\Omega_\Lambda$ from phase saturation on the Recognition Science 8-tick cycle. The core claim is $\Omega_\Lambda = 11/16 - \alpha/\pi \in (0.680, 0.700)$, matching Planck 2018 within the predicted band.
Step 1 of the derivation fixes a mode budget: the 8-tick DFT carries 44 frequency modes, of which 11 are $Q_3$-symmetric. Those 11 come from three spatial degrees of freedom plus the gauge sector under the $[4,2,2]$ Gray-code asymmetry (or, hierarchically, $4+4+2+1$). The denominator 16 is the 4-bit addressing of the 8-tick cycle ($2^4$), so the raw saturated fraction is exactly $11/16$.
Upstream, tick is the RS-native time quantum $\tau_0 = 1$; one octave is eight ticks. That tick structure supplies the addressing bits that make the denominator 16, against which this numerator is measured.
proof idea
Pure definition: the constant is set equal to the natural number 11. No tactics, no lemmas. Downstream equalities such as $11 = 4+4+2+1$ and $11/16$ are proved separately by decide and norm_num after unfolding this constant and the addressing denominator.
why it matters
This integer is the combinatorial numerator of the entire $\Omega_\Lambda$ derivation. It is unfolded in omega_raw (raw fraction before EM correction), omega_raw_eq ($= 11/16$), omega_lambda_one_measured_input (combinatorics minus measured-$\alpha$ correction), and omega_lambda_canonical_form ($\Omega_\Lambda = 11/16 - \alpha_{\mathrm{CODATA}}/\pi$). The sibling q3_mode_count records the hierarchical split $4+4+2+1$.
In framework terms it sits on the eight-tick octave (T7) and the three spatial dimensions (T8): the $Q_3$ count is the mode content of $D=3$ plus gauge under Gray-code asymmetry. The module deliberately keeps one measured input (CODATA $\alpha$); the seed $4\pi\cdot 11$ in the $\alpha$ construction is treated as an identification, not a free derivation of the coupling. Closing a zero-parameter certificate would require a separate derivation of $\alpha$, not a change to this mode count.
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