N_modes_total
plain-language theorem explainer
The total phase-mode budget of the eight-tick DFT is the constant 44. Cosmology derivations of Ω_Λ from Recognition phase saturation cite this as the denominator side of the mode count. It is a bare natural-number definition, not a counting proof.
Claim. The total number of phase modes in the eight-tick discrete Fourier structure is the natural number $N_{\mathrm{modes}}=44$.
background
The module derives the dark-energy fraction $\Omega_\Lambda$ from phase saturation on the Recognition eight-tick cycle (primer landmark T7: period $2^3$). The core claim is $\Omega_\Lambda=11/16-\alpha/\pi\in(0.680,0.700)$, matching Planck 2018 within the stated band.
Step 1 of that derivation is a phase-mode budget: the 8-tick DFT is assigned $N_{\mathrm{modes}}=44$ frequency modes, attributed to the 44-mode structure forced by the $w_8$ gap-weight derivation. Of these, 11 are $Q_3$-saturated (spatial plus gauge degrees under the $[4,2,2]$ Gray-code asymmetry). The saturated combinatorial fraction is then $11/16$, with $16=2^4$ the 4-bit addressing of the cycle.
This declaration simply names the total mode count used in that budget. Upstream edges touch foundation bridges and gauge-history measure structure; they do not compute 44 here.
proof idea
Definitional constant: the body is the literal natural 44. No tactics, no lemmas, no reduction. Downstream siblings (saturated count, raw $\omega$, EM correction, $\Omega_\Lambda$ bounds) consume the name as a fixed budget.
why it matters
Without a fixed total mode count, the saturated fraction $11/16$ and the final formula $\Omega_\Lambda=11/16-\alpha/\pi$ have no denominator. The module places 44 as the 8-tick DFT budget forced by the $w_8$ gap-weight story, then subtracts the one-loop EM piece $\alpha/\pi$.
This sits in the cosmology layer that turns the eight-tick octave (T7) and gauge/Gray-code mode counting into a sharp $\Omega_\Lambda$ interval. Sibling theorems bound $\Omega_\Lambda$ in $(0.683,0.686)$ after the correction. No used_by edges are recorded outside the module, so its role is local scaffolding for that certificate.
It does not close the open ledger-floor bridge noted upstream; it only freezes the integer the saturation arithmetic needs.
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