L7 - Complex Amplitudes and the Quantum Sector of the Apex

Situation. L0 (https://doi.org/10.5281/zenodo.20423985) named the apex generator \( M_x(β,c)\) of the structured categorical simplex as a Fourier-Mellin transform on \( Z/N\) of the centered log-ledger of a positive real ledger \(x∈ R>0^N\). L1–L6 worked the real-positive sector: the amplitude sphere \(SN-1_+⊂ R>0^N\), Born closure \(p_i=x_i^2/∑_jx_j^2\), the cumulant tower, the radical Helmert ladder. L0 Proposition 3.7 established that on the real-positive sector the \(n\)-th-power Kummer cover is trivial, with the deck-group action collapsing because \(μ_n∩ R>0=\{1\}\). L0 explicitly scoped the complex-amplitude extension — where the cover becomes non-trivial and the apex acquires phase structure — as a forward door (§6.1). Gap. The complex-amplitude extension is not a cosmetic generalization. Replacing \( R>0^N\) by \( C^N\) does four structurally distinct things at once: (i) it enlarges the closure symmetry from \(S_N\) permutations to the unitary group \(U(N)\); (ii) it activates the Kummer cover \(μ_n⊂ S^1\) on each coordinate, giving a non-trivial deck group \((μ_n)^N/μ_n\) for the \(n\)-th-power closure; (iii) it introduces a phase fiber \(TN-1=U(1)^N/U(1)\) over each simplex point, with the real-positive sector recovered as the decoherent (phase-averaged) face; (iv) it forces the Fisher–Rao metric to be the real-positive restriction of a strictly larger projective metric, the Fubini–Study metric on \(CPN-1\). The real-positive sector is one slice through a four-axis space (amplitude, phase, conductor, gauge). L1–L6 worked the slice; L7 builds the ambient. Action. We construct the complex apex on \( C^N\). The complex amplitude \(z∈ C^N\) is positioned at three nested layers — the amplitude sphere \(S2N-1⊂ C^N\), the projective space \(CPN-1=S2N-1/U(1)\), and the probability simplex \(ΔN-1=CPN-1/TN-1\) — connected by the Hopf and toric fibrations. The Born closure \[ πBorn: C^N\{0\}ΔN-1, z(|z_i|^2/∑_j|z_j|^2)ᵢ₌₁^N \] factors as \(πBorn=μtoric\) through these layers, with \(μtoric\) the moment map of the toric action of \(TN-1\) on \(CPN-1\). We define the complex apex generator \[ M_z(β,c):=1/N∑s∈ Z/Neβ y_s(z)\,χ_c(s)̄, y_s(z):=log z_s-1/N{∑_j}log z_j, \] phrased precisely below as a holomorphic function of \(β∈ C\) on each Riemann-surface branch of the complex log of \(z\). We then prove the decoherence theorem identifying L0's real-positive apex with the modulus-restriction of \( M_z\) from the complex amplitude to its modulus ledger, and the Kummer-cover theorem identifying the integer-\(β\) samples on the complex side with sections of an explicit \(μ_N\)-twisted line bundle over \(CPN-1\). Result. Six core claims of L7: Three-layer factorization of Born closure. The map \(πBorn: C^N\{0\}→ΔN-1\) factors uniquely as \(μtoric\) (applied right-to-left): radial scaling to the sphere \(S2N-1\), Hopf projection to \(CPN-1\) (modding out global phase \(U(1)\)), then the toric moment map to the simplex (modding out relative phases \(TN-1\)). Each layer has a well-defined symmetry quotient. Fubini–Study restricts to Fisher–Rao on the decoherent face (re-derivation). The Fubini–Study metric \(gFS\) on \(CPN-1\) restricts to one-quarter of the Fisher–Rao metric on the real-positive face under the square-root section. This is the classical FS / quantum-Fisher relation (Provost–Vallée 1980; Braunstein–Caves 1994; Bengtsson–Życzkowski 2017); we re-derive it in the categorical-simplex language to fix the attribution of the factor of \(2\) in L1's \(dFR=2∑√p_iq_i\) to the Born / square-root embedding \(p↦√ p\) (in agreement with L6 §2), not the Hopf fibration: on the decoherent face the Hopf/global-phase quotient acts as an isometry, \(dFS([√ p],[√ q])\) equals the unit-sphere geodesic distance \(∑√p_iq_i\), and the doubling lives entirely in the square root, at all \(N\). Complex apex generator. Define \( M_z(β,c)\) on \( C^N\) via complex centered log-ledger \(y_s=log z_s-1/N∑_jlog z_j\) (with a canonical branch on the positive sector and analytic continuation onto the cut Riemann surface of the complex log). \( M_z\) is holomorphic in \(β\) on each branch, and the real-positive evaluation recovers L0's \( M_x\). Decoherence theorem (modulus-phase reduction). The complex apex generator splits termwise as \[ M_z(β,c)=1/N∑_s e^{β y(|z|)_s}· e^{iβ y( z)_s}\,χ_c(s)̄, \] with \(y(|z|)_s\) the L0 centered log-ledger of the modulus ledger \(|z|∈ R>0^N\) and \(y( z)_s\) the centered argument (Proposition 3.6). The modulus-restriction \(z↦|z|\) — the operational projection of complex amplitude to measurement-basis modulus — sets every phase factor to \(1\) and reduces \( M_z(β,c)\) to L0's \( M|z|(β,c)\) exactly. The real-positive sector is exactly the locus \(U(1)· R>0^N⊂Ω_N\) on which the phase deformation is trivial for generic \(β\) (Proposition 3.6). Kummer-cover theorem. The closure diagonal \( M_z(m,c)\) at integer \(m\) has \((μ_m)^N/μ_m\) deck-group symmetry: replacing each \(z_s\) by \(ζ_s z_s\) for \(ζ_s∈μ_m\) leaves \(|z_s|²ᵐ\) invariant but multiplies the apex generator by \(∏_sζ_s±\), encoding the Galois action on the moment diagonal as character-twisting on the cover. Qubit anchor at \(N=2\). The qubit case \(CP^1 S^2\) realizes the apex on the Bloch sphere, with the real-positive sector being the equatorial great circle \(\{|0,|1\}\). The full Lee–Yang lattice of L0 Proposition 2.5 lifts to a phase-twisted lattice in the complex \(β\)-plane parameterized by the Bloch-vector orientation. Harvest. The phase structure of pure-state finite-dimensional quantum mechanics on \( C^N\) is not an extra principle: it is the cover of the categorical-simplex apex — the same Fourier-Mellin transform, lifted from \( R>0^N\) to \( C^N\) via the canonical Hopf-and-toric fibrations of complex projective space. We use the slogans decoherence is closure of phase and quantization is the cover in a deliberately scoped sense: the cover captures the pure-state phase redundancy — global \(U(1)\), relative \(TN-1\), and the \(μ_N\) branch ambiguity of root closures — and not the non-commutativity of observables, measurement back-action, unitary dynamics, or tensor-product/entanglement structure, which sit in the forward doors of §7. Within that scope the real-positive sector is the decoherent face and the complex sector is the full bundle, and L6 Conjecture 4 ("complex amplitude extension") is now a structure theorem rather than a conjecture.