L7 - Complex Amplitudes and the Quantum Sector of the Apex

Situation. L0 (https: //doi. org/10. 5281/zenodo. 20423985) named the apex generator \ (Mₓ (, 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>₀N\). L1–L6 worked the real-positive sector: the amplitude sphere \ (S^N-1_+ R>₀N\), Born closure \ (pᵢ=xᵢ²/ⱼxⱼ²\), 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 \ (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>₀N\) by \ (CN\) does four structurally distinct things at once: (i) it enlarges the closure symmetry from \ (SN\) permutations to the unitary group \ (U (N) \) ; (ii) it activates the Kummer cover \ (ₙ S¹\) on each coordinate, giving a non-trivial deck group \ ( (ₙ) N/ₙ\) for the \ (n\) -th-power closure; (iii) it introduces a phase fiber \ (T^N-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 \ (CP^N-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 \ (CN\). The complex amplitude \ (z CN\) is positioned at three nested layers — the amplitude sphere \ (S^2N-1 CN\), the projective space \ (CP^N-1=S^2N-1/U (1) \), and the probability simplex \ (^N-1=CP^N-1/T^N-1\) — connected by the Hopf and toric fibrations. The Born closure \ ₁₎ₑ₍: CN\0\^N-1, z (|zᵢ|²ⱼ|zⱼ|²) ₈=₁N \ factors as \ (₁₎ₑ₍=ₓ₎ₑ₈₂\) through these layers, with \ (ₓ₎ₑ₈₂\) the moment map of the toric action of \ (T^N-1\) on \ (CP^N-1\). We define the complex apex generator \ Mᵦ (, c): =1Nₒ ₙ/₍e^ yₛ (z) \, c (s), yₛ (z): = zₛ-1Nⱼ zⱼ, \ phrased precisely below as a holomorphic function of \ (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 \ (Mᵦ\) from the complex amplitude to its modulus ledger, and the Kummer-cover theorem identifying the integer-\ (\) samples on the complex side with sections of an explicit \ (N\) -twisted line bundle over \ (CP^N-1\). Result. Six core claims of L7: Three-layer factorization of Born closure. The map \ (₁₎ₑ₍: CN\0\^N-1\) factors uniquely as \ (ₓ₎ₑ₈₂\) (applied right-to-left): radial scaling to the sphere \ (S^2N-1\), Hopf projection to \ (CP^N-1\) (modding out global phase \ (U (1) \) ), then the toric moment map to the simplex (modding out relative phases \ (T^N-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 \ (g₅ₒ\) on \ (CP^N-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 \ (d₅ₑ=2ᵢqᵢ\) 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, \ (d₅ₒ (p, q) \) equals the unit-sphere geodesic distance \ (ᵢqᵢ\), and the doubling lives entirely in the square root, at all \ (N\). Complex apex generator. Define \ (Mᵦ (, c) \) on \ (CN\) via complex centered log-ledger \ (yₛ= zₛ-1Nⱼ zⱼ\) (with a canonical branch on the positive sector and analytic continuation onto the cut Riemann surface of the complex log). \ (Mᵦ\) is holomorphic in \ (\) on each branch, and the real-positive evaluation recovers L0's \ (Mₓ\). Decoherence theorem (modulus-phase reduction). The complex apex generator splits termwise as \ Mᵦ (, c) =1Nₛ e^ y^{ (|z|) ₛ} e^i y^{ (z) ₛ}\, c (s), \ with \ (y^ (|z|) ₛ\) the L0 centered log-ledger of the modulus ledger \ (|z| R>₀N\) 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 \ (Mᵦ (, c) \) to L0's \ (M|ₙ| (, c) \) exactly. The real-positive sector is exactly the locus \ (U (1) R>₀NN\) on which the phase deformation is trivial for generic \ (\) (Proposition 3. 6). Kummer-cover theorem. The closure diagonal \ (Mᵦ (m, c) \) at integer \ (m\) has \ ( (ₘ) N/ₘ\) deck-group symmetry: replacing each \ (zₛ\) by \ (ₛ zₛ\) for \ (ₛₘ\) leaves \ (|zₛ|^2m\) invariant but multiplies the apex generator by \ (ₛₛ^\), encoding the Galois action on the moment diagonal as character-twisting on the cover. Qubit anchor at \ (N=2\). The qubit case \ (CP¹ S²\) 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 \ (\) -plane parameterized by the Bloch-vector orientation. Harvest. The phase structure of pure-state finite-dimensional quantum mechanics on \ (CN\) is not an extra principle: it is the cover of the categorical-simplex apex — the same Fourier-Mellin transform, lifted from \ (R>₀N\) to \ (CN\) 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 \ (T^N-1\), and the \ (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.