LF4: POVM Born weight as a dilated rank-1 block sum (P.3a) #
Category: 3-Local (LF4 POVM volume reading).
This is P.3a of the POVM tranche (specs/povm-plan.md): the block
decomposition that turns the Naimark Born transfer (born_transfer, P.2) into a
sum of dilated computational-basis (rank-1) Born weights:
pᵢ(ψ) = ⟨ψ, Eᵢ ψ⟩ = ⟨Vψ, Πᵢ (Vψ)⟩ = ∑ₙ ‖⟨e_{(n,i)}, Vψ⟩‖².
The ancilla-i projector Πᵢ = I_N ⊗ |i⟩⟨i| is a coarse (rank-N) outcome —
the union of the N computational-basis cells {(n, i) : n} on the dilated
space. So the POVM weight is the sum, over that block, of the dilated rank-1
Born weights, each of which the achieved general-N result reads as a
Fubini–Study volume on ℂℙ^{N·|ι|−1} (the FS-volume identification, P.3b, sits
on top of this via the Fin N × ι ≃ Fin (N·|ι|) reindex).
The reindex isometry preserves computational-basis Born weights. For the
piLpCongrLeft e reindex isometry L, ‖⟨e_{e p}, L w⟩‖² = ‖⟨e_p, w⟩‖² — the cell
p maps to the coordinate e p with the Born weight unchanged.
‖⟨e_p, w⟩‖² = ‖w_p‖² on the dilated EuclideanSpace ℂ (Fin N × ι).
Block decomposition. The projective Born weight of w against the
ancilla-i projector is the sum of the rank-1 computational-basis Born weights
over the i-th block:
re ⟨w, Πᵢ w⟩ = ∑ₙ ‖⟨e_{(n,i)}, w⟩‖².
POVM Born weight as a dilated rank-1 block sum (P.3a). Composing the
Naimark Born transfer with the block decomposition: the POVM weight pᵢ(ψ) is the
sum, over the i-th ancilla block {(n, i) : n}, of the dilated
computational-basis Born weights of Vψ. Each summand is a rank-1 projective Born
weight on ℂℙ^{N·|ι|−1}, read as a Fubini–Study volume by the achieved
general-N result (P.3b).
POVM Born weight as a sum of Fubini–Study volumes (P.3b). Given a reindex
e : Fin N × ι ≃ Fin (M+1) of the dilated index (concretely finProdFinEquiv
after ι ≃ Fin |ι|, so N·|ι| = M+1) and the induced reindex isometry
L = piLpCongrLeft e, write ψ' = L (Vψ) for the reindexed dilated state on
ℂℙ^M. When ψ' is a unit, fully-generic preparation (all M+1 amplitudes
nonzero — the dilation genericity condition), the POVM Born weight is the sum,
over the i-th ancilla block, of the genuine Fubini–Study typicality volumes
of the dilated barycentric cells:
pᵢ(ψ) = ∑ₙ μ_FS( bornRegion ψ' (e (n, i)) ).
This is the headline of the ontic POVM reading: the (non-projective) POVM Born
weight is a Kähler volume on the dilated configuration space Σ' = ℂℙ^{N·|ι|−1},
carving-free and Gleason-free — it composes the Naimark Born transfer with the
achieved general-N Born = FS-volume result (bornRegion_fs_measure), no
busch_effect_gleason. Honest scope: the dilation V is supplied (P.2), so this
relocates POVM Born onto a larger posited configuration space (system + ancilla);
genericity (hpos) excludes dilated states with a vanishing amplitude.
An hpos-free form is available: povm_born_eq_dilated_volume_uncond
(BornRegionUncond.lean).
POVM empirical frequencies as a dilated Kähler-volume convergence (P.4).
The observable capstone. For i.i.d. trials drawing microstates from the
Fubini–Study typicality measure on the dilated ontic Σ' = ℂℙ^{N·|ι|−1} (the
reindexed dilated state ψ' = L (Vψ) being a unit, fully-generic preparation), the
empirical frequency of the i-th POVM outcome — the sum, over the i-th ancilla
block, of the dilated barycentric-cell frequencies — converges, on a single
almost-sure event, to the POVM Born weight pᵢ(ψ).
Composes the general-N per-cell convergence born_frequency_convergence_N (joint
a.s. over all M+1 cells) with the finite block sum (tendsto_finsetSum), landing
the limit on pᵢ(ψ) via the P.3a block decomposition. Carving-free, Gleason-free —
the empirical → Born chain for a general (non-projective) POVM runs entirely on the
ontic FS-volume derivation, no busch_effect_gleason. Honest scope: dilation
supplied; the block frequency is the sum of the cells' frequencies (the cells are
the rank-1 dilated outcomes), and genericity (hpos) is assumed.
An hpos-free form is available: povm_born_frequency_volume_uncond
(BornRegionUncond.lean).