Empirical/CSD: arbitrary projective measurement contexts as derived Kähler volumes #
Category: 3-Local (CSD-ontic layer; genuine volume derivation, not a transport tag, and not conditional on any preparation bundle).
The context-generic surfacing of LF4.born_frequency_convergence_N_uncond:
measuring a pure state ψ on ℂℙ^M in any orthonormal-basis (rank-1
projective) context B, the outcome Born weights ‖⟨B i, ψ⟩‖² are genuine
Fubini–Study typicality volumes on the ontic Σ = ℂℙ^M. Carving-free,
Gleason-free, unconditional — every unit preparation, eigenstates of the
context included (no genericity hypothesis; hpos-free since the 2026-06-11
call-site migration onto LF4/BornRegionUncond.lean). In the spirit of
Empirical/CSD/BellVolume.lean, Empirical/CSD/GHZVolume.lean, and
Empirical/CSD/HardyVolume.lean, but parameterised over the context rather
than a fixed state.
The key reduction (no new geometry) #
Measuring ψ in the orthonormal context B is, coordinate-for-coordinate,
measuring the rotated coordinate vector B.repr ψ in the computational basis:
⟨B i, ψ⟩ = (B.repr ψ) i = ⟨eᵢ, B.repr ψ⟩ (OrthonormalBasis.repr_apply_apply
+ EuclideanSpace.inner_single_left)
so ‖⟨B i, ψ⟩‖² = ‖⟨eᵢ, B.repr ψ⟩‖². Since B.repr is a LinearIsometryEquiv,
‖B.repr ψ‖ = ‖ψ‖, so the rotated state inherits norm-one. Instantiating
born_frequency_convergence_N_uncond at B.repr ψ therefore lands exactly on
the context Born weights ‖⟨B i, ψ⟩‖², with the Born = ontic-volume content
(fs_born_volume_ratio_N_uncond / _apex_uncond) already discharged for the
rotated state — vanishing rotated amplitudes give FS-null cells, which is
the Born weight 0. No Busch, no carving.
Why this is the grounding for contextuality #
The Kochen–Specker / Mermin–Peres no-go statements (Empirical/QM/) turn on
the fact that the outcome weights a measurement assigns are context-dependent:
no single non-contextual hidden-variable assignment of 0/1 values to all rays
reproduces the quantum statistics across overlapping contexts. This file realises
each such context's rank-1 outcome weight as a genuine Fubini–Study typicality
volume on the same ontic Σ = ℂℙ^M — the context enters only through which
orthonormal frame B carves the moment regions, not through any extra ontic
structure. The context-dependence the KS/MP theorems exploit is, on the CSD
ontology, the dependence of the carved volume regions on the measurement frame.
Honest scope #
No genericity hypothesis: the capstones cover every unit preparation ψ,
including eigenstates of the context (ψ = B i, the Kochen–Specker-interesting
boundary preparations) — resolving the earlier limitation, recorded at the
a9c3a50 landing, that ψ = B 0 failed the engine-inherited hpos. Zero-weight
outcome cells genuinely collapse to FS-null sets (BornRegionUncond per-cell
dichotomy); they are not redefined to hit Born values. The LHV / KS / MP
impossibility itself lives in Empirical/QM/ (KochenSpecker, MerminPeres).
Degenerate-eigenspace extension (rank-1 scope note closed) #
A degenerate projective measurement has outcome projectors Pₐ of rank ≥ 1.
Picking an orthonormal eigenbasis B adapted to the spectral decomposition and a
block labelling blk : Fin (M+1) → ι (the ι-many outcome labels), the outcome-a
projector is Pₐ = ∑_{blk i = a} |B i⟩⟨B i|, so the outcome Born weight is the
block sum of per-ray Born weights:
⟨ψ, Pₐ ψ⟩ = ∑_{blk i = a} ‖⟨B i, ψ⟩‖² (block_born_eq_blockSum)
a finite sum of Fubini–Study typicality volumes on the same ontic Σ = ℂℙ^M.
The block (degenerate-outcome) empirical frequency is the finite sum of the per-ray
frequencies (the per-ray barycentric regions are disjoint, so summing the
frequencies is the frequency of their union, the block outcome region), and it
converges to the block Born weight: block_born_frequency_volume. This closes the
rank-1 scope note above: degenerate contexts — including the two-qubit Mermin–Peres
rank-2 eigenspace observables and any other rank ≥ 1 projective context — are now
grounded as block sums of FS volumes. block_born_eq_blockSum writes the block
Born weight via the explicit rank-1-sum projector image Pₐ ψ = ∑_{blk i = a} ⟨B i, ψ⟩ • B i; the equivalent reading is ∑_{blk i = a} ‖⟨B i, ψ⟩‖² = ‖orthogonalProjection (span {B i : blk i = a}) ψ‖² (Parseval over the orthonormal
sub-family), which is the standard "projection onto the eigenspace" statement.
Honest scope unchanged from the rank-1 case: this is a faithful realisation,
not a derivation (Φ = id, the FS regions carved in the rotated frame); the
contextuality / KS / MP no-go stays at the QM-validity layer (Empirical/QM/).
The context-rotation identity (step 1) #
Context ↔ rotated-state Born identity. The Born weight of outcome i when
measuring ψ in the orthonormal context B equals the i-th computational Born
weight of the rotated coordinate vector B.repr ψ. Pure inner-product geometry:
⟨B i, ψ⟩ = (B.repr ψ) i = ⟨eᵢ, B.repr ψ⟩ (OrthonormalBasis.repr_apply_apply
Rotated-state norm and genericity transport #
B.repr is a LinearIsometryEquiv, so the rotated state preserves the norm.
The rotated state is nonzero (it has norm one).
Genericity transports along the rotation: if every context Born weight is
strictly positive, so is every computational Born weight of B.repr ψ.
No longer consumed by the capstones below (they route through the hpos-free
born_frequency_convergence_N_uncond); retained for callers of the conditional
engine born_frequency_convergence_N.
The context volume-frequency capstone #
Any projective measurement context's outcome Born weights as derived Kähler
volumes. For i.i.d. trials drawing microstates from the Fubini–Study typicality
measure on the ontic Σ = ℂℙ^M, the empirical frequencies of the N = M+1
barycentric Born outcome regions (carved in the rotated frame B.repr ψ)
converge, on a single almost-sure event, to the context Born weights
‖⟨B i, ψ⟩‖² of measuring the norm-one preparation ψ in the orthonormal
context B.
Carving-free, Gleason-free, unconditional — no busch_effect_gleason, no carved
regions, no preparation bundle, no genericity hypothesis. The proof is the
rotation reduction: instantiate born_frequency_convergence_N_uncond at
B.repr ψ (norm one by repr_norm), then rewrite the computational Born
weights back to the context weights via context_born_eq_rotated.
This grounds every rank-1 projective measurement context at every unit
preparation — eigenstates of the context included (ψ = B i: the other rays'
cells are FS-null and their frequencies converge to 0) — the reusable
contextuality primitive (Kochen–Specker, the rank-1 Mermin–Peres parts): each
context-dependent outcome weight that a non-contextual hidden-variable assignment
cannot jointly reproduce is here a genuine Fubini–Study typicality volume on the
fixed ontic Σ. Honest scope: rank-1 (non-degenerate) contexts; degenerate
eigenspaces via block_born_frequency_volume (see the module docstring).
Degenerate-eigenspace blocks (step 1: the block Born weight) #
Degenerate-outcome Born weight = block sum of per-ray Born weights. For an
orthonormal eigenbasis B adapted to a block labelling blk, the rank-≥1
eigenspace projector for outcome a is Pₐ = ∑_{blk i = a} |B i⟩⟨B i|, so
Pₐ ψ = ∑_{blk i = a} ⟨B i, ψ⟩ • B i. Sandwiching against ψ (real part) gives the
block sum of per-ray Born weights ∑_{blk i = a} ‖⟨B i, ψ⟩‖², the outcome-a Born
weight ⟨ψ, Pₐ ψ⟩.
Delivered in the block-sum-direct form (the projector applied to ψ written as
its explicit rank-1 sum), not the orthogonalProjection-over-subfamily-span form:
the latter requires constructing an orthonormal basis of the span submodule from the
sub-family of B, which is span-of-subfamily friction in Mathlib with no payoff here.
The equivalent projector reading is
∑_{blk i = a} ‖⟨B i, ψ⟩‖² = ‖orthogonalProjection (span ℂ {B i : blk i = a}) ψ‖²
(Parseval over the orthonormal sub-family). Proof: inner_sum + inner_smul_right +
inner_conj_symm + RCLike.mul_conj, termwise.
The degenerate-context block volume-frequency capstone (step 2) #
Degenerate projective measurement context's block (eigenspace) Born weight as a
derived sum of Kähler volumes. For i.i.d. trials drawing microstates from the
Fubini–Study typicality measure on the ontic Σ = ℂℙ^M, the empirical frequency of
the degenerate outcome a — the finite sum of the per-ray empirical frequencies
over the block {i : blk i = a} — converges, on a single almost-sure event, to the block
Born weight ∑_{blk i = a} ‖⟨B i, ψ⟩‖² = ⟨ψ, Pₐ ψ⟩ (see block_born_eq_blockSum).
(This sum-of-per-ray-frequencies equals the frequency of the union of the block's
barycentric per-ray regions, because those regions are pairwise disjoint — the
barycentric subdivision is a genuine partition, now formalised as
CSD.LF4.bornRegion_pairwiseDisjoint (LF5-F engine half). The union-event
restatement is block_born_frequency_volume_event below; the sum form proved here
needs only additivity of limits.)
Carving-free, Gleason-free, unconditional — no genericity hypothesis (every unit
preparation, eigenvectors of any block included).
Proof: take the single a.s. event from context_born_frequency_volume (which gives
every ray i's convergence simultaneously) and sum the block's per-ray
convergences via tendsto_finsetSum. The per-ray frequency summand is verbatim the
conclusion of context_born_frequency_volume, so the per-ray limits feed in directly.
This closes the rank-1 scope note of context_born_frequency_volume: degenerate-
eigenspace contexts (the two-qubit Mermin–Peres rank-2 observables and any rank ≥ 1
projective context) are grounded as block sums of FS typicality volumes on the fixed
ontic Σ. Honest scope unchanged: realisation not derivation (Φ = id, FS regions
carved in the rotated frame); the KS / MP no-go stays at the QM-validity layer.
The union-event restatement (the owed aeece86 form) #
Degenerate-outcome block frequency, as the frequency of a single union
event. The aeece86-owed restatement of block_born_frequency_volume: the
empirical frequency of the degenerate outcome a stated as the frequency of the
single union event ⋃_{blk i = a} bornRegion … (the eigenspace outcome
region), converging to the block Born weight ∑_{blk i = a} ‖⟨B i, ψ⟩‖². The
union form is now available because the per-ray barycentric cells are pairwise
disjoint (CSD.LF4.bornRegion_pairwiseDisjoint, LF5-F engine half), so the
indicator of the union equals the sum of the per-ray indicators
(CSD.LF4.indicator_iUnion_disjoint). The sum form (block_born_frequency_volume)
is untouched.
Concrete degenerate (rank-2) witness: the two-qubit parity Z⊗Z #
Concrete degenerate (rank-2) witness: the two-qubit parity observable Z⊗Z.
The parity / Z⊗Z outcome Born weight realised as a block sum of Fubini–Study
typicality volumes — the Mermin–Peres rank-2 observable case made explicit.
Z⊗Z on EuclideanSpace ℂ (Fin 4) has eigenvalues ±1, each eigenspace of
rank 2; its eigenbasis is the computational basis (no rotation needed), so we
take B := EuclideanSpace.basisFun (Fin 4) ℂ (whose i-th vector is
EuclideanSpace.single i 1, EuclideanSpace.basisFun_apply). The block labelling
blk = ![0,1,1,0] : Fin 4 → Fin 2 is the parity grouping of the two-qubit strings:
indices {0,3} (|00⟩, |11⟩, even parity, eigenvalue +1) ↦ outcome 0; indices
{1,2} (|01⟩, |10⟩, odd parity, eigenvalue −1) ↦ outcome 1.
Instantiating block_born_frequency_volume at B, blk, a = 0 and collapsing the
even-parity block {0,3} (Finset.sum_pair, decide on the filter), the empirical
frequency of the even-parity (Z⊗Z = +1) outcome converges almost surely to the
Z⊗Z = +1 Born weight ‖⟨e₀, ψ⟩‖² + ‖⟨e₃, ψ⟩‖² = ⟨ψ, P₊ ψ⟩, a block sum of two
FS typicality volumes on the fixed ontic Σ = ℂℙ³. Carving-free, Gleason-free,
foundational-triple-only.
Honest scope unchanged from the generic degenerate case (block_born_frequency_volume):
this is a faithful realisation of the rank-2 outcome weight as a sum of ontic
volumes, not a derivation (Φ = id, FS regions carved in the computational frame);
the Mermin–Peres / Kochen–Specker no-go itself stays at the QM-validity layer
(Empirical/QM/). No genericity hypothesis: every unit two-qubit preparation is
covered, the computational eigenstates |00⟩, |01⟩, |10⟩, |11⟩ (and the Bell
states with vanishing components) included.