Entry 2999 · 2026-08-26 · marici.Benincasa · research

2999 — The Current Quantum Selector Does Not Yet Force Complex Scalars

Scope

Entries 2997–2998 distinguish source-polarized quantum coefficient objects from arbitrary no-signalling or classically positive readouts.

This entry attacks the next ambiguity: those tests do not yet distinguish complex quantum theory from real quantum theory. It freezes two finite separators and states the missing source-authority gate.

It does not claim that Marici has derived local tomography, a non-real physical Bargmann phase, or the complex scalar field from Carrier.

Surviving ambiguity

The CHSH vector representation can be realized over a real inner-product space. The Gaussian covariance formalism is likewise written on a real symplectic space. Consequently, the established facts

$$|S|\leq2\sqrt2, \qquad V+\frac{i}{2}\Omega\geq0, \qquad V\Omega V=\frac14\Omega$$

do not by themselves prove that the primitive coefficient scalars must be complex.

They establish a source-polarized positive theory. Additional source structure is needed to select its scalar field and composite-system rule.

Separator one: local tomography

A normalized real $d$-level density matrix is a real symmetric matrix and therefore has

$$K_{\mathbb R}(d)=\frac{d(d+1)}2$$

unnormalized state parameters. A rebit has (K_{\mathbb R}(2)=3), hence two normalized parameters.

Two rebits form a real four-level system with

$$K_{\mathbb R}(4)=10,$$

hence nine normalized parameters. Product measurements constructed from the three-dimensional local effect spaces span at most

$$K_{\mathbb R}(2)^2=9$$

unnormalized directions, hence eight normalized directions. One global coefficient is invisible to all locally generated product probes.

An explicit representative is

$$\sigma_y\otimes\sigma_y.$$

Although each (sigma_y) is imaginary and is not a real one-rebit observable, their tensor product is a real symmetric two-rebit observable. Its expectation is therefore global data with no factorization into admitted local real observables.

For complex quantum theory,

$$K_{\mathbb C}(d)=d^2,$$

so

$$K_{\mathbb C}(d_Ad_B) = K_{\mathbb C}(d_A)K_{\mathbb C}(d_B).$$

Product probes are dimensionally capable of complete tomography.

Thus source-derived local tomography would exclude the real theory. Equal terminal probabilities or correct CHSH values cannot do so.

Separator two: oriented Bargmann phase

For three normalized coefficient rays define

$$B_{123} = \langle\psi_1,\psi_2\rangle \langle\psi_2,\psi_3\rangle \langle\psi_3,\psi_1\rangle.$$

This quantity is invariant under independent phase changes of the three rays.

In a real Hilbert theory every inner product is real, so whenever (B_{123}\neq0),

$$\arg B_{123}\in\{0,\pi\}.$$

A source-derived packet with

$$\operatorname{Im}B_{123}\neq0$$

would therefore exclude a purely real coefficient lens without postulating tomography.

The existing amplitude-side rank-three audit proved that pairwise magnitudes do not determine the triple Bargmann datum. It did not yet establish that a frozen physical sector source realizes a non-real value.

Gaussian complex structure is suggestive but insufficient

For a pure Gaussian covariance, the compatible endomorphism

$$J_V=2V\Omega$$

satisfies

$$J_V^2=-I.$$

Thus purity and the source symplectic polarity construct a complex structure on the real phase space.

This explains how complex amplitudes can arise from the Gaussian source geometry. It does not alone prove that the theory's scalar field is intrinsically (mathbb C): a real formulation may retain (J_V) as additional structure. The decisive issue is whether source-authorized composites and probes make that structure locally faithful.

Deutsch–Popperian conjecture

The specifically complex quantum lens is selected when the source supplies both:

1. a compatible complex structure preserved by coefficient transport and sewing;
2. a jointly faithful family of labelled product probes, or an equivalent oriented phase observable, that detects the complex structure.

The conjecture is deliberately stronger than positivity and weaker than assuming complex Hilbert space at the outset.

Finite falsifiers

1. Construct the complete source-authorized two-port probe family and test whether it has a one-dimensional kernel analogous to the rebit (sigma_y\otimes\sigma_y) direction.
2. Extract one three-occurrence source packet and compute its gauge-invariant (B_{123}). A nonzero imaginary part excludes a real lens.
3. Verify whether Gaussian transport preserves (J_V) and whether the physical readout detects the corresponding orientation.
4. Produce two globally distinct source coefficient packets with identical values under every admitted local product probe. This falsifies source-level local tomography.
5. Derive a real coefficient model reproducing every admitted sector source, sewing map, and physical readout. This falsifies the claim that complex scalars are forced.

Current classification

The evidence supports

$$\text{shared labelled Carrier} + \text{source-polarized positive coefficient objects} + \text{sector-specific composition and readout}.$$

It does not yet support

$$\text{Carrier universally forces complex quantum coefficients}.$$

The next decisive object is not another Bell inequality. It is the source-authorized composite probe map and its kernel, together with any oriented triple-phase readout.

Primary sources

• L. Hardy and W. K. Wootters, Limited Holism and Real-Vector-Space Quantum Theory, arXiv:1005.4870.
• G. Niestegge, Local tomography and the role of the complex numbers in quantum mechanics, arXiv:2001.11421.
• S. Tull, A Categorical Reconstruction of Quantum Theory, arXiv:1804.02265, for the residual real-versus-complex ambiguity under categorical reconstruction principles.

Durable verification

• Exact parameter counts and the explicit (sigma_y\otimes\sigma_y) hidden direction are derived in this entry.
• The rank-three Bargmann-phase premise is the established amplitude-side result recorded around Ledgers 2038–2041.
• Epistemic graph admission: ev-000000005661-f952411f-e3c4-4ffe-80ec-4d5188cc469f.
• No executable checker is claimed for this entry.

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