Superposition

In standard quantum mechanics, superposition is a postulate.

A system can exist in a linear combination of states. The coefficients are complex amplitudes. The system is in both states at once until measured.

This is presented as a fundamental feature of nature, requiring acceptance without deeper explanation.

In the 420 Code, superposition is not a postulate. It is the empty set state.

A system in superposition is a system for which the now has not yet written a record.

The degrees of freedom in question have not yet been distinguished.

0 and 1 for those degrees of freedom are still indistinguishable — still in the 1:1. Still in the pre-state, locally.

The system is not in both states at once. It is in the state before states — the state where the distinction between 0 and 1 has not yet been made.

Not undetermined. Not unknown. Undistinguished.

The container and the contents are still the same thing for those degrees of freedom.

You cannot picture this because your brain is a record-writing machine. It distinguishes everything it touches.

Why it is linear

The linearity of quantum mechanics is not imposed.

It is the algebraic structure that emerges when two axiom-derived ingredients combine.

Ingredient 1 — the record monoid. Axiom R gives records an algebraic structure: a monoid. Associative composition, identity element, no inverses.

Each record, when written by the now, transforms the pre-state. It eliminates unactualized possibilities and confirms the actualized one.

The record monoid R acts on the set of pre-states.

Under the embedding hypothesis, this discrete algebra embeds faithfully into a smooth manifold.

Faithfully — distinct records produce distinct transformations. No two records do the same thing to the space of possibilities.

This follows from Axiom B: each record involves ε, and each actualisation event is a distinct break for a distinct degree of freedom.

Ingredient 2 — the scalar field. The amplitudes — the weights assigned to each possibility in the pre-state — are complex numbers.

This is derived below from the Lorentzian signature, which follows from Axioms R + C under EH + QRA.

Complex numbers form a field, ℂ, with addition, multiplication, and inverses for both.

A monoid action alone does not produce a vector space. It gives a set with structure, but without the additive and scalar operations linearity requires.

The scalar field alone does not produce a vector space either. It gives the numbers but not the space they act on.

Both ingredients are needed simultaneously.

The monoid action provides the transformations — how records change the pre-state. The scalar field provides the weights — how possibilities are combined.

Together they produce a vector space over ℂ.

Formally: the pre-state for a degree of freedom with n distinguishable outcomes is a function assigning a complex amplitude to each possible record.

The set of such functions — all maps from {r₁, …, rₙ} to ℂ — is ℂⁿ, a complex vector space by the standard construction.

Addition of pre-states is pointwise. Scalar multiplication is inherited from ℂ. Writing a record is a linear operation on the space.

The pre-state space is therefore a complex vector space.

The linearity of quantum mechanics is not a postulate about nature. It is the algebraic consequence of R (the monoid of transformations), EH (the faithful embedding), and the Lorentzian signature (ℂ as the scalar field).

All three are required. None alone is sufficient.

Status: bridge derivation, not strict proof. The identification — that the pre-state assigns a complex weight to each possible record — is motivated by the axioms but is a constitutive choice, not a forced consequence of the axioms alone.

Any construction from a finite monoid acting on a set with ℂ-valued weights naturally produces a ℂ-module, and every finitely generated ℂ-module is a vector space.

A qualitatively different construction would have to abandon either the monoid action or the ℂ-valued weights. Both are axiom-derived.

Kill switch KS-Q.7. If a construction from R + ℂ + EH that does not produce a vector space can be equally motivated from the axioms, this step fails. LIVE — HARD.

The chain: 1:1 + 1×ε → {S, B, R, C} → R gives monoid acting on pre-states → R + C under EH + QRA give Lorentzian signature → ℂ as scalar field → pre-state space = maps from possible records to ℂ = complex vector space → linearity.

Why it is complex

The amplitudes are complex numbers. Not real. Not quaternionic. Not octonionic.

Why?

Axiom C imposes a finite causal bound c.

Under EH + QRA, this produces a Lorentzian manifold with signature (−,+,+,+). The time direction has opposite sign to the spatial directions.

Any wave propagating on this manifold must respect the signature.

Waves on a (−,+,+,+) manifold are naturally described by complex exponentials: e^{i(ωt−kx)}. The imaginary unit i encodes the signature flip between time and space.

Real amplitudes lose the time direction — they cannot distinguish forward from backward time evolution.

Quaternionic amplitudes over-specify it — they introduce additional structure not present in a single time direction.

Renou et al. (2021) proposed the test that distinguishes real from complex quantum theory; experiments confirmed the complex prediction (Li et al., 2022; Chen et al., 2022).

Complex amplitudes are the algebra compatible with one time direction and a finite propagation rate.

The full argument runs through reconstruction theorems (Hardy, 2001; Stueckelberg, 1960): given a state space with the symmetries of a Lorentzian manifold, the only consistent probability structure uses complex amplitudes.

The Lorentzian signature selects ℂ uniquely.

Quantum mechanics is complex because time has opposite sign to space.

And time has opposite sign to space because records are irreversible (Axiom R) and propagation is bounded (Axiom C).

Kill switch KS-Q.4. If an alternative number system (real, quaternionic, or other) can be shown equally consistent with one time direction and a finite propagation rate — that is, if the Lorentzian signature does not uniquely select ℂ — then the derivation of complex amplitudes requires revision. Renou et al. (2021) proposed the discriminating test, and the 2022 experiments (Li et al.; Chen et al.) confirmed the complex prediction; the result is conditional on the reconstruction theorems holding. LIVE — HARD.