The Formal Mapping

Let |m⟩ be a state in the visible manifold (ℒ-sector). Let σ̂ represent the σ-involution as an operator on quantum states. By Theorem 1, σ̂ = CPT at the event horizon.

σ̂|m⟩ = |m̄⟩

where |m̄⟩ is the σ-conjugated state residing inside the horizon. By Theorem 1, σ̂ maps every quantum number to its conjugate: charge → −charge, baryon number → −baryon number, lepton number → −lepton number.

For every proton in the visible universe, there is an antiproton behind a horizon. For every electron, a positron.

Proposition 2 (Global quantum number balance). For every observable quantum number Q that is σ-odd, the total over the complete system (exterior + all horizon interiors) satisfies: Q_total = Q(ε), where Q(ε) is the quantum number of the splinter.

Proof. By Theorem 1, the exterior manifold is the ℒ-sector and the interior of every horizon is the 𝒫-sector.

By Axiom S, σ is a bijection between ℒ and 𝒫 (σ(ℒ) = 𝒫, σ(𝒫) = ℒ): every element in ℒ has a unique image in 𝒫 and conversely.

If Q is σ-odd, then Q(x) + Q(σ(x)) = 0 for every paired element x. The sum over all paired elements vanishes. By Proposition 1, the only unpaired element is ε.

Therefore Q_total = ΣQ = Q(ε).

The total state of a galactic system is: |galaxy⟩ = |visible matter⟩ + |interior of all BHs⟩. The global quantum numbers of the total state sum to Q(ε).

For practical purposes, |Q(ε)| is negligible compared to the total number of particles. The asymmetry is real but structurally minimal.

Note on mass balance: Proposition 2 requires that the total antibaryonic content inside ALL horizons (not just SMBHs — also stellar black holes, primordial black holes, and any collapsed structure where Axiom C forces the 0-condition) must equal the total baryonic content of the visible universe, minus the splinter’s contribution.

This is the quantitative content of KS-47. The observed mass inside known black holes (total SMBH mass ~ 10⁴³ kg, total stellar BH mass ~ 10⁴² kg) falls roughly ten orders of magnitude short of the visible baryonic mass (~ 10⁵³ kg).

This is a real quantitative tension that the paper does not resolve.

The Schwarzschild mass of a black hole includes all interior energy, not just rest mass; and unobserved primordial BHs below current detection thresholds may contribute significantly.

But until KS-60 (AP26) provides a quantitative calculation, the mass balance remains an open question with teeth: if the total mass inside all horizons cannot, even in principle, account for the antibaryonic content, KS-47 kills the paper.

It is a quantitative prediction with teeth — a near-falsification the paper discloses rather than resolves.

Note on σ̂ and CPT: The standard CPT theorem is a statement about Lorentz-invariant quantum field theories on a fixed background spacetime.

The σ̂-operator has a different origin: it is a consequence of Axiom S — the two-sector structure of the pre-state. Theorem 1 shows that the two coincide at the event horizon.

The CPT theorem, in this argument, is not an accidental symmetry of QFT but a consequence of the universe’s axiomatic geometry expressed at the boundary between the two sectors.

A note on scale: Proposition 2 mirrors the entire visible baryon number — the whole ~10⁸⁰, not merely the one-in-10⁹ asymmetry. This is the strong claim KS-47 tests, and the source of its mass-accounting tension; §3’s distinction between transient local antimatter and the net asymmetry keeps the machinery consistent, but the quantity behind every horizon is the full baryon count, not the surplus alone.