The Self-Consistency Conditions

4.1 — The overdetermined system

There are more equations than unknowns. The six faces give six readings. The corpus provides at least three independent algebraic relations between them (The Lock, The Keys, the fine-structure definition). Plus the Planck boundary condition. One unknown (ε). Multiple constraints. If the system is consistent, it has at most one solution.

4.2 — Why consistency is guaranteed

The system is not constrained by accident. It is constrained by identity. The six readings come from one object. They must be consistent.

The question is not “is the system consistent?” — the break happened, the universe exists, the constants have the values they have. The question is: does the formalisation capture the identity correctly?

If the formalisation gives zero solutions. The formalisation is wrong (it has introduced a false distinction between faces that are actually identical).

If it gives one solution. The formalisation is correct and the value of α_em is determined.

If it gives multiple solutions. The formalisation is incomplete (there are consistency conditions it has not captured).

4.3 — The fixed-point structure

The self-consistency has a deeper structure than a system of equations. It is a fixed point.

ε determines m_e (what escaped). m_e determines the scale at which ε is read (the electromagnetic scale). The scale determines the value of ε at that scale (the coupling α_em). And α_em IS ε read as coupling.

Define: f(ε) = the value of the leakage at the scale determined by ε. The fixed point: ε = f(ε).

This is self-referential but not circular. It is a fixed-point equation. Fixed-point equations have definite solutions — typically zero or one in well-behaved systems. If exactly one: that fixed point is α_em.

The self-referential structure is not a defect of the analysis. It is the identity asserting itself. The leakage determines the scale. The scale determines the leakage. They are the same thing. The fixed point is the value where the two readings agree — where the instrument and the measurement coincide.

You have tuned a guitar string. You play the harmonic. You adjust until the harmonic matches the fundamental. That match — that moment when the two readings coincide — is the fixed point. α_em is the note where the universe’s string is in tune with itself.

4.4 — What the fixed-point equation requires

To write f(ε) explicitly requires expressing all six faces as functions of one dimensionless parameter. The dimensional structure must be handled with care.

ε is dimensionless. G, c, m_e have dimensions. The connection is made through the Planck units, which are built from the faces themselves. This is not circular — it is the identity. The Planck units ARE the natural scale of the leakage.

The Planck boundary condition (ε → O(1) at the Planck scale) provides the normalisation. At the Planck scale, the break is total: the boundary between absorber and radiation dissolves. This is where ε = 1. All other values of ε are readings at scales below Planck.

The running of the coupling between the Planck scale and the electron scale is determined by the particle content — but the particle content is itself determined by the axioms. The spectrum of ε-compositions (which particles exist, with what charges and masses) is not free. It follows from Axioms S, B, R, C acting on ε. The full derivation of the particle spectrum is beyond the scope of this paper but is structurally determined.

[Epistemic status: the fixed-point structure is DERIVED (the self-referential nature follows from the identity). Writing f(ε) explicitly requires either the full particle spectrum or a self-consistency shortcut that bypasses it. Both are OPEN. This is debt D1.]