Forward Direction: The Planck Constraint

Conjecture, non-load-bearing. If every claim in section 8 is wrong, sections 1–7 are unaffected. Section 8 sketches the forward direction that Notebook IV’s subsequent chapters take up.

8.1 — G depends on c through ε

AP03 established the scaling G = κ/(εΛ²), where κ is the backreaction coupling. (This ε⁻¹ scaling form is superseded by AP28, which derives the gravitational coupling α_G = α_em²¹(1 + 1/π) — the canonical G–ε relation, G ∝ ε²¹, verified to 0.69%. The κ/(εΛ²) expression is retained here as a parametrisation of the backreaction, not as a competing scaling claim.) Theorem 3.1 establishes (in the analysed channels) that η > 0 whenever c and coupling are finite: ε = η(c, γ). Substituting:

G = κ / (η(c, γ) × Λ²) (8.1)

G is a function of c through ε. They are not independent. The conjugacy conjectured in AP03 — c and G as two expressions of one symmetry-breaking event — now has a proposed physical mechanism: the leakage mediates the relationship.

8.2 — Consequences for the Planck scale

If G is not independent of c, the Planck units are built from two independent constants (c and ℏ), not three. The ratio G/c³ = κ/(εΛ²c³) sets the Planck length. In the leakage reading: the Planck scale is where ε ∼ O(1) — where the leakage becomes total and the boundary between absorber and radiation dissolves.

8.3 — The constraint path

Increase c → α decreases → coupling loosens → ε increases (more leakage) → G decreases. Decrease c → α increases → coupling tightens → ε decreases (less leakage) → G increases.

8.4 — The multidimensional residual

Conjecture C7 (Multidimensional residual). G, c, α (temporal stiffness), β (spatial stiffness), α_em (≈ 1/137), and m_e are not independent constants. They are readings of a single multidimensional residual — the structure the substrate acquired after the symmetry break. G is the geometry face (how tightly the condensate holds). c is the propagation face (c² = β/α). α_em is the coupling face (how strongly the electron interacts with the substrate). m_e is what escaped (the minimum viable excitation). The Planck scale marks where ε ∼ O(1). AP24 section 2 (Six Faces) formalises this conjecture into the six independent scalar readings of ε. [CONJECTURE, non-load-bearing]