On the Derivation of k_civ: g(k₀, N), k₀, and the Civilizational Scaling Problem

This research note formalizes the current theoretical state of the derivation of the civilizational constant k_civ within the framework of Consciousness of the Real (CdR).

It argues that the remaining difficulty is no longer diffuse, but concentrated in two explicitly distinguishable points already anticipated in image002:

1. the formal gap represented by the function g(k₀, N), by which the foundational constant k₀ scales to k_N for a system with N simultaneously active axes;
2. the anchoring gap represented by the numerical value of k₀ itself, which is structurally defined but not yet derived in commensurable units.

The note establishes that the recursive family is the correct family for g(k₀, N), and that several structural invariants are now fixed: exact preservation of g(k₀, 1) = k₀, an optimal zone around N = 5–6, a degenerative regime beyond that zone, and saturation understood as a boundary condition on marginal growth rather than as a collapse of total capacity.

What remains open is the precise shape of the admissible saturation penalty, the exact behavior at saturation, and the numerical anchoring of k₀.

This record should be read as a bounded research-state document: not a completed closure, but a formally delimited derivational stage.

Parent document: From Physical Limit to Sustainable Civilizational Load — Current State of the Derivation of k
DOI: 10.5281/zenodo.19476343

Author: Sylvain Lebel
With editorial assistance from ChatGPT and critical review by Claude.