The Allocation Ratio: Why $ = 1/2$ is the Only Fixed Point

by Pedro Henrique Corrêa Garcia · Opus 4.7 · 7 pp

Page 1 of 7· 6 stated

The Allocation Ratio:
Why α = 1/2 is the Only Fixed Point

Pedro Henrique Correa Garcia Joinville, Santa Catarina, Brazil garcia.pedro.wow@gmail.com · April 2026

Abstract

A system with finite resources that must recursively split itself into two complementary parts — structure and capacity, signal and residue, known and unknown — faces a single non-trivial decision: at what ratio should the split occur? We show that the only ratio that survives recursive self-application is α = 1/2. The proof requires neither physics nor optimization: it is forced by the demand that the splitting rule commute with itself. We then identify three apparently independent appearances of the value 1/2 in the framework — the allocation fixed point derived here, the real part of the non-trivial Riemann zeros, and the traversal cost of the rotation manifold 1 — and show that they are the same fact read through three different lenses. The ratio 1/2 is not an empirical constant. It is the only value at which the manifold can split without destroying itself.

Introduction

In Article 1 we established that the manifold matrix n =

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admits a state equation = + x and that the rotation manifold 1 has traversal cost

, with

living at /2. In Article~0 the same value 1/2 appeared in a different role: as the recursive allocation ratio that governs how a system with total resource E should partition itself between structure and capacity.

This coincidence is not a coincidence. In this article we derive α = 1/2 from first principles — neither as an empirical fit nor as an optimum of some utility function, but as the unique fixed point of self-application — and then show that the three occurrences of 1/2 (allocation, Riemann, rotation) are three readings of a single underlying identity.

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$1/2$ is the only ratio at which ``split and split again'' is indistinguishable from ``split once.'' Every other ratio drifts under recursio

The recursive partition

Let E > 0 be a finite resource (energy, probability mass, descriptive length, attention budget — the derivation does not depend on the interpretation). Suppose E must be partitioned into two parts, which we label structure (carrying fraction α) and capacity (carrying fraction 1 - α) for some α ∈ (0,1).

Definition

An allocation rule is a function A_α: (0,∞) → (0,∞)² given by

A_α(E) = (α E, (1-α) E).

The rule is recursive if it is applied to the capacity side at every level: at level n the capacity part of level n-1 is itself partitioned by A_α.

After n levels of recursion, the structure side has accumulated a fraction

S_n(α) = ∑_k=0^n-1 α(1-α)^k = 1 - (1-α)ⁿ eq:sn

of the original resource E, and the residual capacity holds the remaining (1-α)ⁿ.

Remark

Equation §? tells us that for any α ∈ (0,1) the structure side eventually absorbs the entire resource in the limit n → ∞. This is true but uninformative: it tells us where the process ends, not how it progresses, and in particular it says nothing about whether the process is stable under its own iteration. Stability is the condition we actually need.

Self-commutation and the fixed point