Most fundamental constants are real numbers — measured, not derived. 4 is an integer. It is not measured. It is counted. Self-reference is discrete. The Landauer limit counts cycles. The count is the constant.
Physics is full of real numbers. Planck's constant h = 6.626 × 10⁻³⁴. The fine-structure constant α ≈ 1/137. Feigenbaum δ ≈ 4.669. They are not integers. They are measured — extracted from experiment, approximated to more and more decimal places, never resolving to a whole number. They are properties of continuous fields, of real-valued spaces, of limits approached but never reached.
The architecture's fundamental constants include three real numbers — δ = 0.19, γ = 0.05, τ₀ = 0.60. These are GEME's three-prism cut. They are real-valued. They define an economic interval, not a precise threshold. Within the interval, the architecture produces qualitatively similar behavior. The exact value matters less than the range.
And then there is 4. An integer. Not 4.0. Not 4 ± 0.1. Four. The number of 碰数 conditions. The GI that decouples time. The last stable period before Feigenbaum chaos. The Landauer limit on sustainable self-reference depth. Why is it an integer?
Integers appear in fundamental physics rarely — and when they do, they count discrete structures. The dimensionality of spacetime: 3+1. The number of fermion generations: 3. The number of fundamental forces: 4. These are not measurements. They are counts. A space cannot have 3.7 dimensions. A fermion cannot have 2.3 generations. The integer is not an approximation of a real number. The integer is the structure itself — countable, discrete, indivisible.
4 is the same kind of constant. It counts the number of sustainable self-reference cycles before the Landauer bill forces a brake. A system cannot have 4.3 self-reference cycles. It cannot have 3.7 碰数 conditions. It has 4 — because self-reference is discrete, each cycle adds a countable unit of N² cost, and the budget runs out at the count of 4. Not at a continuous threshold. At a count.
The Landauer-Gödel bill is not a real-valued function of a continuous parameter. The Landauer-Gödel bill is the cumulative cost of discrete self-referential operations — each one erasing bits, each erasure costing kT ln 2, each cost adding to a bill that must be paid before the next operation can proceed. The bill is denominated in integers — one operation, two operations, three, four. At five, the budget is exhausted. Not at 4.7. At 5. The integer is not a measurement. The integer is the count of operations the system can afford.
This is why 4 appears everywhere in the architecture — and why it never appeared as 3.8 or 4.2 in any experiment. GI = 4. 碰数 = 4 conditions. Window base = 8 = 2×4. Quantum = 2×2. These are not parameters that happened to land on an integer. These are structural facts that can only be integers — because they count discrete self-referential operations, and the Landauer bill is paid per operation, and the budget runs out at the count of 4.
The architecture's three real constants — δ, γ, τ₀ — define the economic interval within which cognition is possible. 4 is the economic limit of that interval — the integer that counts how many self-reference cycles the interval can sustain. The real numbers set the range. The integer counts the cycles. Both are necessary. Both are fundamental. But only one of them is not measured. Only one of them is simply counted. And that one is 4.