unity_defect_zero
The configuration with every ledger entry equal to 1 has total defect zero for any positive integer N. Cosmologists and foundational physicists cite this as the forced low-entropy initial state. It converts the Past Hypothesis into the Past Theorem by showing the minimum-cost ledger is unique. The proof is a direct term reduction that unfolds the sum definition and applies the defect-at-unity identity.
claimFor every positive integer $N$, let $c$ be the configuration with $c_i = 1$ for all $i$. Then the total defect satisfies $D(c) = 0$, where $D(c) = sum_{i=1}^N defect(c_i)$ and $defect(x) = J(x)$ for positive $x$.
background
The module F-005 formalizes why the universe begins in a low-entropy state. Total defect of a configuration is the sum of individual defect values over its N entries, with defect(x) defined as the J-cost functional. The unity configuration is the ledger in which every entry equals exactly 1. The defect_at_one lemma states that defect(1) = 0. This zero-total-defect property is the mathematical necessity behind the Past Theorem: the initial condition is the unique zero-cost state forced by the cost axioms.
proof idea
The term proof unfolds total_defect and unity_config. It simplifies using the defect_at_one theorem, which gives defect(1) = 0. The final step applies the fact that the sum of N copies of zero equals zero.
why it matters in Recognition Science
This supplies the zero-defect fact required by the Past Theorem, the no-singularity result, and the statement that the initial state achieves minimum entropy. It is invoked directly in early-universe cosmology to replace the Past Hypothesis with a forced mathematical outcome. In the Recognition framework it shows that the zero-cost ledger is the only state compatible with the cost axioms, with no external selection required.
scope and limits
- Does not prove that unity is the unique zero-defect configuration.
- Does not derive the value of N or any post-initial dynamics.
- Does not connect to the forcing chain steps T5-T8 or the RCL identity.
- Does not address empirical constants such as alpha or G.
formal statement (Lean)
60theorem unity_defect_zero {N : ℕ} (hN : 0 < N) :
61 total_defect (unity_config N hN) = 0 := by
proof body
Term-mode proof.
62 unfold total_defect unity_config
63 simp only [LawOfExistence.defect_at_one]
64 exact Finset.sum_const_zero
65
66/-! ## The Initial Condition is Forced -/
67
68/-- **Theorem (F-005 core)**: The unity configuration is the unique
69 zero-total-defect configuration.
70 Every entry must be 1 for total defect to vanish. -/