unity_defect_zero
plain-language theorem explainer
The theorem shows that the ledger configuration with every entry equal to 1 has exactly zero total defect. Cosmologists and foundational physicists cite it when deriving the forced low-entropy initial state from the cost axioms. The proof is a direct term reduction that unfolds the sum definition and replaces each defect term by zero.
Claim. Let $N$ be a positive natural number and let $C_N$ be the configuration of $N$ ledger entries in which every entry equals 1. Then the total defect of $C_N$ equals zero.
background
Module F-005 formalizes the low-entropy initial condition as a mathematical necessity rather than a contingent hypothesis. Total defect of a configuration is the sum over all ledger entries of the individual defect, where defect(x) equals the J-cost function. The unity configuration is defined to have every entry exactly 1. Upstream, defect_at_one states that defect(1) = 0, and total_defect is shown non-negative in a sibling result.
proof idea
Term-mode proof that unfolds total_defect and unity_config, applies the simp lemma defect_at_one to each summand, and invokes the fact that the Finset sum of N zero terms is zero.
why it matters
This supplies the zero value used by the Past Theorem, initial_state_minimum_entropy, no_singularity, and unity_is_global_minimum. It converts the Past Hypothesis into the Past Theorem by showing the initial state must be the unique zero-cost ledger configuration. The result anchors the claim that the Big Bang is the first nonzero defect tick rather than a singularity, consistent with the Recognition cost axioms and the eight-tick octave structure.
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