past_theorem
For any positive integer N the configuration of N ledger entries with every entry equal to 1 is the unique configuration whose total defect vanishes, and it is the global minimizer of total defect. Cosmologists and philosophers of physics working on the Past Hypothesis would cite the result to replace a contingent initial condition with a forced mathematical necessity. The proof is a term-mode construction that assembles the uniqueness witness, the zero-defect equality, and the minimality inequality directly from the unity lemmas.
claimFor every positive integer $N$, there exists a unique configuration $c$ of $N$ positive real ledger entries such that the total defect (sum of individual defects) equals zero; this configuration is the one in which every entry equals 1; and the total defect of this configuration is less than or equal to the total defect of any other configuration of the same size.
background
In the InitialCondition module a Configuration of size N is a structure whose entries field maps each index in Fin N to a positive real number. The total_defect function sums the individual defect values (from LawOfExistence) over those entries. The unity_config definition sets every entry to 1, and the module proves that this choice yields total defect zero while every other choice yields strictly positive defect.
proof idea
The term proof refines the goal into a triple: the unique-existence witness built from unity_config, the equality total_defect(unity_config)=0 supplied by unity_defect_zero, and the inequality supplied by unity_is_global_minimum. A short tactic block then invokes zero_defect_iff_unity to obtain that zero total defect forces every entry to equal 1, which matches the unity configuration exactly and establishes uniqueness.
why it matters in Recognition Science
This is the Past Theorem (F-005) that converts the low-entropy initial state from an assumption into a theorem forced by the cost axioms. The module documentation states that it resolves Penrose's question on the specialness of the Big Bang and Albert's Past Hypothesis by showing that maximum-entropy (maximum-defect) states are infinitely costly while the zero-defect state is the unique minimum. It therefore supplies the initial-condition step required by the broader Recognition Science ontology.
scope and limits
- Does not address time evolution or dynamics of configurations.
- Does not fix a numerical value for N or link to physical scales.
- Does not incorporate additional physical constraints beyond total defect.
- Does not derive constants such as the fine-structure constant.
formal statement (Lean)
149theorem past_theorem {N : ℕ} (hN : 0 < N) :
150 (∃! c : Configuration N, total_defect c = 0) ∧
151 total_defect (unity_config N hN) = 0 ∧
152 (∀ c : Configuration N, total_defect (unity_config N hN) ≤ total_defect c) := by
proof body
Term-mode proof.
153 refine ⟨⟨unity_config N hN, unity_defect_zero hN, ?_⟩, unity_defect_zero hN,
154 unity_is_global_minimum hN⟩
155 intro c hc
156 have h_entries : ∀ i, c.entries i = 1 :=
157 (zero_defect_iff_unity hN c).mp hc
158 have h_u_entries : ∀ i, (unity_config N hN).entries i = 1 := fun _ => rfl
159 have h_eq : c.entries = (unity_config N hN).entries :=
160 funext fun i => by rw [h_entries i, h_u_entries i]
161 exact Configuration.mk.injEq .. |>.mpr h_eq
162
163end InitialCondition
164end Foundation
165end IndisputableMonolith