breath_cycle_pos
Establishes that the breath cycle constant equals 1024 and is strictly positive over the reals. Memory thermodynamics researchers cite it when bounding decay rates of the form base_decay_rate times memory_cost divided by breath_cycle. The argument is a direct term-mode reduction that unfolds the constant definition and invokes norm_num.
claimThe breath cycle period $B$ satisfies $0 < B$ in the reals, where $B := 1024$.
background
The MemoryLedger module treats memory as a thermodynamic system in which retention competes with free-energy decay under Recognition Science rules. breath_cycle supplies the fixed natural number 1024 that appears in decay-rate expressions such as base_decay_rate * memory_cost / breath_cycle. Cost is the abbreviation Quantity CostUnit drawn from RSNative.Core and used to measure nonnegative J-costs of memory traces.
proof idea
Unfolds the definition of breath_cycle to expose the literal 1024, then applies norm_num to discharge the inequality 0 < 1024. No external lemmas are invoked; the proof is a pure computational check.
why it matters in Recognition Science
Supplies the positivity needed by five downstream theorems: emotional_forgets_slower, emotional_reduces_cost, forgetting_decreases, low_temp_bistable, and memory_cost_nonneg. These results establish monotonicity of forgetting, emotional cost reduction, and low-temperature bistability. It closes a required positivity gap inside the fully-proven thermodynamic memory chain and aligns with the eight-tick octave scaling of the Recognition framework.
scope and limits
- Does not derive the numerical value 1024 from the forcing chain or phi-ladder.
- Does not extend positivity to continuous or variable cycle lengths.
- Does not connect the cycle to specific physical time scales or units.
formal statement (Lean)
57theorem breath_cycle_pos : 0 < (breath_cycle : ℝ) := by unfold breath_cycle; norm_num
proof body
Term-mode proof.
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59/-! ## Memory J-Cost -/
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