total_dissipation_bound
plain-language theorem explainer
The total recognition cost of a ledger state is bounded from below by half the sum of squared logarithms of its bond multipliers. Researchers deriving entropy production from the Landauer principle in information thermodynamics or Recognition Science would cite this aggregate bound. The proof unfolds the cost definition, rewrites the target as a scaled sum, applies the per-bond Landauer inequality to each active bond, and closes the comparison with linear arithmetic.
Claim. For a ledger state $s$ with active bonds and positive multipliers, the recognition cost satisfies $RecognitionCost(s) ≥ (1/2) ∑_{b ∈ active bonds of s} (log(bond multiplier of b))^2$, where recognition cost is the sum of J-costs over active bonds.
background
The module Phase 7.5.2 formalizes the link between Recognition Science cost and thermodynamic entropy via the Landauer limit. LedgerState records a Finset of active bonds together with positive real multipliers for each; RecognitionCost sums the J-cost function (tied to J-uniqueness from the forcing chain) over those bonds. Upstream results supply the fundamental tick as the discrete time quantum and the eight-tick period as the basic evolution cycle.
proof idea
Unfold RecognitionCost to expose the sum of J-costs. Rewrite the right-hand side via Finset.mul_sum. Apply Finset.sum_le_sum to reduce to a per-bond inequality. Invoke landauer_bound_holds on the chosen bond, simplify the resulting statement, and finish with linarith.
why it matters
The theorem supplies the total dissipation bound inside the Landauer Limit & 8-Tick Dissipation module, directly supporting the 8-tick dissipation limit. It anchors per-bond recognition cost to the Landauer principle and aligns with the eight-tick octave (T7) in the forcing chain. No downstream uses are recorded yet; the result closes the step from individual bond mismatches to aggregate entropy production.
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