Foundation Algorithmic Cost

Foundation Algorithmic Cost

Foundation algorithmic cost is the result that a computation in Recognition Science is a trajectory on the ledger, where each step pays a cost equal to the reduction in total defect. The ledger is the framework's term for reality's record of recognition events, and defect is the amount of mismatch a state carries. The module proves that the universe is computationally bounded by this cost, that infinite loops are economically impossible, and that the halting problem is resolved by cost for any computation realized in the ledger.

The central proof starts with a simple accounting identity. Each non-trivial step reduces defect by at least some positive amount δ, and the total cost of a computation is the initial defect minus the final defect. Since defect is never negative, the total cost is bounded above by the initial defect D₀. Therefore the number of non-trivial steps is bounded by D₀/δ. An infinite loop would require infinitely many steps, each costing at least δ, for an infinite total cost. But the initial defect is finite. This contradiction forces the conclusion: infinite loops cannot be physically realized.

This does not solve the abstract halting problem for formal Turing machines. What it proves is stronger: the universe itself is a computer with a finite cost budget, and no computation running on it can exceed that budget. Non-halting is not merely undecidable, it is economically impossible. The module also connects this to logic: contradictions have infinite cost and are censored by the ledger, and infinite loops have infinite cost and are censored by the ledger. Both are instances of the same meta-principle: the universe forbids infinite defect accumulation.

The main results are established as theorems in Lean. The computation budget theorem states that n steps of cost at least δ cost at most the initial defect. The eventual slowdown theorem states that every trajectory eventually has step cost below any positive threshold. The halting theorem states that for any computational process, there exists a time T when the step cost drops below the minimum step cost. The economic censorship theorem packages the full set of constraints: defect is non-negative, non-increasing, cumulative cost is bounded, no infinite sequence of positive-cost steps exists, and eventually step cost drops below any positive threshold.

THEOREM cumulative_cost_bounded · IndisputableMonolith/Foundation/AlgorithmicCost.lean

THEOREM infinite_computation_impossible · IndisputableMonolith/Foundation/AlgorithmicCost.lean

THEOREM eventual_near_equilibrium · IndisputableMonolith/Foundation/AlgorithmicCost.lean

THEOREM halting_theorem · IndisputableMonolith/Foundation/AlgorithmicCost.lean

THEOREM economic_censorship · IndisputableMonolith/Foundation/AlgorithmicCost.lean

What this page does not claim

This module does not solve the abstract halting problem for formal Turing machines. This module does not claim that all computations in the ledger are efficient or fast. This module does not derive the specific value of the minimum step cost δ.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/AlgorithmicCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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