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94 lines
3.6 KiB
Text
94 lines
3.6 KiB
Text
/-
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CognitiveLoad.lean - Formal Cognitive Load Theory (CLT) Bindings
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Ports rows 2-11 from MATH_MODEL_MAP.tsv (Python → Lean).
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All values are Q16.16 fixed-point. 1.0 = 0x00010000 = 65536.
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ε = 1 (smallest nonzero Q16.16 unit) to prevent division by zero.
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-/
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import Semantics.Bind
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import Semantics.FixedPoint
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namespace Semantics.CognitiveLoad
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open Q16_16
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-- ε = 1 LSB in Q16.16 (prevents division by zero)
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def epsilon : Q16_16 := ⟨1⟩
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structure LoadVector where
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intrinsic : Q16_16 -- L_I: germane schema processing
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extraneous : Q16_16 -- L_E: irrelevant processing
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germane : Q16_16 -- L_G: schema construction effort
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routing : Q16_16 -- L_R: inter-node routing overhead
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memory : Q16_16 -- L_M: working memory pressure
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deriving Repr, Inhabited, DecidableEq
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-- Row 2: L_I(x) — intrinsic load (direct field access)
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def intrinsicLoad (v : LoadVector) : Q16_16 := v.intrinsic
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-- Row 3: L_E(x) — extraneous load
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def extraneousLoad (v : LoadVector) : Q16_16 := v.extraneous
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-- Row 4: L_G(x) — germane load
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def germaneLoad (v : LoadVector) : Q16_16 := v.germane
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-- Row 5: L_R(x) — routing load
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def routingLoad (v : LoadVector) : Q16_16 := v.routing
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-- Row 6: L_M(x) — memory load
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def memoryLoad (v : LoadVector) : Q16_16 := v.memory
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-- Row 7: L_total(x) = L_I + L_E + L_G + L_R + L_M
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def totalLoad (v : LoadVector) : Q16_16 :=
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add (add (add (add v.intrinsic v.extraneous) v.germane) v.routing) v.memory
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-- Row 8: η(x) = L_I / (L_total + ε)
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-- Cognitive efficiency: ratio of useful intrinsic load to total
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def cognitiveEfficiency (v : LoadVector) : Q16_16 :=
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let total := add (totalLoad v) epsilon
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div v.intrinsic total
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-- Row 9: L_ρ(x) = L_total · (1 + ρ / ρ_max)
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-- Regret-adjusted load where ρ is BPB regret signal
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def regretAdjustedLoad (v : LoadVector) (regret regretMax : Q16_16) : Q16_16 :=
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let regretRatio := div regret (add regretMax epsilon)
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let factor := add one regretRatio
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mul (totalLoad v) factor
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-- Row 10: L(x|B) = L_I + L_E + L_R (basin-specific routing replaces germane)
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def basinConditionalLoad (lI lE lR_basin : Q16_16) : Q16_16 :=
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add (add lI lE) lR_basin
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-- Row 11: P_w(x_i | x_{<i}) = Σ w_j · P_{m_j}(x_i | x_{<i})
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-- MoE predictor distribution: weighted mixture
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-- weights and predictions are parallel arrays of Q16.16 values
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-- Result is Q16.16-normalized weighted sum
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def moePredictorDistribution (weights predictions : Array Q16_16) : Q16_16 :=
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if weights.size == 0 || weights.size != predictions.size then zero
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else
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let weightedSum := Array.foldl (fun acc i =>
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if i < predictions.size then
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add acc (mul weights[i]! predictions[i]!)
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else acc
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) zero (Array.range weights.size)
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let totalWeight := Array.foldl add zero weights
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if totalWeight.val == 0 then zero else div weightedSum totalWeight
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-- Invariant string for bind witnesses
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def loadInvariant (v : LoadVector) : String :=
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s!"load:I={v.intrinsic.val},E={v.extraneous.val},G={v.germane.val}"
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-- Bind: computes informational cost between two load states
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def loadDeltaCost (a b : LoadVector) (_m : Metric) : Q16_16 :=
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let da := totalLoad a
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let db := totalLoad b
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Q16_16.ofNat (abs (sub da db)).val.toNat
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def cognitiveLoadBind (a b : LoadVector) (m : Metric) : Bind LoadVector LoadVector :=
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informationalBind a b m loadDeltaCost loadInvariant loadInvariant
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-- Verify
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#eval totalLoad { intrinsic := ⟨32768⟩, extraneous := ⟨16384⟩, germane := ⟨8192⟩, routing := ⟨4096⟩, memory := ⟨2048⟩ }
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#eval cognitiveEfficiency { intrinsic := ⟨32768⟩, extraneous := ⟨16384⟩, germane := ⟨8192⟩, routing := ⟨4096⟩, memory := ⟨2048⟩ }
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end Semantics.CognitiveLoad
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