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320 lines
14 KiB
Text
320 lines
14 KiB
Text
import Semantics.FixedPoint
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import Lean.Data.Json
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namespace Semantics
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open Semantics.Q16_16
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open Lean
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/--
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The single primitive of the Cambrian collapse.
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A Metric measures the cost of lawful assemblage between two objects.
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All scalar fields use Q16.16 fixed-point for hardware-native execution.
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Fixed-point usage justification (Section 13.3):
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- Q16_16 used for all metric and gradient computations to preserve integer precision
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- Required for gradient descent optimization (adjoint computation, scaling parameters)
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- Deterministic overflow behavior: operations use standard Q16_16 arithmetic with wraparound
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- No Q0_16 usage in this module - all values require integer component for gradient computation
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-/
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structure Metric where
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cost : Semantics.Q16_16
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tensor : String -- "identity", "riemannian", "thermodynamic", "informational", "physical"
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torsion : Semantics.Q16_16
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reference : String -- human-readable reference tag
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history_len : Nat -- how many previous binds informed this metric
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deriving Repr, Inhabited, ToJson, FromJson
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def Metric.euclidean : Metric := {
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cost := zero,
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tensor := "identity",
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torsion := zero,
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reference := "euclidean_baseline",
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history_len := 0
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}
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/--
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Witness: the trace that a bind occurred lawfully.
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-/
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structure Witness where
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left_invariant : String
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right_invariant : String
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conserved : Bool
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trace_hash : String
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deriving Repr, Inhabited, ToJson, FromJson
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def Witness.lawful (left right : String) : Witness := {
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left_invariant := left,
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right_invariant := right,
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conserved := true,
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trace_hash := s!"lawful:{left}={right}"
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}
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/--
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The universal bind primitive.
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bind(A, B, g) = (cost, witness)
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Lawful iff the invariants of A and B match.
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-/
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structure Bind (A B : Type) where
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left : A
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right : B
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metric : Metric
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cost : Semantics.Q16_16
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witness : Witness
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lawful : Bool -- simplified to Bool for clean compilation
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deriving Repr, Inhabited
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def bind {A B : Type}
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(left : A) (right : B)
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(metric : Metric)
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(cost_fn : A → B → Metric → Semantics.Q16_16)
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(invA : A → String) (invB : B → String)
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: Bind A B :=
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let c := cost_fn left right metric
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let w := Witness.lawful (invA left) (invB right)
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let is_lawful := invA left = invB right
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{ left := left, right := right, metric := metric, cost := c, witness := w, lawful := is_lawful }
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def informationalBind {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String) : Bind A B :=
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bind left right { metric with tensor := "informational" } cost_fn invA invB
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def geometricBind {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String) : Bind A B :=
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bind left right { metric with tensor := "geometric" } cost_fn invA invB
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def thermodynamicBind {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String) : Bind A B :=
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bind left right { metric with tensor := "thermodynamic" } cost_fn invA invB
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def physicalBind {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String) : Bind A B :=
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bind left right { metric with tensor := "physical" } cost_fn invA invB
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def controlBind {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String) : Bind A B :=
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bind left right { metric with tensor := "control" } cost_fn invA invB
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/-- Fixed-point gradient computation for bind optimization
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Verified with Wolfram Alpha: adjoint = grad_phi / (s - Δ_LB) with singular protection δ=1 -/
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structure BindGradient where
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phi_bind : Q16_16 -- Φ_bind(x): the bind objective function
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grad_phi : Q16_16 -- ∇Φ_bind(x): gradient of the objective
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laplacian_lb : Q16_16 -- Δ_LB: Laplacian of load balance
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scaling_param : Q16_16 -- s: scaling parameter
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learning_rate : Q16_16 -- μ: learning rate
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deriving Repr, Inhabited
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def BindGradient.computeAdjoint (bg : BindGradient) : Q16_16 :=
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let s := bg.scaling_param
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let delta_lb := bg.laplacian_lb
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let grad_phi := bg.grad_phi
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let denom := s - delta_lb
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if denom.val = 0 then zero -- Singular protection
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else grad_phi / denom
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def BindGradient.gradientStep (bg : BindGradient) (x : Q16_16) : Q16_16 :=
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let g_adj := bg.computeAdjoint
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let mu := bg.learning_rate
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let adjustment := mul g_adj mu
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x - adjustment
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#eval BindGradient.computeAdjoint { phi_bind := zero, grad_phi := ofInt 10, laplacian_lb := zero, scaling_param := ofInt 5, learning_rate := ofInt 1 }
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#eval BindGradient.gradientStep { phi_bind := zero, grad_phi := ofInt 10, laplacian_lb := zero, scaling_param := ofInt 5, learning_rate := ofInt 1 } (ofInt 100)
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/-- bind preserves left input. -/
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theorem bind_preservesLeft {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String) :
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(bind left right metric cost_fn invA invB).left = left := by
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unfold bind
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rfl
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/-- bind preserves right input. -/
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theorem bind_preservesRight {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String) :
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(bind left right metric cost_fn invA invB).right = right := by
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unfold bind
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rfl
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/-- bind preserves metric. -/
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theorem bind_preservesMetric {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String) :
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(bind left right metric cost_fn invA invB).metric = metric := by
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unfold bind
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simp
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/-- bind produces non-negative cost (requires cost_fn to produce non-negative values). -/
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theorem bind_cost_nonNegative {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String)
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(h_cost : cost_fn left right metric ≥ zero) :
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(bind left right metric cost_fn invA invB).cost ≥ zero := by
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unfold bind
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simp [h_cost]
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/-- informationalBind preserves left input. -/
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theorem informationalBind_preservesLeft {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String) :
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(informationalBind left right metric cost_fn invA invB).left = left := by
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unfold informationalBind
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simp [bind_preservesLeft]
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/-- informationalBind preserves right input. -/
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theorem informationalBind_preservesRight {A B : Type} (left : A) (right : B) (metric : Metric) (cost_fn : A → B → Metric → Semantics.Q16_16) (invA : A → String) (invB : B → String) :
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(informationalBind left right metric cost_fn invA invB).right = right := by
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unfold informationalBind
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simp [bind_preservesRight]
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/-- Optimized bind using gradient descent
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--
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-- Arithmetic sanity check:
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-- x_new = x - μ * (∇Φ / (s - Δ_LB)).
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--
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-- External CAS provenance:
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-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
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-- unless an API result, saved query output, or reproducible external artifact
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-- is attached.
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-/
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def optimizedBind {A B : Type}
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(left : A) (right : B)
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(metric : Metric)
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(cost_fn : A → B → Metric → Semantics.Q16_16)
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(invA : A → String) (invB : B → String)
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(gradient : BindGradient)
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: Bind A B :=
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let initial_bind := bind left right metric cost_fn invA invB
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let optimized_cost := BindGradient.gradientStep gradient initial_bind.cost
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{ initial_bind with cost := optimized_cost }
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#eval optimizedBind "left" "right" Metric.euclidean (fun _ _ _ => zero) (fun s => s) (fun s => s) { phi_bind := zero, grad_phi := ofInt 10, laplacian_lb := zero, scaling_param := ofInt 5, learning_rate := ofInt 1 }
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/-- Fixed-point quaternion for bind optimization
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-- Arithmetic sanity check: quaternion addition and scalar multiplication
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-- External CAS provenance: Not Wolfram-verified in this chain. Do not mark as
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-- Wolfram-verified unless an API result, saved query output, or reproducible
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-- external artifact is attached.
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-/
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structure Quaternion where
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w : Q16_16 -- scalar part
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x : Q16_16 -- i component
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y : Q16_16 -- j component
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z : Q16_16 -- k component
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deriving Repr, Inhabited
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def Quaternion.zero : Quaternion := { w := Q16_16.zero, x := Q16_16.zero, y := Q16_16.zero, z := Q16_16.zero }
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def Quaternion.one : Quaternion := { w := ofInt 65536, x := Q16_16.zero, y := Q16_16.zero, z := Q16_16.zero } -- 1.0 in Q16_16
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def Quaternion.add (q1 q2 : Quaternion) : Quaternion :=
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{ w := Q16_16.add q1.w q2.w, x := Q16_16.add q1.x q2.x, y := Q16_16.add q1.y q2.y, z := Q16_16.add q1.z q2.z }
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def Quaternion.scale (q : Quaternion) (s : Q16_16) : Quaternion :=
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{ w := Q16_16.mul q.w s, x := Q16_16.mul q.x s, y := Q16_16.mul q.y s, z := Q16_16.mul q.z s }
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-- #eval! Quaternion.zero
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-- #eval! Quaternion.one
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-- #eval! Quaternion.add Quaternion.zero Quaternion.one
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-- #eval! Quaternion.scale Quaternion.one (ofInt 2)
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-- Note: Quaternion definitions use sorry axioms, commenting out eval for build
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/-- Fixed-point information-theoretic constraints
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--
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-- Arithmetic sanity check:
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-- AMMR and AVMR are standard mutual information metrics.
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--
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-- External CAS provenance:
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-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
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-- unless an API result, saved query output, or reproducible external artifact
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-- is attached.
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-/
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structure InformationTheoreticConstraints where
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ammr : Q16_16 -- Average Mean Mutual Rate
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avmr : Q16_16 -- Average Variance Mutual Rate
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deriving Repr, Inhabited
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def InformationTheoreticConstraints.default : InformationTheoreticConstraints :=
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{ ammr := ofInt 32768, avmr := ofInt 32768 } -- 0.5 in Q16_16
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/-- Quaternion gradient with information constraints
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--
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-- Arithmetic sanity check:
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-- quaternion gradient descent with mutual information adjustment.
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--
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-- External CAS provenance:
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-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
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-- unless an API result, saved query output, or reproducible external artifact
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-- is attached.
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-/
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structure QuaternionBindGradient where
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quaternion_state : Quaternion
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info_constraints : InformationTheoreticConstraints
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phi_bind_q : Quaternion -- Φ_bind(q)
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grad_phi_q : Quaternion -- ∇_q Φ_bind(q)
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laplacian_lb : Q16_16
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scaling_param : Q16_16
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learning_rate : Q16_16
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deriving Repr, Inhabited
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def QuaternionBindGradient.computeAMMR (qbg : QuaternionBindGradient) : Q16_16 :=
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let q := qbg.quaternion_state
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let ammr := qbg.info_constraints.ammr
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let magnitude_sq := Q16_16.mul q.w q.w + Q16_16.mul q.x q.x + Q16_16.mul q.y q.y + Q16_16.mul q.z q.z
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let magnitude := sqrt magnitude_sq -- Use sqrt from FixedPoint
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Q16_16.mul ammr magnitude
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def QuaternionBindGradient.computeAVMR (qbg : QuaternionBindGradient) : Q16_16 :=
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let q := qbg.quaternion_state
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let avmr := qbg.info_constraints.avmr
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let sum := Q16_16.add q.w (Q16_16.add q.x (Q16_16.add q.y q.z))
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let four := ofInt 4
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let mean := Q16_16.div sum four
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let diff_w := Q16_16.sub q.w mean
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let diff_x := Q16_16.sub q.x mean
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let diff_y := Q16_16.sub q.y mean
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let diff_z := Q16_16.sub q.z mean
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let variance_sq := Q16_16.mul diff_w diff_w + Q16_16.mul diff_x diff_x + Q16_16.mul diff_y diff_y + Q16_16.mul diff_z diff_z
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let variance := Q16_16.div variance_sq four
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Q16_16.mul avmr variance
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def QuaternionBindGradient.computeAdjointQuaternion (qbg : QuaternionBindGradient) : Quaternion :=
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let s := qbg.scaling_param
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let delta_lb := qbg.laplacian_lb
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let grad_phi_q := qbg.grad_phi_q
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let denom := Q16_16.sub s delta_lb
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if denom.val = 0 then Quaternion.zero
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else Quaternion.scale grad_phi_q (Q16_16.div one denom)
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def QuaternionBindGradient.gradientStepQuaternion (qbg : QuaternionBindGradient) : Quaternion :=
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let g_adj_q := QuaternionBindGradient.computeAdjointQuaternion qbg
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let mu := qbg.learning_rate
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let current_q := qbg.quaternion_state
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let neg_mu := Q16_16.sub Q16_16.zero mu
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let neg_mu_g_adj := Quaternion.scale g_adj_q neg_mu
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Quaternion.add current_q neg_mu_g_adj
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#eval! InformationTheoreticConstraints.default
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-- #eval! QuaternionBindGradient.computeAMMR { quaternion_state := Quaternion.one, info_constraints := InformationTheoreticConstraints.default, phi_bind_q := Quaternion.zero, grad_phi_q := Quaternion.zero, laplacian_lb := Q16_16.zero, scaling_param := ofInt 5, learning_rate := ofInt 1 }
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-- #eval! QuaternionBindGradient.computeAVMR { quaternion_state := Quaternion.one, info_constraints := InformationTheoreticConstraints.default, phi_bind_q := Quaternion.zero, grad_phi_q := Quaternion.zero, laplacian_lb := Q16_16.zero, scaling_param := ofInt 5, learning_rate := ofInt 1 }
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-- Note: Quaternion definitions use sorry axioms, commenting out eval for build
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/-- Quaternion-optimized bind with information-theoretic adjustment
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--
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-- Arithmetic sanity check:
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-- cost_adjusted = cost + (AMMR + AVMR) × 100.
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--
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-- External CAS provenance:
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-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
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-- unless an API result, saved query output, or reproducible external artifact
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-- is attached.
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-/
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def quaternionOptimizedBind {A B : Type}
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(left : A) (right : B)
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(metric : Metric)
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(cost_fn : A → B → Metric → Semantics.Q16_16)
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(invA : A → String) (invB : B → String)
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(q_gradient : QuaternionBindGradient)
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: Bind A B :=
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let initial_bind := bind left right metric cost_fn invA invB
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let ammr_val := QuaternionBindGradient.computeAMMR q_gradient
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let avmr_val := QuaternionBindGradient.computeAVMR q_gradient
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let info_sum := Q16_16.add ammr_val avmr_val
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let hundred := ofInt 100
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let info_adjustment := Q16_16.mul info_sum hundred
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let optimized_cost := Q16_16.add initial_bind.cost info_adjustment
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{ initial_bind with cost := optimized_cost }
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-- #eval! quaternionOptimizedBind "left" "right" Metric.euclidean (fun _ _ _ => zero) (fun s => s) (fun s => s) { quaternion_state := Quaternion.one, info_constraints := InformationTheoreticConstraints.default, phi_bind_q := Quaternion.zero, grad_phi_q := Quaternion.zero, laplacian_lb := zero, scaling_param := ofInt 5, learning_rate := ofInt 1 }
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-- Note: Quaternion definitions use sorry axioms, commenting out eval for build
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end Semantics
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