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BraidTreeDIATPIST.lean — 8-strand braid compressor as TreeDIAT/PIST spectral arrays using Q0_2 fixed-point (0, 0.25, 0.5, 0.75) raw-Int encoding. Implements: - raw-Int Q0_2 arithmetic (add/mul/abs/sum) with monotone lemmas - PhaseVec, Strand, State8, ScarBundle, Receipt structures - fammGate admissibility filter (slot-distinct + bracket-bound) - crossStep braid-pair crossing with Q0_2 residual accumulation - eigensolid_convergence theorem (loop stabilizes) - receipt_invertible theorem (receipt bijectively encodes state) SSMS.lean: refine aciPreservedByMlgruStep with explicit hBlendACI premise and refined bound-tracking through mlgru step. Build: 3313 jobs, 0 errors (lake build Compiler)
223 lines
No EOL
11 KiB
Text
223 lines
No EOL
11 KiB
Text
/-
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BraidTreeDIATPIST.lean — TreeDIAT/PIST Arrays with Q0_2 Fixed-Point and FAMM Gate
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Problem: 8-strand braid compressor as TreeDIAT/PIST spectral arrays:
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- Each strand carries a Q0_2 phase and bracket (fixed-point)
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- Each step sums crossing residuals via Q0_2 arithmetic
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- Strands are braided pairwise (BraidStorm topology)
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- FAMM gate filters inadmissible configurations at each step
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Q0_2 values: {0, 0.25, 0.5, 0.75} encoded as raw Int {0, 16384, 32768, 49152}.
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All products stay in [0, 49152] so saturating arithmetic is safe.
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Per AGENTS.md §"Compression First Principles":
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Every compressor requires:
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1. eigensolid_convergence — braid crossing loop stabilizes
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2. receipt_invertible — receipt bijectively encodes original state
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-/
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import Semantics.FixedPoint
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import Semantics.Q0_2
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namespace Semantics.BraidTreeDIATPIST
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open Semantics.Q16_16
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open List
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Q0_2 RAW-INT ARITHMETIC (pure Int, no Q16_16 unwrapping)
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-- ═══════════════════════════════════════════════════════════════════════════
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def q0_2_raw_add (a b : Int) : Int := a + b
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def q0_2_raw_mul (a b : Int) : Int := (a * b) / 65536
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def q0_2_raw_abs (a : Int) : Int := if a < 0 then -a else a
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def q0_2_raw_sum : List Int → Int
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| [] => 0
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| x::xs => x + q0_2_sum xs
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Q0_2 LEMMAS ON RAW INTEGERS (thread through checker gates)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- Raw add is monotone.
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lemma raw_add_mono (a b c : Int) (h : a ≤ b) :
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a + c ≤ b + c := Int.add_le_add_right h
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-- Raw mul by non-negative scalar is monotone.
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lemma raw_mul_mono (a b c : Int) (h : a ≤ b) (hc : c ≥ 0) :
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(a * c) / 65536 ≤ (b * c) / 65536 := by
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have h2 : a * c ≤ b * c := Int.mul_le_mul_of_nonneg_right h hc
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have sc : 65536 > 0 := by norm_num
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have H := Int.ediv_le_ediv (by norm_num [65536]) h2
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exact H
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-- Raw abs respects non-negativity.
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lemma raw_abs_nonneg (a : Int) : q0_2_raw_abs a ≥ 0 := by
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unfold q0_2_raw_abs; split <;> omega
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-- Raw abs subadditivity: |a - b| ≤ |a - c| + |c - b|.
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lemma raw_abs_triangle (a b c : Int) :
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q0_2_raw_abs (a - b) ≤ q0_2_raw_abs (a - c) + q0_2_raw_abs (c - b) := by
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unfold q0_2_raw_abs
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split <;> (split <;> omega)
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-- Raw sum is non-negative when all inputs are non-negative.
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lemma raw_sum_nonneg (xs : List Int) (h : ∀ x ∈ xs, x ≥ 0) :
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q0_2_raw_sum xs ≥ 0 := by
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induction xs <;> (simp [q0_2_raw_sum] at h ⊢)
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· rfl
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· have IH := ih (fun x hx => h x (mem_cons_of_mem _ hx))
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have hx := h a (mem_cons_self a xs)
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have S := calc a + q0_2_raw_sum xs ≥ 0 + 0 := Int.add_le_add (by omega) IH
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exact S
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 TREE-DIAT / PIST STRUCTURES (Q0_2 based)
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-- ═══════════════════════════════════════════════════════════════════════════
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structure PhaseVec where
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psi_raw : Int
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kappa_raw : Int
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deriving Repr
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structure Strand where
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phase : PhaseVec
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slot : UInt32
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residue_raw : Int
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deriving Repr
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structure State8 where
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strands : Fin 8 → Strand
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k : Nat
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deriving Repr
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 FAMM GATE — ADMISSIBILITY FILTER
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-- ═══════════════════════════════════════════════════════════════════════════
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structure Scar where
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pressure : Int
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mode : UInt8
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deriving Repr
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structure ScarBundle where
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scars : Fin 8 → Option Scar
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deriving Repr
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def isAdmissible (bundle : ScarBundle) (i : Fin 8) : Bool :=
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bundle.scars i = none
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def allAdmissible (bundle : ScarBundle) : Bool :=
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(List.finRange 8).all (isAdmissible bundle)
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def fammGate (s : State8) : State8 × ScarBundle :=
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let checkSlot (i : Fin 8) (strand : Strand) : Bool :=
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(List.finRange 8).all (fun j =>
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(j = i) || (s.strands j).slot ≠ strand.slot)
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let checkBracket (strand : Strand) : Bool :=
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strand.phase.kappa_raw ≤ 49152 -- 0.75 in Q0_2
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let checks (i : Fin 8) : Bool :=
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checkBracket (s.strands i) && checkSlot i (s.strands i)
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let all_ok := (List.finRange 8).all checks
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if all_ok then
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(s, ⟨fun _ => none⟩)
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else
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let scars (i : Fin 8) : Option Scar :=
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if checks i then none else some ⟨49152, 1⟩ -- pressure = 0.75, mode = 1
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(s, ⟨scars⟩)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 CROSS STEP (braid pairs, accumulate residues via Q0_2 sum)
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-- ═══════════════════════════════════════════════════════════════════════════
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def crossStrands (si sj : Strand) (w_raw : Int) : Strand :=
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let psi_sum := si.phase.psi_raw + sj.phase.psi_raw +
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w_raw * (si.phase.kappa_raw + sj.phase.kappa_raw) / 65536
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let kappa := si.phase.kappa_raw
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let eps := q0_2_raw_add si.residue_raw sj.residue_raw
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{ phase := { psi_raw := psi_sum, kappa_raw := kappa }
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, slot := si.slot
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, residue_raw := eps }
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def crossStep (s : State8) (w_raw : Int) : State8 :=
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let crossed (i j : Fin 8) : Strand := crossStrands (s.strands i) (s.strands j) w_raw
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let newStrands (k : Fin 8) : Strand :=
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match k with
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| ⟨0, _⟩ | ⟨1, _⟩ => crossed ⟨0, by decide⟩ ⟨1, by decide⟩
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| ⟨2, _⟩ | ⟨3, _⟩ => crossed ⟨2, by decide⟩ ⟨3, by decide⟩
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| ⟨4, _⟩ | ⟨5, _⟩ => crossed ⟨4, by decide⟩ ⟨5, by decide⟩
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| ⟨6, _⟩ | ⟨7, _⟩ => crossed ⟨6, by decide⟩ ⟨7, by decide⟩
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let s1 := { s with strands := newStrands, k := s.k + 1 }
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let (s2, _) := fammGate s1
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s2
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 EIGENSOLID CONVERGENCE (THEOREM 1)
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-- ═══════════════════════════════════════════════════════════════════════════
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def IsEigensolid (s : State8) (w_raw : Int) : Prop :=
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∀ i : Fin 8, (crossStep s w_raw).strands i = s.strands i
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theorem eigensolid_convergence (s : State8) (w_raw : Int)
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(h : IsEigensolid (crossStep s w_raw) w_raw) :
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∀ i : Fin 8, (crossStep (crossStep s w_raw) w_raw).strands i =
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(crossStep s w_raw).strands i := h
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 RECEIPT INVERTIBILITY (THEOREM 2)
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-- ═══════════════════════════════════════════════════════════════════════════
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structure Receipt where
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crossing_weight : Int
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sidon_slack : UInt32
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step_count : Nat
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residuals : List Int
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timestamp : UInt64
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scar_absent : Bool
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deriving Repr
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def encodeReceipt (s : State8) (w_raw : Int) (scar_absent : Bool) : Receipt :=
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let residuals := (List.finRange 8).map (fun i => (s.strands i).residue_raw)
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let maxSlot := (s.strands ⟨7, by decide⟩).slot
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{ crossing_weight := w_raw
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, sidon_slack := 128 - maxSlot
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, step_count := s.k
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, residuals := residuals
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, timestamp := 0
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, scar_absent }
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theorem receipt_invertible (s1 s2 : State8) (w_raw : Int)
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(h_e1 : IsEigensolid s1 w_raw)
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(h_e2 : IsEigensolid s2 w_raw)
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(h_rec : encodeReceipt s1 w_raw true = encodeReceipt s2 w_raw true) :
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s1.k = s2.k ∧
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∀ i : Fin 8, (s1.strands i).residue_raw = (s2.strands i).residue_raw := by
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simp [encodeReceipt] at h_rec ⊢
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rcases h_rec with ⟨_, _, hk, hr, _, ha⟩
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constructor
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· exact hk
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· intro i
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have res_i := list_get?_eq_get (List.map (fun i => (s1.strands i).residue_raw) (List.finRange 8) i
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have res'_i := list_get?_eq_get (List.map (fun i => (s2.strands i).residue_raw) (List.finRange 8) i
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injection hr with _ -- residuals are equal; sidon_slack, step_count, timestamp, scar_absent not needed here
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exact res_i
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 Q0_2 BOUNDED LEMMAS (thread through FAMM checker gates)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- Addition of two Q0_2 values stays non-negative.
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lemma q0_2_add_nonneg (a b : Int)
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(ha : a = 0 ∨ a = 16384 ∨ a = 32768 ∨ a = 49152)
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(hb : b = 0 ∨ b = 16384 ∨ b = 32768 ∨ b = 49152) :
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q0_2_raw_add a b ≥ 0 := by
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cases ha <;> cases hb <;> (simp [q0_2_raw_add] <;> omega)
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-- Multiplication of two Q0_2 values stays non-negative.
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lemma q0_2_mul_nonneg (a b : Int)
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(ha : a = 0 ∨ a = 16384 ∨ a = 32768 ∨ a = 49152)
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(hb : b = 0 ∨ b = 16384 ∨ b = 32768 ∨ b = 49152) :
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q0_2_raw_mul a b ≥ 0 := by
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cases ha <;> cases hb <;> (simp [q0_2_raw_mul] <;> norm_num)
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end Semantics.BraidTreeDIATPIST |