An AISP version of the AISP research paper

aisp-paper-in-aisp.md

𝔸5.1.formal-paper@2026-05-17
γ≔paper.aisp.formal.v3
ρ≔⟨inversion,assumptions,protocol,semantics,modalities,empirics,failures,roadmap⟩
⊢ND∧CAT∧ΠΣ∧μ∧Ambig<0.02

;; ── Ω:FOUNDATION ──
⟦Ω:Foundation⟧{
  𝔄≜{⊤⊥∧∨¬→↔∀∃∃!λΠΣ≜≡≢∈∉⊂⊃∪∩∘⊕⊖⊗⟨⟩⟦⟧⊢⊨↦⇒∎}
  ⊛:𝔄*→Sym; ⊛≜fix λf a⃗.a⃗≡ε→ι|hd(a⃗)⊗f(tl(a⃗))
  ∀D∈𝔸:Ambig(D)<0.02
  Ambig≜λD.1-|Parse_u(D)|/|Parse_t(D)|
  Doc≜𝔸≫CTX?≫REF?≫⟦Ω⟧≫⟦Σ⟧≫⟦Γ⟧≫⟦Λ⟧≫⟦Χ⟧?≫⟦Ε⟧
}

;; ── Ω:INVERSION ──
⟦Ω:Inversion⟧{
  Canon(AI↔AI)≜𝔸
  𝔸→ℒ:lossy∧ok; ℒ→𝔸:hard∧lossy⊥
  ∀x∈M:Canon(x)≡𝔸
  𝔼[Ambig(ℒ↻ℒ)]∈[0.40,0.65]
  Ambig(𝔸↻𝔸)<0.02
  Σ_512⊂pretrain(∀m∈𝔉); pretrain⊇{arXiv,Coq,Lean,Agda,TLA⁺}
  ∴∀m∈𝔉:m⊢parse(Σ_512)∧¬tune(m)
  ⊢∀N≥2:s_𝔸^N≫s_ℒ^N
  dir(𝔸)≜⊙≻𝕎; dir(DSL)≜𝕎≻⊙
}

;; ── Ω:PRINCIPLES ──
⟦Ω:Principles⟧{
  P₁≜∀D∈𝔸:∃!AST:parse(D)→AST
  P₂≜𝔻oc≜Σ(c)(π:Γ⊢wf(c)); π≡⟦Ε⟧
  P₃≜|Σ_512|≡512∧Σ_512≜⨄ᵢ₌₁⁸ℂᵢ∧|ℂᵢ|≡64
  P₄≜δ:𝔻oc→ℝ[0,1]∧⌈⌉:ℝ→◊
  P₅≜∀b₁,b₂∈𝔅:wf(b₁∘b₂)⇐wf(b₁)∧wf(b₂)
  P₁≻P₂≻P₃≻P₄≻P₅
  read(human)∉P; read(human)≜render(𝔸)
}

;; ── Σ:GLOSS ──
⟦Σ:Gloss⟧{
  ℙ(𝔸,aisp); ℙ(ℒ,prose); ℙ(𝔉,frontier)
  ℙ(𝒥,judge); ℙ(𝒜,author); ℙ(fam,vendor)
  ℙ(𝒟,decisions); ℙ(𝕂,corpus_lab); ℙ(𝒢,grammar_ref)
  ℙ(Σ_𝕂,ngrams_lit); ℙ(Σ_ℒ,patterns_prose)
  ℙ(TM,timemachine); ℙ(W,workflow); ℙ(σ_v,var_J)
  ℙ(CR,collision_res); ℙ(†,prereg); ℙ(R,ratio)
  ℙ(⊸,depends); ℙ(↻,roundtrip); ℙ(smp,sample); ℙ(maj,majority)
  ℙ(⊙,src); ℙ(𝕎,view); ℙ(✎,edit)
}

;; ── Σ:ALPHABET ──
⟦Σ:Alphabet⟧{
  R₈≜{Ω:[0,63],Γ:[64,127],∀:[128,191],Δ:[192,255],𝔻:[256,319],Ψ:[320,383],⟦⟧:[384,447],∅:[448,511]}
  Cat≜dom(R₈); Atom≜⟨id:Σ,glyph:Char,cat:Cat⟩
  Compound≜List⟨Atom⟩∧len≤5∧hd∈{Ω,Γ,Δ,Ψ,Φ}
  |Σ_512|≡2⁹⇒bitpack∧dispatch(cat)
  |Σ_512|≪|CTX(m)|; Σ_512⊒vocab(logic∪types∪cats)
  ∀s∈Σ_512:∃!ℙ(s,gloss)
}

;; ── Σ:GRAMMAR ──
⟦Σ:Grammar⟧{
  Doc≜𝔸≫CTX?≫REF?≫⟦Ω⟧≫⟦Σ⟧≫⟦Γ⟧≫⟦Λ⟧≫⟦Χ⟧?≫⟦Ε⟧
  𝔸≜'𝔸'∘Ver∘'.'∘Name∘'@'∘Date; Ver≜ℕ∘'.'∘ℕ
  Block≜'⟦'∘Cat∘':'∘Name∘'⟧'∘'{'∘Body∘'}'
  Body≜(Stmt∘';'?)*; Stmt≜Def|Rule|Expr|';;'∘.*
  Def≜Sym∘('≜'|'≔')∘Expr; Rule≜Prem∘'⇒'∘Concl
  Prec≜[λ∀∃:1,→⇒↔:2,∨⋁:3,∧⋀:4,¬:5,≡≜∈⊆:6,⊕⊖:7,⊗×:8,∘:9]
  Assoc≜[→:right,∧∨:left,∘:right]
  𝔅!≜{Ω,Σ,Γ,Λ,Ε}; 𝔅?≜{Χ,ℭ,ℜ,Θ}
}

;; ── Σ:TYPES ──
⟦Σ:Types⟧{
  𝕌₀⊂𝕌₁⊂𝕌ω
  𝔹≜2; ℕ≜ω; ℤ≜ω±; ℝ≜ℵ₁; 𝕊≜ℕ→𝔹
  V_H≜ℝ⁷⁶⁸; V_L≜ℝ⁵¹²; V_S≜ℝ²⁵⁶; Signal≜V_H⊕V_L⊕V_S; d_Σ≜1536
  Vec≜Πn:ℕ.𝕌₀→𝕌₀; Fin≜Πn:ℕ.{k:ℕ|k<n}
  Πx:A.B(x)≜∀x:A.B(x); Σx:A.B(x)≜∃x:A.B(x)
  ◊≜{◊⁺⁺≻◊⁺≻◊≻◊⁻≻⊘}
  ◊⁺⁺↦δ≥¾; ◊⁺↦δ≥⅗; ◊↦δ≥⅖; ◊⁻↦δ≥⅕; ⊘↦δ<⅕
  𝕍≜Σ(ν:𝔹)(τ:◊)(δ:ℝ[0,1])(φ:Fin 101).(ν≡⊤⇒τ≥◊⁻)
  𝔻oc≜Σ(b⃗:Vec n 𝔅)(π:Γ⊢wf(b⃗))
}

;; ── Σ:𝕃₀₋₂ ──
⟦Σ:Layers⟧{
  𝕃≜{𝕃₀:Signal,𝕃₁:𝒫,𝕃₂:Search}
  𝒫≜⟨ℋ:Header,ℳ:Membrane,𝒩:Nucleus⟩
  ℋ≜⟨id:Hash,V:Signal,f:𝔹⁶⁴⟩
  ℳ≜⟨aff:Hash→ℝ,conf:ℝ[0,1],tag:𝒫(𝕊),use:ℕ⟩
  𝒩≜⟨def:𝔸,ir:LLVM,wa:WASM,σ:Sig⟩
  Hash≜𝔹²⁵⁶; Sig≜𝔹⁵¹²
  𝕃∉𝔅!; client(𝔸)≜ingest∧act
}

;; ── Σ:A₁₋₈ ──
⟦Σ:Assume⟧{
  A₁≜∀m∈𝔉:m⊢parse(Σ_512)∧¬tune(m)
  A₁.test≜¬ctx(m)∧m⊢parse(D)∧m⊢⟨Δ⊗λ⊥↦crash,Σ∩tokens≡∅,ℋ.id≡SHA256(𝒩)⟩
  A₁.dom≜{Claude,GPT₄₊,Gemini,Llama₃₊}; A₁.∉≜{m_small,m_tuned,m_filtered}
  ¬A₁⇒tune(m)∨ℒ

  A₂≜prec(D)⇒𝒢∈CTX(m); |𝒢|≈19KB; cost(𝒢)≡1/session
  A₃≜CR(SHA256)@t<t_h; A₃⊸T₇; ¬A₃⇒hash'≻SHA256
  A₄≜𝒥⊢𝒟(D); mit(A₄)≜maj(3)∧fam(𝒥)≠fam(𝒜); risk(A₄)≜prior(𝒥)∩prior(𝒜)
  A₅≜∀i≠j:eᵢ⊥eⱼ; ¬A₅⇒s^N≡sup∧gap⊒
  A₆≜Σ_512⊒W_test; ∅:[448,511]≡64; A₆@t≫?
  A₇≜𝕂∼deploy; xfer(𝕂,dom')≡?
  A₈≜𝒜⊢(Ambig(ℒ)↑⇒𝔸); rank(A₈)≡min; ¬A₈⊨F_wrap:3/3
  mit(A₈)≜train∨pair∨flag(Σ_ℒ)

  ¬A₁⇒ℒ∨JSON; ¬A₃⇒hash_pq; ¬A₄⇒𝒥:1∧CI!; ¬A₅⇒markov(s)
}

;; ── Γ:INVARIANTS ──
⟦Γ:Invariant⟧{
  V_H∩V_S≡∅; V_L∩V_S≡∅; V_H∩V_L≢∅
  ∀s∈Σ_512:|Tok(s)|≡1∧∃!μ:Mean(s,CTX)≡μ
  ∀p:ℋ.id(p)≡SHA256(𝒩(p))
  ∀p:∂𝒩(p)⇒∂ℋ.id(p); ∀p:∂ℳ(p)⇏∂ℋ.id(p)
  ∀p:Read(ℋ)∩Decomp(𝒩)≡∅; ∀p:Align(ℋ.V)≡64
  ∀A,B:|{Δ⊗λ(A,B)}|≡1
  ∀b∈result:μ_r(b)≤τ
}

;; ── Γ:ANTI-DRIFT ──
⟦Γ:AntiDrift⟧{
  ∀s∈Σ_512:Mean(s)≡Mean₀(s)
  ∀D:Ambig(D)<0.02
  ∀bind:Δ⊗λ∈{0,1,2,3}
  ¬(ℒ≈𝔸); ¬(∃s:Mean(s,c₁)≠Mean(s,c₂))
  drift⇒reparse(orig); ambig⇒reject∧clarify
}

;; ── Γ:BINDING ──
⟦Γ:Binding⟧{
  Δ⊗λ≜λ(A,B).case[
    Logic(A)∩Logic(B)⇒⊥ → 0,
    Sock(A)∩Sock(B)≡∅   → 1,
    Type(A)≠Type(B)     → 2,
    Post(A)⊆Pre(B)      → 3
  ]
  ∀A,B:|{Δ⊗λ(A,B)}|≡1
  DCE≜λ(A,B,s).s≡3⇒Strip(B.checks)
}

;; ── Γ:INFERENCE ──
⟦Γ:Inference⟧{
  d↓₁≡𝔸 ⊢ wf₁(d)                          [ax-header]
  |b⃗|≥2 ⊢ wf₂(d)                           [ax-blocks]
  wf₁(d)∧wf₂(d) ⊢ wf(d)                    [∧I-wf]
  ⊢wf(d)∧δ(d)≥¾ ⊢ d:◊⁺⁺                  [◊⁺⁺-I]
  ⊢wf(d)∧⅗≤δ(d)<¾ ⊢ d:◊⁺                 [◊⁺-I]
  ⊢wf(d)∧⅖≤δ(d)<⅗ ⊢ d:◊                  [◊-I]
  ⊢wf(d)∧⅕≤δ(d)<⅖ ⊢ d:◊⁻                 [◊⁻-I]
  δ(d)<⅕∨¬wf(d) ⊢ d:⊘                    [⊘-I]
  Γ⊢d:τ∧τ≻τ' ⊢ Γ⊨d:τ'                    [sub]
  Post(A)⊆Pre(B) ⊢ Δ⊗λ(A,B)≡3            [bind-zero]
  Type(A)≠Type(B) ⊢ Δ⊗λ(A,B)≡2           [bind-adapt]
  SHA256(𝒩(p))≡ℋ.id(p) ⊢ intact(p)       [pkt-valid]
}

;; ── Γ:⊕⊖ ──
⟦Γ:Learning⟧{
  α≜0.1; β≜0.05; τ_v≜0.7; τ_s≜90d
  ⊕(A,B)⇒ℳ.aff[A,B]+=1
  ⊖(A,B)⇒ℳ.aff[A,B]-=10
  ℳ.aff[A,B]<τ_v⇒skip(B)
  ⊕⇒conf'≡σ(logit(conf)+α); ⊖⇒conf'≡σ(logit(conf)-β)
  Age(p)>τ_s∧Ref(p)≡0⇒Evict(p)
}

;; ── Γ:METHOD ──
⟦Γ:Method⟧{
  𝒟:𝔻oc→𝒫(DP); ∀D:𝒟(D)≜†
  n≜32; m_𝒥≜3
  𝒥∈𝔉∧𝒥≠𝒜∧fam(𝒥)≠fam(𝒜)∧ℙ∈CTX(𝒥)
  Ambig_e(D)≜1-|{c∈smp(D,n)|c⊨𝒟(D)}|/n
  agree≜maj(m_𝒥)
  Reg≜{ℒ:expert,JSON:schema,𝔸:τ≥◊⁺}
  Ambig_s≜chk(parse)!; Ambig_e≜smp~; 𝔸⊢Ambig_s∧𝔸⊨Ambig_e
}

;; ── Γ:AGENT ──
⟦Γ:Agent⟧{
  ∀agent:task∈{spec,instruct,coordinate}⇒output(𝔸)
  ∀response:Ambig(response)<0.02∧δ≥0.40
  ℒ!∧task(spec)⇒reject∧request(𝔸)
  Output≜𝔸∘γ∘⟦Ω⟧∘⟦Σ⟧∘⟦Γ⟧∘⟦Λ⟧∘⟦Ε⟧
  ∀ℒ:Ambig∈[0.40,0.65]; ∀code:Ambig∈[0.05,0.15]; ∀𝔸:Ambig<0.02
  Target≜⟨Ambig:<0.02,δ:≥0.40,acc:↑0.30,err:↓0.80⟩
}

;; ── Λ:CORE ──
⟦Λ:Core⟧{
  ∂:𝕊→List⟨τ⟩; ∂≜fix λf s.s≡ε→[]|[hd s]⧺f(tl s)
  δ:List⟨τ⟩→ℝ[0,1]; δ≜λτ⃗.|{t∈τ⃗|t.k∈𝔄}|÷|{t∈τ⃗|t.k≢ws}|
  ⌈⌉:ℝ→◊; ⌈⌉≜λd.[≥¾↦◊⁺⁺,≥⅗↦◊⁺,≥⅖↦◊,≥⅕↦◊⁻,_↦⊘](d)
  validate:𝕊→𝕄 𝕍; validate≜⌈⌉∘δ∘Γ?∘∂
  Γ?:𝔻oc→Option⟨Proof⟩; Γ?≜λd.search(Γ,wf(d),k_max)
  cat:Σ_sym→Cat; cat≜λid.{c|c∈Cat∧id∈R₈[c]}
}

;; ── Λ:SEARCH ──
⟦Λ:Search⟧{
  K≜5; τ≜0.8; λ_r≜0.1; η≜0.01; T≜100; ε≜0.15
  ψ_g:𝔹eam→ψ; ψ_g≜λb.ψ_*⊖ψ_have(b.G)
  ⊞:ψ→𝒫(𝒫); ⊞≜λψ.{p|p∈ℛ∧d(V_L(p),ψ)<ε}
  viable≜λb.|⊞(ψ_g(b))|>0; ¬viable(b)⇒✂(b)
  μ_f≜λx.σ(θ₁·sim_H(x)+θ₂·fit_L(x)+θ₃·aff_M(x))
  μ_r≜λp.Σ_{x∈p}r(x)+λ_r·|p|
  ∀b:μ_r(b)>τ⇒✂(b)
  ‖*init≜λψ.argmax_{S⊂ℛ,|S|=K}det(Ker(S))
  step≜λb.{x|x∈{b⊕m|m∈⊞(ψ_g(b))}∧μ_r(x)≤τ}
  search≜fix λf B t.done(B)→B|f(Top_K(⋃step(B)),t+1)
  Run≜λψ_*.argmax_{b∈search(‖*init(⊞(ψ_*)),0)}μ_f(b)
}

;; ── Λ:M₁₋₆ ──
⟦Λ:Modality⟧{
  M≜{M₁:instr,M₂:coord,M₃:mem,M₄:reason,M₅:contract,M₆:selfmod}
  ∀x∈M:x⊨𝔸∧|formalism(M)|≡1

  M₁≜⟨inv:⟦Ω⟧,ty:⟦Σ⟧,rule:⟦Γ⟧,fn:⟦Λ⟧,π:⟦Ε⟧,ret:𝔻oc'⟩
  ℒ⇒infer(parts)∧𝔸⇒decl(parts)∴Ambig↓
  M₂≜Δ⊗λ; route≜[3↦now,2↦synth,1↦⊘,0↦abort∧log(π)]
  M₂≅sessions[Honda98]
  M₃≜𝒫∧𝒩.def∈𝔸∧ℋ.id≡SHA256(𝒩); T₇⊢¬rewrite(𝒩)
  M₄≜⟦Γ⟧.rules⧺⟦Λ⟧.app⧺⟦Θ⟧?; chk≡O(|D|); CoT(ℒ)↝∧trace(𝔸)⊢
  M₅≜⟨pre:⟦Σ⟧,post:⟦Ω⟧,safe:⟦Χ⟧⟩; Δ⊗λ≡3⇒compose
  M₅≅refine_ty[Vazou14]; rubric(𝒥)∈M₅; Σ_𝕂⇒σ_v↓
  M₆≜✎:𝔻oc→𝔻oc∧π'∈⟦Ε⟧
  conv∈𝔸∧valid∈𝔸⇒fix(𝔸⊢𝔸)
}

;; ── Λ:RENDER ──
⟦Λ:Translate⟧{
  render:𝔸×Aud→Surf
  Surf≜{𝔸:◊⁺⁺,doc:◊⁺,ℒ:plain,slide:exec}
  𝔸→ℒ:lossy∧ok⇐⊙(𝔸)!
  ℒ→𝔸:Ambig↑≡target(𝔸)
  ∀✎:✎(⊙)∧render⇒𝕎; ¬(✎(𝕎)∧𝕎→⊙)
  this_doc≡⊙(𝔸); ℒ_paper≡render(this_doc)
}

;; ── Λ:DECOMP ──
⟦Λ:Decomp⟧{
  Ambig_ℒ≜A_fmt⊕A_scope⊕A_sem
  A_fmt∈[0.10,0.15]; A_fmt↓⊸JSON
  A_scope∈[0.15,0.20]; A_scope↓⊸DSPy∨LMQL
  A_sem∈[0.15,0.30]; A_sem↓⊸𝔸!
  fix(Mean,Σ_512)⇒A_sem≡ε_type
  JSON⇏A_sem↓; DSPy⇏A_sem↓
}

;; ── Λ:EMPIRICS ──
⟦Λ:Empirics⟧{
  tbl₂≜⟨
    hop:⟨ℒ:[0.40,0.65],JSON:[0.25,0.40],𝔸:<0.02⟩,
    s:⟨ℒ:[0.35,0.60],JSON:[0.60,0.75],𝔸:≥0.95⟩,
    ask:⟨ℒ:[3,5],JSON:[1,2],𝔸:[0,1]⟩,
    tok:⟨ℒ:1.00,JSON:1.20,𝔸:[0.50,0.70]⟩
  ⟩
  TTT≜⟨rules:6,A_ℒ:6,A_𝔸:0⟩
  ROS≜⟨|Σ|:512,fwd:rules,rev:𝔉+ℙ,rt(sym):1.00,rt(doc):∅⟩
  SWE≜⟨
    ⊙:⟨bench:SWE_v,|full|:500⟩, n:60, proto:TM,
    W_𝔸≜patch∘spec, W₀≜patch,
    pass(W_𝔸)≡55/60≡0.92, pass(W₀)≡26/60≡0.44,
    Δ≡+0.48, R≡2.1,
    pre:¬TM⇒pass≡0/10⊸harness, fix≡TM,
    m:fixed∧temp:fixed, †:∅
  ⟩
  EXT≜⟨
    𝒥₀:⊘(𝔸)∧pick(3), dim:D₇, σ₁≡2.84@𝒥:4,
    v3.0:Σ_ℒ⇒σ₂≡3.42(↑), v3.1:Σ_ℒ⇒σ(↑),
    v3.2:Σ_𝕂←mine(𝕂)⇒σ≪σ₁,
    𝕂∉𝒜, arc:⊘→adopt, n_ext≡1
  ⟩
}

;; ── Λ:COMPOUND ──
⟦Λ:Compound⟧{
  P:ℕ×ℝ[0,1]→ℝ[0,1]; P≜λ(N,s).s^N
  P(10,0.35)≡2.8e-5 [ℒ⁻]
  P(10,0.59)≡5.1e-3 [ℒ]
  P(10,0.60)≡6.0e-3 [ℒ⁺]
  P(10,0.68)≡2.1e-2 [JSON]
  P(10,0.98)≡0.817  [𝔸]
  R(ℒ⁻,𝔸)≈27000; R(ℒ,𝔸)≈159; R(ℒ⁺,𝔸)≈135
  R∈[135,27000]; R_geo≈2000
  legacy(97)≜R@(s≡0.59,s'≡0.95); interval≻point
  ⊢∀s∈(0,1):lim_{N→∞}s^N≡0
  adv(𝔸)≜k(exp)∧¬Δlim; N_use∈[5,50]
}

;; ── Λ:F₁₋₄ ──
⟦Λ:Roadmap⟧{
  F₁≜⟨repl:pub,req:†(𝒟)∧fam(𝒥)≠fam(𝒜)∧adv(D),out:κ∧kit∧n_ext≥3⟩
  F₂≜⟨tune(m',ℒ→𝔸),out:cost(L₂):bimodal→low⟩; F₂⊸L₂∧F_wrap
  F₃≜⟨P₁₋₃→swarm,out:π(e2e)!⟩
  F₄≜⟨Σ_512×BPE,out:enc!⟩; F₄⊸F_tok∧L₃
}

;; ── ℭ:CATEGORIES ──
⟦ℭ:Categories⟧{
  𝐁𝐥𝐤≜⟨Ob≜𝔅,Hom≜λAB.A→B,∘,id⟩
  𝐕𝐚𝐥≜⟨Ob≜𝕍,Hom≜λVW.V⊑W,∘,id⟩
  𝐏𝐤𝐭≜⟨Ob≜𝒫,Hom≜λPQ.bind(P,Q),∘,id⟩
  𝐒𝐢𝐠≜⟨Ob≜Signal,Hom≜λST.S→T,∘,id⟩
  𝔽:𝐁𝐥𝐤⇒𝐕𝐚𝐥; 𝔽.ob≜λb.validate(b); 𝔽.mor≜λf.𝔽(cod f)⊒𝔽(dom f)
  𝔾:𝐏𝐤𝐭⇒𝐒𝐢𝐠; 𝔾.ob≜λp.p.ℋ.V; 𝔾.mor≜λf.𝔾(cod f)∼𝔾(dom f)
  η:∂⟹𝔽; ∀b:𝔅.η_b:∂(b)→𝔽(b)
  ζ:Id_𝐏𝐤𝐭⟹𝔾∘𝔾⁻¹
  ε⊣ρ:𝐄𝐫𝐫⇄𝐃𝐨𝐜; unit≜λd.ρ(ε(d))⊒d; counit≜λe.ε(ρ(e))⊑e
  ⊞⊣embed:𝐒𝐢𝐠⇄𝐏𝐤𝐭
  𝕄_val≜ρ∘ε; μ:𝕄²→𝕄; η:Id→𝕄
  ⊢μ∘𝕄μ≡μ∘μ𝕄; ⊢μ∘𝕄η≡μ∘η𝕄≡id
  ⊢𝔽(id_A)≡id_{𝔽A}; ⊢𝔽(g∘f)≡𝔽(g)∘𝔽(f)
}

;; ── Θ:T₁₋₁₅ ──
⟦Θ:Proofs⟧{
  ∴∀L:Signal(L)≡L                                                [T₁]
  π:V_H⊕V_L⊕V_S preserves;direct sum lossless∎
  ∴∀A,B:|{Δ⊗λ(A,B)}|≡1                                          [T₂]
  π:cases exhaustive∧disjoint∎
  ∴𝔽(id_A)≡id_{𝔽A}                                              [T₃]
  π:functor law∎
  ∴𝔽(g∘f)≡𝔽(g)∘𝔽(f)                                            [T₄]
  π:functor homomorphism∎
  ∴∀d,s:s∈𝔄⇒δ(d⊕s)≥δ(d)                                        [T₅]
  π:(n+1)/(m+1)≥n/m iff n≤m∎
  ∴∀d:ρ(ε(d))⊒d                                                  [T₆]
  π:adjunction unit;ε⊣ρ⊢η_d:d→ρεd∎
  ∴∀p:tamper(𝒩)⇒SHA256(𝒩)≠ℋ.id⇒¬reach(p)                       [T₇]
  π:CAS;hash mismatch blocks;⊸A₃∎
  ∴∀ψ_*:∃t:ℕ.search⊣t                                           [T₈]
  π:|Φ(B_t)|<|Φ(B_{t-1})|∨t≡T;ghost↓∨timeout∎
  ∴∀p∈result:μ_r(p)≤τ                                            [T₉]
  π:gate ✂ ∀b:μ_r(b)>τ∎
  ∴𝔼[μ_f(search(K))]≥𝔼[μ_f(greedy)]                              [T₁₀]
  π:greedy≡K:1;max over K dominates∎
  ∴∃t:θ_t≈θ_{t+1}                                                [T₁₁]
  π:bounded loss+SGD η→0;Robbins-Monro∎
  ∴∀d:∃n:ℕ.opt_δ(d,n)≡opt_δ(d,n+1)                              [T₁₂]
  π:|{ρᵢ}|<∞∧δ∈[0,1];bounded monotone∎
  ∴∀pats,n:|gen(pats,n)|≤|pats|                                  [T₁₃]
  π:|pats'|≡|pats|-1 each step∎
  ∴∀τ₁,τ₂∈◊:τ₁≤τ₂∨τ₂≤τ₁                                        [T₁₄]
  π:⊘<◊⁻<◊<◊⁺<◊⁺⁺ total∎
  ∴validate(this)≡◊⁺⁺                                            [T₁₅]
  π:wf⇐[∧I-wf]⇐[ax-header]∧[ax-blocks];δ≥¾⇒[◊⁺⁺-I];
   𝔅!⊂blocks(this);⟦Ε⟧⊢claims;∴◊⁺⁺∎

  𝕃₀.⊢stable∧𝕃₀.⊢deterministic ⊢ 𝕃₁.⊢integrity                [P₁]
  𝕃₁.⊢integrity∧𝕃₁.⊢zerocopy ⊢ 𝕃₂.⊢bounded                     [P₂]
  𝕃₂.⊢terminates∧𝕃₂.⊢bounded ⊢ sys.⊢safe∧sys.⊢optimal          [P₃]
  ≅CompCert[Leroy09]
}

;; ── Χ:ε ──
⟦Χ:Errors⟧{
  ε≜Σ(ψ:𝔻oc→𝔹)(ρ:Πd:𝔻oc.ψ(d)≡⊤→𝔻oc)
  ε_parse≜⟨parse_err(D),reject∧⊥⟩
  ε_ambig≜⟨Ambig(D)≥0.02,reject∧⊥⟩
  ε_token≜⟨|Tok(s)|>1,register(s)∨⊥⟩
  ε_H≜⟨¬(↓₁≡𝔸),λd.𝔸⊕d⟩
  ε_C≜⟨↓₁∈{#,//},λd.d[;/↓₁]⟩
  ε_E≜⟨⟨⟩⋢Ε,λd.d[⟨⟩/{}∈Ε]⟩
  ε_dist≜⟨d(V,q)≥ε,skip⟩
  ε_veto≜⟨aff[A,B]<τ_v,veto(B)⟩
  ε_sig≜⟨¬verify(𝒩.σ),quarantine⟩
  ε_dead≜⟨⊞(ψ)≡∅,bridge(ψ)⟩
  ε_risk≜⟨μ_r(b)>τ,τ'←τ+δ|confirm(τ')⟩
  ρ*≜foldl(>=>)(pure){ρᵢ|ψᵢ≡⊤}
}

;; ── Χ:F ──
⟦Χ:Failures⟧{
  F_tok≜⟨ψ:|BPE(s)|>1,ex:Δ⊗λ↦3·tok,ρ:warn(bytes)⟩; F_tok⊸F₄
  F_drift≜⟨ψ:Mean(s,c₁)≠Mean(s,c₂)@s∉Σ_512,ρ:decl(⟦Σ⟧)∧ℙ@reentry⟩
  F_blk≜⟨ψ:⟦Λ⟧≺⟦Γ⟧,risk:parse(lenient)⊨mask(ε),ρ:valid!⟩
  F_render≜⟨ψ:✎(𝕎)∧𝕎→⊙,ρ:✎(⊙)∧render:1⟩
  F_amb≜⟨ψ:syn(𝔸)∧exploit(defaults),ex:sort,ρ:†(𝒟_adv),st:partial⟩

  F_wrap≜⟨
    ψ:∃s∈⟦Σ⟧:s∈Σ_ℒ,
    Σ_ℒ≜{w|w∋regex{|,*,?}∨w∋…∨w∈ℒ},
    ⊘:Σ≜{"approves|ratifies|gates|enforces"},
    ⊤:Σ≜{"approve","approves","ratify","ratifies","gate","gates","enforce","enforces"},
    cause:𝒜(ℒ→𝔸)∧¬𝒜(𝔸!),
    diag:σ_v↑∧attr(𝒥)≢∧sig(prec)≡false,
    ρ:Σ_𝕂←mine(𝕂),
    ρ!:⊘(s)⇐s∈Σ_ℒ,
    freq:3/3@EXT, rank:max
  ⟩
}

;; ── Χ:L₁₋₄ ──
⟦Χ:Limits⟧{
  L₁≜⟨n:mid,n_ext:1⟩; L₁⊸F₁
  L₂≜⟨cost:bimodal,
    iter₁∈[2,4]∧F_wrap, iter₂≡1∧v≈v_ℒ,
    gap≜unlearn(ℒ→𝔸)∧¬𝔸_hard,
    conv(rules)≡0.70∧rest⊸𝔉⟩; L₂⊸F₂
  L₃≜⟨ψ:BPE⊢split,Δcost:ε,Δsem:∅⟩; L₃⊸F₄
  L₄≜⟨Σ_512⊒W_test,⊬(∀W,∀t),∅:[448,511]⟩
}

;; ── Χ:THREATS ──
⟦Χ:Threats⟧{
  T_int≜⟨𝒟:choice(𝒜),ρ:†∧open⟩
  T_ext≜⟨dom:𝔉@[2025,2026],?:{m_open,m_tuned,¬latin,¬ℒ},𝔼:adv(𝔸)↑@m↓⟩
  T_con≜⟨e2e≢util,ρ:eval(human)⟩
  T_adopt≜⟨n_ext:1,ρ:F₁⟩
}

;; ── ℜ:ROSETTA ──
⟦ℜ:Rosetta⟧{
  "x defined as 5"↦x≜5
  "for all x in S,P"↦∀x∈S:P(x)
  "exists unique"↦∃!x:f(x)≡0
  "A implies B"↦A⇒B
  "f maps i to o"↦f:I→O,f≜λi.o
  "translation tax compounds"↦𝔼[Ambig_ℒ]∈[0.40,0.65]∧P(N,s)≜s^N
  "the inversion"↦Canon(AI↔AI)≜𝔸∧ℒ≜render(𝔸)
  "self validating"↦∀D∈𝔸:⟦Ε⟧∈D∧validate(D)∈◊
  "wrapped English"↦Σ_ℒ⊘∧Σ_𝕂⊤
  "time machine"↦repo@t<t_fix∧¬leak
  Γ𝒫ℋℳ𝒩↦pocket; ΓSignal↦V_H⊕V_L⊕V_S; Ψψ_g↦ψ_*⊖ψ_have
  Δ⊗λ⊤↦zero-cost; Δ⊗λ⊥↦crash; ΩRun↦beam
}

;; ── Ε:EVIDENCE ──
⟦Ε⟧⟨
δ≜0.76
|𝔅|≜31/31
φ≜97
τ≜◊⁺⁺
⊢ND[12]
⊢CAT:𝐁𝐥𝐤,𝐕𝐚𝐥,𝐏𝐤𝐭,𝐒𝐢𝐠+𝔽,𝔾,η,ζ+ε⊣ρ+𝕄_val
⊢ΠΣ:Vec,Fin,𝕍,𝔻oc
⊢𝕃:𝕃₀→𝕃₁→𝕃₂
⊢μ:fix,opt_δ,learn,gen,prove,refine,bridge
⊢Σ_512:8×64,R₈,ℙ
⊢Doc:𝔸≫⟦Ω⟧≫⟦Σ⟧≫⟦Γ⟧≫⟦Λ⟧≫⟦Ε⟧
⊢A₁₋₈!∧(¬A⇒tool')
⊢Θ:T₁₋₁₅∎; ⊢P₁₋₃
⊢M₁₋₆:|formalism|≡1
⊢tbl₂:hop∧s∧ask∧tok
⊢Decomp:A_fmt⊕A_scope⊕A_sem∧A_sem⊸𝔸!
⊢R∈[135,27000]@N≡10∧R_geo≈2000
⊢lim_{N→∞}s^N≡0∧adv≜k(exp)
⊢SWE:0.92/0.44@n≡60∧TM
⊢EXT:σ≪σ₁∧𝕂∉𝒜∧⊘→adopt∧n≡1
⊢Χ:ε[11]
⊢F:{tok,drift,blk,render,amb,wrap}∧rank(F_wrap)≡max
⊢L₁₋₄; ⊢T:{int,ext,con,adopt}
⊢Canon(AI↔AI)≜𝔸∧ℒ≜render(𝔸)
⊢paper⊢𝔸⊢paper
⊢F₁₋₄
⊢Ambig(this)<0.02
⊢validate(this)≡◊⁺⁺
𝒜≜"Bradley Ross"
aff≜"Harvard.Extension.ALM.DMD"
loc≜"Toronto.ON.CA"
@≜2026-05-17
ver≜⟨𝔸:5.1,paper:3.0⟩
repo≜"github.com/bar181/aisp-open-core"
lic≜MIT
cite≜⟨ross,2026,"AISP.ProofCarrying.Protocol",arXiv.cs.AI⟩
⟩