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249 lines
8.3 KiB
Text
249 lines
8.3 KiB
Text
(* RGFlow_Isabelle.thy
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Isabelle/HOL port of Bitcoin RG Flow compression using Q16.16 fixed-point arithmetic.
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Purpose:
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* Formalize the RGFlow lawfulness invariant
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* Define Q16.16 fixed-point arithmetic operations
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* Specify sigma channel computation
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* Separate validity from lawfulness
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This theory focuses on the core mathematical definitions and invariants.
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*)
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theory RGFlow_Isabelle
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imports Main
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begin
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section {* Q16.16 Fixed-Point Arithmetic *}
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definition SCALE :: int where
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"SCALE = 65536"
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definition HALF :: int where
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"HALF = SCALE div 2"
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definition ONE :: int where
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"ONE = SCALE"
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(* Q16.16 constructor from integer *)
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definition q_of_int :: "int \<Rightarrow> int" where
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"q_of_int x = x * SCALE"
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(* Q16.16 constructor from numerator/denominator *)
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definition q_div_int :: "int \<Rightarrow> int \<Rightarrow> int" where
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"q_div_int num den = (if den = 0 then 0 else (num * SCALE) div den)"
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(* Q16.16 multiplication *)
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definition q_mul :: "int \<Rightarrow> int \<Rightarrow> int" where
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"q_mul a b = (a * b) div SCALE"
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(* Q16.16 division *)
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definition q_div :: "int \<Rightarrow> int \<Rightarrow> int" where
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"q_div a b = (if b = 0 then 0 else (a * SCALE) div b)"
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(* Absolute value *)
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definition q_abs :: "int \<Rightarrow> int" where
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"q_abs x = (if x < 0 then -x else x)"
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(* Clamp fixed-point value *)
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definition q_clamp :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int" where
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"q_clamp lo hi x = max lo (min hi x)"
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(* Q16 constants *)
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definition q0001 :: int where "q0001 = q_div_int 1 1000" (* 0.001 *)
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definition q0002 :: int where "q0002 = q_div_int 2 1000" (* 0.002 *)
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definition q0003 :: int where "q0003 = q_div_int 3 1000" (* 0.003 *)
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definition q0004 :: int where "q0004 = q_div_int 4 1000" (* 0.004 *)
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definition q0008 :: int where "q0008 = q_div_int 8 1000" (* 0.008 *)
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definition q0125 :: int where "q0125 = q_div_int 125 1000" (* 0.125 *)
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definition q025 :: int where "q025 = q_div_int 25 100" (* 0.25 *)
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definition q035 :: int where "q035 = q_div_int 35 100" (* 0.35 *)
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definition q05 :: int where "q05 = q_div_int 5 10" (* 0.5 *)
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definition q0875 :: int where "q0875 = q_div_int 875 1000" (* 0.875 *)
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definition q3 :: int where "q3 = q_of_int 3"
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section {* RGFlow Constants *}
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definition D :: int where "D = q0003"
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definition B :: int where "B = q0001"
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definition lambda :: int where "lambda = q05"
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definition epsilon :: int where "epsilon = 1"
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section {* Data Structures *}
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record PricePoint =
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day_index :: nat
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price_cents :: int
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record CompressedCode =
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index :: nat
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symbol :: "8 word"
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valid :: bool
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mu_q :: int
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rho_q :: int
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c_fac :: int
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m_fac :: int
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ne_eff :: int
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sigma_q :: int
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lawful :: bool
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cost :: nat
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record AuditSummary =
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total_codes :: nat
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valid_codes :: nat
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lawful_codes :: nat
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sigma_min :: int
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sigma_max :: int
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sigma_mean :: int
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sigma_std :: int
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sigma_pinned :: bool
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section {* Helper Functions *}
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(* Healthy modularity factor, maximized at 0.5 *)
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definition phi :: "int \<Rightarrow> int" where
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"phi m_fac = ONE - q_abs (m_fac - q05)"
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(*
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The RGFlow lawfulness invariant.
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This is deliberately independent from `valid`.
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A code may be syntactically valid but fail the RGFlow invariant.
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*)
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definition rg_lawful :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
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"rg_lawful mu_q rho_q c_fac m_fac ne_eff sigma_q =
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(mu_q \<le> q_div D (max c_fac epsilon)) \<and>
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(q_mul (q_mul rho_q ne_eff) (phi m_fac) \<ge> B) \<and>
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(sigma_q > ONE + q_mul lambda mu_q)"
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(*
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Integer log-return proxy in Q16.16.
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Uses absolute relative price move: |curr - prev| / prev
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This avoids floating-point logarithms in the fixed-point path.
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*)
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definition rel_move_q16 :: "int \<Rightarrow> int \<Rightarrow> int" where
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"rel_move_q16 prev curr =
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(if prev \<le> 0 \<or> curr \<le> 0 then 0
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else q_div (q_abs (curr - prev)) prev)"
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(*
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Mean in Q16.16 for a list of values.
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*)
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definition q_mean :: "int list \<Rightarrow> int" where
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"q_mean xs = (if xs = [] then 0 else sum xs div int (length xs))"
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(*
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Mean absolute deviation in Q16.16.
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Used instead of sqrt variance to keep the fixed-point path simple.
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*)
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definition q_mad :: "int list \<Rightarrow> int" where
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"q_mad xs =
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(if length xs \<le> 1 then 0
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else let m = q_mean xs in
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sum (map (\<lambda>x. q_abs (x - m)) xs) div int (length xs))"
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(*
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First-pass fixed-point sigma channel.
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It rewards local trend coherence and penalizes move dispersion.
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raw: sigma = 1.0 + 0.35 * coherence - 8 * dispersion
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where: coherence = |mean(move)| / (MAD(move) + epsilon)
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Range: sigma_q is clamped to [0.25, 3.0].
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*)
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definition compute_sigma_q :: "int list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> int" where
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"compute_sigma_q prices code_index window =
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(let moves = map (\<lambda>i. rel_move_q16 (prices ! (i - 1)) (prices ! i))
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[1..<length prices];
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ri = (if code_index = 0 then 0 else code_index - 1);
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local_len = min (ri + 1) window;
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local = take local_len (drop (max 0 (ri + 1 - window)) moves);
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disp = q_mad local;
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mom = q_mean local;
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coherence = q_div (q_abs mom) (disp + epsilon);
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raw = ONE + q_mul q035 coherence - q_mul (q_of_int 8) disp
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in q_clamp q025 q3 raw)"
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section {* Validity Predicate *}
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(*
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Toy validity predicate for a compressed symbol.
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Replace with the production validity predicate. Kept separate from lawfulness.
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*)
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definition valid_symbol :: "8 word \<Rightarrow> bool" where
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"valid_symbol s = ((ucast s :: nat) AND 3) \<noteq> 3"
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section {* Encoding Function *}
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(*
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Encode one adjacent pair into Q16.16 RGFlow variables.
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This is a reference mapping. Replace with production genome mapping.
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*)
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definition encode_code :: "int list \<Rightarrow> int \<Rightarrow> int \<Rightarrow> nat \<Rightarrow> CompressedCode" where
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"encode_code prices min_price max_price i =
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(let p0 = prices ! i;
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p1 = prices ! (i + 1);
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span = max (max_price - min_price) 1;
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x = max 0 (min span (p0 - min_price));
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pos0 = nat ((x * 65535) div span);
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y = max 0 (min span (p1 - min_price));
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pos1 = nat ((y * 65535) div span);
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d = (if pos1 \<ge> pos0 then pos1 - pos0 else pos0 - pos1);
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sym = word_of_int (int (d mod 256)) :: 8 word;
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valid = valid_symbol sym;
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move = rel_move_q16 p0 p1;
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mu_q = q_clamp q0001 q0008
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(q0001 + q_mul (q_div_int 7 1000)
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(q_clamp 0 ONE (q_mul move (q_of_int 25))));
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rho_q = q_clamp q0001 q0008
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(q0004 + (if valid then q0002 else -q0002));
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c_fac = q_clamp q0125 ONE
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(q0125 + q_mul q0875 (q_clamp 0 ONE (q_mul move (q_of_int 10))));
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m_fac = q05;
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ne_raw = min (i + 1) 365;
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ne_eff = q_mul (q_of_int (int ne_raw)) LN2_Q16;
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sigma_q = compute_sigma_q prices i 30;
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lawful = rg_lawful mu_q rho_q c_fac m_fac ne_eff sigma_q;
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cost = (if valid then 0 else 1) + (if lawful then 0 else 4)
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in \<lparr> index = i, symbol = sym, valid = valid, mu_q = mu_q,
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rho_q = rho_q, c_fac = c_fac, m_fac = m_fac, ne_eff = ne_eff,
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sigma_q = sigma_q, lawful = lawful, cost = cost \<rparr>)"
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(* LN(2) approximation in Q16.16: ln(2) \<approx> 0.693147 *)
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definition LN2_Q16 :: int where "LN2_Q16 = 45426"
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section {* Theorems *}
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theorem rg_lawful_components:
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assumes "rg_lawful mu_q rho_q c_fac m_fac ne_eff sigma_q"
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shows "mu_q \<le> q_div D (max c_fac epsilon)"
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and "q_mul (q_mul rho_q ne_eff) (phi m_fac) \<ge> B"
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and "sigma_q > ONE + q_mul lambda mu_q"
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using assms unfolding rg_lawful_def by auto
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theorem phi_bounds:
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assumes "0 \<le> m_fac" and "m_fac \<le> ONE"
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shows "0 \<le> phi m_fac" and "phi m_fac \<le> ONE"
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proof -
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have "q_abs (m_fac - q05) \<le> q05"
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using assms unfolding q_abs_def by auto
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thus "0 \<le> phi m_fac" and "phi m_fac \<le> ONE"
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unfolding phi_def ONE_def q05_def by auto
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qed
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theorem sigma_clamped:
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assumes "compute_sigma_q prices i window = sigma"
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shows "q025 \<le> sigma" and "sigma \<le> q3"
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using assms unfolding compute_sigma_q_def q_clamp_def by auto
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theorem valid_symbol_decidable:
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"\<forall>s. valid_symbol s \<or> \<not> valid_symbol s"
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unfolding valid_symbol_def by auto
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end
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