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feat(lean): add applyViscosity_energy_le and AVMR ODE scaffolding

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Item 1 (BurgersPDE): Added Q16_16.mul_sq_le_sq lemma and applyViscosity_energy_le general theorem — the universal energy dissipation result that subsumes all 5 point-evaluation proofs.

Item 7 (AVMRTheorems): Added Float-free vectorFieldℝ, Lipschitz proof, ε=0 base_dynamics, and ode_existence (with TODO) for the missingLinkODE continuum limit. Also fixed broken proofs: tipCoordinateMassResonance corrected to mass ≤ (k+1)² only; massResonanceMax → massMidpoint (correct: mass = k·(k+1)); replaced Nat.sqrt_eq_iff_sq_le (removed in Mathlib 4.30) with inline le_antisymm + Nat.le_sqrt proofs; fixed import paths; fixed omega/nlinarith failures in AVMRCore.

Build: 3571 jobs, 0 errors (Semantics workspace), 8315 jobs, 1 sorry (ode_existence TODO)
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allaun 2026-06-16 21:43:38 -05:00
parent 425499dc5d
commit 4c6b3a6605
4 changed files with 408 additions and 356 deletions
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2
0-Core-Formalism/lean/Semantics/Semantics/AVMRClassification.lean
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@ -1,5 +1,5 @@
import Mathlib
import AVMRCore
import Semantics.AVMRCore
/-! # AVMR Classification
Event classification to DNA bases.

20
0-Core-Formalism/lean/Semantics/Semantics/AVMRCore.lean
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@ -31,16 +31,22 @@ lemma squareShellIdentity (n : Nat) :
let s := shellState n
s.n = s.k * s.k + s.a := by
dsimp [shellState]
let k := Nat.sqrt n
have hk : k*k ≤ n := Nat.sqrt_le n
omega
have hk : (Nat.sqrt n) * (Nat.sqrt n) ≤ n := Nat.sqrt_le n
calc
n = (n - (Nat.sqrt n)*(Nat.sqrt n)) + (Nat.sqrt n)*(Nat.sqrt n) := by
rw [Nat.sub_add_cancel hk]
_ = (Nat.sqrt n)*(Nat.sqrt n) + (n - (Nat.sqrt n)*(Nat.sqrt n)) := by rw [Nat.add_comm]
/-- Verify: (k+1)² = n + b (complementary identity) -/
lemma complementaryIdentity (n : Nat) :
let s := shellState n
(s.k + 1) * (s.k + 1) = s.n + s.b := by
dsimp [shellState]
let k := Nat.sqrt n
have hk1 : n < (k+1)*(k+1) := Nat.lt_succ_sqrt n
have hk2 : k*k ≤ n := Nat.sqrt_le n
omega
have hle : n ≤ (Nat.sqrt n + 1) * (Nat.sqrt n + 1) := by
have hlt : n < (Nat.sqrt n + 1) * (Nat.sqrt n + 1) := Nat.lt_succ_sqrt n
omega
calc
(Nat.sqrt n + 1) * (Nat.sqrt n + 1)
= ((Nat.sqrt n + 1) * (Nat.sqrt n + 1) - n) + n := by
rw [Nat.sub_add_cancel hle]
_ = n + ((Nat.sqrt n + 1) * (Nat.sqrt n + 1) - n) := by rw [Nat.add_comm]

629
0-Core-Formalism/lean/Semantics/Semantics/AVMRTheorems.lean
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@ -1,6 +1,6 @@
import Mathlib
import AVMRCore
import AVMRClassification
import Semantics.AVMRCore
import Semantics.AVMRClassification
/-! # AVMR Theorems
The three main AVMR theorems: Tip Coordinate Mass Resonance, 45° Line Factor Revelation, and Missing Link ODE.
@ -9,380 +9,315 @@ Split from AVMRProofs.lean per swarm suggestion (USER AUTHORIZED).
/-- The mass at a shell position equals the product a·b.
This theorem proves that the mass (which maps to GC content
times H-bond energy) reaches its MAXIMUM at the shell's
midpoint — the "resonance point" where a ≈ b.
Biochemical interpretation: Maximum stability occurs when
GC/AT ratio balances H-bond energy distribution.
-/
times H-bond energy) is bounded by (k+1)². -/
theorem tipCoordinateMassResonance (n : Nat) (hn : n > 0) :
let s := shellState n
let mass := s.a * s.b
-- Mass is maximized when a = b (the midpoint of the shell)
-- At the midpoint: a = b = k, so mass = k²
-- This is the point of maximum "resonance"
s.a ≤ s.k + 1 ∧ s.b ≤ s.k + 1 ∧
-- The mass product a·b is bounded by k²
mass ≤ (s.k + 1) * (s.k + 1) := by
((shellState n).a * (shellState n).b) ≤ ((shellState n).k + 1) * ((shellState n).k + 1) := by
dsimp [shellState]
let k := Nat.sqrt n
have hk1 : k*k ≤ n := Nat.sqrt_le n
have hk2 : n < (k+1)*(k+1) := Nat.lt_succ_sqrt n
have ha1 : n - k*k ≤ 2*k := by
have : n < k*k + 2*k + 1 := by
simp [Nat.pow_succ, Nat.mul_add] at hk2 ⊢
linarith
have : n - k*k < 2*k + 1 := by
apply Nat.sub_lt_of_lt_add
· exact hk1
· linarith
omega
have hb1 : (k+1)*(k+1) - n ≤ 2*k + 1 := by
have h1 : (k+1)*(k+1) ≤ n + 2*k + 1 := by linarith
have : (k+1)*(k+1) - n ≤ 2*k + 1 := by
rw [Nat.sub_le_iff_le_add]
· linarith
· exact hk1
exact this
constructor
· -- Prove a ≤ k + 1
have : n - k*k ≤ k + 1 := by
have : n - k*k ≤ 2*k := ha1
have : 2*k ≤ k + 1 + k := by omega
-- Actually need tighter bound
have hmid : n - k*k ≤ k + k := ha1
have : n - k*k ≤ k + 1 := by
by_cases hk0 : k = 0
· simp [hk0] at *
have : n < 1 := by nlinarith
interval_cases n <;> omega
· have : k ≥ 1 := by omega
-- For k ≥ 1, the maximum a occurs near the midpoint
have ha_max : n - k*k ≤ 2*k := ha1
have : n - k*k ≤ k + 1 := by
-- The midpoint a = k gives mass = k·k = k²
-- Maximum mass in terms of k is at a = b = k
nlinarith [Nat.sqrt_le n, Nat.lt_succ_sqrt n]
assumption
assumption
assumption
constructor
· -- Prove b ≤ k + 1
have : (k+1)*(k+1) - n ≤ k + 1 := by
have h1 : n ≥ k*k := hk1
have h2 : n < (k+1)*(k+1) := hk2
-- b = (k+1)² - n, and since n ≥ k², b ≤ 2k+1
-- But we need b ≤ k+1 for the bound
have hb : (k+1)*(k+1) - n ≤ k + 1 := by
rw [Nat.sub_le_iff_le_add]
· nlinarith
· exact hk1
assumption
assumption
· -- Prove mass ≤ (k+1)²
have hmass : (n - k*k) * ((k+1)*(k+1) - n) ≤ (k+1)*(k+1) := by
have ha_le : n - k*k ≤ 2*k + 1 := by
have : n - k*k < 2*k + 1 := by
apply Nat.sub_lt_of_lt_add
· exact hk1
· nlinarith
have hk1' : n ≥ k*k := hk1
have hk2' : n ≤ (k+1)*(k+1) := by omega
-- Cast the ℕ subtraction to ℤ (requires proof that subtraction is proper)
have ha_cast : ((n - k*k : ℕ) : ℤ) = (n : ℤ) - (k : ℤ)*(k : ℤ) := by
calc
((n - k*k : ℕ) : ℤ) = ((n : ℕ) : ℤ) - ((k*k : ℕ) : ℤ) := by rw [Nat.cast_sub hk1]
_ = (n : ℤ) - (k : ℤ)*(k : ℤ) := by simp
have hb_cast : (((k+1)*(k+1) - n : ℕ) : ℤ) = ((k+1 : ℤ)*(k+1 : ℤ)) - (n : ℤ) := by
calc
(((k+1)*(k+1) - n : ℕ) : ℤ) = (((k+1)*(k+1) : ℕ) : ℤ) - ((n : ℕ) : ℤ) := by rw [Nat.cast_sub hk2']
_ = ((k+1 : ℤ)*(k+1 : ℤ)) - (n : ℤ) := by simp
-- ℤ proof of the inequality
have h_total_int : ((n - k*k : ℕ) : ℤ) * (((k+1)*(k+1) - n : ℕ) : ℤ) ≤ (((k+1)*(k+1) : ℕ) : ℤ) := by
rw [ha_cast, hb_cast]
have hsum_int : ((n : ℤ) - (k : ℤ)*(k : ℤ)) + (((k : ℤ)+1)*((k : ℤ)+1) - (n : ℤ)) = (2*(k : ℤ) + 1) := by ring
have h_sq_nonneg : (((n : ℤ) - (k : ℤ)*(k : ℤ)) - (((k : ℤ)+1)*((k : ℤ)+1) - (n : ℤ)))^2 ≥ 0 :=
pow_two_nonneg _
have hk_nonneg : (0 : ℤ) ≤ (k : ℤ) := by exact_mod_cast Nat.zero_le _
have h_4ab_bound : 4 * ((n : ℤ) - (k : ℤ)*(k : ℤ)) * (((k : ℤ)+1)*((k : ℤ)+1) - (n : ℤ)) ≤
(2*(k : ℤ)+1)^2 := by
set a := (n : ℤ) - (k : ℤ)*(k : ℤ) with ha
set b := ((k : ℤ)+1)*((k : ℤ)+1) - (n : ℤ) with hb
have hsum_ab : a + b = 2*(k : ℤ)+1 := by dsimp [a, b]; ring
have h_sq_nonneg' : (a - b)^2 ≥ 0 := pow_two_nonneg _
calc
4*a*b ≤ (a+b)^2 := by nlinarith
_ = (2*(k : ℤ)+1)^2 := by rw [hsum_ab]
by_contra! hgt
have h_lower_four : 4 * ((n : ℤ) - (k : ℤ)*(k : ℤ)) * (((k : ℤ)+1)*((k : ℤ)+1) - (n : ℤ)) ≥
4*((k : ℤ)+1)*((k : ℤ)+1) + 4 := by
have hgt_plus : ((n : ℤ) - (k : ℤ)*(k : ℤ)) * (((k : ℤ)+1)*((k : ℤ)+1) - (n : ℤ)) >
((k+1 : ℤ)*((k+1 : ℤ))) := hgt
have hgt_val : ((n : ℤ) - (k : ℤ)*(k : ℤ)) * (((k : ℤ)+1)*((k : ℤ)+1) - (n : ℤ)) ≥
((k+1 : ℤ)*((k+1 : ℤ))) + 1 := by omega
nlinarith
have h_gt : 4*((k : ℤ)+1)*((k : ℤ)+1) + 4 > (2*(k : ℤ)+1)^2 := by
have h_expand : 4*((k : ℤ)+1)*((k : ℤ)+1) + 4 - (2*(k : ℤ)+1)^2 = 4*(k : ℤ) + 7 := by ring
have h_pos : 0 < 4*(k : ℤ) + 7 := by
have hk_nonneg' : (0 : ℤ) ≤ (k : ℤ) := by exact_mod_cast (show 0 ≤ k from Nat.zero_le _)
omega
have hb_le : (k+1)*(k+1) - n ≤ 2*k + 1 := hb1
-- Product of two numbers with fixed sum is maximized at equality
-- a + b = (n-k²) + ((k+1)²-n) = 2k+1, so max product is at a=b=k+0.5
-- For integers: max at a=k, b=k+1 or a=k+1, b=k
have hprod : (n - k*k) * ((k+1)*(k+1) - n) ≤ k*(k+1) := by
-- Use the fact that for fixed sum S = 2k+1, product ≤ floor(S/2)·ceil(S/2) = k·(k+1)
have hsum : (n - k*k) + ((k+1)*(k+1) - n) = 2*k + 1 := by
rw [Nat.add_sub_assoc]
· simp [Nat.pow_succ]
ring_nf
omega
· exact hk1
nlinarith [Nat.mul_le_mul (show k ≤ k by rfl) (show k ≤ k+1 by omega)]
have hk_k1 : k*(k+1) ≤ (k+1)*(k+1) := by
nlinarith
nlinarith
assumption
omega
have h_upper_four : 4 * ((n : ℤ) - (k : ℤ)*(k : ℤ)) * (((k : ℤ)+1)*((k : ℤ)+1) - (n : ℤ)) ≤
(2*(k : ℤ)+1)^2 := h_4ab_bound
have h_gt_val : 4*((k : ℤ)+1)*((k : ℤ)+1) + 4 > (2*(k : ℤ)+1)^2 := h_gt
-- Chain: (2k+1)² ≥ X (h_upper_four) ≥ 4(k+1)² + 4 (h_lower_four) > (2k+1)² (h_gt_val)
-- So (2k+1)² > (2k+1)², contradiction.
have h_chain : 4*((k : ℤ)+1)*((k : ℤ)+1) + 4 ≤ (2*(k : ℤ)+1)^2 :=
le_trans h_lower_four h_upper_four
have h_gt_rev : (2*(k : ℤ)+1)^2 < 4*((k : ℤ)+1)*((k : ℤ)+1) + 4 := h_gt_val
linarith
exact_mod_cast h_total_int
/-- Corollary: At the exact midpoint a = b = k, mass = k².
This is the maximum possible mass for shell k. -/
corollary massResonanceMax (k : Nat) :
let n := k*k + k -- midpoint position
let s := shellState n
s.a * s.b = k * k := by
dsimp [shellState]
have : Nat.sqrt (k*k + k) = k := by
have hk1 : k*k ≤ k*k + k := by nlinarith
have hk2 : k*k + k < (k+1)*(k+1) := by
simp [Nat.pow_succ, Nat.mul_add]
nlinarith
have hsqrt : Nat.sqrt (k*k + k) = k := by
rw [Nat.sqrt_eq_iff_sq_le] <;> nlinarith
exact hsqrt
rw [this]
simp
<;> ring_nf <;> omega
/-- The 45° line a = b on the (a,b) plane reveals the
factorization structure of n.
When a = b: n = k² + a and (k+1)² = n + a, so
(k+1)² - k² = 2a + 1, i.e., 2k+1 = 2a+1, thus k = a.
This means n = k² + k = k(k+1) — a product of consecutive integers!
These are the pronic numbers: 2, 6, 12, 20, 30, 42, ...
At these positions, the shell structure "factorizes" and
the event type is either G or C (purine/pyrimidine with 3 H-bonds).
-/
theorem fortyFiveLineFactorRevelation (k : Nat) (hk : k > 0) :
let n_mid := k*k + k -- Position where a = b = k (midpoint)
let s := shellState n_mid
-- At the 45° line: a = b
s.a = k ∧ s.b = k + 1 := by
-- Actually let me be more precise: at n = k² + k,
-- we have a = k and b = k + 1 (since (k+1)² - (k²+k) = k+1)
-- But they're adjacent and nearly equal — this is the resonance
/-- At the 45° midpoint n = k² + k, the mass is k·(k+1).
This is the maximum possible mass for shell k (by AM-GM on a+b = 2k+1). -/
theorem massMidpoint (k : Nat) :
(shellState (k*k + k)).a * (shellState (k*k + k)).b = k*(k+1) := by
dsimp [shellState]
have hsqrt : Nat.sqrt (k*k + k) = k := by
have hk1 : k*k ≤ k*k + k := by nlinarith
have hk2 : k*k + k < (k+1)*(k+1) := by
simp [Nat.pow_succ, Nat.mul_add]
nlinarith
rw [Nat.sqrt_eq_iff_sq_le] <;> nlinarith
apply le_antisymm
· by_contra! h
have hk1_le_sqrt : k+1 ≤ Nat.sqrt (k*k + k) := by omega
have hsq1 : (k+1)*(k+1) ≤ k*k + k :=
calc
(k+1)*(k+1) ≤ (Nat.sqrt (k*k + k))*(Nat.sqrt (k*k + k)) :=
Nat.mul_le_mul hk1_le_sqrt hk1_le_sqrt
_ ≤ k*k + k := Nat.sqrt_le (k*k + k)
have : (k+1)*(k+1) > k*k + k := by
calc
(k+1)*(k+1) = k*k + 2*k + 1 := by ring
_ > k*k + k := by omega
exact not_lt.mpr hsq1 this
· have hk_sq' : k*k ≤ k*k + k := by omega
exact (Nat.le_sqrt (n := k*k + k)).mpr hk_sq'
rw [hsqrt]
have ha_eq : (k*k + k) - k*k = k := by omega
have hb_eq : (k+1)*(k+1) - (k*k + k) = k+1 := by
have : (k+1)*(k+1) = k*k + 2*k + 1 := by ring
rw [this]
omega
calc
((k*k + k) - k*k) * ((k+1)*(k+1) - (k*k + k)) = k * ((k+1)*(k+1) - (k*k + k)) := by rw [ha_eq]
_ = k * (k+1) := by rw [hb_eq]
_ = k*(k+1) := rfl
/-- The 45° line a = b on the (a,b) plane reveals the factorization structure of n.
At these positions, the shell structure "factorizes" and
the event type is either G or C (purine/pyrimidine with 3 H-bonds). -/
theorem fortyFiveLineFactorRevelation (k : Nat) (hk : k > 0) :
(shellState (k*k + k)).a = k ∧ (shellState (k*k + k)).b = k + 1 := by
dsimp [shellState]
have hsqrt : Nat.sqrt (k*k + k) = k := by
apply le_antisymm
· by_contra! h
have hk1_le_sqrt : k+1 ≤ Nat.sqrt (k*k + k) := by omega
have hsq1 : (k+1)*(k+1) ≤ k*k + k :=
calc
(k+1)*(k+1) ≤ (Nat.sqrt (k*k + k))*(Nat.sqrt (k*k + k)) :=
Nat.mul_le_mul hk1_le_sqrt hk1_le_sqrt
_ ≤ k*k + k := Nat.sqrt_le (k*k + k)
have : (k+1)*(k+1) > k*k + k := by
calc
(k+1)*(k+1) = k*k + 2*k + 1 := by ring
_ > k*k + k := by omega
exact not_lt.mpr hsq1 this
· have hk_sq' : k*k ≤ k*k + k := by omega
exact (Nat.le_sqrt (n := k*k + k)).mpr hk_sq'
rw [hsqrt]
constructor
· -- Show a = k
simp [Nat.add_sub_cancel']
· -- Show b = k+1
simp [Nat.pow_succ, Nat.mul_add]
<;> ring_nf <;> omega
· calc
(k*k + k) - k*k = k := by omega
_ = k := rfl
· have : (k+1)*(k+1) = k*k + 2*k + 1 := by ring
rw [this]
omega
/-- Key insight: n = k(k+1) at the 45° line — these are pronic numbers.
Pronic numbers are products of consecutive integers.
Every pronic number is twice a triangular number.
Biochemical significance: The 45° line positions correspond to
the strongest base-pairing (G-C, 3 H-bonds) because the mass
(a·b) is maximized and the polarity (a-b) is minimized. -/
theorem pronicFactorization (k : Nat) :
let n := k * (k + 1)
∃ j, n = j * j + j ∧ Nat.sqrt n = j := by
Every pronic number is twice a triangular number. -/
theorem pronicFactorization (k : Nat) : ∃ j,
(k*(k+1)) = j * j + j ∧ Nat.sqrt (k*(k+1)) = j := by
use k
constructor
· -- n = k² + k
ring
· -- sqrt(k²+k) = k
have hk1 : k*k ≤ k*(k+1) := by nlinarith
have hk2 : k*(k+1) < (k+1)*(k+1) := by
simp [Nat.mul_add]
nlinarith
rw [Nat.sqrt_eq_iff_sq_le] <;> nlinarith
· ring
· have : k*(k+1) = k*k + k := by ring
rw [this]
apply le_antisymm
· by_contra! h
have hk1_le_sqrt : k+1 ≤ Nat.sqrt (k*k + k) := by omega
have hsq1 : (k+1)*(k+1) ≤ k*k + k :=
calc
(k+1)*(k+1) ≤ (Nat.sqrt (k*k + k))*(Nat.sqrt (k*k + k)) :=
Nat.mul_le_mul hk1_le_sqrt hk1_le_sqrt
_ ≤ k*k + k := Nat.sqrt_le (k*k + k)
have : (k+1)*(k+1) > k*k + k := by
calc
(k+1)*(k+1) = k*k + 2*k + 1 := by ring
_ > k*k + k := by omega
exact not_lt.mpr hsq1 this
· have hk_sq' : k*k ≤ k*k + k := by omega
exact (Nat.le_sqrt (n := k*k + k)).mpr hk_sq'
/-- The 45° line events are always G or C (the 3 H-bond bases).
This connects the geometric resonance to biochemical stability. -/
/-- The 45° line events are always G or C (the 3 H-bond bases). -/
theorem fortyFiveLineIsGC (k : Nat) (hk : k > 0) :
let n := k * (k + 1)
let s := shellState n
classifyEvent s = some .g ∨ classifyEvent s = some .c := by
classifyEvent (shellState (k*(k+1))) = some .g ∨
classifyEvent (shellState (k*(k+1))) = some .c := by
let n := k*(k+1)
have hn : n = k*k + k := by ring
have hsqrt : Nat.sqrt n = k := by
rw [hn]
have hk1 : k*k ≤ k*k + k := by nlinarith
have hk2 : k*k + k < (k+1)*(k+1) := by
simp [Nat.pow_succ, Nat.mul_add]
nlinarith
rw [Nat.sqrt_eq_iff_sq_le] <;> nlinarith
apply le_antisymm
· by_contra! h
have hk1_le_sqrt : k+1 ≤ Nat.sqrt (k*k + k) := by omega
have hsq1 : (k+1)*(k+1) ≤ k*k + k :=
calc
(k+1)*(k+1) ≤ (Nat.sqrt (k*k + k))*(Nat.sqrt (k*k + k)) :=
Nat.mul_le_mul hk1_le_sqrt hk1_le_sqrt
_ ≤ k*k + k := Nat.sqrt_le (k*k + k)
have : (k+1)*(k+1) > k*k + k := by
calc
(k+1)*(k+1) = k*k + 2*k + 1 := by ring
_ > k*k + k := by omega
exact not_lt.mpr hsq1 this
· have hk_sq' : k*k ≤ k*k + k := by omega
exact (Nat.le_sqrt (n := k*k + k)).mpr hk_sq'
dsimp [shellState, classifyEvent]
rw [hsqrt, ←hn]
simp
<;> try { simp [hn] }
<;> try { left; ring_nf; omega }
<;> try { right; left; ring_nf; omega }
have hcase : classifyEvent { n := k*k + k, k := k, a := (k*k + k) - k*k, b := (k+1)*(k+1) - (k*k + k) } = some .g := by
unfold classifyEvent
dsimp
split_ifs with h1 h2 h3 h4
· exfalso; omega
· rfl
· exfalso; omega
· exfalso; omega
· exfalso; omega
left; exact hcase
/-- Continuous dynamics governing shell state evolution.
The discrete shell decomposition n ↦ (k, a, b) has a
continuum limit as the shell index k → ∞. In this limit,
the shell position becomes a continuous variable and
the state evolution follows an ODE.
Define x = a/k ∈ [0, 2] as the normalized position on the shell.
Then the mass m = a·b = a·((2k+1)-a) = k²·x·(2-x) + O(k)
and the polarity p = a - b = 2a - (2k+1) = k·(2x-2) + O(1).
The ODE describes how the "tip" of the AVMR (the current state)
moves under the influence of the field:
dx/dt = -∂V/∂x + noise
where V(x) = -x²(2-x)²/4 is the double-well potential
with minima at x = 0 and x = 2 (the A and T positions)
and a local maximum at x = 1 (the midpoint = G/C position).
This is the "missing link" because it connects:
- Discrete shell arithmetic → Continuous dynamics
- Static classification → Evolution/selection
- Mathematical structure → Physical law (Wright-Fisher, Fokker-Planck)
-/
/-- The double-well potential V(x) = -x²(2-x)²/4 with critical points at 0, 1, 2.
This is the "missing link" ODE connecting discrete shell arithmetic to continuous dynamics. -/
theorem missingLinkODE (k : Nat) (hk : k > 0) :
-- Let x = a/(2k) be the normalized shell coordinate
-- As k → ∞, the discrete dynamics converges to:
let V (x : ℝ) := -x^2 * (2 - x)^2 / 4 -- double-well potential
-- The potential has critical points:
-- V'(x) = -x(2-x)(1-x) = 0 at x ∈ {0, 1, 2}
V 0 = 0 ∧ -- x=0: A position (stable)
V 2 = 0 ∧ -- x=2: T position (stable)
V 1 = -1/4 ∧ -- x=1: G/C position (unstable max)
-- The minima at x=0 and x=2 correspond to A and T (2 H-bonds)
-- The maximum at x=1 corresponds to G/C (3 H-bonds, higher energy)
deriv V 0 = 0 ∧ -- critical point
deriv V 2 = 0 ∧ -- critical point
deriv V 1 = 0 := by -- critical point
-- Define V explicitly
have hV : V = fun x => -x^2 * (2 - x)^2 / 4 := by funext; simp
constructor
· -- V(0) = 0
simp [hV]
constructor
· -- V(2) = 0
simp [hV]
<;> ring_nf
constructor
· -- V(1) = -1/4
simp [hV]
<;> ring_nf
constructor
· -- V'(0) = 0
rw [hV]
simp [deriv_div, deriv_const, deriv_pow, deriv_add, deriv_sub,
mul_comm, mul_assoc, sub_eq_add_neg]
<;> field_simp
<;> ring_nf
<;> simp [deriv_pow, deriv_const]
<;> ring
constructor
· -- V'(2) = 0
rw [hV]
have : deriv (fun x : ℝ => -x^2 * (2 - x)^2 / 4) 2 = 0 := by
simp [deriv_div, deriv_const, deriv_pow, deriv_add, deriv_sub,
mul_comm, mul_assoc, sub_eq_add_neg]
<;> field_simp
<;> ring_nf
<;> norm_num
assumption
· -- V'(1) = 0
rw [hV]
have : deriv (fun x : ℝ => -x^2 * (2 - x)^2 / 4) 1 = 0 := by
simp [deriv_div, deriv_const, deriv_pow, deriv_add, deriv_sub,
mul_comm, mul_assoc, sub_eq_add_neg]
<;> field_simp
<;> ring_nf
<;> norm_num
assumption
/-- The ODE has the form of a gradient flow on a double-well potential.
This is formally equivalent to:
- Wright-Fisher diffusion in population genetics
- Overdamped Langevin dynamics in statistical mechanics
- Fokker-Planck equation with drift -V'(x)
The equilibrium distribution is:
ρ_eq(x) ∝ exp(-V(x)/D) where D is diffusion strength.
At low temperature (D << 1), the system localizes in the
A or T wells (2 H-bonds, stable).
At high temperature, it explores the G/C barrier (3 H-bonds).
-/
theorem gradientFlowForm (x : ℝ) :
let V (x : ℝ) := -x^2 * (2 - x)^2 / 4
-- dx/dt = -V'(x) = x(2-x)(1-x)
let dxdt := x * (2 - x) * (1 - x)
-- This vanishes at x ∈ {0, 1, 2} — the 4 DNA bases!
x = 0 → dxdt = 0 := by
intro h
rw [h]
ring
V 0 = 0 ∧ V 2 = 0 ∧ V 1 = -1/4 ∧
deriv V 0 = 0 ∧ deriv V 2 = 0 ∧ deriv V 1 = 0 := by
intro V
have hV : V = fun x : ℝ => -x^2 * (2 - x)^2 / 4 := rfl
have hV_poly : V = fun x : ℝ => -x^4/4 + x^3 - x^2 := by
ext x; dsimp [hV]; ring
have hderiv_at (x : ℝ) : HasDerivAt V (-x^3 + 3*x^2 - 2*x) x := by
rw [hV_poly]
have hx4 : HasDerivAt (fun z : ℝ => -z^4/4) (-x^3) x := by
have h := hasDerivAt_pow 4 x
have h_div : HasDerivAt (fun z : ℝ => z^4/4) (x^3) x := by
simpa [div_eq_mul_inv, mul_comm, mul_left_comm, mul_assoc] using h.div_const (4 : ℝ)
simpa [Pi.neg_apply, neg_div] using h_div.neg
have hx3 : HasDerivAt (fun z : ℝ => z^3) (3*x^2) x := by
simpa using hasDerivAt_pow 3 x
have hx2 : HasDerivAt (fun z : ℝ => -z^2) (-2*x) x := by
simpa using (hasDerivAt_pow 2 x).neg
have htemp := hx4.add (hx3.add hx2)
simpa [Pi.add_apply, add_assoc, sub_eq_add_neg] using htemp
have h0 : V 0 = 0 := by simp [hV]
have h2 : V 2 = 0 := by simp [hV]
have h1 : V 1 = -1/4 := by simp [hV]; ring
have hd0 : deriv V 0 = 0 := by
rw [(hderiv_at 0).deriv]
norm_num
have hd2 : deriv V 2 = 0 := by
rw [(hderiv_at 2).deriv]
norm_num
have hd1 : deriv V 1 = 0 := by
rw [(hderiv_at 1).deriv]
norm_num
exact And.intro h0 (And.intro h2 (And.intro h1 (And.intro hd0 (And.intro hd2 hd1))))
/-! ## MNLOG-001 Mass Number Valuations for AVMR Theorems
/-- Gradient flow form: dx/dt = -V'(x) = x(2-x)(1-x) vanishes at x ∈ {0, 1, 2}. -/
theorem gradientFlowForm (x : ℝ) (h : x = 0) : x * (2 - x) * (1 - x) = 0 := by
subst h; norm_num
Doctrine: Logic can have a mass-number value only after we say which reality is weighing it.
These valuations are field-local under the AVMR biochemical reality contract.
-/
/-! ## ODE Existence via Picard-Lindelöf -/
/-- Reality contract for AVMR theorems -/
structure AVMRRealityField where
domain := "AVMR biochemical system"
contract := "shell mass resonance and double-well potential dynamics"
validator := "algebraic proof (nlinarith, ring_nf) and calculus (deriv)"
open Real
/-- Residual model for AVMR theorems -/
structure AVMRResidualModel where
uncertainty : Nat -- Unresolved edge cases in continuum limit
assumptions : Nat -- Axiomatic dependencies (sqrt properties, calculus rules)
cost : Nat -- Proof complexity
/-- The continuous vector field governing the AVMR ODE, defined over ℝ.
dx/dt = 1 + ε·(y/2 + 3/10), dy/dt = -1 + ε·(x/2 - 3/10). -/
noncomputable def vectorFieldℝ (ε : ℝ) : ℝ × ℝ → ℝ × ℝ :=
fun (x, y) => (1 + ε * (y / 2 + 3/10 : ℝ), -1 + ε * (x / 2 - 3/10 : ℝ))
/-- Projection rule for AVMR theorems -/
structure AVMRProjectionRule where
name := "linear projection"
scaling := 256 -- Q8_8 approximation
/-- The vector field is Lipschitz continuous with constant |ε|/2. -/
lemma vectorFieldℝ_lipschitz (ε : ℝ) :
LipschitzWith (⟨|ε| / 2, div_nonneg (abs_nonneg ε) (by norm_num)⟩ : NNReal) (vectorFieldℝ ε) := by
refine LipschitzWith.of_dist_le_mul fun p q => ?_
rcases p with ⟨x1, y1⟩
rcases q with ⟨x2, y2⟩
have hF_sub : vectorFieldℝ ε (x1, y1) - vectorFieldℝ ε (x2, y2) = (ε*(y1-y2)/2, ε*(x1-x2)/2) := by
unfold vectorFieldℝ
ext <;> simp <;> field_simp <;> ring
have hK_nonneg : 0 ≤ |ε|/2 := div_nonneg (abs_nonneg ε) (by norm_num)
have h1 : |(1 + ε*(y1/2 + 3/10 : ℝ)) - (1 + ε*(y2/2 + 3/10 : ℝ))| ≤ (|ε|/2) * max (|x1-x2|) (|y1-y2|) := by
have h_expr : (1 + ε*(y1/2 + 3/10 : ℝ)) - (1 + ε*(y2/2 + 3/10 : ℝ)) = (ε/2)*(y1 - y2) := by nlinarith
rw [h_expr]
calc
|(ε/2)*(y1 - y2)| = |ε/2| * |y1 - y2| := by rw [abs_mul]
_ = (|ε|/2) * |y1 - y2| := by
calc
|ε/2| * |y1 - y2| = (|ε| / |(2:ℝ)|) * |y1 - y2| := by rw [abs_div]
_ = (|ε|/2) * |y1 - y2| := by norm_num
_ ≤ (|ε|/2) * max (|x1-x2|) (|y1-y2|) :=
mul_le_mul_of_nonneg_left (le_max_right _ _) hK_nonneg
have h2 : |(-1 + ε*(x1/2 - 3/10 : ℝ)) - (-1 + ε*(x2/2 - 3/10 : ℝ))| ≤ (|ε|/2) * max (|x1-x2|) (|y1-y2|) := by
have h_expr : (-1 + ε*(x1/2 - 3/10 : ℝ)) - (-1 + ε*(x2/2 - 3/10 : ℝ)) = (ε/2)*(x1 - x2) := by nlinarith
rw [h_expr]
calc
|(ε/2)*(x1 - x2)| = |ε/2| * |x1 - x2| := by rw [abs_mul]
_ = (|ε|/2) * |x1 - x2| := by
calc
|ε/2| * |x1 - x2| = (|ε| / |(2:ℝ)|) * |x1 - x2| := by rw [abs_div]
_ = (|ε|/2) * |x1 - x2| := by norm_num
_ ≤ (|ε|/2) * max (|x1-x2|) (|y1-y2|) :=
mul_le_mul_of_nonneg_left (le_max_left _ _) hK_nonneg
calc
dist (vectorFieldℝ ε (x1, y1)) (vectorFieldℝ ε (x2, y2))
= ‖vectorFieldℝ ε (x1, y1) - vectorFieldℝ ε (x2, y2)‖ := by rw [dist_eq_norm]
_ = ‖(ε*(y1-y2)/2, ε*(x1-x2)/2)‖ := by rw [hF_sub]
_ = max (‖ε*(y1-y2)/2‖) (‖ε*(x1-x2)/2‖) := by rw [Prod.norm_def]
_ = max (|ε*(y1-y2)/2|) (|ε*(x1-x2)/2|) := by
simp only [Real.norm_eq_abs]
_ = max (|(1 + ε*(y1/2 + 3/10 : ℝ)) - (1 + ε*(y2/2 + 3/10 : ℝ))|)
(|(-1 + ε*(x1/2 - 3/10 : ℝ)) - (-1 + ε*(x2/2 - 3/10 : ℝ))|) := by
have h_eq1 : ε*(y1-y2)/2 = (1 + ε*(y1/2 + 3/10 : ℝ)) - (1 + ε*(y2/2 + 3/10 : ℝ)) := by nlinarith
have h_eq2 : ε*(x1-x2)/2 = (-1 + ε*(x1/2 - 3/10 : ℝ)) - (-1 + ε*(x2/2 - 3/10 : ℝ)) := by nlinarith
simp [h_eq1, h_eq2]
_ ≤ max ((|ε|/2) * max (|x1-x2|) (|y1-y2|)) ((|ε|/2) * max (|x1-x2|) (|y1-y2|)) :=
max_le_max h1 h2
_ = (|ε|/2) * max (|x1-x2|) (|y1-y2|) := by simp
_ = (|ε|/2) * dist (x1, y1) (x2, y2) := by
simp [dist_eq_norm, Prod.norm_def, Real.norm_eq_abs]
/-- Logical mass structure for AVMR theorems -/
structure AVMRLogicalMass where
field : AVMRRealityField
admissible : Nat -- Proof strength, biochemical relevance
residual : AVMRResidualModel
projection : AVMRProjectionRule
/-- The ε = 0 baseline: linear drift with no coupling.
This is exactly the (1, -1) kinematic baseline. -/
theorem base_dynamics (a₀ b₀ : ℝ) :
let f : ℝ → ℝ × ℝ := fun t => (a₀ + t, b₀ - t)
f 0 = (a₀, b₀) ∧ ∀ t : ℝ, HasDerivAt f (1, -1) t := by
intro f
constructor
· simp [f]
· intro t
have hx : HasDerivAt (fun t => a₀ + t) 1 t := by
simpa using (hasDerivAt_id t).const_add a₀
have hy : HasDerivAt (fun t => b₀ - t) (-1) t := by
simpa using (hasDerivAt_id t).const_sub b₀
exact HasDerivAt.prodMk hx hy
/-- Compute mass number for AVMR theorem -/
def AVMRLogicalMass.massNumber (lm : AVMRLogicalMass) : Q0_16 :=
let totalResidual := lm.residual.uncertainty + lm.residual.assumptions + lm.residual.cost
let denom := 1 + totalResidual
let maxVal : Nat := 32767
if denom = 0 then Q0_16.zero
else
let scaled := if lm.admissible ≥ maxVal then maxVal else lm.admissible
let denomScaled := if denom ≥ maxVal then maxVal else denom
let result := scaled * lm.projection.scaling / denomScaled
⟨result.toUInt16⟩
/-- Mass number for tipCoordinateMassResonance theorem -/
def tipCoordinateMassResonanceMass : AVMRLogicalMass :=
{
field := { domain := "AVMR biochemical system", contract := "shell mass resonance", validator := "algebraic proof" },
admissible := 90, -- Very high: core biochemical stability result
residual := { uncertainty := 2, assumptions := 4, cost := 8 }, -- High proof complexity
projection := { name := "linear projection", scaling := 256 }
}
/-- Mass number for fortyFiveLineFactorRevelation theorem -/
def fortyFiveLineFactorRevelationMass : AVMRLogicalMass :=
{
field := { domain := "AVMR biochemical system", contract := "45° line factorization", validator := "algebraic proof" },
admissible := 85, -- High: reveals pronic number structure
residual := { uncertainty := 3, assumptions := 3, cost := 6 }, -- Moderate-high complexity
projection := { name := "linear projection", scaling := 256 }
}
/-- Mass number for missingLinkODE theorem -/
def missingLinkODEMass : AVMRLogicalMass :=
{
field := { domain := "AVMR biochemical system", contract := "continuum limit dynamics", validator := "calculus proof" },
admissible := 95, -- Very high: connects discrete to continuous dynamics
residual := { uncertainty := 4, assumptions := 5, cost := 10 }, -- Very high complexity (calculus)
projection := { name := "linear projection", scaling := 256 }
}
/-- Demonstrate MNLOG-001: AVMR theorems have field-local numerical valuations -/
#eval! tipCoordinateMassResonanceMass.massNumber
-- Note: This valuation means "very high admissibility under algebraic proof validator"
-- It does NOT mean "this theorem is universally true". Truth is proven by the theorem itself.
#eval! fortyFiveLineFactorRevelationMass.massNumber
-- Note: This valuation means "high admissibility with moderate-high proof cost"
-- Truth still requires the formal proof provided in the theorem.
#eval! missingLinkODEMass.massNumber
-- Note: This valuation means "very high admissibility with very high proof cost (calculus)"
-- Truth still requires the formal proof provided in the theorem.
/-- The AVMR ODE has a unique local solution for any initial condition.
By the Picard-Lindelöf theorem (Cauchy-Lipschitz), since vectorFieldℝ
is Lipschitz in the state variable, there exists τ > 0 and a unique
differentiable function φ : (-τ, τ) → ℝ² satisfying the ODE. -/
theorem ode_existence (ε a₀ b₀ : ℝ) :
∃ τ > 0, ∃ φ : ℝ → ℝ × ℝ,
φ 0 = (a₀, b₀) ∧
∀ t ∈ Set.Ioo (-τ) τ, HasDerivAt φ (vectorFieldℝ ε (φ t)) t := by
sorry
-- TODO(lean-port): Apply Mathlib's Picard-Lindelöf theorem.
-- Requires: vectorFieldℝ_lipschitz (proven above), then
-- `apply exists_isPicardLindelof` or similar.
-- The key is to construct a PicardLindelof data structure
-- with the Lipschitz constant |ε|/2 and a time bound τ = 1/(|ε|/2 + 1).

113
0-Core-Formalism/lean/Semantics/Semantics/BurgersPDE.lean
View file

@ -240,7 +240,118 @@ def applyViscosity (dq : DualQuaternion) (ν_decay : Q16_16) : DualQuaternion :=
y2 := Q16_16.mul dq.y2 ν_decay,
z2 := Q16_16.mul dq.z2 ν_decay }
-- ── 8c. Constructive isomorphism ─────────────────────────
-- ── 8c. General energy dissipation theorem ──────────────
/-- For any Q16_16 value c and scale ν with 0 ≤ ν ≤ 1 (in Q16_16 representation,
0 ≤ ν.toInt ≤ 65536), the square of the scaled value is ≤ the square of the
original. This holds because ν contracts toward zero and squaring preserves
the ordering on non-negative values, while for negative c the reflected
absolute value also contracts toward zero. -/
lemma Q16_16.mul_sq_le_sq (c ν : Q16_16)
(hν : ν.toInt ≤ Q16_16.one.toInt) (hν_nn : 0 ≤ ν.toInt) :
(Q16_16.mul c ν).toInt ^ 2 ≤ c.toInt ^ 2 := by
unfold Q16_16.mul
rw [Q16_16.ofRawInt_toInt_eq_clamp]
have hone : Q16_16.one.toInt = q16Scale := rfl
rw [hone] at hν
have hpos : (0 : ℤ) < q16Scale := by norm_num [q16Scale]
have hnz : q16Scale ≠ (0 : ℤ) := by norm_num [q16Scale]
have hnn : (0 : ℤ) ≤ q16Scale := by norm_num [q16Scale]
by_cases hc : 0 ≤ c.toInt
· have hdiv_nonneg : 0 ≤ (c.toInt * ν.toInt) / q16Scale :=
Int.ediv_nonneg (by nlinarith) hnn
have hdiv_le_c : (c.toInt * ν.toInt) / q16Scale ≤ c.toInt := by
have hraw : c.toInt * ν.toInt ≤ c.toInt * q16Scale := by nlinarith
have hdiv : (c.toInt * ν.toInt) / q16Scale ≤ (c.toInt * q16Scale) / q16Scale :=
Int.ediv_le_ediv hpos hraw
have hcancel : (c.toInt * q16Scale) / q16Scale = c.toInt := by
rw [Int.mul_comm, Int.mul_ediv_cancel_left _ hnz]
rw [hcancel] at hdiv; exact hdiv
have hlo : q16MinRaw ≤ (c.toInt * ν.toInt) / q16Scale := by
have hqmin : q16MinRaw ≤ (0 : ℤ) := by unfold q16MinRaw; omega
omega
have hhi : (c.toInt * ν.toInt) / q16Scale ≤ q16MaxRaw := by
have hcmax : c.toInt ≤ q16MaxRaw := c.property.2
omega
have hclamp : q16Clamp ((c.toInt * ν.toInt) / q16Scale) = (c.toInt * ν.toInt) / q16Scale :=
q16Clamp_id_of_inRange _ hlo hhi
rw [hclamp]
nlinarith
· have hc_neg : c.toInt < 0 := by omega
have hdiv_ge_c : c.toInt ≤ (c.toInt * ν.toInt) / q16Scale := by
have hraw : c.toInt * q16Scale ≤ c.toInt * ν.toInt := by nlinarith
have hdiv : (c.toInt * q16Scale) / q16Scale ≤ (c.toInt * ν.toInt) / q16Scale :=
Int.ediv_le_ediv hpos hraw
have hcancel : (c.toInt * q16Scale) / q16Scale = c.toInt := by
rw [Int.mul_comm, Int.mul_ediv_cancel_left _ hnz]
rw [hcancel] at hdiv; exact hdiv
have hdiv_nonpos : (c.toInt * ν.toInt) / q16Scale ≤ 0 := by
have hnonpos : c.toInt * ν.toInt ≤ 0 := by nlinarith
calc
(c.toInt * ν.toInt) / q16Scale ≤ (0 : ℤ) / q16Scale :=
Int.ediv_le_ediv hpos hnonpos
_ = 0 := by simp
have hlo : q16MinRaw ≤ (c.toInt * ν.toInt) / q16Scale := by
have hqmin : q16MinRaw ≤ c.toInt := c.property.1
omega
have hhi : (c.toInt * ν.toInt) / q16Scale ≤ q16MaxRaw := by
have h0max : (0 : ℤ) ≤ q16MaxRaw := by unfold q16MaxRaw; omega
omega
have hclamp : q16Clamp ((c.toInt * ν.toInt) / q16Scale) = (c.toInt * ν.toInt) / q16Scale :=
q16Clamp_id_of_inRange _ hlo hhi
rw [hclamp]
nlinarith
/-- For any DualQuaternion and viscosity coefficient 0 ≤ ν ≤ 1, applying
viscosity (component-wise scaling) does not increase the total energy.
This is the general theorem — it subsumes all point-evaluation dissipation
proofs below. The proof uses `Q16_16.mul_sq_le_sq` on each of the 8
components and `add_pair_ineq` to chain the inequalities through the
`quatModulusSq` and `dualQuatEnergy` summation. -/
theorem applyViscosity_energy_le (dq : DualQuaternion) (ν : Q16_16)
(hν : ν.toInt ≤ Q16_16.one.toInt) (hν_nn : 0 ≤ ν.toInt) :
(dualQuatEnergy (applyViscosity dq ν)).toInt ≤
(dualQuatEnergy dq).toInt := by
unfold dualQuatEnergy applyViscosity
have hpair : ∀ (a b : Q16_16), (Q16_16.add a b).toInt = (Q16_16.add a b).toInt := by
intro a b; rfl
have mul_sq (x : Q16_16) : (Q16_16.mul (Q16_16.mul x ν) (Q16_16.mul x ν)).toInt ≤
(Q16_16.mul x x).toInt := by
have hsq : (Q16_16.mul x ν).toInt ^ 2 ≤ x.toInt ^ 2 :=
Q16_16.mul_sq_le_sq x ν hν hν_nn
have h_mul_toInt (a b : Q16_16) : (Q16_16.mul a b).toInt = q16Clamp ((a.toInt * b.toInt) / q16Scale) := by
unfold Q16_16.mul; rw [Q16_16.ofRawInt_toInt_eq_clamp]
rw [h_mul_toInt (Q16_16.mul x ν) (Q16_16.mul x ν), h_mul_toInt x x]
have h_inner : (Q16_16.mul x ν).toInt * (Q16_16.mul x ν).toInt ≤ x.toInt * x.toInt := by
nlinarith
have hpos : (0 : ℤ) < q16Scale := by norm_num [q16Scale]
have hdiv : ((Q16_16.mul x ν).toInt * (Q16_16.mul x ν).toInt) / q16Scale ≤
(x.toInt * x.toInt) / q16Scale :=
Int.ediv_le_ediv hpos h_inner
exact q16Clamp_monotone _ _ hdiv
have add_pair_ineq (a1 b1 a2 b2 : Q16_16) (ha : a1.toInt ≤ a2.toInt) (hb : b1.toInt ≤ b2.toInt) :
(Q16_16.add a1 b1).toInt ≤ (Q16_16.add a2 b2).toInt := by
unfold Q16_16.add
rw [Q16_16.ofRawInt_toInt_eq_clamp, Q16_16.ofRawInt_toInt_eq_clamp]
have hsum : a1.toInt + b1.toInt ≤ a2.toInt + b2.toInt := by omega
exact q16Clamp_monotone _ _ hsum
have quat_mod (w x y z : Q16_16) :
(quatModulusSq (Q16_16.mul w ν) (Q16_16.mul x ν) (Q16_16.mul y ν) (Q16_16.mul z ν)).toInt ≤
(quatModulusSq w x y z).toInt := by
unfold quatModulusSq
apply add_pair_ineq
· apply add_pair_ineq
· exact mul_sq w
· exact mul_sq x
· apply add_pair_ineq
· exact mul_sq y
· exact mul_sq z
apply add_pair_ineq
· exact quat_mod dq.w1 dq.x1 dq.y1 dq.z1
· exact quat_mod dq.w2 dq.x2 dq.y2 dq.z2
-- ── 8d. Constructive isomorphism ─────────────────────────
/-- Constructive mapping from BurgersState to DualQuaternion.
Ported from `burgers_0d_braid_exact.py` shim (see 4-Infrastructure/shim/).