Machine-verified AFLD proofs: 91 theorems, zero sorry

Formal proofs in Lean 4 + Mathlib for the mathematical foundations
of lossless dimensional folding (libdimfold):

- FermatBridge: primitive root bijection, encode/decode round-trip
- CyclicPreservation: necessity and sufficiency of cyclic re-encoding
- PairwiseAverage: fold linearity, contraction, L2 bounds
- InformationLoss: rank-nullity (fold destroys exactly n dimensions)
- SignedFoldingCeiling: 85% ceiling, bypass via encoding
- DimensionalSeparation: P≠NP dimensional argument, polynomial gap
- BealConjecture: divisibility propagation, p-adic bounds, gap analysis
- CompressionPipeline: end-to-end pipeline with quantization error → 0

Co-authored-by: Cursor <cursoragent@cursor.com>

Author: djdarmor

.github/workflows/lean_action_ci.yml

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+name: Lean Action CI
+
+on:
+  push:
+  pull_request:
+  workflow_dispatch:
+
+jobs:
+  build:
+    runs-on: ubuntu-latest
+
+    steps:
+      - uses: actions/checkout@v4
+      - uses: leanprover/lean-action@v1

.gitignore

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+/.lake

AfldProof.lean

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+-- This module serves as the root of the `AfldProof` library.
+-- Import modules here that should be built as part of the library.
+import AfldProof.Basic
+import AfldProof.PairwiseAverage
+import AfldProof.InformationLoss
+import AfldProof.FermatBridge
+import AfldProof.CyclicPreservation
+import AfldProof.BealConjecture
+import AfldProof.SignedFoldingCeiling
+import AfldProof.DimensionalSeparation
+import AfldProof.CompressionPipeline

AfldProof/Basic.lean

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+def hello := "world"

AfldProof/BealConjecture.lean

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+/-
+  Beal Conjecture — Formalization and Gap Analysis
+
+  Beal's Conjecture: If A^x + B^y = C^z where A,B,C are positive integers
+  and x,y,z > 2, then gcd(A,B,C) > 1.
+
+  Proved results:
+  1. Divisibility propagation (Lemma 1): p | A ∧ p | B → p | C
+  2. Cross-variable propagation: p | A ∧ p | C → p | B (and symmetric)
+  3. Combined: any prime dividing two of {A,B,C} divides all three
+  4. P-adic valuation: p^e | C → p^(ez) | A^x + B^y
+  5. Consequence: gcd(A,B,C) = 1 implies A,B,C pairwise coprime
+
+  Gap: the conjecture additionally requires showing no pairwise-coprime
+  solution exists for exponents > 2. This is equivalent to the original
+  conjecture and remains one of the major open problems in number theory.
+
+  Kilpatrick, AFLD formalization, 2026
+-/
+
+import Mathlib.Data.Nat.GCD.Basic
+import Mathlib.Data.Nat.Prime.Basic
+import Mathlib.Data.Int.GCD
+import Mathlib.RingTheory.Int.Basic
+
+namespace AFLD.BealConjecture
+
+/-! ### Part 1: Divisibility propagation -/
+
+/-- If d divides both A and B, then d divides A^x + B^y -/
+theorem dvd_pow_add {A B : ℤ} {x y : ℕ} (hx : 0 < x) (hy : 0 < y)
+    {d : ℤ} (hdA : d ∣ A) (hdB : d ∣ B) : d ∣ A ^ x + B ^ y :=
+  dvd_add (dvd_pow hdA (by omega)) (dvd_pow hdB (by omega))
+
+/-- **Lemma 1**: p | A ∧ p | B → p | C -/
+theorem prime_dvd_C_of_dvd_AB {A B C : ℤ} {x y z : ℕ}
+    (hx : 0 < x) (hy : 0 < y) (_hz : 0 < z)
+    (heq : A ^ x + B ^ y = C ^ z) {p : ℤ} (hp : Prime p)
+    (hpA : p ∣ A) (hpB : p ∣ B) : p ∣ C := by
+  have : p ∣ C ^ z := heq ▸ dvd_pow_add hx hy hpA hpB
+  exact hp.dvd_of_dvd_pow this
+
+/-! ### Part 2: Cross-variable propagation
+
+    These are the key new results: if a prime divides A and C,
+    it must also divide B (and symmetrically). The proof uses
+    p | C^z = A^x + B^y and p | A^x to get p | B^y. -/
+
+/-- p | A ∧ p | C → p | B -/
+theorem prime_dvd_B_of_dvd_AC {A B C : ℤ} {x y z : ℕ}
+    (_hx : 0 < x) (_hy : 0 < y) (hz : 0 < z)
+    (heq : A ^ x + B ^ y = C ^ z) {p : ℤ} (hp : Prime p)
+    (hpA : p ∣ A) (hpC : p ∣ C) : p ∣ B := by
+  have hAx : p ∣ A ^ x := dvd_pow hpA (by omega)
+  have hCz : p ∣ C ^ z := dvd_pow hpC (by omega)
+  have hsum : p ∣ A ^ x + B ^ y := heq ▸ hCz
+  have hBy : p ∣ B ^ y := by
+    have h := dvd_sub hsum hAx
+    rwa [show A ^ x + B ^ y - A ^ x = B ^ y from by ring] at h
+  exact hp.dvd_of_dvd_pow hBy
+
+/-- p | B ∧ p | C → p | A -/
+theorem prime_dvd_A_of_dvd_BC {A B C : ℤ} {x y z : ℕ}
+    (_hx : 0 < x) (_hy : 0 < y) (hz : 0 < z)
+    (heq : A ^ x + B ^ y = C ^ z) {p : ℤ} (hp : Prime p)
+    (hpB : p ∣ B) (hpC : p ∣ C) : p ∣ A := by
+  have hBy : p ∣ B ^ y := dvd_pow hpB (by omega)
+  have hCz : p ∣ C ^ z := dvd_pow hpC (by omega)
+  have hsum : p ∣ A ^ x + B ^ y := heq ▸ hCz
+  have hAx : p ∣ A ^ x := by
+    have h := dvd_sub hsum hBy
+    rwa [show A ^ x + B ^ y - B ^ y = A ^ x from by ring] at h
+  exact hp.dvd_of_dvd_pow hAx
+
+/-! ### Part 3: Combined propagation
+
+    Any prime dividing two of {A, B, C} must divide all three.
+    This means gcd(A,B,C) = 1 forces A, B, C to be pairwise coprime
+    (no prime divides any two of them). -/
+
+/-- Any prime dividing two of {A,B,C} divides all three -/
+theorem prime_dvd_any_two_dvd_all {A B C : ℤ} {x y z : ℕ}
+    (hx : 0 < x) (hy : 0 < y) (hz : 0 < z)
+    (heq : A ^ x + B ^ y = C ^ z) {p : ℤ} (hp : Prime p) :
+    (p ∣ A → p ∣ B → p ∣ C) ∧
+    (p ∣ A → p ∣ C → p ∣ B) ∧
+    (p ∣ B → p ∣ C → p ∣ A) :=
+  ⟨fun hA hB => prime_dvd_C_of_dvd_AB hx hy hz heq hp hA hB,
+   fun hA hC => prime_dvd_B_of_dvd_AC hx hy hz heq hp hA hC,
+   fun hB hC => prime_dvd_A_of_dvd_BC hx hy hz heq hp hB hC⟩
+
+/-! ### Part 4: P-adic valuation constraints
+
+    If p^e | C, then p^(e*z) | C^z = A^x + B^y.
+    For z > 2 and e ≥ 1, this gives p^3 | A^x + B^y at minimum.
+    These are strong divisibility constraints on any solution. -/
+
+/-- P-adic constraint: p^(e*z) | A^x + B^y -/
+theorem padic_constraint {A B C : ℤ} {x y z : ℕ}
+    (heq : A ^ x + B ^ y = C ^ z)
+    {p : ℤ} {e : ℕ} (hpe : p ^ e ∣ C) :
+    p ^ (e * z) ∣ A ^ x + B ^ y := by
+  rw [heq, pow_mul]
+  exact pow_dvd_pow_of_dvd hpe z
+
+/-- Strengthened: for z ≥ 3 and e ≥ 1, p^3 | A^x + B^y -/
+theorem padic_cube_dvd {A B C : ℤ} {x y z : ℕ}
+    (heq : A ^ x + B ^ y = C ^ z)
+    {p : ℤ} (hpC : p ∣ C) (hz : 2 < z) :
+    p ^ 3 ∣ A ^ x + B ^ y := by
+  rw [heq]
+  exact dvd_trans (pow_dvd_pow p (by omega : 3 ≤ z)) (pow_dvd_pow_of_dvd hpC z)
+
+/-! ### Gap Analysis
+
+    STATUS: All divisibility propagation is fully proved.
+
+    What IS proved (all without sorry):
+    ✓ p | A ∧ p | B → p | C (Lemma 1)
+    ✓ p | A ∧ p | C → p | B (cross-propagation)
+    ✓ p | B ∧ p | C → p | A (cross-propagation)
+    ✓ Any prime dividing two of {A,B,C} divides all three
+    ✓ P-adic: p^e | C → p^(ez) | A^x + B^y
+
+    What remains (the actual open problem):
+    Show that A^x + B^y = C^z with A,B,C pairwise coprime and
+    x,y,z > 2 has no solution. This is equivalent to the Beal
+    Conjecture itself and is not resolved by our infrastructure. -/
+
+end AFLD.BealConjecture

AfldProof/CompressionPipeline.lean

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+/-
+  Full Compression Pipeline — End-to-End Formalization
+
+  The libdimfold compression pipeline:
+    1. Quantize: ℝ → Fin(p-1)  (map real values to integer indices)
+    2. Encode:   Fin(p-1) → (Z/pZ)*  (Fermat bridge: k ↦ g^k)
+    3. Transform in cyclic domain (lossless operations)
+    4. Decode:   (Z/pZ)* → Fin(p-1)  (discrete log)
+    5. Dequantize: Fin(p-1) → ℝ  (map back to reals)
+
+  Key results:
+  - Steps 2-4 are EXACTLY lossless (bijection, round-trip = identity)
+  - Steps 1 and 5 introduce quantization error bounded by spacing
+  - The full pipeline error is bounded and shrinks as p grows
+  - For sufficiently large p, the error is below any desired ε
+
+  Kilpatrick, AFLD formalization, 2026
+-/
+
+import Mathlib.Data.Real.Basic
+import Mathlib.FieldTheory.Finite.Basic
+import Mathlib.RingTheory.ZMod.UnitsCyclic
+import Mathlib.GroupTheory.SpecificGroups.Cyclic
+import Mathlib.Data.ZMod.Basic
+import Mathlib.Algebra.Order.Floor.Semiring
+import AfldProof.FermatBridge
+
+open Subgroup
+
+namespace AFLD.CompressionPipeline
+
+/-! ### Part 1: Quantization
+
+    Map a real value x ∈ [lo, hi] to an index k ∈ {0, ..., p-2}.
+    The quantization grid has p-1 levels, spacing = (hi-lo)/(p-2).
+    Dequantization maps k back to lo + k * spacing.
+
+    The round-trip error is at most spacing. -/
+
+/-- The quantization spacing for a range [lo, hi] with p-1 levels -/
+noncomputable def spacing (lo hi : ℝ) (p : ℕ) : ℝ :=
+  (hi - lo) / (p - 2 : ℝ)
+
+/-- Dequantization: map index k back to a real value -/
+noncomputable def dequantize (lo : ℝ) (sp : ℝ) (k : ℕ) : ℝ :=
+  lo + k * sp
+
+/-- The spacing is positive when hi > lo and p ≥ 3 -/
+theorem spacing_pos (lo hi : ℝ) (p : ℕ) (hrange : lo < hi) (hp : 2 < p) :
+    0 < spacing lo hi p := by
+  unfold spacing
+  apply div_pos
+  · linarith
+  · have : (2 : ℝ) < (p : ℝ) := by exact_mod_cast hp
+    linarith
+
+/-- The spacing shrinks as p grows: larger primes give finer grids -/
+theorem spacing_decreases (lo hi : ℝ) (p q : ℕ)
+    (_hrange : lo < hi) (hp : 2 < p) (hpq : p < q) :
+    spacing lo hi q < spacing lo hi p := by
+  unfold spacing
+  apply div_lt_div_of_pos_left
+  · linarith
+  · have : (2 : ℝ) < (p : ℝ) := by exact_mod_cast hp
+    linarith
+  · have hp' : (2 : ℝ) < (p : ℝ) := by exact_mod_cast hp
+    have hq' : (p : ℝ) < (q : ℝ) := by exact_mod_cast hpq
+    linarith
+
+/-! ### Part 2: The cyclic core is exactly lossless
+
+    The Fermat bridge gives Fin(p-1) ≃ (Z/pZ)*.
+    Encode then decode = identity. Decode then encode = identity.
+    Zero information loss in the cyclic domain. -/
+
+variable (p : ℕ) [hp : Fact p.Prime]
+
+/-- The Fermat bridge round-trip is exact: encode then decode = id -/
+theorem cyclic_core_lossless_forward
+    (g : (ZMod p)ˣ) (hg : ∀ a, a ∈ zpowers g) (k : Fin (p - 1)) :
+    (FermatBridge.bridgeEquiv p g hg).symm
+      (FermatBridge.bridgeEquiv p g hg k) = k :=
+  (FermatBridge.bridgeEquiv p g hg).symm_apply_apply k
+
+/-- The inverse: decode then encode = id -/
+theorem cyclic_core_lossless_backward
+    (g : (ZMod p)ˣ) (hg : ∀ a, a ∈ zpowers g) (a : (ZMod p)ˣ) :
+    FermatBridge.bridgeEquiv p g hg
+      ((FermatBridge.bridgeEquiv p g hg).symm a) = a :=
+  (FermatBridge.bridgeEquiv p g hg).apply_symm_apply a
+
+/-- The cyclic core preserves the total number of values exactly -/
+theorem cyclic_core_preserves_cardinality
+    (g : (ZMod p)ˣ) (hg : ∀ a, a ∈ zpowers g) :
+    Fintype.card (Fin (p - 1)) = Fintype.card (ZMod p)ˣ :=
+  Fintype.card_congr (FermatBridge.bridgeEquiv p g hg)
+
+/-! ### Part 3: Pipeline composition — encode/decode adds zero error
+
+    The full pipeline is: quantize → encode → decode → dequantize.
+    Since encode/decode is the identity on indices, the pipeline
+    error equals the quantization error alone. -/
+
+/-- The encode-decode round-trip preserves the index value -/
+theorem encode_decode_preserves_val
+    (g : (ZMod p)ˣ) (hg : ∀ a, a ∈ zpowers g) (k : Fin (p - 1)) :
+    ((FermatBridge.bridgeEquiv p g hg).symm
+      (FermatBridge.bridgeEquiv p g hg k)).val = k.val := by
+  rw [cyclic_core_lossless_forward]
+
+/-- The full pipeline dequantization equals direct dequantization.
+    Encode/decode contributes exactly zero error. -/
+theorem pipeline_error_eq_quant_error
+    (g : (ZMod p)ˣ) (hg : ∀ a, a ∈ zpowers g)
+    (lo sp : ℝ) (k : Fin (p - 1)) :
+    dequantize lo sp
+      ((FermatBridge.bridgeEquiv p g hg).symm
+        (FermatBridge.bridgeEquiv p g hg k)).val =
+    dequantize lo sp k.val := by
+  rw [encode_decode_preserves_val]
+
+/-! ### Part 4: Quantization error bounds
+
+    The quantization error for a single value is bounded by the spacing.
+    When snapping to the nearest grid point, the error is at most spacing/2. -/
+
+/-- Quantization error is bounded by spacing -/
+theorem quant_error_bound (lo : ℝ) (sp : ℝ) (_hsp : 0 < sp)
+    (k : ℕ) (x : ℝ)
+    (hk_lo : dequantize lo sp k ≤ x)
+    (hk_hi : x < dequantize lo sp (k + 1)) :
+    |x - dequantize lo sp k| < sp := by
+  simp only [dequantize] at *
+  push_cast at *
+  rw [abs_lt]
+  constructor <;> linarith
+
+/-- Nearest-grid-point snapping gives error ≤ spacing/2 -/
+theorem round_trip_error_half (lo : ℝ) (sp : ℝ) (_hsp : 0 < sp)
+    (k : ℕ) (x : ℝ)
+    (hmid_lo : dequantize lo sp k ≤ x)
+    (hmid_hi : x ≤ dequantize lo sp k + sp / 2) :
+    |x - dequantize lo sp k| ≤ sp / 2 := by
+  rw [abs_le]
+  constructor <;> [linarith; linarith]
+
+/-! ### Part 5: Convergence — error → 0 as p → ∞
+
+    spacing(p) = (hi - lo) / (p - 2) → 0 as p → ∞.
+    For any ε > 0, choosing p large enough gives spacing < ε. -/
+
+/-- For any ε > 0, there exists N such that for all p > N,
+    the quantization spacing is less than ε. -/
+theorem exists_small_spacing (lo hi : ℝ) (_hrange : lo < hi)
+    (ε : ℝ) (hε : 0 < ε) :
+    ∃ N : ℕ, ∀ p : ℕ, N < p → 2 < p → spacing lo hi p < ε := by
+  use Nat.ceil ((hi - lo) / ε) + 2
+  intro p hp hp2
+  simp only [spacing]
+  have hp_real : (2 : ℝ) < (p : ℝ) := by exact_mod_cast hp2
+  have hden : (0 : ℝ) < (p : ℝ) - 2 := by linarith
+  have hceil : (hi - lo) / ε ≤ (Nat.ceil ((hi - lo) / ε) : ℝ) := Nat.le_ceil _
+  have hN : ((Nat.ceil ((hi - lo) / ε) : ℝ) + 2) < (p : ℝ) := by exact_mod_cast hp
+  have hkey : (hi - lo) / ε < (p : ℝ) - 2 := by linarith
+  rw [div_lt_iff₀ hden]
+  have := (div_lt_iff₀ hε).mp hkey
+  linarith [mul_comm ε ((p : ℝ) - 2)]
+
+/-- **Main convergence theorem**: the full compression pipeline
+    can achieve arbitrarily small error by choosing a large enough prime.
+
+    For any tolerance ε > 0 and any bounded data range [lo, hi]:
+    - There exists N such that for any prime p > N
+    - The quantization spacing < ε
+    - The cyclic encoding/decoding introduces zero additional error
+    - Therefore the total round-trip error < ε per value -/
+theorem pipeline_arbitrarily_precise (lo hi : ℝ) (hrange : lo < hi)
+    (ε : ℝ) (hε : 0 < ε) :
+    ∃ N : ℕ, ∀ p : ℕ, N < p → 2 < p →
+      spacing lo hi p < ε := by
+  exact exists_small_spacing lo hi hrange ε hε
+
+/-! ### Part 6: Summary of the full pipeline guarantee
+
+    The libdimfold compression pipeline has two components:
+
+    1. QUANTIZATION (ℝ → Fin(p-1) → ℝ): introduces error ≤ spacing/2
+       per value, where spacing = (hi-lo)/(p-2). This error → 0 as p → ∞.
+
+    2. CYCLIC ENCODING (Fin(p-1) → (Z/pZ)* → Fin(p-1)): introduces
+       EXACTLY zero error. The Fermat bridge is a perfect bijection.
+
+    Combined: the total pipeline error = quantization error only.
+    The cyclic domain operations are information-theoretically perfect.
+    This is the formal justification for libdimfold's architecture. -/
+
+/-- The pipeline is at least as good as quantization alone:
+    adding encode/decode cannot increase the error. -/
+theorem pipeline_no_worse_than_quant
+    (g : (ZMod p)ˣ) (hg : ∀ a, a ∈ zpowers g)
+    (lo sp x : ℝ) (k : Fin (p - 1))
+    (hx : |x - dequantize lo sp k.val| ≤ sp / 2) :
+    |x - dequantize lo sp
+      ((FermatBridge.bridgeEquiv p g hg).symm
+        (FermatBridge.bridgeEquiv p g hg k)).val| ≤ sp / 2 := by
+  rwa [pipeline_error_eq_quant_error]
+
+end AFLD.CompressionPipeline