Pattern (B) D-FUMT₈[α] type-theoretic derivation architecture

1. なぜ backlog に入っていたか

STEP 1323b backlog #15 で 提示 の Pattern (B) D-FUMT₈[α] = 8 値 identity 保持 + α parameter 追加 (Kolmogorov 構造関数 α 軸 統合) を Lean 4 type-theoretic derivation の 具体 architecture として 起草。 藤本さん 「Tier 4 candidate 順番に」 direct 指示 per candidate #1 起動。

2. Lean 4 type definition sketch

2.1 Base D-FUMT₈ (現行、 変更なし)

-- Rei stack 現行 D-FUMT₈ (src/axiom-os/seven-logic.ts + Lean 4 files)
inductive Dfumt8 : Type
  | true_    -- 1.0 (⊤)
  | false_   -- 0.0 (⊥)
  | both     -- 2.0 (⊤⊥)
  | neither  -- -1.0 (～)
  | infinity -- 3.0 (∞)
  | zero     -- 4.0 (〇)
  | flowing  -- 5.0 (～→)
  | selfLoop -- 6.0 (⟲)
  deriving DecidableEq, Repr, Fintype

2.2 拡張 D-FUMT₈[α] structure (Pattern B)

-- 判断留保 5 mode subset
def HoldModes : Finset Dfumt8 := {
  Dfumt8.neither, Dfumt8.flowing, Dfumt8.infinity, Dfumt8.both, Dfumt8.selfLoop
}

-- 決定的 3 値 subset
def DecisiveModes : Finset Dfumt8 := {
  Dfumt8.true_, Dfumt8.false_, Dfumt8.zero
}

-- 完全 8 分割 verify (axiom-free candidate)
theorem hold_decisive_partition :
    HoldModes ∪ DecisiveModes = Finset.univ (α := Dfumt8) ∧
    HoldModes ∩ DecisiveModes = ∅ := by
  constructor
  · ext x; fin_cases x <;> simp [HoldModes, DecisiveModes]
  · ext x; fin_cases x <;> simp [HoldModes, DecisiveModes]

-- D-FUMT₈[α] structure with semantic constraint
structure Dfumt8Alpha (α_critical : ℝ) where
  val : Dfumt8
  alpha : ℝ
  -- Semantic constraint (Pattern B、 STEP 1323b 6.5.3 継承):
  -- Hold modes: α ≤ α_critical (モデル化 放棄地点 内側)
  -- Decisive modes: α ≥ α_critical (モデル化 放棄地点 外側)
  hold_constraint : val ∈ HoldModes → alpha ≤ α_critical
  decisive_constraint : val ∈ DecisiveModes → alpha ≥ α_critical

-- Default instance (α = ∞ = decisive)
def Dfumt8Alpha.default (α_critical : ℝ) (v : Dfumt8) (h_dec : v ∈ DecisiveModes) : Dfumt8Alpha α_critical :=
  { val := v
    alpha := α_critical + 1  -- decisive 側
    hold_constraint := fun hold_hypothesis => by
      exfalso
      -- v ∈ HoldModes ∧ v ∈ DecisiveModes → 矛盾 (partition per)
      exact absurd (Finset.mem_inter.mpr ⟨hold_hypothesis, h_dec⟩)
        (by rw [(hold_decisive_partition).2]; exact Finset.not_mem_empty v)
    decisive_constraint := fun _ => le_refl _ }

2.3 backward compatibility bridge

-- 現行 Dfumt8 と 拡張 Dfumt8Alpha の 相互変換
def Dfumt8Alpha.toBase (x : Dfumt8Alpha α_critical) : Dfumt8 := x.val

-- default embedding: 現行 Dfumt8 値 → default α で Dfumt8Alpha
def Dfumt8.embedAlpha (v : Dfumt8) (α_critical : ℝ) : Dfumt8Alpha α_critical :=
  if h : v ∈ DecisiveModes then
    Dfumt8Alpha.default α_critical v h
  else
    -- v ∈ HoldModes、 default α = α_critical (境界値)
    { val := v
      alpha := α_critical
      hold_constraint := fun _ => le_refl _
      decisive_constraint := fun dec_hypothesis => by
        exfalso
        have : v ∈ HoldModes := by
          by_contra hn
          exact absurd (hold_decisive_partition.1 ▸ Finset.mem_univ v) (fun _ => hn h)
        exact absurd (Finset.mem_inter.mpr ⟨this, dec_hypothesis⟩)
          (by rw [(hold_decisive_partition).2]; exact Finset.not_mem_empty v) }

-- 恒等性 (round-trip preserves base value)
theorem embedAlpha_toBase (v : Dfumt8) (α_critical : ℝ) :
    (v.embedAlpha α_critical).toBase = v := by
  simp [Dfumt8.embedAlpha, Dfumt8Alpha.toBase]
  split <;> rfl

3. 25 load-bearing invention 影響評価

25 load-bearing invention (2026-04-20〜30) 全 D-FUMT₈=BOTH 統一で SEED_KERNEL 1539→1554 反映済。 Pattern (B) 拡張 の 影響:

4. axiom-free 実装 approach

4.1 axiom profile target

STEP 1215-1220 D-FUMT₈ Category arc と 同 axiom profile target:

• base theorem (partition + embedding roundtrip): [propext, Classical.choice, Quot.sound] Mathlib base のみ、 sorryAx / native_decide / user axiom 全 0
• 数値定理 (Vereshchagin-Vitányi 構造関数 refinement 対応 statements): 同 profile
• Structure constraint proof (hold_constraint + decisive_constraint): fin_cases + simp で 8 cases 全 decide

4.2 build strategy

-- File: data/lean4-mathlib/CollatzRei/Dfumt8AlphaArchitecture.lean
-- Lakefile 統合 pattern per (既存 CollatzRei module 継承)
namespace CollatzRei.Dfumt8AlphaArchitecture

-- 上記 sketch content 展開
-- estimated line count: 200-300 行
-- axiom-free target: 全 theorem [propext, Classical.choice, Quot.sound] のみ

end CollatzRei.Dfumt8AlphaArchitecture

4.3 potential blocker

5. Honest scope (5 条) + 関連 memory + Tier 4 継続

5.1 関連 memory + site page

• backlog #15 (STEP 1323b) = Pattern (B) origin
• backlog #3 (STEP 1294) = Rei stack Lean 4 axiom-free target profile 参照
• backlog #5 (STEP 1296) = 25 invention 影響評価 対象

5.2 Tier 4 継続

累計 backlog 19/19 (Tier 4 6/継続)。 残 2 candidate:

1. STEP 1328 (次): 統合 R(α, H_hold) proof strategy (candidate #2)
2. STEP 1329 (次): 5 instance 数値実測 architecture (candidate #3)