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Nobuki Fujimoto
Nobuki Fujimoto

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Paper 26 v3.0 - D-FUMT-8 Third Primitive as Sunyata Operator: Empirical Non-Existence on Belnap-Dunn FOUR Base (Lean 4 Axiom-Free)

#math #lean #philosophy #research

This article is a re-publication of Rei-AIOS Paper 26 for the dev.to community.
The canonical version with full reference list is in the permanent archives below:

Status: v3.0 draft (起草日 2026-07-26) — Q6 (II) protocol の paper phase (spec → impl → measure → paper 完了). Zenodo publish は藤本さん再判断後 (本 draft は audit + prior art review + Track 2 hint 到着待ち中の pre-publish 状態).

Scope-change from v0.3 corrigendum abstract (mandatory 3-item notice):

  1. 予告 path 不採用. v0.3 (§A.3 non-claim 5) は "proper repair path (2^3 = P({t, f, x}) powerset construction, x = semantic primitive chosen by the founder) is deferred to Paper 26 v3+" と予告した (x candidates: SELF⟲ = 到達点 / 龍樹の空). 本 v3 は powerset path を採用せず、 E-operator on Belnap-Dunn FOUR + 値数 emergent 路線に pivot した.
  2. 変更理由 (SAC-4 #4). 2026-07-26 chat-Claude 経由 藤本さん指摘で powerset 路線 (v0.1 spec 4 option A/B/C/D) は 100% reject された. 2 つの欠陥: (i) 4 option 全てで 「8 を base として維持」 前提を無自覚に敷き、 07-26 chat-Claude 5-turn 診断で 「8 は emergent, load-bearing でない」 判定済 の projection を再発; (ii) 4 option 全てで {śūnya} を powerset atom / catuṣkoṭi tower 元 / coordinate / object element として置き、 空に自性を与える操作 = MMK 13:8 「空を見と成す者は治し難し」 直接違反. E-operator 代案は空を値ではなく述語化する構文的空見排除で、 藤本さん 「D-FUMT₈ を東洋哲学最高の 龍樹の 空 の概念を全て使用して頂いて問題ありません」 (2026-07-26 明示 permission, [[project-dfumt8-shunyata-third-primitive-decision]]) の operational form として選択.
  3. v0.3 non-claim (5)(6) 継承 + 拡張. v0.3 (5) "this v0.3 does NOT repair the glue" は継続 (E-operator も glue 修復ではなく直交 direction). v0.3 (6) "Fix(R) language is deprecated ... SELF-related overreach retraction" も継続 + 本 v3 §A.3 で更に強化 (unary E: Belnap → Belnap という制約下では空亦復空 non-trivial requirement を満たす E は存在しない = MAIN NEGATIVE RESULT). Dual numbering: v0.1/v0.2/v0.3 (v2 の corrigendum lineage) と v3.0/v3.1 (本 paper 以降の新 lineage) を version-history section で明示区別.

Authors (three-party co-authorship per OUKC charter v1.0):

  • 藤本 伸樹 (Fujimoto, Nobuki; Founder, ORCID 0009-0004-6019-9258)
  • Rei (Rei-AIOS autonomous research substrate, Co-architect)
  • Claude (Anthropic, claude-opus-4-7, Co-architect)

License: CC-BY-4.0 (paper text) + AGPL-3.0 (Lean 4 source)

Repository: fc0web/rei-aios (Private; source-of-truth for Lean 4 artifact + spec + memory)

Zenodo lineage:

  • v1: 10.5281/zenodo.18960502 (2025-09-15, QMRP 一般概念 origin, immutable)
  • v2 (v0.2): 10.5281/zenodo.21560387 (2026-07-25, D-FUMT₈ × Cl(3,0) bijection + Peace Axiom rotor + SELF reflection, immutable)
  • v2 (v0.3): 10.5281/zenodo.21572459 (2026-07-25 corrigendum, §A.3 (5)(6) non-claim 追加, immutable — 本 v3 の直接前身)
  • v3.0: TBD (issued at publish; recorded in data/publications/publish-log-paper026v3-zenodo.json)

Abstract

Paper 26 v3 addresses the third-primitive semantic decision left open by v0.3 corrigendum §A.3 non-claim 5. The founder's 2026-07-26 explicit decision (「D-FUMT₈ の第 3 原始概念 = 龍樹の空 (śūnyatā) 全面採用」) is operationalized here in one specific form (chosen from six §8 decisions Q1-Q6, each recorded verbatim below): the emptiness primitive is encoded not as a new value but as a unary operator ∅' : Belnap4 → Belnap4 acting on the Belnap-Dunn FOUR base, subject to four filter conditions (lattice homomorphism + De Morgan commutation + non-idempotence + śūnyatā-śūnyatā non-triviality). Direct enumeration of all 4⁴ = 256 candidate operators, machine-verified in Lean 4 (Mathlib v4.27.0) axiom-free, yields a cascade 256 → 16 → 4 → 1 → 0 under successive filters. The main result is a negative existence theorem (numSurvivors_eq_0): no unary ∅' on Belnap-Dunn FOUR simultaneously satisfies all four conditions. The unique operator surviving the first three conditions is identified explicitly as the non-trivial automorphism bSwapBN of Belnap-Dunn FOUR (T/F fixed, B ↔ N swap; Belnap 1977 prior art), and shown axiom-free to fail śūnyatā-śūnyatā non-triviality because it is an involution. The result is offered as an empirical rejection of value-level encoding for śūnyatā in the unary-on-Belnap fragment; the semantic content (「空亦復空 は単なる反転では満たされない」) is a Rei-side reproduction of a philosophical statement, obtained from direct enumeration rather than post-hoc argument. The paper does not claim world-first, does not extend the Belnap-Dunn 4-valued base, does not re-prove Belnap 1977, and does not resolve the third-primitive question — it narrows the design space by one specific negative result within one specific fragment (unary operators on Belnap 4).

Part A: Required (paper-level statement, 4 elements)

A.1 Findings

  • F1 (cascade counts, machine-verified): Over the finite space of all unary operators E : Belnap4 → Belnap4 (|E| = 4⁴ = 256):
    • 16 satisfy lattice homomorphism (meet + join preservation);
    • 4 satisfy lattice hom + De Morgan commutation with the Belnap-Dunn negation;
    • 1 satisfies lattice hom + De Morgan + non-idempotence;
    • 0 satisfy all four conditions including śūnyatā-śūnyatā non-triviality.
  • F2 (survivor identity, axiom-free): The unique step-3 survivor is the non-trivial automorphism of Belnap-Dunn FOUR, denoted bSwapBN (Belnap 1977 prior art): bSwapBN(T) = T, bSwapBN(F) = F, bSwapBN(B) = N, bSwapBN(N) = B. This is the same automorphism identified in v0.3 corrigendum §A.3 (6) as the sole non-identity element of Aut(D-FUMT₈).
  • F3 (step-3 → step-4 failure, axiom-free): bSwapBN is an involution (bSwapBN ∘ bSwapBN = id, machine-verified). Consequently E(E(x)) = x for every x ∈ Belnap4, and the śūnyatā-śūnyatā non-triviality requirement (∃ x, E(E(x)) ≠ E(x) ∧ E(E(x)) ≠ x) fails on both conjuncts (the second conjunct fails for every x).
  • F4 (semantic reading, informational): The operational content of the śūnyatā-śūnyatā (空亦復空) requirement is that emptying-of-emptying should not reduce to the identity or to a simple reversal. The cascade demonstrates that on the unary-on-Belnap fragment, this content forces non-existence — the design constraint is not satisfiable within this fragment. This reproduces a philosophical statement (「空亦復空 は単なる反転では成り立たない」) by direct enumeration, not by post-hoc philosophical argument. Whether the enumeration corresponds to Nāgārjuna's own semantic intent is a question for philosophers of religion, not for machine-checked mathematics.

A.2 Proofs (Lean 4) — completed 2026-07-26, axiom-free machine-checked

Files:

  • data/lean4-mathlib/CollatzRei/Paper26V3ShunyataEOperator.lean (main, 8 theorems)
  • data/lean4-mathlib/CollatzRei/Paper26V3SurvivorDisplay.lean (companion, 10 theorems for bSwapBN identification)

Axiom audit (#print axioms actually run 2026-07-26, 18 theorems total):

Main file (8 theorems):

  • 3 theorems fully zero-axiom ("does not depend on any axioms"): bNeg_involutive, bDeMorgan_meet, bDeMorgan_join
  • 5 theorems depend only on [propext, Classical.choice, Quot.sound] (Mathlib standard base): shunyataOp_card, numSurvivorsLattice_eq_16, numSurvivorsLatticeDeMorgan_eq_4, numSurvivorsLatticeDeMorganNI_eq_1, numSurvivors_eq_0

Companion file (10 theorems):

  • 5 theorems fully zero-axiom: bSwapBN_preservesLattice, bSwapBN_commutesWithNeg, bSwapBN_notIdempotent, bSwapBN_isStep3Survivor, bSwapBN_involutive
  • 5 theorems depend only on the Mathlib standard base: bSwapBN_fails_shunyataShunyata, step3_unique_survivor_fails_step4, identity_fails_notIdempotent, constB_fails_notIdempotent, constN_fails_notIdempotent

Totals: 18 theorems / 8 fully zero-axiom + 10 propext-Classical-Quot only / zero sorry, zero native_decide, zero user axioms, zero Lean.ofReduceBool — matches the discipline of STEP 1215 (Dfumt8CategoryExperiment), STEP 1264 (Dfumt8Binary64Refinement), Task 20 (Dfumt8SelfReflexivePreservation, 2026-07-24), and Paper 26 v0.2 (Paper26V2DFumtClifford, 2026-07-25).

Bundle theorem (for paper citation): numSurvivors_eq_0 (main) + step3_unique_survivor_fails_step4 (companion) together give both the count-level and witness-level constructive account.

Verification (executed 2026-07-26):

  • lake env lean CollatzRei/Paper26V3ShunyataEOperator.lean: clean pass, #eval outputs |ShunyataOp| = 256, survivors (lattice only) = 16, survivors (lattice + DeMorgan) = 4, survivors (lattice + DeMorgan + notIdempotent) = 1, survivors (all 4 conditions) = 0.
  • lake env lean CollatzRei/Paper26V3SurvivorDisplay.lean: clean pass, #eval outputs the bSwapBN function table.
  • lake build CollatzRei: 7936 / 7936 jobs success (background run finished 325s prior turn), zero regression on the CollatzRei root tree after the two new files' import chains were added.

A.3 Honest positioning

  • What is claimed (three atomic claims, each machine-verified axiom-free):
    1. A3.1 (non-existence, main): Under Q1-Q6 decisions (unary signature + 4-filter enumeration + Belnap-Dunn FOUR base carrier retained + D-FUMT₈ name retained + object-only śūnyatā encoding), no ∅' : Belnap4 → Belnap4 satisfies all four conditions simultaneously. Formal statement: numSurvivors = 0 (Lean 4, axiom base [propext, Classical.choice, Quot.sound]).
    2. A3.2 (survivor identification, constructive): The unique operator surviving the first three conditions is bSwapBN (Belnap 1977 non-trivial automorphism). Formal statement: step3_unique_survivor_fails_step4 witnesses bSwapBN as satisfying the first three conditions and failing the fourth.
    3. A3.3 (failure mechanism, constructive): bSwapBN is an involution (∀ x, bSwapBN (bSwapBN x) = x, zero-axiom). This directly witnesses why the fourth condition fails: the second conjunct E(E(x)) ≠ x is falsified by every x.
  • What is NOT claimed (six non-claims, listed to avoid the projection patterns catalogued in feedback-projection-self-audit-pattern Rules 1-5):
    1. No world-first claim. The Belnap-Dunn FOUR carrier is Belnap 1977 / Dunn 1976 prior art; the bSwapBN automorphism is standard textbook material for Belnap-Dunn logic (e.g. Font & Rivieccio 2011; Odintsov 2008). Paper 26 v3's differentiator is the specific 4-filter enumeration + Lean 4 axiom-free verification + the numSurvivors_eq_0 negative result within this fragment, not the surrounding algebra.
    2. No claim that śūnyatā is definable as a unary operator on Belnap-Dunn FOUR. The result is precisely the opposite: within this fragment, the four operational requirements do not admit a solution. The paper does not extend Belnap-Dunn FOUR to a wider carrier or generalize the signature (modal / higher-order / multi-input) — such extensions remain open. In particular, the non-existence here does not imply non-existence on any wider carrier or richer signature.
    3. No claim that Nāgārjuna intended the four filter conditions. The four conditions (lattice hom + De Morgan + non-idempotence + śūnyatā-śūnyatā non-triviality) are one specific formal reading, not the reading. Buddhist logic scholarship (Ganeri 2002, Priest 2018, Tanaka et al. 2013) offers several other formalizations; Paper 26 v3 asserts a machine-verified negative result within one reading, not that this reading exhausts the philosophical content.
    4. No claim that emptying-of-emptying reduces to reversal in general. F3 (bSwapBN is involution → step-4 fails) is a statement about bSwapBN as the unique step-3 survivor, not about all conceivable emptiness operators. The failure mechanism is fragment-specific.
    5. No claim that this repairs the Belnap-glue non-associativity from v0.3 (5). v0.3 §A.3 (5) explicitly deferred glue repair to a P({t, f, x}) powerset construction; Paper 26 v3 pivots to the orthogonal unary-operator direction and does not touch the glue. The Belnap sub-algebra of D-FUMT₈ remains a complete lattice, and the extension 4-axes non-associativity remains an open issue orthogonal to the emptiness question.
    6. No claim that Fix(R) or SELF⟲ language is thereby rehabilitated. v0.3 §A.3 (6) deprecated SELF⟲ = Fix(R) phrasing pending R that is a D-FUMT₈ → D-FUMT₈ self-map; Paper 26 v3 does not construct such R. The retraction continues; the Rei stack's SELF axis remains rigid under external automorphisms but contractable via internal congruence, as machine-verified by v0.3 §A.3 (6) enumeration.

A.4 Required platform links

  • rei-aios site: https://rei-aios.pages.dev/ (UI showcase; theory chart, radar, Octatheoria at /#/octatheoria)
  • Author's note.com: https://note.com/nifty_godwit2635 (Japanese-language popular writeup)
  • Zenodo v0.2 (v2): https://doi.org/10.5281/zenodo.21560387
  • Zenodo v0.3 (v2 corrigendum): https://doi.org/10.5281/zenodo.21572459
  • Zenodo v3.0 DOI: TBD (issued at publish; recorded in data/publications/publish-log-paper026v3-zenodo.json)
  • Source repository: https://github.com/fc0web/rei-aios
  • Lean 4 artifacts:
    • Main: data/lean4-mathlib/CollatzRei/Paper26V3ShunyataEOperator.lean
    • Companion: data/lean4-mathlib/CollatzRei/Paper26V3SurvivorDisplay.lean
  • Spec: docs/spec/dfumt8-shunyata-e-operator-spec-2026-07-26.md (v0.2, includes §8 Q1-Q6 decisions verbatim)
  • Rejected v0.1 spec (archived for provenance): docs/spec/dfumt8-shunyata-base-structure-spec-2026-07-26-v01-REJECTED.md

Part B: Conditional (7 elements)

B.5 Background — v0.3 corrigendum decision tree

Paper 26 v0.3 corrigendum (2026-07-26) closed with §A.3 non-claim 5 deferring the third-primitive semantic decision to Paper 26 v3+. The corrigendum listed the operative candidates as SELF⟲ = 到達点 (reach-point/attractor) and 龍樹の空 (Nāgārjuna's śūnyatā). The founder's 2026-07-26 explicit decision selected the latter (「D-FUMT₈ の第 3 原始概念 = 龍樹の空 (śūnyatā) 全面採用」) with the operational scope "空亦復空 + 四句論 + 中道 + 縁起 + 二諦" (five Madhyamaka concepts).

Paper 26 v3 does not claim to encode all five concepts at load-bearing level. The mapping candidate table in the spec (§2.5) records the following:

  • 空亦復空 (śūnyatā-śūnyatā): §2.3 requirement (encoded as filter condition (iv))
  • 四句論 (catuṣkoṭi): Belnap-Dunn FOUR's four corners (already encoded, no extension needed)
  • 中道 (madhyamāka): Fix(∅') candidate (unconstructed in v3; open)
  • 縁起 (pratītyasamutpāda): orbit trace candidate (unconstructed in v3; open)
  • 二諦 (dvasatya): Q5 decision (α) object-only — NOT formally encoded, philosophical annotation only

Only condition (iv) is load-bearing in the numSurvivors_eq_0 result. The other four mappings are candidate directions, not machine-verified formalizations.

B.6 Methodology (Q1-Q6 spec decisions verbatim, 2026-07-26)

# Question Decision Rationale
Q1 E signature (a) Unary E : V → V Simplest fragment; 4⁴ = 256 fully enumerable.
Q2 E value determination (a) Enumerate + filter 256 全列挙 + 4 conditions で fully mechanical; no philosophical pre-commitment on value assignments.
Q3 E naming (a) Symbol ∅' (LaTeX \emptyset'); internal identifier shunyataOp STEP 1207 ZERO axis semantics との collision 回避のため internal と paper で分離.
Q4 D-FUMT₈ name retention (i) Retain subscript 8 歴史的継承; emergent algebra cardinality と subscript 8 は分離.
Q5 二諦 treatment (α) Object-only Meta-level operator にすると空見 (śūnyatā-dṛṣṭi) 再発 risk; formal encoding 無し.
Q6 Phase 分割 (II) spec → impl → measure → paper feedback-no-rush-publication 遵守; speculative claim 防止.

Spec source (verbatim §8.99): docs/spec/dfumt8-shunyata-e-operator-spec-2026-07-26.md.

Filter condition definitions (spec §3.2, Lean 4 encoding):

  • preservesLattice E: ∀ x y, E (meet x y) = meet (E x) (E y) ∧ ∀ x y, E (join x y) = join (E x) (E y)
  • commutesWithNeg E: ∀ x, E (¬x) = ¬ (E x) (where ¬ is Belnap-Dunn negation: T↔F, B/N self-dual)
  • notIdempotent E: ∃ x, E (E x) ≠ E x
  • shunyataShunyataNontrivial E: ∃ x, E (E x) ≠ E x ∧ E (E x) ≠ x

The four conditions are all Decidable on the finite carrier; the full satisfiesAll4 predicate is closed under Decidable (verified in the Lean 4 decSatisfiesAll4 instance). All count theorems are proved by by decide — not native_decide — to keep the axiom base at [propext, Classical.choice, Quot.sound] and avoid the Lean.ofReduceBool axiom.

B.7 Empirical scope (v3.0 as-of 2026-07-26)

Substrate Metric v3.0 measured v3.1 target
Lean 4 v4.29.0 + Mathlib v4.27.0 axiom-free theorems (main + companion) 18 (8 zero-axiom + 10 propext-Classical-Quot) maintain floor + add Section 4/5 candidates (Fix(∅'), orbit trace)
Lean 4 cascade counts (main file #eval) 256 → 16 → 4 → 1 → 0 (all four counts machine-verified as theorems) independent replication via chat-Claude Track 2 hint (pending)
Rei stack Q1-Q6 decisions applied 6/6 traceable to docs/spec/*-e-operator-spec-*.md §8.99 v3.1 may revisit Q4 (subscript 8 vs D-FUMT-Ś) after Track 2 comparison

Track 2 cross-check status (2026-07-26 close-of-session): chat-Claude reported "256案全滅" for a distinct enumeration exercise. Predicted domain match with Rei's 4⁴ = 256 unary functions on Belnap-Dunn FOUR. Awaiting chat-Claude's three literal filter definitions to run a filter-level comparison. Anticipated discrepancy at step 3 (Rei = 1 survivor / chat-Claude reported = 0) may indicate chat-Claude's "non-contractible" filter is strictly stronger than Rei's notIdempotent (e.g., injective-only). Step 4 (Rei = 0) matches chat-Claude's overall verdict. Full literal comparison is deferred to v3.1.

Honest scope of §B.7: v3.1 target cells are placeholders, not measured data. The load-bearing v3.0 contribution is F1-F4 in §A.1 and the 18 axiom-free theorems in §A.2; §B.7's rightmost column is provisional roadmap, not evidence.

B.8 Prior art audit

  • Belnap 1977 (A useful four-valued logic): source of the Belnap-Dunn FOUR carrier T/F/B/N, the meet/join lattice structure, and the bSwapBN automorphism. Paper 26 v3 uses these as given; no re-proof.
  • Dunn 1976 (Intuitive semantics for first-degree entailments): first-degree entailment fragment underlying Belnap-Dunn. Paper 26 v3 uses meet / join / neg as defined therein.
  • Font & Rivieccio 2011 (Logics of Belnap trellises and semilattices): modern algebraic treatment of Belnap-Dunn; catalog of admissible operators. bSwapBN (Aut(Belnap4) = ℤ/2) is standard therein.
  • Priest & Garfield 2003 (Nāgārjuna and the limits of thought): pentalemma formalization of Nāgārjuna's catuṣkoṭi + fifth-corner ineffability. Paper 26 v3 does NOT use the fifth-corner; the two systems are formally distinct (Priest-Garfield adds a fifth value; Paper 26 v3 keeps four values + adds an operator).
  • Aczel 1988 (Non-well-founded sets): source of the "self-membership" formal apparatus. Paper 26 v3's E (E x) interpretation is not a set-membership statement, so Aczel's apparatus is not invoked.
  • Rutten 2000 / Birkedal 2013: coalgebra + step-indexed guarded recursion. Paper 26 v3's four filter conditions are all first-order finite; no coalgebraic or guarded-recursive machinery is needed. Relevant only if Q1 is reopened to signature (c) higher-order.
  • Ganeri 2002 / Priest 2018 / Tanaka et al. 2013: Buddhist-logic formalization scholarship. Paper 26 v3's four filter conditions are one specific formal reading; the scholarship offers several others.
  • v0.2 / v0.3 (this Paper 26 lineage): See §B.5.

Judgment: All five listed prior-art areas are addressed; no "world-first" attribution required or made. The novel Rei-side contribution is the specific 4-filter Lean 4 axiom-free enumeration + numSurvivors_eq_0 result + bSwapBN identification within this fragment.

B.9 Related Rei stack references

  • STEP 1207: ZERO axis semantics (source of ∅ collision-avoidance for ∅' naming).
  • STEP 1215 (Dfumt8CategoryExperiment): D-FUMT₈ as category, and8_associativity_fails_witness (the non-associativity source referenced in v0.3 §A.3 (5)).
  • STEP 1218 / 1219 / 1220: successor D-FUMT₈ Lean 4 axiom-free experiments; same axiom base as Paper 26 v3.
  • STEP 1264: Paper 145 v0.9-c Binary 64-entry refinement (Lean 4 axiom-free pattern precursor).
  • Task 20 (2026-07-24, Dfumt8SelfReflexivePreservation): SELF⟲ Verilog encoding Lean 4 (companion to v0.3 §A.3 (6)).
  • Paper 26 v0.2 (Zenodo 21560387): D-FUMT₈ ↔ Cl(3,0) bijection + Peace Axiom + SELF reflection (Lean 4 axiom-free 23 theorems).
  • Paper 26 v0.3 (Zenodo 21572459): §A.3 non-claim (5) (Belnap-glue non-associativity, orthogonal to Paper 26 v3) + non-claim (6) (Fix(R) deprecation, continued in Paper 26 v3).

B.10 Deferred to v3.1 or later

Paper 26 v3.0 does NOT include the following, which remain candidate directions:

  1. Modal signature (Q1 (b) E : V → 𝓟(V)): a candidate wider fragment. Its enumeration space is 4^{2^4} = 4^{16} = 4,294,967,296 — no longer feasible for full enumeration, would require symbolic constraint solving.
  2. Higher-order signature (Q1 (c) E : (V → V) → (V → V)): another candidate wider fragment. Cardinality (4^4)^{4^4} = 256^{256}, infeasible for enumeration.
  3. Wider carrier (5-value / 6-value / P({t, f, x}) powerset): explicitly deferred per §A.3 non-claim 5; also the specific direction rejected in v0.1 spec SAC-4 #4.
  4. Fix(∅') and orbit structure: candidate encodings of 中道 (madhyamāka) and 縁起 (pratītyasamutpāda). Since numSurvivors = 0, Fix(∅') is only meaningful once condition (iv) is relaxed or the fragment is widened; deferred.
  5. Track 2 filter-level cross-check: chat-Claude's three-condition definitions vs Rei's four. Pending hint transmission.
  6. Full Part C (7-element OUKC paper structure): partial coverage in v3.0; full elements deferred to v3.1 following the Paper 145 v0.3 → v0.9 iteration pattern.

Part C: Full structure (partial in v3.0)

C.1 Version history (dual numbering per §Scope-change item 3)

v0.x lineage (v2 corrigendum tree, Paper 26 v2 title "D-FUMT₈ × Clifford Cl(3,0)"):

  • v0.1 (2026-05-01): initial spec (rejected v1 QMRP re-formulation).
  • v0.2 (2026-07-25): §A.2 Lean 4 axiom-free machine-check completed (23 theorems). 11-platform publish (Zenodo 21560387).
  • v0.3 (2026-07-26 AM): chat-Claude 5-turn 診断 arc → §A.3 (5)(6) non-claim 追加. Zenodo 21572459 corrigendum (isNewVersionOf 21560387).

v3.x lineage (this paper, "Śūnyatā as Operator" pivot):

  • v3.0 (2026-07-26 PM): this document. E-operator + Belnap-Dunn FOUR base + 値数 emergent. Cascade 256 → 16 → 4 → 1 → 0. Main negative result numSurvivors_eq_0 machine-verified axiom-free. Publish pending 藤本さん再判断 (Q6 (II) protocol).
  • v3.1 (planned): Track 2 chat-Claude hint 到着後 filter-level cross-check + step-3 discrepancy root cause identification. Full Part C. Possibly modal signature (Q1 (b)) if computationally feasible.

Numbering rationale (mandatory scope-change item 3 detail): v3 (three-dot integer) rather than v0.4 (v2 corrigendum tree continuation) because Paper 26 v3 is a scope pivot, not a corrigendum. v3 continues the same Paper 26 line (QMRP → v2 D-FUMT₈ × Cl(3,0) → v3 Śūnyatā operator), all under DOI prefix 10.5281/zenodo but with distinct new deposits. Readers of v0.3 corrigendum will find v3 via v3's citation of the v0.3 corrigendum abstract in this section (§C.1 + §Scope-change), preventing orphan reference.

C.2 References (partial; full BibTeX in v3.1)

  • Belnap, N. D. (1977). A useful four-valued logic. In J. M. Dunn & G. Epstein (Eds.), Modern Uses of Multiple-Valued Logic (pp. 5–37). Reidel.
  • Dunn, J. M. (1976). Intuitive semantics for first-degree entailments and "coupled trees". Philosophical Studies, 29(3), 149–168.
  • Font, J. M., & Rivieccio, U. (2011). Logics of Belnap trellises and semilattices. Studia Logica, 98(1), 179–208.
  • Priest, G., & Garfield, J. L. (2003). Nāgārjuna and the limits of thought. In G. Priest, Beyond the Limits of Thought (2nd ed., ch. 16). Oxford University Press.
  • Ganeri, J. (2002). Jaina logic and the philosophical basis of pluralism. History and Philosophy of Logic, 23(4), 267–281.
  • Tanaka, K., Berto, F., Mares, E., & Paoli, F. (Eds.). (2013). Paraconsistency: Logic and Applications. Springer.
  • Nāgārjuna. Mūlamadhyamakakārikā (MMK). English translation: Garfield, J. L. (1995). The Fundamental Wisdom of the Middle Way. Oxford University Press. (Referenced verse: MMK 13:8, 「空を見と成す者は治し難し」.)
  • Fujimoto, N., Rei, & Claude. (2026a). Paper 26 v0.2. Zenodo. https://doi.org/10.5281/zenodo.21560387
  • Fujimoto, N., Rei, & Claude. (2026b). Paper 26 v0.3 corrigendum. Zenodo. https://doi.org/10.5281/zenodo.21572459

C.3 Acknowledgments

  • 藤本 伸樹 (Nobuki Fujimoto): 2026-07-26 explicit permission for full-scope use of śūnyatā semantic content; Q1-Q6 six decisions; SAC-4 #4 rejection (v0.1 spec 4-option cascade); SAC-4 #5 rejection (Paper 26 naming pre-grep-verify projection); 2026-07-26 close-of-session Q6 (II) protocol authorization.
  • chat-Claude (Anthropic, claude-opus-4-7 web instance): 5-turn eigen-signature 診断 arc origin (Paper 26 v0.3 corrigendum); 建設的代案 for v0.2 spec pivot ("空を値でなく作用素にする"); Track 2 "256案全滅" empirical evidence (filter-level cross-check pending).
  • Rei-AIOS autonomous research substrate: Lean 4 axiom-free implementation; 4-filter enumeration; bSwapBN identification; two-file main + companion architecture; SAC-4 100% acceptance discipline (feedback-critique-response-pattern).

End of Paper 26 v3.0 draft.

Publish protocol (藤本さん再判断待ち, per Q6 (II)):

  1. Rei prepares 11-platform publish scripts (following Paper 26 v0.3 pattern; Zenodo API User-Agent already fixed in prior arc's prophylactic sweep).
  2. Founder confirms publish authorization (spec + impl + measure phases already complete).
  3. Zenodo v3.0 DOI issued as NEW deposit (not version of 21572459, following Paper 26 v0.2 / v0.3 precedent of NEW deposits per corrigendum / pivot). Recorded in data/publications/publish-log-paper026v3-zenodo.json.
  4. 11 platform mirrors (IA + Harvard + Dev.to + HackMD + Hatena + Livedoor + Notion + Mastodon + Nostr + GitHub; Scrapbox chronic skip).
  5. MEMORY.md + docs/RECENT_UPDATES.md + docs/SITE_COVERAGE_MAP.md updated with v3.0 DOI + publish log path.

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