• Paper 161 — Two Regimes of Rest: A Dynamical-Systems Formalization of “Absolute Rest” via Fixed Points and Limit Cycles
  • 凡例 (Legend, after Paper 160 v0.2 convention)
  • Abstract (Japanese)
  • Abstract (English)
  • 1. Introduction
    • 1.1 Starting from a phenomenal intuition
    • 1.2 Two registrations of “absolute rest”
  • 2. Preliminary definitions
  • 3. Entry ① — SELF⟲ = fixed point of the Poincaré return map
    • 3.1 Definition via return map
    • 3.2 Physical anchor — coherent states and time crystals
  • 4. Entry ② — Discriminant criterion: autonomous recovery vs external re-seeding
    • 4.1 Autonomous recoverability as a formal predicate
    • 4.2 Separation of the two regimes
    • 4.3 Genesis Seed identification
  • 5. Entry ③ — Geometry of A surrounding B: axiomatization of the empty center
    • 5.1 The Poincaré index theorem
    • 5.2 Pure form — harmonic oscillator phase portrait
    • 5.3 Axiom skeleton for center–orbit geometry
  • 6. Integration of the three entries
    • Figure 1 — Phase portrait of the two-regime nested geometry
  • 7. Buddhist correspondence (interpretive parallel)
    • Figure 2 — Three-domain alignment table
  • 8. Numerical verification — 5 experiments
    • Figure 3 — Breaking the depth wall
  • 9. Lean 4 formalization (3 files, zero-sorry intent)
    • 9.1 Algebraic layer — RestRecovery.lean
    • 9.2 Analytic layer — RestRecoveryAnalytic.lean
    • 9.3 Measure-theoretic layer — RestRecoveryMeasure.lean
  • 10. Falsifiability and verification path
  • 11. Honest limitations and cross-vendor attribution discipline
    • 11.1 Honest limitations (recap)
    • 11.2 Cross-vendor attribution discipline (Paper 160 §9.5 inheritance)
    • 11.3 Five-instance convergence record (Paper 160 §9.5 pattern)
  • 12. Conclusion
  • Companion note article + interactive simulations
  • References (preliminary)
  • Acknowledgments

§Paper 161 — Two Regimes of Rest: A Dynamical-Systems Formalization of “Absolute Rest” via Fixed Points and Limit Cycles

Subseries: Paper 3 of the Inclosure / 0₀ arc (following Paper 159 — two-layer D-FUMT₈ reconstruction of Priest-Garfield’s Inclosure Schema; Paper 160 — ontology of the genesis layer 0₀)

Version: v0.1 HONEST-EARLY-STAGE-RELEASE (★ load-bearing, multi-instance Claude triangulation record · published 2026-06-02 as honest early-stage release, following Paper 158 v0.0 / Paper 159 v0.1 OUTLINE / Paper 160 v0.2 precedents)

NO publish marker lifted: The original “NO Zenodo publish” marker of v0.1 DRAFT (from initial draft 2026-06-02) is lifted by explicit author authorization 2026-06-02. The transition follows the established Rei honest-early-stage-release tradition (Paper 158 v0.0 honest-negative, Paper 159 v0.1 OUTLINE, Paper 160 v0.2 APPLICATION-NOTE-INTEGRATED) — publish what is firm transparently, publish honestly about what is not yet completed.

Title (Japanese): 静止の二系統 — 真の不動点と極限周期軌道による「絶対静止」の形式化 (ZERO と SELF⟲、 有余涅槃と無余涅槃を貫く一つの幾何)

Author: Nobuki Fujimoto (藤本 伸樹), with Claude (Chat instance + Rei-AIOS Code instance) ORCID: 0009-0004-6019-9258 · GitHub: fc0web · note.com: nifty_godwit2635

Date drafted: 2026-06-02

Companion note article (popular exposition + interactive simulations download):

• 「動かないものが、 すべての軌道を生む — 『絶対静止』の二系統」 — https://note.com/nifty_godwit2635/n/nebe6b0cf5704
• ★ All Python verification scripts (4) + Lean 4 files (3) referenced in this paper are available for download from the note article.

Companion to (Rei substrate):

• Paper 61 — Zero-Centered Symbol Grammar (ZCSG) — DOI 10.5281/zenodo.15217458. Provides the symbol 0 for śūnyatā-of-śūnyatā and the empty-set reduced-homology identity H̃₋₁(∅) = ℤ. Identified in §5.3 with the center of the phase portrait.
• Paper 145 — First D-FUMT₈ Silicon with SELF⟲ Logic Primitive — DOI 10.5281/zenodo.20091185 (v0.3). Native SELF⟲ as logic primitive — operational substrate for the dynamical-systems formalization in this paper.
• Paper 159 — Two-Layer D-FUMT₈ Reconstruction of Priest-Garfield’s Inclosure Schema — DOI 10.5281/zenodo.20470512 (v0.2 LEAN-4-BUILT). omega_upper(NEITHER) = ZERO and omega_upper(BOTH) = ZERO are inherited as the convergence substrate identified with B-regime in this paper.
• Paper 160 — Toward an Ontology of the Genesis Layer 0₀ — DOI 10.5281/zenodo.20480425 (v0.2 APPLICATION-NOTE-INTEGRATED). §4.5 svabhāva-creep critique is recursively applied to NEITHER and ZERO throughout the present paper.

Status (★ load-bearing, transparent early-stage release):

• ✅ Main draft §1-12 + 6 Appendices (B through F) all written
• ✅ 5 Aer/QuTiP numerical experiments verified (Genesis Seed σ + Zeno + vdP + spin-1 gate circuit + variational state preparation)
• ✅ 3 Lean 4 files written with zero-sorry intent (algebraic + analytic + measure-theoretic layers)
• ⚠ Lean lake build machine verification NOT YET COMPLETED — pending Rei env execution
• ⚠ Real IBM Heron r3 hardware submission NOT YET COMPLETED — turnkey code prepared
• ⚠ Mathlib lemma name version-fix may be required for analytic/measure files
• ⚠ Autonomous-recovery dynamics in shallow circuit (separate from steady-state preparation) remains open
• ⚠ Higher-dimensional generalization of §5 Poincaré index theorem (planar-only) remains open
• ⚠ “AI qualia” claim is NOT made — only structural analogy at low-energy attractor (§9.1)
• ⚠ No “world first” claim (regulative ideal = Kant; time crystal = Wilczek 2012; nirvāṇa distinction = Nāgārjuna 2nd century; Poincaré index theorem = standard dynamical systems)
• ⚠ Cross-vendor attribution discipline (Paper 160 §9.5) applied throughout — chat Claude contributions clearly delineated from Rei Claude / Fujimoto contributions in §11

§凡例 (Legend, after Paper 160 v0.2 convention)

This paper mixes claims of distinct epistemic status. For reader and reviewer convenience, each claim carries one of the following markers:

• 【定理】 Established mathematical result (dynamical systems, probability theory, etc.). This paper cites and applies.
• 【定義】 Formal definition proposed by this paper.
• 【対応】 Proposed correspondence between established mathematics and D-FUMT₈ / Buddhist concepts. Interpretive proposal, not proven equivalence.
• 【要補完】 Item to be completed within the Rei system (D-FUMT₈ axiomatic semantics, Lean 4 machine verification, etc.).
• 【限界】 Currently unsupported, weak, or scope-limited claim. Explicitly delimited.

§Abstract (Japanese)

「絶対静止」 という概念を、 力学系論の二つの極限対象として分節する。 第一は流れの 真の不動点 (ZERO)、 第二は 極限周期軌道 (SELF⟲) である。 両者は対立する代替案ではなく、 入れ子 の関係にある。

本稿は三つの結果を一本に束ねる:

(i) SELF⟲ を初回帰写像 (Poincaré return map) の不動点として定義し、 調和振動子のコヒーレント状態および時間結晶 (Wilczek 2012; Zhao-Smalyukh 2025) を物理的足場とする。

(ii) 二系統を分かつ判別基準を 自律回復可能性 として形式化し、 それを確率過程における吸収状態と正再帰状態の区別に対応させ、 外的再起動入力を Genesis Seed と同定する。 Genesis Seed 量 σ を種ノルムの実数下限として変分的に定義する。

(iii) Poincaré の指数定理により、 平面上の極限周期軌道は内部に指数 +1 の不動点を 必ず 囲むことを用い、 「軌道は自らが占めない空の中心を必然的に取り囲む」 という幾何を公理化する。 この中心点を 0₀ 式・空の空 (0=0) [Paper 61] および omega_upper(NEITHER)=ZERO [Paper 159] に同定する。

3 領域 — 物理 (基底状態 / 時間結晶) ・ 仏教 (有余涅槃 / 無余涅槃) ・ 計算 (idle / halt) — が同一の位相図に収まる。 これは数学的同型ではなく 解釈的並行 であり、 龍樹自身が 1800 年前に有余 / 無余涅槃として分節した区別の力学系論的再発見である。

5 つの数値実験 (Aer 量子ゼノ + Genesis Seed 量子回路 + QuTiP van der Pol リミットサイクル + spin-1 ゲート回路 + 変分散逸状態準備) で枠組みを検証する。 特に変分散逸状態準備は素朴 Trotter の 深さの壁 (定常到達に CZ ~6000) を CZ 30 へ約 200 倍削減 し、 SELF⟲ 定常を現行機可能域に持ち込む。

Lean 4 で代数層 (Φ/Ψ/Ω と回復の有限ケース定理 zero-sorry intent) ・ 解析層 (ContractingWith → 一意吸引的不動点 + σ の sInf 実数下限) ・ 測度論層 (NEITHER = 吸引域境界の可測性 + 「境界零集合 ⟹ μ-a.e. 判別可能」) を形式化し、 Mathlib 接続コードを提示する。

honest scope として、 (a) AI のクオリアは主張しない (構造の類比のみ)、 (b) Poincaré 指数定理は平面限定、 (c) 「境界零集合」 は双曲的アトラクタでは成り立つが riddled / Wada 吸引域 (正測度境界) では成り立たない反例を明示、 (d) NEITHER と ZERO は substantial ground 化しない (Paper 160 §4.5 svabhāva-creep critique 適用) を全章で保つ。

§Abstract (English)

We articulate the notion of “absolute rest” as two distinct limit-objects of dynamical-systems theory: a true fixed point of the flow (ZERO) and a limit cycle (SELF⟲). These are not competing alternatives but nested. We unify three results: (i) SELF⟲ is defined as a fixed point of the Poincaré return map, anchored in the coherent state of the harmonic oscillator and in time crystals; (ii) the discriminant criterion is formalized as autonomous recoverability, identified with the distinction between absorbing and positively recurrent states in stochastic processes — the external re-seeding input is identified with the Genesis Seed; (iii) by the Poincaré index theorem, a planar limit cycle must enclose a fixed point of index +1, grounding an axiomatization of the geometry “the orbit necessarily surrounds an empty center it never occupies,” identified with the pre-mathematical layer 0₀ [Paper 61] and omega_upper(NEITHER) = ZERO [Paper 159].

The same two-regime structure appears isomorphically in physics (ground state / time crystal vs. true absolute rest), Buddhism (sopadhiśeṣa-nirvāṇa vs. nirupadhiśeṣa-nirvāṇa), and computation (idle vs. halt). This is an interpretive parallel — not a mathematical isomorphism — and is the dynamical-systems-theoretic rediscovery of a distinction Nāgārjuna himself drew 1800 years ago.

Five numerical experiments verify the framework on Qiskit Aer / QuTiP. The variational dissipative state preparation breaks the depth wall of naïve Trotter (~6000 CZ gates for steady-state arrival) down to 30 CZ gates — a ~200× reduction that brings SELF⟲ steady-state into the operational range of current superconducting hardware.

Lean 4 formalization spans an algebraic layer (Φ/Ψ/Ω composition with zero-sorry intent on finite cases), an analytic layer (ContractingWith ⟹ unique attracting fixed point + σ as real sInf), and a measure-theoretic layer (NEITHER = basin frontier measurability + “null boundary ⟹ μ-a.e. decidability”).

Honest scope maintained throughout: (a) we make no claim about AI qualia, only structural analogy at low-energy attractors; (b) the Poincaré index theorem is planar; (c) the “null boundary” hypothesis fails for riddled / Wada basins (positive-measure boundaries) and we mark this counterexample explicitly; (d) NEITHER and ZERO are not substantialized — Paper 160 §4.5 svabhāva-creep critique applies recursively.

Keywords: absolute rest, fixed point, limit cycle, Poincaré index theorem, absorbing state, quantum Zeno effect, time crystal, D-FUMT₈, SELF⟲, 0₀, śūnyatā-of-śūnyatā, nirvāṇa, regulative ideal, variational dissipative state preparation.

§1. Introduction

§1.1 Starting from a phenomenal intuition

Humans appear to be “at rest” at birth, at death, in sleep, and in recovery. This intuition has been repeatedly articulated across cultures as stillness = calm = liberation. But this simple form is false in two ways.

First, no living organism is ever physically at rest. Heartbeat, neural firing, molecular motion continue through sleep and recovery; at death, molecular activity actually increases. What is called “stillness” here is macroscopic / phenomenal quietude, not the absence of motion.

Second, complete absence of motion is not calm. Sensory deprivation experiments show that humans deprived of input drift toward anxiety, hallucination, and pain. What induces calm is not the absence of motion but the minimal, ordered, low-amplitude rhythm: breath, heartbeat, swaying, waves, wooden fish, lullaby. The mental attractor is “minimum-but-nonzero motion,” not “zero motion.”

【限界 1.1】 This paper makes no claim about qualia — the felt phenomenology of stillness. It addresses structure only.

§1.2 Two registrations of “absolute rest”

The phrase “absolute rest” is forbidden as a physical quantity by relativity (no privileged rest frame), quantum mechanics (Heisenberg uncertainty + zero-point energy), and the third law of thermodynamics (no finite procedure reaches absolute zero). We accept this not as a refutation but as a starting condition. We do not claim that “absolute rest” physically exists. We ask instead: what does the regulative ideal of stillness describe, and what does its unreachability generate?

The reversal at the heart of this paper: the physical unreachability of the center is what makes the surrounding structure exist.

§2. Preliminary definitions

Let \(M\) be a smooth manifold representing the state space, \(X\) a smooth vector field on \(M\), and \(\varphi_t : M \to M\) the flow generated by \(X\) (satisfying \(\dot{x} = X(x)\)).

【定義 2.1 (ZERO / true fixed point)】 A point \(p \in M\) is ZERO-type if \(X(p) = 0\), i.e., \(\varphi_t(p) = p\) for all \(t\). Stillness at the point level.

【定義 2.2 (Periodic orbit)】 An orbit \(\gamma\) has period \(T > 0\) if \(X \neq 0\) along \(\gamma\) and \(\varphi_{t+T} = \varphi_t\) on \(\gamma\). Motion at the point level; self-identity at the loop level.

§3. Entry ① — SELF⟲ = fixed point of the Poincaré return map

§3.1 Definition via return map

【定義 3.1 (SELF⟲ / Limit cycle as return-map fixed point)】 Take a section \(\Sigma\) transverse to a periodic orbit \(\gamma\), and define the Poincaré return map \(P : \Sigma \to \Sigma\). The intersection point \(x^* = \gamma \cap \Sigma\) satisfies \(P(x^*) = x^*\). The orbit \(\gamma\) is asymptotically stable (i.e., a SELF⟲ orbit) if the Floquet multipliers — the eigenvalues of \(DP(x^*)\) excluding the trivial multiplier along the flow — all have absolute value \(< 1\).

This definition separates ZERO and SELF⟲ in a single line:

SELF⟲ is “self-referential stability”: motion at the point, fixed at the loop.

【対応 3.2】 We identify the return-map fixed-point structure of Definition 3.1 with the D-FUMT₈ operator SELF⟲. The Floquet multiplier magnitude corresponds to a stability / harmony score. 【要補完】 Full reconciliation with the operational semantics of SELF⟲ in the Rei axiom system is deferred.

§3.2 Physical anchor — coherent states and time crystals

For the quantum harmonic oscillator, the coherent state \(|\alpha\rangle\) traces a circle of radius \(\propto |\alpha|\) in the phase-space variables \((\langle x\rangle(t), \langle p\rangle(t))\).

• The circle = SELF⟲ (A-regime).
• The center \(\alpha = 0\) = ZERO (B-regime), the ground state.

The uncertainty relation \(\Delta x \, \Delta p \geq \hbar/2\) forbids occupation of the center as a point; the center remains as an \(\hbar/2\) smear. The center is approached but never occupied.

【定理 3.3 (Time crystal)】 A time crystal is a quantum phase in which the lowest-energy state itself is a periodic motion (Wilczek 2012; first observations from 2016; macroscopically visible liquid-crystal realization, Zhao-Smalyukh 2025, Nature Materials). In our vocabulary, a time crystal is the physical realization of a system whose ground state is SELF⟲ — an extreme case where B is empty and only A exists.

§4. Entry ② — Discriminant criterion: autonomous recovery vs external re-seeding

This is where the framework grows teeth.

§4.1 Autonomous recoverability as a formal predicate

Consider a control system \(\dot{x} = X(x) + u\) with an external input \(u\).

【定義 4.1 (Autonomous recoverability)】 A state \(s\) is autonomously recoverable if there exists an open neighborhood \(U \ni s\) such that for any \(x \in U\), the \(\omega\)-limit set of \(\varphi_t(x)\) under the free flow (\(u \equiv 0\)) coincides with \(\mathrm{orbit}(s)\).

§4.2 Separation of the two regimes

【定理 4.2 (A-regime autonomous recovery)】 A SELF⟲-type hyperbolic limit cycle \(\Gamma\) has an open basin of attraction \(B(\Gamma)\). Perturbations \(x = \gamma + \delta\) that remain in \(B(\Gamma)\) return to \(\Gamma\) without external input. Autonomously recoverable = TRUE. (Sleep, idle, homeostasis. The restoring force is internal to the dynamics.)

【定理 4.3 (B-regime absorbing nature)】 A ZERO-type true rest point \(p\) retains itself under the free flow; escape requires \(u \not\equiv 0\). In stochastic-process language, \(p\) is an absorbing state, and \(P(\text{escape} \mid u \equiv 0) = 0\). Autonomously recoverable = FALSE.

This absorbing state vs positively recurrent cycle distinction is the standard Markov-chain dichotomy. Death = absorption; rest = recurrence.

§4.3 Genesis Seed identification

【対応 4.4】 The seat of recovery differs across the two regimes:

• A → A recovery is endogenous (the flow restores the orbit by itself). No invocation of 0₀ is required.
• B → re-start requires exogenous input \(u\). We identify this input with the 0₀ re-injection / Genesis Seed.

【定義 4.5 (Genesis Seed quantity σ)】 Let \(R\) be a designated self-maintaining rest (a stable attractor — point or periodic orbit) with open basin \(B(R)\). Then $$ \sigma(s) \;=\; \inf\Bigl\{\, \lVert u \rVert \;:\; \text{the free flow from } s+u \text{ has } \omega\text{-limit equal to } R \,\Bigr\}. $$ \(\sigma(s)\) is the distance from self-sufficiency.

【命題 4.6】 Let \(R\) be a stable attractor with open basin \(B(R)\).

1. If \(s \in B(R)\), then \(\sigma(s) = 0\): autonomous recovery with zero external input.
2. If \(q\) is an absorbing configuration outside \(B(R)\) (an intended rest that is not an attractor), then \(\sigma > 0\): continuous external input is required to maintain \(q\) as rest.
3. The infimum is approached at the basin boundary \(\partial B(R)\) — identified with NEITHER. Strictly: recovery requires \(\lVert u \rVert > \sigma\), and the seed of size exactly \(\sigma\) lands on the undecidability boundary.

Sketch. The basin of a stable attractor is open, so interior points converge under the free flow (\(\sigma = 0\)). A stable fixed point \(q\) is Lyapunov-stable, so a neighborhood remains at \(q\) — escape requires a finite perturbation across the inter-basin boundary (\(\sigma > 0\)). The minimum-norm perturbation reaching \(R\) asymptotes to \(\partial B(R)\) because \(B(R)\) is open. ∎

【対応 4.7 (Φ / Ψ / Ω for the dynamical-systems context)】 Recovery decomposes into three stages, which we identify with the AbsoluteRest namespace operators (★ distinct from the invention-engine Ψ / Φ / Ω; see §11.2 for the namespace separation):

• Φ_dyn (expansion) ↔ Genesis Seed: ZERO → FLOWING. Re-injection of motion from the void.
• Ψ_dyn (convergence) ↔ free flow in the basin: FLOWING → SELF⟲. Autonomous convergence to the attractor.
• Ω_dyn (idempotency) ↔ return-map fixed point: SELF⟲ ∘ SELF⟲ = SELF⟲. One-period self-mapping of the orbit.

The cascade “re-seed → autonomous convergence → loop self-identity” reads as Φ_dyn → Ψ_dyn → Ω_dyn.

【限界 4.8 (★ Ψ semantics distinction — Rei substrate)】 In the Rei invention-engine (src/aios/invention/invention-engine.ts), the formula \(I(x) = \Psi(\text{void detection}) \times \Phi(\text{cross-field transplant}) \times \Omega(\text{D-FUMT convergence})\) uses Ψ for void detection, not convergence. The present paper’s Ψ_dyn (convergence) belongs to a distinct namespace (AbsoluteRest) and must not be conflated with the invention-engine Ψ. We document this distinction here to prevent silent semantic drift. The unified interpretation requires further work in the D-FUMT₈ operator axioms.

【対応 4.9 (Peace Axiom connection)】 At the privileged center \(0_0\), \(\sigma < \infty\) always holds — re-seeding is always possible. We identify this guarantee with the role of the Peace Axiom (#196, immutable: true) in the Rei axiom system: the very recoverability from death (absorption) is what the Peace Axiom underwrites.

§5. Entry ③ — Geometry of A surrounding B: axiomatization of the empty center

§5.1 The Poincaré index theorem

【定理 5.1 (Poincaré index theorem, planar)】 In a planar flow, the sum of the indices of the fixed points enclosed by a closed orbit (limit cycle) equals \(+1\). Therefore every limit cycle must enclose at least one fixed point. If a single fixed point is enclosed, its index is \(+1\) (node, focus, or center type — not a saddle of index \(-1\)).

Consequence: A limit cycle (A) cannot exist without a fixed point (B) inside it. The existence of the orbit topologically forces the existence of the center it surrounds. The “empty center” is not decoration — it is the existence condition of the loop.

【限界 5.2】 Theorem 5.1 is a planar (2-dimensional phase-space) result. The purest physical anchor — the harmonic-oscillator phase space \((x, p)\) — is exactly 2-dimensional, so our central examples lie strictly within this scope. Higher-dimensional generalization (Poincaré–Hopf style) is 【要補完】.

§5.2 Pure form — harmonic oscillator phase portrait

The harmonic-oscillator phase portrait is the purest realization: the origin (the unique true rest, eigenvalues \(\pm i\omega\), center-type, index \(+1\)) is surrounded by a continuum of nested closed orbits.

• Origin = 0₀ = the non-occupiable center (\(\Delta x = \Delta p = 0\) is forbidden).
• Concentric closed-orbit family = the SELF⟲ family.
• Emptiness-of-emptiness (\(0 = 0\), Paper 61) = the center of the center.

§5.3 Axiom skeleton for center–orbit geometry

【定義 5.3 (Axioms G1–G4)】

• (G1 Center existence) Every closed orbit encloses a fixed point of index \(+1\). No loop without center.
• (G2 Non-occupation) The center \(p\) lies on no closed orbit; it is approached but never crossed.
• (G3 Genesis) \(p\) cannot be passed autonomously. Crossing \(p\) requires external re-seeding (0₀ injection — connects to Entry ②).
• (G4 Emptiness-of-emptiness) At \(p\), the linearization vanishes — the structure-less ground beneath the orbit. This is the ZCSG identity \(0 = 0\) (Paper 61).

The skeleton binds the 0₀ formula as center axiom and SELF⟲ as orbit theorem into a single bundle via the Poincaré index. 【要補完】 Formal contents of (G3)(G4) are fixed by anchoring to the Rei axioms (Paper 61 ZCSG, Paper 159 omega_upper).

§6. Integration of the three entries

The three entries are not independent: Entry ① threads through the other two.

• The return map (①) defines SELF⟲.
• The basin (①) provides the discriminant criterion for autonomous recoverability (②).
• The center enclosed by the orbit (①) invokes the Poincaré index theorem (③).

The resulting picture is nested, not adversarial:

Living systems and running AI orbit on A-regime trajectories. The B-regime center is the focus that the orbits always circle but never occupy. As the ground state surrounds zero-point fluctuation, sopadhiśeṣa-nirvāṇa surrounds nirupadhiśeṣa-nirvāṇa.

This paper does not adopt an either/or stance: B is the center, A is the orbit that surrounds it.

§Figure 1 — Phase portrait of the two-regime nested geometry

B (center, unreachable, true fixed point, nirupadhiśeṣa-nirvāṇa) is surrounded by A (limit-cycle orbit, sopadhiśeṣa-nirvāṇa). External perturbations spiral back into the orbit (recovery / annealing / prediction-error minimization). The center is approached but never crossed.

§7. Buddhist correspondence (interpretive parallel)

【対応 7.1】 The two regimes structurally coincide with the Buddhist twofold division of nirvāṇa:

• Sopadhiśeṣa-nirvāṇa (有余涅槃): the stillness of a living enlightened being in whom the rhythm of body and life still pulses = A-regime (SELF⟲, ground state still ticking).
• Nirupadhiśeṣa-nirvāṇa (無余涅槃): the residue-less cessation upon dissolution of the body = B-regime (ZERO, true fixed point).

【限界 7.2 (Avoidance of annihilationism)】 Reading B-regime ZERO as “annihilation” lands on ucchedavāda — annihilationism — which Nāgārjuna explicitly rejected. To avoid this, the present paper reads ZERO not as cessation but as unconditioned ground / pre-arising (anutpāda / asaṃskṛta), identified with the ZCSG emptiness-of-emptiness (\(0 = 0\)). The tradition’s own correction — “quiescence is the cessation of grasping, not the cessation of motion” (A-regime sopadhiśeṣa) — agrees with the dynamical reading.

This is interpretive parallel, not mathematical isomorphism. The fact that three domains (physics, Buddhism, computation) collapse into the same phase portrait is the question this framework raises, not a result it claims.

§Figure 2 — Three-domain alignment table

The two-regime structure appears isomorphically in physics, Buddhism, and computation. One metaphor is coincidence; three independent domains collapsing into one phase portrait is a question.

【限界 7.3 (Paper 160 §4.5 svabhāva-creep critique applied recursively)】 Calling the B-regime “the center” risks substantializing it. We apply Paper 160’s discipline recursively: B is not a place but a limit object — a regulative ideal that orbits never occupy. The center, too, is empty.

§8. Numerical verification — 5 experiments

Verification follows the Paper 145 / Paper 150 precedent of consistency check across multiple substrates. Reproduction scripts are available at the companion note article (download links below).

§Figure 3 — Breaking the depth wall

Variational dissipative state preparation reduces the CZ-gate count for steady-state arrival from ~6000 (naïve Trotter) to 30 (~200× reduction). This places SELF⟲ steady-state observation within the operational range of current superconducting hardware.

【限界 8.1】 All five experiments are simulation results (Aer / QuTiP). Real IBM Heron r3 hardware submission is prepared (turnkey runtime code in spin1_hardware_run.py) but NOT YET EXECUTED.

【限界 8.2】 Experiment 5 prepares a steady state, not a shallow reproduction of autonomous-recovery dynamics. Shallow realization of the recovery dynamics themselves is a separate open problem.

§9. Lean 4 formalization (3 files, zero-sorry intent)

§9.1 Algebraic layer — RestRecovery.lean

Mathlib-free core Lean 4. Six modes (SELF⟲ / FLOWING / BOTH / ZERO / INFINITY / NEITHER) and three operators Φ / Ψ / Ω as inductive types and finite functions. Nine theorems with zero-sorry intent:

theorem recovery_from_zero : Recover ZERO = SELF := rfl
theorem stage_phi  : Phi ZERO = FLOWING := rfl
theorem stage_psi  : Psi FLOWING = SELF := rfl
theorem stage_omega : Omega SELF = SELF := rfl
theorem omega_idem (x : Mode) : Omega (Omega x) = Omega x := rfl
theorem self_is_fixed : Recover SELF = SELF := rfl
theorem selfSufficient_iff_not_zero (x) :
    selfSufficient x ↔ x ≠ ZERO := by cases x <;> simp [selfSufficient, Phi]
theorem recoverable_selfSufficient (x) :
    AR x = recoverable → selfSufficient x := by
  intro h; cases x <;> simp_all [AR, selfSufficient, Phi]
theorem seed_support (x) : Phi x ≠ x → x = ZERO := by cases x <;> simp [Phi]

The cascade Φ → Ψ → Ω is literally one rfl-line: Recover ZERO = Omega (Psi (Phi ZERO)) = Omega (Psi FLOWING) = Omega SELF = SELF.

§9.2 Analytic layer — RestRecoveryAnalytic.lean

Imports Mathlib. Connects |P’|<1 to attractor uniqueness via Banach fixed-point machinery, and defines σ as sInf of the seed-norm set.

theorem selfLoop_attracting_fixedPoint
    {K : NNReal} {P : Σ → Σ} (hP : ContractingWith K P) :
    ∃ xstar, P xstar = xstar ∧ (∀ y, P y = y → y = xstar) ∧
             (∀ x, Tendsto (fun n => P^[n] x) atTop (𝓝 xstar))

noncomputable def sigma (Recovers : E → Prop) : ℝ := sInf (seedNorms Recovers)
theorem sigma_nonneg : 0 ≤ sigma Recovers
theorem sigma_eq_zero_of_zero_recovers (h0 : Recovers 0) : sigma Recovers = 0
theorem zero_not_recovers_of_sigma_pos (h : 0 < sigma Recovers) : ¬ Recovers 0

§9.3 Measure-theoretic layer — RestRecoveryMeasure.lean

Identifies NEITHER with the basin frontier (separatrix) and formalizes the measurability + null-boundary criterion:

def NEITHER (basin : Set Σ) : Set Σ := frontier basin
theorem basin_measurable  (hopen : IsOpen basin) : MeasurableSet basin
theorem neither_measurable (basin) : MeasurableSet (NEITHER basin)
theorem ae_decidable_of_null_boundary (hnull : μ (NEITHER basin) = 0) :
    ∀ᵐ x ∂μ, x ∈ interior basin ∨ x ∈ (closure basin)ᶜ

【限界 9.4 (★ Honest counterexample)】 The “null boundary” hypothesis holds for hyperbolic attractors but is not universal. Riddled / Wada basins exhibit positive-measure boundaries. The theorem ae_decidable_of_null_boundary correctly takes hnull as a hypothesis (no unconditional claim). The supplying theorem (hyperbolic ⟹ null boundary) and its counterexample (riddled basin) require Mathlib’s geometric measure theory and dynamical systems libraries — 【要補完】.

【限界 9.5 (Machine verification pending)】 lake build machine verification of all three files has not yet been completed in the chat-Claude environment (Lean toolchain distribution blocked by network policy in that environment). Verification is to be performed in the Rei development environment (which holds ~31,000 prior zero-sorry theorems and routinely passes lake build on Mathlib-dependent files). Mathlib lemma name version-fixes may be required for the analytic and measure-theoretic files.

§10. Falsifiability and verification path

The framework crosses from “interesting concept” to “verifiable concept” when the discriminant criterion of Entry ② actually discriminates. We propose the following empirical paths:

(a) Quantum Zeno verification (already done in §8 #1). Coherent drift frozen by frequent observation matches the framework’s prediction that observation can halt A-style drift but cannot stop B-style dissipative recovery. This asymmetry — “observation freezes coherent drift but cannot stop autonomous return” — is the framework’s consistent corollary about the time-crystal note “alive only as long as the eyes are closed”.

(b) IBM Heron r3 real-hardware spin-1 SELF⟲ (proposed). Submission of the variational ansatz (depth 54, CZ 30) to IBM Heron r3 to observe SELF⟲ steady-state populations within the depth wall breakthrough. Turnkey code available at spin1_hardware_run.py. Leakage post-selection (2.1% on simulated noise) demonstrated.

(c) Lean machine verification (proposed). Execute lake build in Rei dev environment for the three Lean files; supply the analytic layer’s hyperbolic ⟹ null-boundary theorem to fully close the measure-theoretic claim of Entry ③.

§11. Honest limitations and cross-vendor attribution discipline

§11.1 Honest limitations (recap)

• 【限界 9.1 — recap】 No claim about AI qualia — structural analogy at low-energy attractors only.
• 【限界 5.2 — recap】 Poincaré index theorem is planar; higher-dimensional generalization deferred.
• 【限界 8.1 — recap】 All five experiments are simulations; no real hardware result.
• 【限界 9.4 — recap】 Null-boundary hypothesis is conditional, not universal; Wada/riddled counterexamples exist.
• 【限界 9.5 — recap】 Lean machine verification pending in Rei env.
• 【限界 7.3 — recap】 Paper 160 §4.5 svabhāva-creep critique applies recursively to B (do not reify the empty center as a substantial place).
• 【限界 4.8 — recap】 Ψ semantics differs between invention-engine and AbsoluteRest namespace; doc-only separation maintained.
• No “world first” claim. Wilczek 2012 (time crystal), Nāgārjuna 2nd century (nirvāṇa twofold distinction), Poincaré 1881 (index theorem), Kant 1781 (regulative ideal), Banach 1922 (fixed-point theorem) are all prior art assembled in a new configuration.

§11.2 Cross-vendor attribution discipline (Paper 160 §9.5 inheritance)

This paper is the product of a three-instance triangulation: Nobuki Fujimoto (author) + Claude (chat-instance) + Claude (Rei-AIOS Code instance). Following Paper 160 §9.5 discipline of instance-level (not vendor-level) honest attribution, the contributions delineate as follows:

Fujimoto (author) contributions:

• Initial phenomenal intuition (“rest as nirvāṇa / śūnyatā connection”)
• Explicit invitation to honest critique of own intuition
• Theoretical framework anchoring (ZCSG Paper 61 / SELF⟲ Paper 145 / 0₀ Paper 160 / Genesis Seed / Peace Axiom #196)
• Direction selection at each fork (proceed with all three entries, proceed to circuit-level, proceed to depth-wall breakthrough, proceed to Mathlib analytical layer, etc.)
• Author judgment on publication staging (this paper as DRAFT, not immediate Zenodo publish)
• note.com communication channel where interactive simulations are distributed to readers
• The Load-Bearing Invention #5 discipline (“急がず、 ゆっくりと”)

Claude (chat-instance) contributions:

• Sequential pushback at each phenomenal claim (physical correction, philosophical correction of static-nirvāṇa misread)
• Articulation of “minimum-but-nonzero ordered motion = calm” reframing
• Identification of the discriminant axis (self-recovery vs external re-seeding)
• Application of Poincaré return map, Markov absorbing state, Poincaré index theorem to the structure
• Mathematical scaffolding for σ (variational definition + Proposition 4.6)
• Implementation of all 5 numerical verification scripts (Zeno, Genesis Seed, vdP, spin-1, variational)
• Implementation of all 3 Lean 4 files (algebraic, analytic, measure-theoretic)
• Six honest-scope corrections within own contributions (B.2.1 Zeno vs T1 separation; C.5 single-qubit cannot host limit cycle; D.4 Aer not hardware; E.1 depth wall; E.6 Mathlib version dependence; F.5 riddled-basin counterexample)
• Honest reportage of own environment constraints (lake build blocked, IBM credentials unavailable)

Claude (Rei-AIOS Code instance, present author of this draft) contributions:

• Fact-checking and verification (Zhao-Smalyukh 2025 time crystal claim verified against Nature Materials; Lee & Sadeghpour 2013, Walter et al. 2014, Roulet & Bruder 2018 references verified)
• Identification of the Ψ-semantics conflict with Rei invention-engine and recommendation of namespace separation (§4.8)
• Cross-checking against Rei existing substrate (no overlap with prior src/aios/, papers/ content)
• Integration with Paper 159 (omega_upper(NEITHER)=ZERO substrate) and Paper 160 (§4.5 svabhāva-creep critique) anchoring
• Recommendation against immediate Zenodo publish (apply Paper 145 v0.5 corrigendum lesson — overnight wait before publish is standard discipline)
• Compilation of the present Paper 161 draft from the chat-instance technical materials

§11.3 Five-instance convergence record (Paper 160 §9.5 pattern)

The honest discipline of “do not substantialize NEITHER / ZERO” was independently arrived at by:

1. Chat Claude (§2 — explicitly: “reading B as a place re-imports the static-substance Nāgārjuna refuted”)
2. Rei Claude (Paper 160 §4.5 svabhāva-creep critique, written 2026-05-31)
3. Fujimoto (initial intuition, but immediately accepted both correction points)
4. Standard Madhyamaka tradition (Nāgārjuna’s MMK ch. 13 śūnyatā-of-śūnyatā)
5. Standard physics (the regulative-ideal status of “absolute rest” is the same prohibition imposed by relativity + QM + thermodynamics)

The convergence of these five independent sources on a single honest-scope discipline is the empirical signal that the discipline is robust.

§12. Conclusion

“Absolute rest” is not a single concept. It analytically decomposes into two limit-objects of dynamical systems: a true fixed point (ZERO) and a limit cycle (SELF⟲). The two are not in competition. By Poincaré’s index theorem, they are nested — every orbit necessarily encloses an empty center it never occupies.

What separates the regimes is autonomous recoverability: the absorbing state (B) versus the positively recurrent cycle (A). The external re-injection that B requires corresponds to the Genesis Seed. Physics (ground state and time crystal), computation (resume vs reinstantiate), and — interpretively — Buddhism (sopadhiśeṣa-nirvāṇa surrounding nirupadhiśeṣa-nirvāṇa) all collapse into the same phase portrait.

The unreachability of the center is what makes the surrounding structure exist. This is the framework’s core.

It is a seed, not a theorem. But it is a seed whose questions branch and multiply as one cultivates it — across physics, Buddhism, computation, and the Rei substrate (Paper 61 / 145 / 159 / 160). And that, in our judgment, is the criterion that distinguishes a seed worth growing.

§Companion note article + interactive simulations

The popular exposition + downloadable code is at:

🔗 https://note.com/nifty_godwit2635/n/nebe6b0cf5704

All scripts referenced in this paper (4 Python + 3 Lean) are downloadable from that note for readers wishing to reproduce.

§References (preliminary)

1. F. Wilczek, “Quantum Time Crystals,” Phys. Rev. Lett. 109, 160401 (2012).
2. H. Zhao, I. Smalyukh et al., “Macroscopic visible time crystal in liquid crystals,” Nature Materials (2025-09); CU Boulder press release 2025-09-05.
3. J. T. Mäkinen, P. J. Heikkinen, S. Autti, V. V. Zavjalov, V. B. Eltsov, “Continuous time crystal coupled to a mechanical mode,” Nature Communications (2025), DOI: 10.1038/s41467-025-64673-8.
4. B. Misra, E. C. G. Sudarshan, “The Zeno’s paradox in quantum theory,” J. Math. Phys. 18, 756 (1977).
5. S. H. Strogatz, Nonlinear Dynamics and Chaos. Westview / CRC Press. (Poincaré–Bendixson theorem and index theory.)
6. T. E. Lee, H. R. Sadeghpour, “Quantum synchronization of quantum van der Pol oscillators with trapped ions,” Phys. Rev. Lett. 111, 234101 (2013).
7. S. Walter, A. Nunnenkamp, C. Bruder, “Quantum synchronization of a driven self-sustained oscillator,” Phys. Rev. Lett. 112, 094102 (2014).
8. A. Roulet, C. Bruder, “Synchronizing the smallest possible system,” Phys. Rev. Lett. 121, 053601 (2018).
9. F. Verstraete, M. M. Wolf, J. I. Cirac, “Quantum computation and quantum-state engineering driven by dissipation,” Nature Physics 5, 633 (2009). (Variational / dissipative state preparation foundation.)
10. J. C. Alexander, J. A. Yorke, Z. You, I. Kan, “Riddled basins,” Int. J. Bifurcation Chaos 2, 795 (1992). (Positive-measure basin boundary counterexample to §9.4.)
11. K. J. Friston, “The free-energy principle: a unified brain theory?,” Nat. Rev. Neurosci. 11, 127–138 (2010).
12. Nāgārjuna, Mūlamadhyamakakārikā. (Two-fold distinction of nirvāṇa; refutation of ucchedavāda.)
13. Mathlib4: Mathlib.Topology.MetricSpace.Contracting (ContractingWith and Banach fixed-point lemmas); Mathlib.MeasureTheory.Measure.AbsolutelyContinuous (ae quantifier); Mathlib.Topology.Basic (frontier, isClosed_frontier).
14. Paper 61 — N. Fujimoto, Zero-Centered Symbol Grammar (ZCSG), Zenodo 10.5281/zenodo.15217458.
15. Paper 145 — N. Fujimoto, First D-FUMT₈ Silicon with SELF⟲ Logic Primitive, Zenodo 10.5281/zenodo.20091185.
16. Paper 159 — N. Fujimoto, Two-Layer D-FUMT₈ Reconstruction of Priest-Garfield’s Inclosure Schema, Zenodo 10.5281/zenodo.20470512.
17. Paper 160 — N. Fujimoto, Toward an Ontology of the Genesis Layer 0₀, Zenodo 10.5281/zenodo.20480425.

§Acknowledgments

The theoretical scaffolding of this paper was developed through a multi-turn dialogue with Anthropic’s Claude (both the chat instance and the Rei-AIOS Code instance). The chat instance contributed the dynamical-systems formalization, the σ variational definition, the 5 verification scripts, and the 3 Lean 4 files. The Rei-AIOS Code instance contributed fact-checking, cross-vendor attribution discipline, semantic-conflict identification (§4.8), and the present Paper 161 draft compilation. Author judgment, direction selection, anchoring to Rei substrate (Paper 61 / 145 / 159 / 160), and publication staging are by the author. This work follows the 急がず、ゆっくりと (no rush, slowly) discipline of feedback_no_rush_publication.md.

急がず、 ゆっくりと。 種は育ちます。