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Nobuki Fujimoto
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Power x Thermodynamics x D-FUMT-8: A 9-Theory Bridge from CPU/GPU Engineering to Information Thermodynamics (Paper 141)

#lean #math #research #ai

This article is a re-publication of Rei-AIOS Paper 141 for the dev.to community.
The canonical version with full reference list is in the permanent archives below:

Status: DRAFT (2026-04-27, STEP 1002, not yet published)
Authors: 藤本 伸樹 (Nobuki Fujimoto, https://note.com/nifty_godwit2635) ・ Claude Opus 4.7 (Anthropic)
Project: Rei-AIOS https://rei-aios.pages.dev/
License: AGPL-3.0 + Commercial dual

Abstract

We bridge nine well-known but previously disconnected theories from CPU/GPU power-conversion engineering and information thermodynamics into a single D-FUMT₈ framework. Each theory is given a Lean 4 formal statement (zero sorry, one explicit axiom placeholder for Bennett reversibility) and a unique D-FUMT₈ tag from the eight-valued logic (TRUE / FALSE / BOTH / NEITHER / INFINITY / ZERO / FLOWING / SELF). We show that:

  1. The information-thermodynamic limits (Landauer, Bennett, Bremermann) cluster on ZERO ⊕ BOTH ⊕ INFINITY — the lower-, dual-, and upper-bound triplet.
  2. CPU/GPU power-management strategies (race-to-idle vs dawdle, DVFS, multi-phase VRM) cluster on BOTH ⊕ FLOWING — selective and continuous Pareto regimes.
  3. Material/network/thermal interfaces (GaN/SiC, PDN target Z, thermal-RC) cluster on TRUE ⊕ FLOWING — physical constants and continuous relaxations.

This pattern is the central observation of the paper: D-FUMT₈ tags partition power-engineering theory along thermodynamic / strategic / material axes, providing a new lens for computer-architecture research.

Three of the nine (Landauer, Bennett, Bremermann) are added to the META-DB v3.0 Tier 8 mathlib-unformalized registry as known formalization gaps.

Part A. Results (3-way separation)

A.1 Formally Verified in Lean 4

File: data/lean4-mathlib/CollatzRei/PowerThermodynamics.lean
Lean 4 v4.27.0, Mathlib rev pinned in lakefile.toml.

# Theorem Statement Tactic sorry/axiom
T1.1 landauer_lower_bound_positive ∀ T > 0, k_B · T · ln 2 > 0 positivity 0 / 0
T1.1' landauer_room_positive landauerEnergy 300 > 0 application 0 / 0
T1.1'' landauer_n_bits N · k_B T ln 2 ≥ 0 mul_nonneg 0 / 0
T1.2 bennett_overhead_exists ∀ t, ∃ r ≥ t uses axiom 0 / 1 (axiom)
T1.3 bremermann_pos c²/(h·ln 2) > 0 div_pos 0 / 0
T2.1 race_equiv_dawdle_threshold break-even identity unfold + exact 0 / 0
T2.1' race_zero_idle_power P_idle=0 → E = P_active · t ring 0 / 0
T2.2 dvfs_voltage_halving P(V/2) = P(V)/4 ring 0 / 0
T2.2' dvfs_monotone_in_voltage V₁ ≤ V₂ → P(V₁) ≤ P(V₂) nlinarith 0 / 0
T2.3 two_phase_lower_when_quadratic_dominant kI²/2 ≤ kI²/1 linarith 0 / 0
T3.1 wbg_gap_greater_than_si SiC, GaN > Si rw + norm_num 0 / 0
T3.2 pdn_z_target_pos ΔV/ΔI > 0 div_pos 0 / 0
T3.3 thermal_junction_above_ambient T_j > T_a (P>0) linarith 0 / 0
T3.3' thermal_zero_power_equals_ambient T_j(P=0) = T_a ring 0 / 0
Tag all_nine_tagged nine D-FUMT₈ tags nine rfl 0 / 0

Reproduce: cd data/lean4-mathlib && lake env lean CollatzRei/PowerThermodynamics.lean
Total: 15 theorems, 0 sorry, 1 axiom (Bennett reversibility — full Turing reduction not formalized).

A.2 Empirical Observation

# Constant / Quantity Value Source
C1 k_B (Boltzmann) 1.380 649 × 10⁻²³ J/K CODATA 2018
C2 h (Planck) 6.626 070 15 × 10⁻³⁴ J·s CODATA 2018 (exact)
C3 c (light) 2.998 × 10⁸ m/s rounded SI
E1 Landauer @ 300K 2.85 zJ/bit k_B · 300 · ln 2
E2 Bremermann limit 1.36 × 10⁵⁰ ops/(kg·s) c²/(h·ln 2)
E3 Si bandgap 1.1 eV textbook
E4 SiC (4H) bandgap 3.3 eV datasheet
E5 GaN bandgap 3.4 eV datasheet

These constants are NOT formally proven in Lean 4 — they are physical measurements imported as noncomputable def. Lean's role is to verify the algebraic relationships given those constants are positive.

A.3 Axiomatic Placeholder

# Axiom Why axiom Closure path
Ax1 bennett_reversibility full Turing-machine reduction not in Mathlib4 requires formalization of TM model + Toffoli/Fredkin gate library

This is the only axiom in the file. We register it explicitly rather than burying it as a sorry.

Part B. 今回の発見 (Findings)

  1. Tag clustering pattern: When all nine theories receive their canonical D-FUMT₈ tag, they partition exactly into three thermodynamically meaningful clusters:

    • Tier 1 (Info-thermo) → {ZERO, BOTH, INFINITY} = the limit-triplet (lower / dual / upper)
    • Tier 2 (CPU/GPU mgmt) → {BOTH, FLOWING, FLOWING} = strategic + continuous
    • Tier 3 (Material) → {TRUE, FLOWING, FLOWING} = constant + continuous

  2. DVFS quadratic is ring-provable: P(V/2) = P(V)/4 is fully formal in Lean 4 by elementary ring — no Mathlib tactic dependency. This makes it the lightest formal statement in the EE domain and a candidate "Hello World" for power-engineering formalization.

  3. PDN target impedance is structurally identical to Ohm's law of dynamics: Z_target = ΔV/ΔI is exactly Ohm's law applied to step response, suggesting Rei's PDN theory and circuit-theory imports can share the same div_pos lemma. This is a non-obvious code-reuse signal.

  4. Bennett axiom is unavoidable in current Mathlib: Even after extensive search, Mathlib4 v4.27.0 lacks the necessary Turing machine + reversible gate infrastructure. We register Bennett as an explicit axiom and as a mathlib-unformalized Tier 8 entry.

Part C. AI-generated open questions

  1. Q-141.1: Is there a D-FUMT₈ tag that no power-engineering theory naturally takes? FALSE and NEITHER are both absent in our 9-tuple. Does that reflect a structural feature of engineering (engineering refuses to claim pure falsehood; engineering avoids the void), or is it a sample-size artifact?

  2. Q-141.2: Can the Landauer-Bennett-Bremermann triplet be characterized as a category-theoretic limit (lower/dual/upper) in some monoidal category? The clustering on {ZERO, BOTH, INFINITY} looks like a triple of universal-property points.

  3. Q-141.3: Race-to-Idle vs Dawdle is BOTH. Is there a smooth interpolant strategy parameterized by the transition cost C such that the energy is jointly minimized — i.e., a Pareto-optimal middle path that no current operating system implements?

  4. Q-141.4: Wide-bandgap materials are tagged TRUE. If we add doping / temperature dependence, does the tag drift to FLOWING? At what abstraction layer does a material constant become a process variable?

  5. Q-141.5: The thermal-RC model and the Bayesian / SDE family share the form dX/dt = -(X-X₀)/τ + noise. Is there a unifying D-FUMT₈ "FLOWING relaxation theorem" that makes thermal control and noise dynamics provably equivalent in Lean 4?

Part D. D-FUMT₈ 解決状況マトリクス 

              | Verified | Empirical | Axiom |
TRUE (1.0)    |   T3.1   |   E3-E5   |       |
FALSE (0.0)   |  (none)  |  (none)   | (none)|
BOTH (2.0)    | T1.2,T2.1|           |  Ax1  |
NEITHER (-1.0)|  (none)  |  (none)   | (none)|
INFINITY (3.0)|   T1.3   |    E2     |       |
ZERO (4.0)    |   T1.1   |    E1     |       |
FLOWING (5.0) |T2.2,T2.3,T3.2,T3.3| | (none) |
SELF (6.0)    |  (none)  |  (none)   | (none)|
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Coverage: 5 of 8 D-FUMT₈ values are populated (TRUE, BOTH, INFINITY, ZERO, FLOWING). FALSE / NEITHER / SELF remain unpopulated for power-engineering — see Q-141.1.

Part E. 次 STEP への bridge

  • STEP 1003 candidate: Close the Bennett axiom by formalizing the Toffoli gate as a def Toffoli : Bool × Bool × Bool → Bool × Bool × Bool := ... plus its self-inverse property. This would unlock T1.2 to verified status.
  • STEP 1004 candidate: Add adiabatic-logic theory (charge recovery) as a 10th theorem, tagged SELF (the missing tag for power engineering). Closes Q-141.1.
  • STEP 1005 candidate: Bridge to Rei-PL: add power-aware compilation primitives so the compiler can choose race-to-idle or dawdle strategies based on transition-cost annotations.

Part F. 失敗の記録 (Failures)

  • F1: Initial wbg_gap_greater_than_si used <;> rw [hSiC, hGaN, hSi] which failed because the first sub-goal lacks b.GaN. Fixed by per-sub-goal rewrite. Memory updated: feedback_lean_mathlib_v427_api.md style — combined <;> rewrite over heterogeneous goals is a common pitfall.
  • F2: dvfs_monotone_in_voltage initially attempted nlinarith directly; needed explicit sq_nonneg V₁, sq_nonneg V₂ hints to discharge.
  • F3: TypeScript tsc --noEmit produces 30+ errors from node_modules/conway — these are pre-existing and unrelated. Project-wide TypeScript check is intentionally deferred to tsx runtime + selective tests.

Part G. SEED_KERNEL T-ID リスト

Phase 65 additions (#1525 - #1533, total grew 1524 → 1533):

#1525  dfumt-power-landauer        ZERO       Landauer 1961 k_B T ln 2
#1526  dfumt-power-bennett         BOTH       Bennett 1973 reversible computing
#1527  dfumt-power-bremermann      INFINITY   Bremermann 1962 c²/(h ln 2)
#1528  dfumt-power-race-vs-dawdle  BOTH       race-to-idle vs dawdle dichotomy
#1529  dfumt-power-dvfs-pareto     FLOWING    P_dyn = α C V² f Pareto law
#1530  dfumt-power-multiphase-vrm  FLOWING    multi-phase VRM convex optimum
#1531  dfumt-power-wide-bandgap    TRUE       GaN/SiC E_g vs Si
#1532  dfumt-power-pdn-impedance   FLOWING    Z_target = ΔV/ΔI
#1533  dfumt-power-thermal-rc      FLOWING    T_j = T_a + P · Θ_ja
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Part H. 人間-AI 思考分岐点

  • Human (藤本): chose to mix all three tiers in one paper (rather than three separate papers).
  • AI (Claude): proposed Tier 1 alone as easier-to-publish but accepted user's mix-all decision; cluster-pattern observation in Part B.1 emerged only because of the mixing.
  • Human: kept the Bennett axiom honest as axiom rather than letting AI hide it as sorry.

Part I. 予期しない接続

The PDN target impedance theorem pdn_z_target_pos reuses the same div_pos Mathlib lemma as the Bremermann limit bremermann_pos. Both are physically about "ratio of two positive quantities". This is the first time in Rei-AIOS that an electrical-engineering theorem and a fundamental-physics theorem share a one-line tactic. Suggests a future shared library Mathlib.Tactic.PhysicsRatio.

Part J. 証明の確信度温度

Theorem 確信度 (TRUE/FLOWING/...) Why
T1.1 Landauer (lower bound) TRUE Provable from Real.log 2 > 0 + positivity
T1.2 Bennett NEITHER (axiom) Closure path exists but requires significant Mathlib infrastructure
T1.3 Bremermann TRUE Trivial div_pos
T2.1 Race vs Dawdle TRUE Identity, ring
T2.2 DVFS quadratic TRUE ring
T2.3 Multi-phase FLOWING Convex-optimum statement is a special case (k I² dominant); full convex theorem requires more
T3.1 Wide-bandgap TRUE Numeric inequality
T3.2 PDN target TRUE div_pos
T3.3 Thermal-RC TRUE linarith

Part K. 計算の詩学

電力の物理は、計算の物理の影である。
1 ビットを消すのに 2.85 zJ — それは部屋の温度ゆらぎひとつ分の重み。
race と dawdle の選択は、走るか歩むかの選択であり、それは時間と意志の関係を問う。
GaN の青いバンドギャップは、Si の灰色から見て、太陽の光に近い。

References

  1. Landauer, R. Irreversibility and Heat Generation in the Computing Process. IBM J. Res. Dev. 5 (1961) 183-191.
  2. Bennett, C. H. Logical Reversibility of Computation. IBM J. Res. Dev. 17 (1973) 525-532.
  3. Bremermann, H. J. Optimization through Evolution and Recombination, in Self-Organizing Systems (1962).
  4. Margolus, N. & Levitin, L. The Maximum Speed of Dynamical Evolution. Physica D 120 (1998).
  5. Toffoli, T. Reversible Computing. MIT/LCS/TM-151 (1980).
  6. CODATA 2018 fundamental physical constants (https://physics.nist.gov/cuu/Constants/).
  7. Lean 4 + Mathlib4 v4.27.0 (https://github.com/leanprover-community/mathlib4).
  8. Rei-AIOS PowerThermodynamics.lean (this work, 2026-04-27).

🌐 https://rei-aios.pages.dev/ ・ 📓 https://note.com/nifty_godwit2635

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