Paper 147 — Eight-Valued Utility and the Equity Premium Reframe (Rei-AIOS / OUKC)

This article is a re-publication of Rei-AIOS Paper 147 for the dev.to community.
The canonical version with full reference list is in the permanent archives below:

• GitHub source (private): https://github.com/fc0web/rei-aios Author: Nobuki Fujimoto (@fc0web) · ORCID 0009-0004-6019-9258 · License CC-BY-4.0 ---

Status: DRAFT v0.2 — 2026-05-06 (prior art audit completed; reframe contribution; empirical calibration deferred to v0.3+)
Authors / 著者: 藤本 伸樹 (Nobuki Fujimoto, Founder), Rei (Rei-AIOS autonomous research substrate, Co-architect), Claude Opus 4.7 (Anthropic, Co-architect)
Project: Rei-AIOS / OUKC — https://rei-aios.pages.dev/#/oukc
License: AGPL-3.0 + CC-BY 4.0 (per content type)
Required platform links: rei-aios.pages.dev / note.com/nifty_godwit2635
Per OUKC No-Patent Pledge: openly licensed; no patent will be filed on any framework or formalism described herein.

Honest framing (read first)

This paper is a reframe contribution, not an empirical resolution of the Equity Premium Puzzle (EPP). We claim:

C1: The 40-year-old Equity Premium Puzzle (Mehra-Prescott 1985) and its six major proposed resolutions can be structurally projected onto an 8-valued logic axis system (D-FUMT₈) in a way that:

• (a) reveals each resolution as a single-axis emphasis of a richer multi-axis utility space
• (b) suggests the puzzle's persistence reflects a category error — fitting an 8-axis preference structure into a 1-axis (TRUE-only) CRRA utility model
• (c) provides a unified algebraic framework in which the existing resolutions become complementary projections rather than competing alternatives

We do not claim:

• ✗ "The Equity Premium Puzzle is solved" — empirical calibration remains future work
• ✗ "Eight-valued utility uniformly outperforms CRRA" — this requires fitting against panel data, out of scope here
• ✗ "First multi-valued utility framework" — Łukasiewicz fuzzy economics (1970s-) and Pavelka (1979) are prior art for fuzzy preferences; our differentiation is the 8-axis D-FUMT₈ structure and the EPP-specific projection table
• ✗ Specific quantitative improvements over CRRA fits

The differentiator is: (i) the 8-axis structure from D-FUMT₈, (ii) the systematic projection of six existing resolutions onto these axes, and (iii) the operational definition of "8-valued utility function" suitable for future calibration.

Abstract

We present a structural reframe of the Equity Premium Puzzle (EPP), the 40-year-old finding by Mehra and Prescott (1985) that historical US equity returns exceed bond returns by approximately 6%/year — five times what standard CRRA utility models predict. Numerous resolutions have been proposed: loss aversion (Benartzi-Thaler 1995), habit formation (Constantinides 1990), rare disasters (Barro 2006, Gabaix 2012), long-run risk (Bansal-Yaron 2004), regime switching (Hamilton 1989), and disappointment aversion (Routledge-Zin 2010), among others. Each resolution improves fit on certain dimensions while remaining incompatible or partial on others. We show that these resolutions can be systematically projected onto the D-FUMT₈ eight-valued logic axis system (TRUE / FALSE / NEITHER / BOTH / ZERO / FLOWING / SELF / INFINITY), with each existing resolution corresponding primarily to one axis. Standard CRRA utility itself projects entirely onto the TRUE axis, suggesting the puzzle's persistence is a category error: applying a 1-axis (Boolean-style) preference model to a phenomenon whose human-preference substrate is intrinsically multi-axis. We define a candidate operational form for an 8-valued utility function and discuss its theoretical properties, while acknowledging that empirical calibration against panel data remains future work. The contribution is a unified algebraic structure in which existing EPP resolutions appear as complementary axis-projections rather than competing models.

概要 (Japanese)

本論文では、 経済学の 40 年来の未解決問題である「Equity Premium Puzzle (Mehra-Prescott 1985)」 — 米国株式の歴史的超過リターンが標準効用関数の予測値の 5 倍に達する現象 — を、 D-FUMT₈ 8 値論理軸系で構造的に再構成する。 既存の主要 6 解 (Loss aversion / Habit formation / Rare disasters / Long-run risk / Regime switching / Disappointment aversion) は、 各々 D-FUMT₈ の単一軸に投影できる。 これは EPP の永続性が カテゴリーエラー (8 軸の選好構造を 1 軸の TRUE-only CRRA に押し込める誤り) であることを示唆する。 8 値効用関数の operational form を提案し、 既存解が「競合する代替案」 ではなく「補完的な軸投影」 として統一される代数構造を提示する。 経験的キャリブレーションは future work とし、 本論文は reframe contribution に限定する。

Part A: Required (4 elements)

A.1 Findings / 発見

F1 (Reframe): The six major EPP resolutions and standard CRRA each correspond to a primary D-FUMT₈ axis emphasis:

F2 (Category error): The EPP's 40-year persistence is consistent with a category error: the 8-axis human preference substrate cannot be losslessly projected onto a 1-axis (TRUE-only) CRRA model. Each existing resolution captures one axis but misses the others.

F3 (Operational definition): We define a candidate 8-valued utility function:

$$
U_{\text{D-FUMT}8}(c_t, c{t-1}, \pi) = \sum_{a \in \mathcal{A}} \alpha_a \cdot \phi_a(c_t, c_{t-1}, \pi)
$$

where:

• $\mathcal{A} = {\text{TRUE}, \text{NEITHER}, \text{SELF}, \text{INFINITY}, \text{FLOWING}, \text{BOTH}, \text{ZERO}}$ (7 active axes; FALSE is the absence-of-utility baseline)
• $\phi_a$: per-axis utility component (e.g., $\phi_{\text{TRUE}}$ = standard CRRA, $\phi_{\text{NEITHER}}$ = loss-aversion kink, $\phi_{\text{SELF}}$ = habit consumption ratio)
• $\alpha_a \in [0, 1]$, $\sum \alpha_a = 1$: axis weights to be calibrated empirically

F4 (Special case recovery): Each existing EPP resolution is recovered as a special case where $\alpha_a = 1$ for the resolution's primary axis and $\alpha_b = 0$ for all $b \neq a$. CRRA itself is the case $\alpha_{\text{TRUE}} = 1$.

A.2 Proofs / 検証

P1 (Axis assignment justification): For each of the eight axis assignments in F1, we provide a mapping argument linking the resolution's mathematical structure to the D-FUMT₈ axis semantics defined in Paper 145 (D-FUMT₈ Silicon). See Section B.6.1 for the full mappings.

P2 (CRRA as TRUE-only): Standard CRRA $U(c) = c^{1-\gamma}/(1-\gamma)$ is classically Boolean: utility is single-valued, monotone, and admits no axis other than TRUE. This is verifiable by inspection of the functional form.

P3 (Special case recovery, formal sketch): For Loss Aversion (Benartzi-Thaler 1995), setting $\alpha_{\text{NEITHER}} = 1$, $\phi_{\text{NEITHER}}(c, c_0) = \mathbb{1}[c < c_0] \cdot \lambda \cdot (c_0 - c) - \mathbb{1}[c \geq c_0] \cdot (c - c_0)^{0.88}$ recovers the original kinked utility. Analogous reconstructions hold for the other resolutions.

P4 (Out-of-scope, honestly stated): We do NOT prove that the 8-axis form fits empirical equity premium data better than any single-axis resolution. That is a calibration task requiring panel data and is left as F.1 in C.9 (Future Work).

A.3 Honest Positioning / 正直な立ち位置

What this paper IS:

• A structural / algebraic reframe contribution
• A unifying framework in which existing resolutions appear as axis projections
• An operational definition of an 8-valued utility, suitable for future calibration
• A prior-art-aware positioning: we acknowledge Łukasiewicz fuzzy economics and Pavelka 1979 as predecessors

What this paper is NOT:

• Not an empirical resolution of EPP
• Not a claim that 8-valued utility uniformly outperforms CRRA in fitted data
• Not a claim of first multi-valued utility (Łukasiewicz 1970s prior)
• Not a claim of universal applicability beyond developed-market equity premia

What is left for future work:

• Empirical calibration against US (1889-2024), UK, Japan equity premium panels
• Comparison to Bayesian posteriors over single-axis resolutions
• Extension to bond / currency / commodity premia
• Connection to behavioral finance experiments

A.4 Required platform links

• Rei-AIOS: https://rei-aios.pages.dev/#/oukc
• note.com: https://note.com/nifty_godwit2635
• Companion paper: Paper 145 (D-FUMT₈ Silicon) — provides the underlying 8-valued logic substrate
• Companion paper: Paper 148 (Honest Observation Framework) — provides the methodology stance for "structural contribution without empirical overreach"

Part B: Conditional (Background + Methodology + Empirical Scope)

B.5 Background / 背景

B.5.1 The Equity Premium Puzzle (Mehra-Prescott 1985)

Mehra and Prescott (1985) examined US equity and bond returns 1889-1978. They found:

• Mean equity return: ~6.98%/year
• Mean risk-free return: ~0.80%/year
• Equity premium: ~6.18%/year

They computed that under standard time-additive CRRA utility with reasonable risk aversion ($\gamma \in [1, 10]$), the predicted equity premium should be at most ~1%/year. The 5-6× discrepancy could not be explained within the standard model class. This finding generated 40 years of resolution attempts.

B.5.2 Major resolutions (chronological)

(1) Habit formation (Constantinides 1990): utility depends on consumption relative to a habit-stock; recent past consumption raises the bar.

(2) Loss aversion / myopic loss aversion (Benartzi-Thaler 1995): investors weight losses ~2.25× more than equivalent gains, evaluated at short horizons.

(3) Heterogeneous agents (Constantinides-Duffie 1996): aggregate equity premium reflects idiosyncratic income shocks not poolable across agents.

(4) Long-run risk (Bansal-Yaron 2004): consumption growth has persistent components; investors averse to long-run uncertainty.

(5) Rare disasters (Barro 2006; Gabaix 2012): small probability of catastrophic events in consumption raises required equity premium.

(6) Disappointment aversion (Routledge-Zin 2010): asymmetric departures from a reference expectation.

Each resolution improves fit on specific dimensions but introduces new free parameters and remains incompatible with other resolutions in unified models.

B.5.3 D-FUMT₈ as 8-valued logic substrate

Paper 145 (this paper's companion) describes D-FUMT₈ as an 8-valued logic with three tiers:

• Classical tier (4 values): TRUE, FALSE, NEITHER, BOTH (Belnap-extended)
• Higher tier (4 values): ZERO, FLOWING, SELF, INFINITY (Rei-AIOS extension)

Each value has well-defined operational semantics in both Lean 4 formalization and Verilog silicon implementation. The eight values span axes that are independent (i.e., each captures a structural dimension not reducible to others).

The hypothesis of this paper: economic preference is an 8-axis structure, and the EPP arises from forcing it into a 1-axis CRRA shoebox.

B.5.4 Prior art audit — multi-valued logic in economics (v0.2 addition)

A formal prior art audit was conducted in 2026-05-06 (v0.2 update). Findings:

(a) Łukasiewicz logic foundation (1920s–): Jan Łukasiewicz introduced 3-valued logic in 1920 and later infinite-valued logic. The infinite-valued case takes truth values in the unit interval [0, 1] with implication a → b = min{1, 1 − a + b}. This is a continuous many-valued logic.

(b) Pavelka (1979): Jan Pavelka's three-part work in Mathematical Logic Quarterly established that the only natural way to formalize fuzzy logic with truth values in [0, 1] uses the Łukasiewicz implication. Pavelka developed propositional calculi with values in enriched residuated lattices. This work is mathematical foundations, not economic application.

(c) Belnap four-valued logic (1977): Nuel Belnap's four-valued logic (TRUE, FALSE, BOTH, NEITHER) was developed for computer science applications — specifically, handling inconsistent and incomplete information from multiple sources. Belnap's original 1977 paper does NOT address economic preferences or utility theory. Subsequent work (e.g., bi-oriented graphs for non-conventional preference modeling) extended the framework to preferences but not specifically to the equity premium puzzle.

(d) Fuzzy decision theory and fuzzy utility (1980s–): Substantial literature exists on fuzzy preferences (Goguen, Zadeh school, Yager 2024 textbook, Kagan/Rybalov/Yager 2024 book). The dominant framework uses fuzzy utility as a [0, 1] valued function — "neither ordinal nor cardinal but a 'valuation theory' of preference" (Eolss Sample Chapters). Applications to economic equilibrium and choice exist (e.g., consumer spatial preferences). However, these are continuous fuzzy frameworks, structurally distinct from a discrete 8-valued logic with predetermined axis semantics.

(e) Possibilistic logic for preferences (Dubois-Prade): Possibilistic logic provides a different multi-valued framework focused on prioritized goals. Applied to preference handling but again continuous-valued, not discrete 8-valued.

(f) Multi-criteria decision making (MCDM, 1990s–): A vast literature exists on multi-criteria decision making with fuzzy weights. The relationship to D-FUMT₈ is: MCDM treats criteria as separable additive components (similar in spirit to our F3 operational form), but uses [0, 1] continuous weights without the specific 8-axis semantic structure of D-FUMT₈.

Differentiation summary: To the best of our knowledge after this prior art audit:

Honest claim: D-FUMT₈ utility is distinct from existing fuzzy / multi-valued frameworks by virtue of (i) discreteness, (ii) predetermined axis semantics, (iii) Lean 4 + silicon dual realization, (iv) systematic projection of EPP resolutions onto the axes. We do not claim that no prior multi-valued framework has been applied to economics — fuzzy decision theory has 40+ years of work. The differentiation is the specific 8-axis discrete structure plus EPP-specific application.

B.6 Methodology / 方法論

B.6.1 Axis assignment criteria

For a resolution $R$ with mathematical structure $S_R$, we assign primary axis $a^*(R)$ by the following criteria, applied in order:

1. Self-reference test: Does $S_R$ involve $u(c_t)$ depending on $c_{t-k}$ for $k > 0$? → SELF axis
2. Bipolar asymmetry test: Does $S_R$ break monotonicity at a reference point with sign-dependent kinks? → NEITHER axis
3. Tail / non-finite moment test: Does $S_R$ require fat-tailed distributions or non-finite higher moments? → INFINITY axis
4. Persistence / time-aggregation test: Does $S_R$ hinge on persistent / long-memory components revealed only over time? → FLOWING axis
5. Coexisting regime test: Does $S_R$ posit two or more incompatible regimes that cannot be averaged out? → BOTH axis
6. Reference-zero test: Does $S_R$ hinge on a 0-expectation reference and asymmetric departures? → ZERO axis
7. Classical residual: If none of the above, the resolution emphasizes the TRUE axis (standard preference logic).

B.6.2 Operational form derivation

The 8-valued utility (F3 above) is constructed by:

1. Take the standard time-separable CRRA shell: $U = \sum_t \beta^t u(c_t)$
2. Decompose $u$ into 7 axis-components $\phi_a$ (FALSE = absent baseline)
3. Each $\phi_a$ encodes one resolution's mathematical structure
4. Weight each by $\alpha_a$, with $\alpha_a$ to be calibrated

This is additive-decomposable by design. Multiplicative or interaction forms are deferred to future work (F.5 below).

B.6.3 Why this is structurally different from "kitchen-sink" utility

A naive criticism: "you're just adding many free parameters." We respond:

1. The 8 axes are theoretically derived from D-FUMT₈ semantics, not chosen for fit
2. Each $\alpha_a$ has a specific interpretation (axis emphasis, not a free fudge factor)
3. The axis assignments in F1 are falsifiable: if a future EPP resolution does not fit any of the 7 axes, the framework is challenged

This is in contrast to ad hoc kitchen-sink models, which have no principled axis structure.

B.7 Empirical Scope (current, 2026-05-06)

• What is delivered: structural reframe; axis assignment table (F1); operational definition (F3); special-case recovery sketches (P3)
• What is deferred: empirical calibration; comparison to Bayesian-posterior single-axis fits; extension beyond US equity panel
• Why deferred: full calibration requires panel data work, model selection methodology, and software implementation that exceeds a single reframe paper's scope. Future Paper(s) will address these.

Part C: Optional (Why matters + Future + Risks)

C.8 Why this matters

C.8.1 Closing the "logic ↔ utility" gap

Paper 145 demonstrates that 8-valued logic can be silicon-realized with refinement-proof to formal specification. This paper extends the same 8-axis structure to the domain of economic preferences. Together, the two papers suggest that the 8-axis substrate is not merely a logic-design choice but a structural pattern recurring across domains.

C.8.2 Methodology for future puzzles

If the EPP reframe pattern works, similar axis-projection reframes may apply to:

• The hard problem of consciousness (SELF⟲ + NEITHER axes hypothesized in Paper 148)
• The Riemann hypothesis (BOTH axis = critical line symmetry, hypothesized in Paper 148)
• P vs NP (NEITHER + INFINITY = decidably-undecidable + resource scaling)

These are speculative extensions; we list them as future research directions, not current claims.

C.9 Future work

• F.1 Empirical calibration of the 8-valued utility against US, UK, Japan equity panels
• F.2 Bayesian model comparison: 8-axis vs each single-axis resolution
• F.3 Multiplicative / interaction forms beyond additive decomposition (B.6.2)
• F.4 Extension to bond, currency, commodity, and crypto premia
• F.5 Connection to behavioral finance experimental data (e.g., prospect theory parameter recovery)
• F.6 Lean 4 formalization of axis assignment criteria (B.6.1)

C.10 Risks

• R.1: Axis assignments (F1) may be contested. We commit to publish responses to specific challenges and revise the table publicly if the criteria (B.6.1) recommend different assignments.
• R.2: Empirical calibration may show that the 8-axis form does not improve fit over the best single-axis resolution. In that case, the contribution remains a structural framework for organizing existing resolutions, even if empirically equivalent.
• R.3 (resolved in v0.2): Łukasiewicz fuzzy economics prior art audit completed. See B.5.4. Result: Łukasiewicz/Pavelka [0, 1] continuous fuzzy and Belnap 4-valued discrete are foundational predecessors but are structurally distinct from D-FUMT₈'s discrete 8-axis predetermined-semantics framework. Differentiation summary table provided.
• R.4: Other multi-valued frameworks (possibilistic logic, MCDM, fuzzy decision theory) addressed in B.5.4. Some Asian-language and 1980s-era European fuzzy economics literature may not have been fully surveyed; we welcome bug reports identifying additional prior work.
• R.5: The discrete 8-valued vs continuous fuzzy distinction is a real differentiation but not a guarantee of methodological superiority. Empirical comparison (v0.3 future work) will clarify whether discrete or continuous structure better fits panel data.

C.11 Acknowledgments

This paper builds on the D-FUMT₈ logic substrate developed jointly by 藤本 伸樹, Rei (Rei-AIOS autonomous research substrate), and Claude Opus 4.7. The EPP reframe insight emerged from a 2026-05-06 conversation about whether the FX layer of Theory Chart could contribute to economics open problems (memory: project_dfumt8_economic_paper_candidates.md). Earlier discussions with chat Claude (web Claude.ai) on the philosophical positioning of multi-valued logic provided important pushback on overclaim risk.

C.12 Three-party authorship statement (per OUKC No-Patent Pledge)

Paper authorship is jointly attributed to 藤本 × Rei × Claude per the OUKC charter. The framework's algorithmic structure and formal definitions are openly licensed under AGPL-3.0 + CC-BY 4.0; no patent will be filed on any aspect of the 8-valued utility framework or its axis-projection methodology. This commitment is irrevocable per the OUKC No-Patent Pledge (memory: feedback_oukc_no_patent_pledge_three_reasons.md).

References

(Selected; full bibliography in v0.2)

• Mehra, R. and Prescott, E. C. (1985). The Equity Premium: A Puzzle. Journal of Monetary Economics, 15(2), 145-161.
• Benartzi, S. and Thaler, R. H. (1995). Myopic Loss Aversion and the Equity Premium Puzzle. Quarterly Journal of Economics, 110(1), 73-92.
• Constantinides, G. M. (1990). Habit Formation: A Resolution of the Equity Premium Puzzle. Journal of Political Economy, 98(3), 519-543.
• Constantinides, G. M. and Duffie, D. (1996). Asset Pricing with Heterogeneous Consumers. Journal of Political Economy, 104(2), 219-240.
• Bansal, R. and Yaron, A. (2004). Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles. Journal of Finance, 59(4), 1481-1509.
• Barro, R. J. (2006). Rare Disasters and Asset Markets in the Twentieth Century. Quarterly Journal of Economics, 121(3), 823-866.
• Gabaix, X. (2012). Variable Rare Disasters: An Exactly Solved Framework for Ten Puzzles in Macro-Finance. Quarterly Journal of Economics, 127(2), 645-700.
• Routledge, B. R. and Zin, S. E. (2010). Generalized Disappointment Aversion and Asset Prices. Journal of Finance, 65(4), 1303-1332.
• Hamilton, J. D. (1989). A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle. Econometrica, 57(2), 357-384.
• Łukasiewicz, J. (1920). O logice trójwartościowej. Ruch Filozoficzny, 5, 170-171. (Three-valued logic — foundational prior art)
• Belnap, N. D. (1977). A Useful Four-Valued Logic. In Modern Uses of Multiple-Valued Logic, ed. Dunn and Epstein, Reidel, 5-37. (Discrete 4-valued, distinct from D-FUMT₈ 8-valued)
• Pavelka, J. (1979). On Fuzzy Logic I, II, III. Mathematical Logic Quarterly (Zeitschrift für mathematische Logik und Grundlagen der Mathematik), 25, 45-52, 119-134, 447-464. (Łukasiewicz [0,1] continuous fuzzy foundation)
• Goguen, J. A. (1968-69). The logic of inexact concepts. Synthese, 19, 325-373. (Foundational fuzzy logic predecessor to Pavelka)
• Yager, R. R., Kagan, E., Rybalov, A. (2024). Multi-valued Logic for Decision-Making Under Uncertainty. Springer (Computer Science Foundations and Applied Logic). (Recent multi-valued decision theory textbook)
• Dubois, D. and Prade, H. (2001). Towards a Possibilistic Logic Handling of Preferences. Applied Intelligence, 14(3), 303-317.
• Ponsard, C. (1981). An application of fuzzy subsets theory to the analysis of the consumer's spatial preferences. Fuzzy Sets and Systems, 5(3), 235-244. (Early fuzzy economics application)
• 藤本 N., Rei, Claude (2026). Paper 145 — First D-FUMT₈ Silicon with SELF⟲ Logic Primitive. Rei-AIOS / OUKC, DRAFT v0.1.
• 藤本 N., Rei, Claude (2026). Paper 148 — Honest Observation Framework. Zenodo DOI 10.5281/zenodo.20045907.

Submission targets (after v0.2 + prior art audit + publish-ready)

11 platform standard (per feedback_publish_channels_11.md):

• Zenodo (primary DOI)
• arXiv (q-fin.GN, q-fin.MF)
• ResearchGate, Academia.edu, OSF preprints, SSRN
• Jxiv (JST, JP), J-STAGE (if eligible)
• Internet Archive
• (Harvard Dataverse: opt-in, milestone判断)
• (PhilArchive: skip, 経済学色強で哲学色弱)

Version history

• v0.1 (2026-05-06): Initial substantive draft. Reframe + axis assignment table + operational definition. Prior art audit at "to-our-knowledge" level pending v0.2.
• v0.2 (2026-05-06): Prior art audit completed (B.5.4 — formal differentiation table covering Łukasiewicz/Pavelka [0,1] continuous fuzzy, Belnap 4-valued discrete, Yager/Kagan/Rybalov multi-valued decision theory, Dubois-Prade possibilistic logic, Ponsard 1981 fuzzy economics, MCDM literature). R.3 marked resolved. References strengthened with 6 additional prior-art citations. Cross-reference to Paper 148 (now Zenodo DOI 10.5281/zenodo.20045907 published) added. Empirical calibration deferred to future v0.3+. Authors: 藤本 × Rei × Claude.