RIMC: Recursive Intelligence Market Cycle

A Framework for α-Drift and Learning Markets

For the Japanese version of this document, see
README_JA.md.

1. What is RIMC?

RIMC (Recursive Intelligence Market Cycle) is a theoretical framework that:

• Treats the market as a learning system with finite-speed information processing, not as a static equilibrium.
• Models the interaction between technological recursion and economic value as a coupled dynamic system.
• Reinterprets CAPM α as a structural drift term generated by observation delay and learning dynamics, rather than as unexplained regression residue.

RIMC is not a price-forecasting tool.
It is a language for describing “how information arrives in the market and how slowly it is absorbed”.

2. Core Idea in One Paragraph

At the heart of RIMC are two coupled layers:

1. A generative layer where technological recursion $R(t)$ drives economic value $V(t)$ via a dynamic system (the “RV equations”).
2. An observational layer where the market only sees a delayed and noisy projection of that value, typically through something CAPM-like.

Because observation is delayed, noisy, and heterogeneous across agents, the market’s estimate of recursion $r_{\text{market}}(t)$ systematically deviates from the underlying $r_{\text{real}}(t)$.
The time-accumulation of that deviation generates a persistent structural bias:

$$ \alpha_{\text{drift}}(t) = \int_{t-T}^{t}\varepsilon_R(\tau)G(\tau)e^{-\lambda(t-\tau)}d\tau $$

where:

• $\varepsilon_R(t) = r_{\text{real}}(t) - r_{\text{market}}(t)$ is the recursion misalignment,
• $G(t)$ is a sensitivity kernel,
• $\lambda$ is a learning / forgetting rate.

This $\alpha_{\text{drift}}$ is what RIMC calls structural α, emerging purely from finite-time learning and observation delay.

3. Mathematical Skeleton (Very High Level)

RIMC is built around three interacting components:

3.1 RV System (Technological Recursion and Value)

A generic form of the value-generation engine:

$$ \frac{dV}{dt} = L(t)A(t)R(t)^{\beta(t)} - \kappa_V(t)V(t)^{\mu(t)} $$

with a complementary recursion equation for $R(t)$, e.g.:

$$ \frac{dR}{dt} = \gamma(t)V(t)^{\delta(t)} - \kappa_R(t)R(t)^{\nu(t)} $$

• $R(t)$: technological recursion rate (R&D, data, organizational learning, etc.).
• $V(t)$: economic value (market cap, value proxy, etc.).
• $L,A,\gamma,\delta,\kappa_V,\kappa_R,\beta,\mu,\nu$: time-varying, data-driven mappings from external series $Z_t$.

RIMC assumes “one firm, one equation” in the sense that these mappings are firm- or strategy-specific, derived from the chosen information set.

3.2 α-Drift Dynamics

Given an observation gap between $r_{\text{real}}$ and $r_{\text{market}}$:

Define the recursion misalignment:

$$ \varepsilon_R(t) = r_{\text{real}}(t) - r_{\text{market}}(t) $$

Define the α-drift as an exponentially weighted memory of that misalignment:

$$ \alpha_{\text{drift}}(t) = \int_{t-T}^{t}\varepsilon_R(\tau)G(\tau)e^{-\lambda(t-\tau)}d\tau $$

Differentiate to obtain the α-drift differential equation:

$$ \frac{d\alpha_{\text{drift}}}{dt} = \varepsilon_R(t)G(t) - \lambda\alpha_{\text{drift}}(t) $$

This looks like a relaxation equation: a balance between forward bias generation and learning-based decay.

3.3 Connection to CAPM (Continuous-Time Generalization)

Starting from continuous-time CAPM:

$$ \frac{dr_i}{dt} = \frac{dr_f}{dt} \quad + \beta_i(t)\frac{d(r_m - r_f)}{dt} \quad + \alpha_i(t) $$

RIMC interprets the “residual” $\alpha_i(t)$ not as pure noise, but as a projection of α-drift generated by information delay.

The observational α extracted from past data:

$$ \alpha_{\text{obs}}(t) = \int_{t-T}^{t} \left[ r_i(\tau) - r_f(\tau) - \beta_i(\tau)\bigl(r_m(\tau) - r_f(\tau)\bigr) \right] e^{-\lambda(t-\tau)}d\tau $$

can be compared to theoretical $\alpha_{\text{drift}}(t)$

4. Alpha Drift Capture Ratio

To bridge theoretical α and observed α, RIMC defines the Alpha Drift Capture Ratio:

$$ \rho_{\alpha}(t) = \frac{\displaystyle \int_{t-T}^{t} \bigl|\alpha_{\text{obs}}(\tau)\bigr|^2d\tau} {\displaystyle \int_{t-T}^{t} \bigl|\varepsilon_R(\tau)G(\tau)\bigr|^2d\tau} $$

• $0 \le \rho_{\alpha}(t) \le 1$ by construction (with appropriate regularization).
• Interpreted as a finite-time efficiency metric:
  • How much of the theoretically available α-drift the market has actually “captured” within a finite window.
• Recasts EMH into a finite-time learning efficiency problem:
  Markets are not required to be “fully efficient”, only to learn with some finite speed.

5. Repository Structure (Suggested)

A typical layout for this project might be:

• README.md — this document.
• 01_introduction_and_motivation.md
  Conceptual overview of RIMC and the motivation.
• 02_foundational_equations_of_rimc.md
  Formal definition of the RV system and core assumptions.
• 03_observation_structure_and_time_lag.md
  Observation structure, α-drift integral and differential forms.
• 04_alpha_cycle_and_market_dynamics.md
  Multi-agent, multi-timescale structure, α-cycles.
• 05_alpha_drift_structure.md
  Observational interpretation and extraction from data.
• 06_from_capm_to_alpha_drift_equation.md
  Continuous-time CAPM, delay extension, and α-drift derivation.
• 07_the_hypothesis_and_its_horizon.md
  Final chapter: learning markets and research agenda.
• appendix_a_practitioners_guide_to_rimc.md
  “How to read RIMC in practice” for quants and researchers.
• references_bibliography.md
  Bibliography (Soros, Fama, Sharpe, Schumpeter, Aghion–Howitt, Bellman, Kuramoto, Shannon, Wiener, Kalman, etc.).

Feel free to rename files; the framework is text-first and tool-agnostic (LaTeX, Markdown, or a mix).

6. How a Quant Might Use RIMC

RIMC is designed as a meta-layer that can sit on top of existing models:

• As a diagnostic:

  • Compute $\alpha_{\text{obs}}(t)$ from your production CAPM / factor stack.
  • Propose a proxy for $\varepsilon_R(t)$ (e.g., anything you believe tracks “true innovation intensity”).
  • Estimate $\lambda$ and $G(t)$ and measure $\rho_{\alpha}(t)$ over rolling windows.
  • Interpret $\rho_{\alpha}$ as a measure of how quickly your universe learns.

• As a modeling lens:

  • Treat your existing factor model or RL-based strategy as a particular choice of value-generation engine.
  • Use RIMC’s observation layer to separate “model misspecification” from “structural delay”.
  • Examine whether persistent α is better explained by informational lag than by missing factors.

• As a research scaffold:

  • Reinterpret macro cycles, sector rotations, or tech regimes as α-cycles around a slowly drifting baseline.
  • Test whether transitions resemble delay-induced bifurcations rather than simple regime switches.

RIMC itself does not prescribe a trading strategy; it provides a language to discuss the time structure of information and learning in any strategy.

7. What RIMC Is Not

To avoid misunderstandings:

• It is not a fully specified asset-pricing model ready for direct implementation.
• It is not a claim that markets are systematically “wrong” or “inefficient” in a normative sense.
• It is not an alpha engine or a backtestable strategy library on its own.

Instead, RIMC should be read as:

• A hypothesis about how structural α arises from finite-speed learning and observation delay.
• A proposal for how to measure that structure using existing market data and factor models.
• A bridge between nonlinear dynamical systems and practical factor-based finance.

8. Status and Roadmap

Current status:

• The repository primarily contains a conceptual and mathematical manuscript.
• No production-grade code is included by default; any future code would be illustrative (simulation, toy examples, etc.), not plug-and-play for live trading.

Potential future work (if this evolves into an applied project):

• Numerical experiments: simulate simple RV systems and α-drift dynamics.
• Toy implementation of α-drift extraction from historical equity data.
• Empirical estimation of finite-time efficiency $\rho_{\alpha}(t)$ across sectors, factors, or regions.
• Comparison with classical EMH and factor models for different horizons.

9. Citation

If you reference RIMC in internal memos, talks, or research notes, a generic citation could be:

Reiya Sodeyama, “RIMC: Recursive Intelligence Market Cycle — A Framework for α-Drift and Learning Markets,” working manuscript, 2025.

When the manuscript has a stable version or DOI, you can update the citation accordingly.

10. License

This project is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.

You are free to share, adapt, and use this work — including for commercial purposes —
as long as appropriate credit is given.

11. Origin and Disclaimer

RIMC is a personal research hypothesis, developed outside of institutional academia or sell-side research, and written in collaboration with a generative AI assistant. Final structure and responsibility, however, remain with the human author.

Nothing in this repository constitutes:

• Investment advice,
• A recommendation to buy or sell any security, or
• A representation of any institution’s official view.

It is offered purely as a theoretical and conceptual tool for those interested in the intersection of:

• Nonlinear dynamics,
• Information theory,
• Technological growth, and
• Quantitative finance.

12. Pronunciation of “RIMC”

Although “RIMC” is an acronym, this project adopts a unified pronunciation for clarity in discussion and citation.

RIMC is pronounced “RIM-see”
/ˈrɪm.siː/

This pronunciation is used throughout the documentation and is recommended when referring to the framework in research conversations, seminars, or internal discussions.

Contact

For research discussions, technical suggestions, or collaboration proposals related to RIMC, you may contact me via the address below:

rimclab.project@gmail.com

If you prefer to submit feedback publicly, feel free to open an Issue or Discussion in this repository.