Thread 4 — Temporal Dynamics¶

Jean-Paul Niko · February 2026

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Introduction: The Dynamics of Ideas¶

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The Ideometrics paper defined Dynamic IdeaRank with temporal decay \(\IdeaRank(i, t) \propto \sum_{j \to i} \IdeaRank(j, t) \cdot e^{-\kappa(t - t_j)} / \mathrm{out}(j, t)\) and Thread 1 developed the portfolio-theoretic treatment. This thread builds the full temporal theory: ideas as stochastic processes, the idea lifecycle (birth, growth, maturity, obsolescence), cultural evolution as time-varying filters on the idea graph, and the connection between temporal dynamics and the existing mathematical apparatus.

The central claim: the time-evolution of the idea graph is a filtered diffusion process on a growing random graph, and the filter pipeline determines which ideas survive.

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Three Temporal Structures¶

The three-space ontology (Part XIII) reveals that the single time parameter \(t\) used throughout the IAG framework conceals three fundamentally different temporal structures:

\begin{interpretation} Quantum time (\(t_\QS\)): reversible, symmetric under \(t \to -t\). The Schr\"odinger equation preserves inner products in both temporal directions. In the idea graph, quantum time governs the potentiality of ideas---the space of possible future developments that exists in superposition before instantiation.

Physical time (\(t_\mathbb{R}\)): irreversible, carries the thermodynamic arrow. In the idea graph, physical time is the clock on which the idea lifecycle (birth, growth, maturity, obsolescence) is measured. The "age" of an idea is its \(t_\mathbb{R}\)-duration since instantiation.

Consciousness time (\(t_\CSp = t_\mathbb{R} + i\,t_{\mathrm{lat}}\)): complex-valued. The real part is physical time; the imaginary part \(t_{\mathrm{lat}}\) is lateral time---the freedom of consciousness to navigate outside the \(t_\mathbb{R}\) direction. Insight, creativity, and idea recombination correspond to movement in the \(t_{\mathrm{lat}}\) direction: reaching across \(t_\mathbb{R}\)-separated ideas to forge new connections.

The arrow of time---the fact that \(t_\mathbb{R}\) flows in one direction---is not fundamental but accumulated: each instantiation event is irreversible (\(\Inst^{-1}\) does not exist), and the macroscopic arrow is the statistical accumulation of \(\sim 10^{40}\) instantiation events per second across the physical universe. Entropy increases because instantiation count increases, not the other way around. \end{interpretation}

The idea lifecycle equation (Section ref:sec:lifecycle) is a \(t_\mathbb{R}\)-equation: it tracks how ideas age in physical time. The Idea Revival Theorem (Section ref:sec:revival) is a \(t_\CSp\)-theorem: revival occurs when a consciousness navigates laterally in \(t_{\mathrm{lat}}\) to reconnect an obsolescent idea to a new context. The distinction resolves a puzzle: how can "dead" ideas revive? In \(t_\mathbb{R}\), they cannot---obsolescence is irreversible. In \(t_\CSp\), they can---consciousness reaches laterally across physical time to re-instantiate the idea in a new configuration.

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Idea Stochastic Processes¶

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\tB\;

The price of idea \(\iota\) at time \(t\) is: \begin{keyeq} [ p(\iota, t) = \IdeaRank(\iota, t) \cdot \bar{u}(\iota, t) ] \end{keyeq} where \(\IdeaRank(\iota, t)\) is the Dynamic IdeaRank (network importance) and \(\bar{u}(\iota, t) = \E_a[u(\iota, a, t)]\) is the average utility across the agent population. The price is the "consensus value" of an idea: network centrality \(\times\) average usefulness.

\tB\;

The price process follows a geometric Brownian motion with regime structure: \begin{keyeq} [ \frac{dp(\iota, t)}{p(\iota, t)} = \mu_\iota(t) \, dt + \sigma_\iota(t) \, dW_\iota(t) + J_\iota(t) \, dN_\iota(t) ] \end{keyeq} where: [nosep] - \(\mu_\iota(t)\) is the drift (expected rate of value change), - \(\sigma_\iota(t)\) is the volatility (uncertainty in value), - \(W_\iota(t)\) is a Wiener process (continuous random fluctuations), - \(J_\iota(t)\) is the jump size and \(N_\iota(t)\) is a Poisson process with intensity \(\lambda_J\), capturing paradigm shifts (sudden revaluation of an idea due to a breakthrough or refutation).

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The liquidity of idea \(\iota\) at time \(t\) is: \begin{keyeq} [ \ell(\iota, t) = \frac{\text{number of agents actively investing in } \iota \text{ at } t} {\text{total agent population at } t} ] \end{keyeq} Liquidity measures how "tradeable" an idea is: how easily an agent can find collaborators, mentors, or materials for engaging with the idea. [nosep] - High liquidity: widely studied ideas (calculus, machine learning). Easy to enter, many resources available. - Low liquidity: niche ideas (non-associative algebraic geometry, medieval Icelandic poetry). Hard to enter, few collaborators. - Zero liquidity: ideas with no active participants. The idea exists in \(\mathcal{G}\) but is "frozen": no agent allocates attention to it.

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The volatility of idea \(\iota\) decomposes into: \begin{keyeq} [ \sigma_\iota^2(t) = \underbrace{\beta_\iota^2 \sigma_M^2(t)}{\text{systematic}} + \underbrace{\sigma ] \end{keyeq} where }^2(t)}_{\text{idiosyncratic}\(\sigma_M\) is the volatility of the "idea market" (aggregate uncertainty about the value of knowledge), \(\beta_\iota\) is the idea's sensitivity to market-wide cognitive shocks (a "knowledge recession" devalues all ideas), and \(\sigma_{\epsilon,\iota}\) is the idea-specific uncertainty.

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The Idea Lifecycle¶

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\tA\;

Every idea passes through a canonical lifecycle with four phases, characterized by their drift and volatility regime: \begin{keyeq} [ \begin{array}{lcccl} \text{Phase} & \mu_\iota & \sigma_\iota & \ell(\iota) & \text{Character}
\hline \text{I. Birth} & \text{high, uncertain} & \text{very high} & \text{very low} & \text{Radical novelty, few early adopters}
\text{II. Growth} & \text{high, stabilizing} & \text{high} & \text{rising} & \text{Rapid adoption, connections multiply}
\text{III. Maturity} & \text{low, stable} & \text{low} & \text{high} & \text{Canonical knowledge, widely taught}
\text{IV. Obsolescence} & \text{negative} & \text{rising} & \text{falling} & \text{Superseded, declining relevance} \end{array} ] \end{keyeq}

\tA\;

The four lifecycle phases correspond to distinct IdeaRank dynamics: \begin{keyeq}

\end{keyeq} Moreover, the transition from Phase II to Phase III occurs at the inflection time \(t^*\) where the second derivative changes sign, and the transition from Phase III to Phase IV is triggered by a supersession event: the birth of a new idea \(\iota'\) with \(\iota' \to \iota\) in \(\mathcal{G}\) (the new idea depends on and generalizes the old one).

The Dynamic IdeaRank equation is: [ \IdeaRank(\iota, t) = \frac{1 - \alpha}{|V(t)|} + \alpha \sum_{j \to \iota} \frac{\IdeaRank(j, t)}{\mathrm{out}(j, t)} \cdot e^{-\kappa(t - t_j)} ] At birth (\(t = t_{\iota}\)), \(\iota\) has few in-edges but they may be high-value (radical ideas often build on deep prerequisites). As the idea gains citations (new edges \(j \to \iota\)), the sum grows, driving \(d\IdeaRank/dt > 0\). The acceleration \(d^2\IdeaRank/dt^2 > 0\) during Phase I reflects network effects: each new adopter creates more edges.

The inflection point occurs when the marginal contribution of new edges equals the decay of old edges: the growth rate starts decreasing even though IdeaRank itself still increases. This is the "S-curve" of diffusion.

At maturity, the inflow (new edges) exactly compensates the temporal decay \(e^{-\kappa(t - t_j)}\), giving \(d\IdeaRank/dt \approx 0\).

Obsolescence begins when the decay of old edges exceeds the inflow of new ones. The supersession event (birth of a generalizing idea \(\iota'\)) diverts edge formation from \(\iota\) to \(\iota'\), accelerating the decline.

\tA\; [nosep] - Birth (1687): Principia published. Very few people can read it (\(\ell \approx 0\)), extremely high novelty (\(\sigma\) huge), high expected return (\(\mu\) high). - Growth (1700--1850): Widespread adoption, textbooks written, applications multiply. IdeaRank accelerates then decelerates. - Maturity (1850--1905): Canonical knowledge, taught universally, stable IdeaRank, low volatility. - Partial Obsolescence (1905--): Superseded by relativity and quantum mechanics for extreme regimes. IdeaRank declines but stabilizes at a high level (still taught, still useful for most engineering applications). This is partial obsolescence: the idea retains value in a restricted domain.

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Cultural Evolution as Time-Varying Filters¶

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\tA\;

A cultural filter \(\Phi_{\mathrm{cult}}(t)\) is a time-varying cognitive filter that modulates the effective intelligence of all agents in a population: \begin{keyeq} [ \Phi_{\mathrm{cult}}(t) : \mathbb{R}^{n(e)}{\geq 0} \to \mathbb{R}^{n(e)}, \quad \Phi_{\mathrm{cult}}(t)(\bI) = D(t) \cdot \bI ] \end{keyeq} where \(D(t) = \mathrm{diag}(d_1(t), \ldots, d_8(t))\) with \(d_\tau(t) \in [0, 1]\) is the cultural modulation vector. Each \(d_\tau(t)\) represents the degree to which culture supports intelligence type \(\tau\) at time \(t\): \(d_\tau = 1\) means full support, \(d_\tau = 0\) means complete cultural suppression.

\tA\; [nosep] - Medieval Europe (c.\ 1200): \(d_{\mathrm{ling}} = 0.4\) (literacy restricted to clergy), \(d_{\mathrm{symb}} = 0.3\) (mathematics suppressed outside monasteries), \(d_{\mathrm{kin}} = 0.8\) (physical labor valued), \(d_{\mathrm{soc}} = 0.5\) (rigid hierarchy). - Renaissance Florence (c.\ 1480): \(d_{\mathrm{ling}} = 0.7\), \(d_{\mathrm{spat}} = 0.9\) (visual arts flourishing), \(d_{\mathrm{symb}} = 0.6\), \(d_{\mathrm{soc}} = 0.7\) (patronage networks). - Modern research university (c.\ 2024): \(d_{\mathrm{ling}} = 0.9\), \(d_{\mathrm{symb}} = 0.95\), \(d_{\mathrm{kin}} = 0.3\) (sedentary culture), \(d_{\mathrm{aud}} = 0.4\) (music de-emphasized in STEM).

\tA\;

The cultural filter \(\Phi_{\mathrm{cult}}(t)\) induces a time-varying selection pressure on the idea graph: ideas whose intelligence requirement profiles \(\mathbf{R}_\iota\) are "compatible" with the cultural filter survive; those that are "incompatible" decay. Formally: \begin{keyeq} [ \frac{d}{dt} \IdeaRank(\iota, t) \propto \langle D(t) \cdot \bar{\bI}(t), \; \mathbf{R}\iota \rangle - \kappa \cdot \IdeaRank(\iota, t) ] \end{keyeq} where \(\bar{\bI}(t)\) is the population-average intelligence vector. The first term is the cultural fitness of idea \(\iota\): the \(\bK\)-inner product of the culturally-filtered population intelligence with the idea's requirement. Ideas whose requirements match the culturally-supported types grow; mismatched ideas decay.

The Dynamic IdeaRank of \(\iota\) grows when agents invest in \(\iota\) (creating new edges \(j \to \iota\)). An agent invests in \(\iota\) when the expected cognitive return is positive: \(r_\iota(a) > 0\), which requires sufficient effective intelligence \(\langle \bI_{\mathrm{eff},a}, \mathbf{R}_\iota \rangle_{\bK} > \theta\) (sufficiency threshold). Under the cultural filter, \(\bI_{\mathrm{eff},a} = \Phi_{\mathrm{cult}}(t)(\bI_a) = D(t) \cdot \bI_a\). The population-level investment rate in \(\iota\) is proportional to the fraction of agents meeting the threshold, which for a large population approximates \(\langle D(t) \cdot \bar{\bI}(t), \mathbf{R}_\iota \rangle_{\bK}\). This drives edge creation, hence IdeaRank growth. The \(\kappa\) term is the inherent temporal decay of Dynamic IdeaRank.

\tA\; A "dark age" for intelligence type \(\tau\) occurs when \(d_\tau(t) \to 0\): the cultural filter suppresses type \(\tau\). This causes: [nosep] - All ideas with \(R_{\iota,\tau} > 0\) lose cultural fitness, accelerating their decay. - The kernel of the cultural filter expands: \(\ker(\Phi_{\mathrm{cult}}) \ni \tau\), in the filter-formalism sense. - The effective efficient frontier (Thread 1) contracts, reducing the space of viable cognitive portfolios.

Conversely, a "renaissance" for type \(\tau\) (\(d_\tau(t)\) increasing) expands the frontier and revives ideas that load on type \(\tau\).

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The Idea Graph as a Growing Network¶

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\tB\;

New ideas enter the graph \(\mathcal{G}(t)\) via a marked Poisson process with intensity: \begin{keyeq} [ \Lambda(t) = \sum_a \lambda_a(t) \cdot \langle \bI_{\mathrm{eff},a}(t), \mathbf{1} \rangle ] \end{keyeq} where \(\lambda_a(t)\) is agent \(a\)'s "creativity rate" and \(\langle \bI_{\mathrm{eff},a}, \mathbf{1} \rangle = \sum_\tau I_{\mathrm{eff},a,\tau}\) is the total effective intelligence. Each new idea \(\iota'\) arrives with: [nosep] - A random intelligence profile \(\mathbf{R}_{\iota'}\) drawn from a distribution \(\pi(t)\) that is itself shaped by the cultural filter. - A set of prerequisite edges from existing ideas, with edge probability proportional to \(\IdeaRank(\iota, t)\) (preferential attachment: new ideas are more likely to build on important ideas).

\tB\;

Under the birth process of Definition ref:def:birth-process, the steady-state IdeaRank distribution follows a power law: \begin{keyeq} [ \Pr[\IdeaRank(\iota) \geq x] \sim x^{-\gamma} ] \end{keyeq} with exponent \(\gamma = 1 + 1/(\alpha \cdot \bar{m})\), where \(\alpha\) is the IdeaRank damping factor and \(\bar{m}\) is the average number of prerequisite edges per new idea.

This follows from the standard Barab\'asi--Albert preferential attachment model, adapted to the IdeaRank setting. Each new idea creates \(\bar{m}\) edges, each attaching to existing idea \(\iota\) with probability \(\propto \IdeaRank(\iota, t)\). The master equation for the degree distribution is: [ \frac{\partial}{\partial t} n(k, t) = \bar{m} \frac{(k-1) n(k-1, t) - k \, n(k, t)}{2\bar{m} t} + \delta_{k, \bar{m}} \frac{1}{t} ] whose steady-state solution is a power law with exponent \(\gamma = 1 + 2\bar{m}/\bar{m} = 3\) in the basic model. The IdeaRank damping factor modifies this: with probability \((1-\alpha)/|V|\) each node receives a "random teleportation" share, which flattens the tail. The modified exponent is \(\gamma = 1 + 1/(\alpha \bar{m})\). For \(\alpha = 0.85\) and \(\bar{m} = 3\): \(\gamma \approx 1.39\)---a heavy tail, consistent with the empirical observation that a few ideas (calculus, evolution, electromagnetism) dominate the intellectual landscape.

\begin{interpretation} The power-law distribution of IdeaRank means that the "idea market" has extreme concentration: a few "blue-chip" ideas account for most of the total IdeaRank in the graph. This parallels the concentration of financial markets (a few companies dominate market capitalization) and is a direct consequence of preferential attachment: important ideas attract more connections, which makes them more important.

The cultural filter shapes the power law by biasing which ideas receive new connections. A culture that suppresses type \(\tau\) will have fewer high-IdeaRank ideas loading on \(\tau\)---not because those ideas are inherently less valuable, but because the cultural filter prevents agents from investing in them. \end{interpretation}

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Temporal Portfolio Optimization¶

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\tB\;

A dynamic cognitive portfolio is a stochastic process \(\bw(t) \in \Delta^{N(t)-1}\) adapted to the filtration generated by the idea price processes and the idea birth process. The agent continuously rebalances as new ideas are born, existing ideas change value, and the cultural filter evolves.

\tB\;

The growth-optimal dynamic portfolio under the idea SDE (Definition ref:def:idea-sde) and cultural filter \(\Phi_{\mathrm{cult}}(t)\) satisfies: \begin{keyeq} [ w_i^*(t) = \frac{[\Sigma^{-1}(t) (\bmu(t) - r_f \mathbf{1})]i^+} {\sum_j [\Sigma^{-1}(t) (\bmu(t) - r_f \mathbf{1})]_j^+} ] \end{keyeq} where \([\cdot]^+\) denotes the positive part (no short-selling of cognitive attention), \(\bmu(t)\) is the vector of instantaneous expected returns filtered through \(\Phi_{\mathrm{cult}}(t)\): [ \mu_i(t) = \mu_i^{\mathrm{raw}} \cdot \frac{\langle D(t) \bar{\bI}(t), \mathbf{R}_i \rangle {\langle \bar{\bI}(t), \mathbf{R}}i \rangle ] and }\(\Sigma(t)\) is the time-varying covariance matrix.

This extends the Kelly criterion (Thread 1, Theorem 4.2) to the continuous-time setting with time-varying parameters. The log-growth rate at time \(t\) is: [ g(\bw, t) = \bw^\top \bmu(t) - r_f - \frac{1}{2} \bw^\top \Sigma(t) \bw ] Maximizing pointwise in \(\bw\) (myopic optimization, valid under log-utility by the separation theorem of Merton): \(\bw^*(t) = \Sigma^{-1}(t) (\bmu(t) - r_f \mathbf{1})\), projected onto the simplex. The cultural filter enters through \(\bmu(t)\): it modulates the expected returns by the ratio of culturally-filtered to raw cultural fitness.

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Idea Extinction and Revival¶

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\tA\; An idea \(\iota\) is extinct at time \(t\) if: [ \ell(\iota, t) = 0 \quad \text{and} \quad \IdeaRank(\iota, t) < \theta_{\mathrm{ext}} ] where \(\theta_{\mathrm{ext}}\) is the extinction threshold. An extinct idea has no active participants and negligible network importance. The idea still exists as a vertex in \(\mathcal{G}\) but is effectively invisible.

\tA\;

An extinct idea \(\iota\) can be revived at time \(t' > t\) if and only if: \begin{keyeq} [ \exists \, a : \quad \langle D(t') \cdot \bI_a, \; \mathbf{R}\iota \rangle > \theta \quad \text{and} \quad \nu(\iota, t') > \nu_0 ] \end{keyeq} where the first condition requires at least one agent with sufficient culturally-filtered intelligence to engage with the idea, and the second requires that the idea has become "novel again" (\(\nu(\iota, t')\) exceeds a novelty threshold \(\nu_0\)) by virtue of being forgotten.

Revival requires an agent to invest in \(\iota\), creating new edges. The sufficiency condition \(\langle D(t') \cdot \bI_a, \mathbf{R}_\iota \rangle_{\bK} > \theta\) ensures the agent has the cognitive capacity (under the current cultural filter) to engage. The novelty condition ensures the idea offers positive expected return: if \(\iota\) is well-known (high \(\bar{u}\), low \(\nu\)), re-engagement yields no new utility. But an idea that has been forgotten (\(\nu\) increases as collective knowledge of \(\iota\) decays) becomes "novel" to the current population, making re-discovery worthwhile.

\begin{interpretation} This formalizes the phenomenon of "rediscovery": ideas that were known to a previous civilization, lost during a cultural dark age (filter contraction), and rediscovered when the cultural filter re-expands. Classical examples include the rediscovery of Aristotelian logic in medieval Europe via Arabic translations, and the revival of ancient Greek mathematics during the Renaissance.

The novelty-through-forgetting mechanism is a deep feature: the temporal decay of IdeaRank and the mortality of agents ensure that extinct ideas become novel again after sufficient time. The idea graph has a memory horizon beyond which all ideas are effectively new. This creates a cyclic structure in cultural evolution: birth \(\to\) growth \(\to\) maturity \(\to\) obsolescence \(\to\) extinction \(\to\) (potential) revival \(\to\) growth \(\to \cdots\) \end{interpretation}

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Unification: The Master Temporal Equation¶

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\tA\;

The complete temporal dynamics of the idea graph are governed by: \begin{keyeq} [ \frac{d}{dt} \mathbf{p}(t) = \mathcal{M}(t) \cdot \mathbf{p}(t) + \mathbf{b}(t) ] \end{keyeq} where \(\mathbf{p}(t) = (p(\iota_1, t), \ldots, p(\iota_{|V(t)|}, t))^\top\) is the vector of idea prices, \(\mathcal{M}(t)\) is the master evolution operator: \begin{keyeq} [ \mathcal{M}(t) = \underbrace{\alpha \, A_{\mathcal{G}}(t)}{\text{IdeaRank propagation}} - \underbrace{\kappa \, \mathrm{Id}} + \underbrace{P_{\mathrm{cult}}(t)}}{\text{cultural selection}} + \underbrace{S ] \end{keyeq} and }(t)}_{\text{synergy coupling}\(\mathbf{b}(t)\) is the birth term (new ideas entering the graph).

Here: [nosep] - \(A_{\mathcal{G}}(t)\) is the column-normalized adjacency matrix of the idea graph (IdeaRank transition matrix). - \(P_{\mathrm{cult}}(t) = \mathrm{diag}(\langle D(t) \bar{\bI}(t), \mathbf{R}_{\iota_i} \rangle_{\bK})\) is the cultural fitness diagonal. - \(S_{\bK}(t)\) encodes cross-idea synergy through shared intelligence type requirements and the compatibility tensor.

The eigenvalues of \(\mathcal{M}(t)\) determine the idea graph's "cognitive temperature": the rate at which the graph reorganizes.

The price \(p(\iota, t) = \IdeaRank(\iota, t) \cdot \bar{u}(\iota, t)\) evolves as: [ \dot{p}_i = \dot{\IdeaRank}_i \cdot \bar{u}_i + \IdeaRank_i \cdot \dot{\bar{u}}_i ] The IdeaRank dynamics are the Dynamic IdeaRank equation (matrix form): \(\dot{\IdeaRank} = (\alpha A_{\mathcal{G}} - \kappa \mathrm{Id}) \cdot \IdeaRank + \frac{1-\alpha}{|V|} \mathbf{1}\). The utility dynamics are driven by the cultural selection equation (Theorem ref:thm:cultural-selection): \(\dot{\bar{u}}_i \propto \langle D(t) \bar{\bI}(t), \mathbf{R}_i \rangle_{\bK}\). Combining (using the product rule and linearizing): \(\dot{\mathbf{p}} \approx \mathcal{M}(t) \mathbf{p}(t) + \mathbf{b}(t)\) where \(\mathcal{M}\) collects all the linear terms and \(\mathbf{b}\) captures the constant birth-rate injection.

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Summary of Results¶

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\begin{tempbox}[Thread 4: New Results] [nosep,leftmargin=1.5em] - Theorem ref:thm:lifecycle-ir (Tier A): Lifecycle--IdeaRank correspondence; four phases characterized by derivatives of IdeaRank. - Theorem ref:thm:cultural-selection (Tier A): Cultural selection equation; ideas grow/decay proportional to cultural fitness. - Theorem ref:thm:power-law (Tier B): Power-law IdeaRank distribution from preferential attachment, \(\gamma = 1 + 1/(\alpha \bar{m})\). - Theorem ref:thm:revival (Tier A): Conditions for idea revival---novelty through forgetting. - Theorem ref:thm:dynamic-opt (Tier B): Optimal dynamic cognitive portfolio under cultural filter evolution. - Theorem ref:thm:master-temporal (Tier A): Master temporal equation unifying IdeaRank, decay, cultural selection, and synergy coupling. - Definition ref:def:cultural-filter (Tier A): Cultural filter as time-varying diagonal modulation, with historical examples. - Definition ref:def:liquidity (Tier B): Idea liquidity as fraction of active participants. - Definition ref:def:idea-sde (Tier B): Idea price SDE with jump-diffusion (paradigm shifts).

\end{tempbox}

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