Thread 1 — IdeaRank Portfolio¶

Jean-Paul Niko · February 2026

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Introduction: Ideas as Financial Assets¶

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In Parts II--III and the Ideometrics paper, we established that ideas are formal objects in the conceptual topos $\mathbf{C}_{\mathrm{co}}$, each characterized by four measures: depth $\delta$, novelty $\nu$, utility $u$, and intelligence profile $\mathbf{R}$. IdeaRank assigns a scalar importance to each idea based on its position in the global idea graph $\mathcal{G}$.

We now observe that the structure of cognitive investment---allocating finite attention and intelligence-time across a portfolio of ideas---is formally identical to the structure of financial portfolio theory. An agent with bounded cognitive resources must choose which ideas to invest in, how deeply, and in what combination. The returns are stochastic (an idea may prove more or less valuable than anticipated), and ideas are correlated (investing in group theory and topology yields synergistic returns).

This section develops the correspondence rigorously, proves three new theorems, and connects the portfolio formalism back to the filter pipeline and the compatibility tensor $\bK$.

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Three-Space Grounding of Portfolio Theory¶

The three-space ontology (Part XIII) gives idea portfolio theory a deeper foundation. Foundation ideas---those with highest IdeaRank---are convergence conditions: $\QS$-structures whose instantiation is prerequisite for accessing more complex $\QS$-regions. The efficient frontier of knowledge is thus not merely a mathematical analogy but reflects the actual geometry of $\QS$: there exist optimal paths through the space of convergence conditions, and deviation from these paths wastes instantiation capacity.

The Kelly criterion for intellectual investment becomes: allocate cognitive resources to the idea whose instantiation maximizes expected access to new $\QS$-regions. Risk is the probability that the instantiation fails (the idea turns out to be wrong or the agent lacks the filter profile to instantiate it). Return is the volume of newly accessible $\QS$-structure.

The Idea Market¶

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Cognitive Asset

\tA\; Let $\mathcal{G} = (V, E)$ be the global idea graph. A cognitive asset is an idea $\iota \in V$ together with: [nosep] - Expected return: $\mu_\iota = \E[r_\iota]$, the expected cognitive return from investing attention in idea $\iota$. Concretely: [ r_\iota = \Delta u(\iota, a, t) = u(\iota, a, t + \Delta t) - u(\iota, a, t) ] where $u$ is the utility function from the ideometrics paper (Def. 3.5), evaluated for agent $a$ over investment period $\Delta t$. - Risk: $\sigma^2_\iota = \Var[r_\iota]$, the variance of cognitive return. High-risk ideas include radical conjectures (high upside, high probability of dead end) and novel domains (high variance in realized utility). - Investment cost: $c_\iota \in \R_{>0}$, the intelligence-time (cog$\cdot$hr) required for meaningful engagement. This is the minimum cognitive expenditure to change the agent's posterior over $\iota$.

Cognitive Return

\tA\;

The cognitive return of idea $\iota$ for agent $a$ with intelligence vector $\bI_a$ is: \begin{keyeq} [ r_\iota(a) = \frac{\Delta u(\iota, a)}{c_\iota} = \frac{u_{\mathrm{post}}(\iota, a) - u_{\mathrm{prior}}(\iota, a)} {\int_0^{\Delta t} \langle \bI_a(s), \mathbf{R}_\iota \rangle \, ds} ] \end{keyeq} where $\mathbf{R}_\iota$ is the intelligence requirement vector of the idea and $\langle \cdot, \cdot \rangle$ is the $\bK$-inner product (Def. 3.1 of Part I). The denominator is the realized cognitive work: intelligence committed weighted by compatibility.

\[\begin{interpretation} This is the "return on cognitive investment" (ROCI). A mathematician investing in a group theory paper has low cost (high $I_{\mathrm{symb}}$ matches $R_{\mathrm{symb}}$) and potentially high return (new tools for existing problems). The same mathematician investing in social psychology has higher cost (mismatch) and uncertain return. \end{interpretation}\]

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Portfolio Construction¶

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Idea Portfolio

\tA\; An idea portfolio for agent $a$ is a weight vector $\bw = (w_1, \ldots, w_N) \in \Delta^{N-1}$ over a universe of $N$ cognitive assets, where $w_i$ is the fraction of total available intelligence-time allocated to idea $\iota_i$.

The constraint $\bw \in \Delta^{N-1}$ (the probability simplex) is the attention budget constraint: the agent cannot invest more total attention than it has. This is the portfolio-theoretic restatement of the attention simplex constraint from Part III.

Remark

The connection is precise: the attention simplex $\Delta^{n-1}$ from Part III allocates attention across intelligence types; the portfolio simplex $\Delta^{N-1}$ allocates attention across ideas. The two are related by the requirement vectors $\mathbf{R}_\iota$: an allocation across ideas induces an allocation across intelligence types via $\lambda_\tau = \sum_{i} w_i \cdot R_{\iota_i, \tau}$ (normalized).

Portfolio Return and Risk

\tA\; For portfolio $\bw$: \begin{keyeq}

[\begin{aligned} \mu_{\bw} &= \bw^\top \bmu = \sum_{i=1}^N w_i \mu_i [6pt] \sigma^2_{\bw} &= \bw^\top \Sigma \bw = \sum_{i,j} w_i w_j \sigma_{ij}

\end{aligned}]

\end{keyeq} where $\bmu = (\mu_1, \ldots, \mu_N)^\top$ is the vector of expected cognitive returns and $\Sigma \in \R^{N \times N}$ is the idea covariance matrix with entries $\sigma_{ij} = \Cov[r_i, r_j]$.

Idea Covariance via $\bK$

\tB\;

The covariance between ideas $\iota_i$ and $\iota_j$ has two sources: \begin{keyeq} [ \Sigma_{ij} = \underbrace{\mathbf{R}i^\top \bK \, \mathbf{R}_j} + \underbrace{\gamma \cdot \mathbf{1}[\iota_i \to \iota_j \text{ or } \iota_j \to \iota_i]}_{\text{graph proximity covariance}} ] \end{keyeq} The first term captures that ideas requiring similar intelligence profiles covary through the compatibility tensor. The second term captures that ideas connected in the idea graph }$\mathcal{G}$ (prerequisite or extension relationships) share structural dependency. The parameter $\gamma > 0$ scales the graph contribution.

\[\begin{interpretation} Learning group theory and ring theory covary highly (both require $I_{\mathrm{symb}}$, and they share prerequisites). Learning group theory and playing piano covary less (orthogonal type profiles, distant in $\mathcal{G}$). A well-diversified cognitive portfolio spreads investment across ideas with low $\Sigma_{ij}$, just as a financial portfolio diversifies across uncorrelated assets. \end{interpretation}\]

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Markowitz Mean-Variance Optimization¶

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Cognitive Efficient Frontier

\tA\; The cognitive efficient frontier is the set of portfolios solving: \begin{keyeq} [ \min_{\bw \in \Delta^{N-1}} \; \bw^\top \Sigma \bw \quad \text{subject to} \quad \bw^\top \bmu \geq \mu_0 ] \end{keyeq} for each target return $\mu_0 \in [\mu_{\min}, \mu_{\max}]$. Equivalently, these are the portfolios maximizing return for each level of risk.

Existence of the Cognitive Efficient Frontier

\tA\;

If $\Sigma$ is positive definite and $\bmu$ is not proportional to $\mathbf{1}$, then the cognitive efficient frontier is a parabola in $(\sigma_{\bw}, \mu_{\bw})$-space, parameterized by: [ \sigma^2_{\bw}(\mu_0) = \frac{A \mu_0^2 - 2B\mu_0 + C}{AC - B^2} ] where $A = \mathbf{1}^\top \Sigma^{-1} \mathbf{1}$, $B = \mathbf{1}^\top \Sigma^{-1} \bmu$, $C = \bmu^\top \Sigma^{-1} \bmu$.

Proof

This is the standard Markowitz result applied to the cognitive setting. The Lagrangian for the constrained minimization is: $\mathcal{L} = \bw^\top \Sigma \bw - \lambda_1 (\bw^\top \bmu - \mu_0) - \lambda_2 (\bw^\top \mathbf{1} - 1)$. Setting $\nabla_{\bw} \mathcal{L} = 0$ gives $\bw^* = \frac{1}{2}\Sigma^{-1}(\lambda_1 \bmu + \lambda_2 \mathbf{1})$. The two constraints yield a $2 \times 2$ system for $(\lambda_1, \lambda_2)$ in terms of $A, B, C$. Substituting back gives the parabolic form. Positive definiteness of $\Sigma$ ensures the minimum is unique; the non-proportionality condition ensures the frontier is not degenerate (a single point).

Minimum-Variance Cognitive Portfolio

\tA\; The minimum-variance portfolio (MVP) is: [ \bw_{\mathrm{MVP}} = \frac{\Sigma^{-1} \mathbf{1}}{\mathbf{1}^\top \Sigma^{-1} \mathbf{1}} ] This is the "safest" cognitive strategy: maximum diversification across ideas, minimizing the total uncertainty in realized cognitive returns.

\[\begin{interpretation} The MVP corresponds to the generalist strategy: spreading cognitive investment evenly across maximally diverse ideas. An agent on the efficient frontier above the MVP is a *specialist*: accepting more risk (variance in cognitive payoff) for higher expected returns. The entire history of academic disciplines can be read as a collective movement along the efficient frontier, with specialization increasing as the idea graph grows and covariance structure becomes richer. \end{interpretation}\]

Cognitive Sharpe Ratio

\tB\;

The cognitive Sharpe ratio of portfolio $\bw$ is: \begin{keyeq} [ S_{\bw} = \frac{\mu_{\bw} - r_f}{\sigma_{\bw}} ] \end{keyeq} where $r_f$ is the risk-free cognitive return---the return from routine maintenance cognition (habitual tasks, already-known material). The tangency portfolio maximizing $S_{\bw}$ is the optimal risky cognitive portfolio.

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Kelly Criterion for Cognitive Betting¶

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The Markowitz framework optimizes a single-period trade-off between return and risk. For sequential cognitive investment---where returns compound and the agent reinvests cognitive capital---the Kelly criterion provides the growth-optimal allocation.

Cognitive Wealth

\tB\; The cognitive wealth of agent $a$ at time $t$ is: [ W(t) = \sum_{\iota \in V_a(t)} u(\iota, a, t) ] where $V_a(t) \subset V$ is the set of ideas the agent has invested in up to time $t$. Cognitive wealth grows as ideas yield returns and decays as knowledge becomes obsolete (the temporal decay of Dynamic IdeaRank).

Kelly Criterion for Ideas

\tB\;

For an agent with log-utility over cognitive wealth, the growth-optimal portfolio satisfies: \begin{keyeq} [ \bw^*_{\mathrm{Kelly}} = \Sigma^{-1} (\bmu - r_f \mathbf{1}) ] \end{keyeq} This maximizes $\E[\log W(t+1)]$, the expected log-growth of cognitive wealth.

Proof

The log-growth rate of the portfolio is: [ g(\bw) = \E[\log(1 + r_{\bw})] \approx \bw^\top \bmu - r_f - \frac{1}{2} \bw^\top \Sigma \bw ] using the second-order Taylor expansion for small returns (appropriate for incremental cognitive investments). The first-order condition $\nabla_{\bw} g = \bmu - r_f \mathbf{1} - \Sigma \bw = 0$ yields the result. The Hessian $-\Sigma$ is negative definite, confirming the maximum.

Kelly vs.\ Markowitz

The Kelly portfolio is typically more concentrated than the Markowitz tangency portfolio because it maximizes long-run growth rather than single-period risk-adjusted return. In cognitive terms: the Kelly-optimal agent is a bold specialist, willing to tolerate short-term variance for maximum compound growth. The Markowitz agent is more conservative, hedging against bad periods.

The "fractional Kelly" $\bw = f \cdot \bw^*_{\mathrm{Kelly}}$ with $f \in (0, 1)$ provides a continuum between the two strategies, and corresponds to a cognitive risk aversion parameter.

The Graduate Student's Dilemma

\tB\; A mathematics graduate student has ideas $\{\iota_1 = \text{thesis problem}, \iota_2 = \text{adjacent conjecture}, \iota_3 = \text{applied collaboration}\}$ with: [ \bmu = \begin{pmatrix} 0.8 0.3 0.5 \end{pmatrix}, \quad \Sigma = \begin{pmatrix} 0.25 & 0.15 & 0.05
0.15 & 0.20 & 0.02
0.05 & 0.02 & 0.10 \end{pmatrix}, \quad r_f = 0.1 ] The Kelly portfolio is: $\bw^*_{\mathrm{Kelly}} = \Sigma^{-1}(\bmu - 0.1 \cdot \mathbf{1}) \propto (0.54, 0.12, 0.34)$ (after normalization to $\Delta^2$).

Interpretation: invest 54% of cognitive time in the thesis, 34% in the applied collaboration (high diversification benefit from low correlation), and only 12% in the adjacent conjecture (too correlated with the thesis to provide much marginal value).

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The Filter--Portfolio Connection¶

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The filter formalism from the Filter Paper directly modifies the portfolio problem through the effective intelligence vector.

Filter-Adjusted Returns

\tA\;

Under the full filter pipeline $\bI_{\mathrm{eff}} = \lambda \odot \eta(\Psi) \odot (\mathrm{Id} + A) \cdot \mathrm{diag}(d) \cdot \min(\bI_{\mathrm{raw}}, \bI_{\max})$, the cognitive return of idea $\iota$ becomes: \begin{keyeq} [ r_\iota^{\mathrm{filtered}}(a) = \frac{\Delta u(\iota, a)} {\int_0^{\Delta t} \langle \bI_{\mathrm{eff}}(s), \mathbf{R}\iota \rangle ] \end{keyeq} The filter pipeline affects the } \, dsdenominator (effective cognitive cost), not the numerator (utility gained). When $\bI_{\mathrm{eff}} < \bI_{\mathrm{raw}}$ (typical: filters reduce capacity), costs increase and returns decrease.

Proof

The cognitive work integral $\int \langle \bI, \mathbf{R} \rangle_{\bK} \, ds$ measures the actual intelligence deployed against the requirement. Under filtering, the deployed intelligence is $\bI_{\mathrm{eff}}$, not $\bI_{\mathrm{raw}}$. The utility gained $\Delta u$ depends on the agent's learning (which may be less efficient under filtering) but is fundamentally about the change in the agent's knowledge state, which is measured by the ideometric utility function independently of how that state was reached. The cost, however, is directly proportional to the effective intelligence available.

Filter-Shifted Efficient Frontier

\tB\; Filtering shifts the entire efficient frontier down and to the right: returns decrease (higher denominator in ROCI) and risk increases (filter-induced variance in $\bI_{\mathrm{eff}}$ propagates into return variance). Specifically, if the somatic filter $\Phi_{\mathrm{som}}$ has random component $\epsilon \sim \mathcal{N}(0, \sigma_{\mathrm{som}}^2)$, the portfolio variance inflates by: [ \Sigma^{\mathrm{filtered}} = \Sigma + \sigma_{\mathrm{som}}^2 \cdot \mathbf{R} \mathbf{R}^\top ] where $\mathbf{R} = (\mathbf{R}_1, \ldots, \mathbf{R}_N)$ is the matrix of requirement vectors.

\[\begin{interpretation} A tired, stressed, or ill agent faces a *worse* efficient frontier than a rested, calm, healthy one. This is the portfolio-theoretic restatement of what the somatic and affective filters already describe: biological state degrades cognitive performance. The portfolio lens adds the insight that the degradation is *not uniform across ideas*---ideas requiring intelligence types most affected by the filter suffer the largest return decrease. The optimal portfolio under filtering is therefore different from the optimal portfolio at baseline: it shifts toward ideas that are "filter-robust" (low sensitivity to the degraded types). \end{interpretation}\]

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IdeaRank as Market Capitalization¶

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IdeaRank--Capitalization Correspondence

\tB\;

In the market analogy, IdeaRank is the analogue of market capitalization: \begin{keyeq} [ \IdeaRank(\iota) \longleftrightarrow \text{Market Cap}(\text{asset}) ] \end{keyeq} Specifically: [nosep] - The market-cap weighted portfolio $w_i \propto \IdeaRank(\iota_i)$ is the cognitive analogue of an index fund: invest in ideas proportional to their global importance. - The equal-weighted portfolio $w_i = 1/N$ is the cognitive analogue of diversified exposure without importance-weighting. - Active management corresponds to deviating from the IdeaRank-weighted portfolio based on private information (the agent's unique intelligence profile, utility function, or filter state).

Cognitive Market Efficiency

\tB\;

The idea market is weakly efficient in the following sense: if all agents have identical intelligence vectors $\bI$, identical filter states, and access to the same idea graph $\mathcal{G}$, then the IdeaRank-weighted portfolio is mean-variance optimal for all agents. Equivalently: there are no excess cognitive returns from active management when agents are homogeneous.

Formally: let $\bw_{\mathrm{IR}}$ be the IdeaRank-weighted portfolio and $\bw^*$ be the Markowitz-optimal portfolio for a representative agent. If $\bI_a = \bI$ and $\Phi_a = \Phi$ for all agents $a$, then $\bw_{\mathrm{IR}} = \bw^*$.

Proof

Under homogeneity, all agents compute the same expected returns $\mu_i$ (identical utility functions and intelligence profiles) and the same covariances $\Sigma_{ij}$ (identical filter states and graph access). The equilibrium portfolio weights in a homogeneous market are proportional to the asset "float"---here, the idea's importance in the network, which is exactly $\IdeaRank(\iota_i)$.

More precisely: in a CAPM equilibrium, the market portfolio is mean-variance efficient. The "market portfolio" in idea space is the aggregate of all agents' holdings, which under homogeneity is proportional to $\IdeaRank$ (since IdeaRank is the fixed point of the collective attention flow on $\mathcal{G}$).

Source of Cognitive Alpha

\tB\; Active management outperforms the IdeaRank-weighted portfolio if and only if the agent has heterogeneous intelligence, filter state, or information. The "cognitive alpha" is: [ \alpha_a = \mu_{\bw_a} - \mu_{\bw_{\mathrm{IR}}} = \bw_a^\top \bmu_a - \bw_{\mathrm{IR}}^\top \bmu_a = (\bw_a - \bw_{\mathrm{IR}})^\top \bmu_a ] This is positive when the agent tilts toward ideas where its private expected returns exceed the market-implied returns.

\[\begin{interpretation} The corollary formalizes why *individual differences matter* in intellectual life. A musician investing in music theory has positive cognitive alpha not because the ideas are objectively "better," but because the musician's intelligence profile ($I_{\mathrm{aud}}$ dominant) makes the returns higher for that particular agent. The intelligence vector is the source of cognitive alpha: each agent's unique profile creates private mispricing relative to the market-cap portfolio. \end{interpretation}\]

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Factor Models and Cognitive Beta¶

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Intelligence-Type Factors

\tB\; Define eight intelligence-type factor portfolios $\bw_\tau$ for $\tau \in \mathcal{T}$ by: [ w_{\tau, i} \propto R_{\iota_i, \tau} ] i.e., the $\tau$-factor portfolio overweights ideas that load heavily on intelligence type $\tau$. Then every idea's return admits a factor decomposition: \begin{keyeq} [ r_\iota = \alpha_\iota + \sum_{\tau \in \mathcal{T}} \beta_{\iota,\tau} \cdot F_\tau + \epsilon_\iota ] \end{keyeq} where $F_\tau$ is the return of the $\tau$-factor portfolio and $\beta_{\iota,\tau}$ is the idea's cognitive beta with respect to type $\tau$.

Cognitive CAPM

\tB\; Under the assumptions of Theorem ref:thm:cog-efficiency, the expected return of any idea satisfies: [ \mu_\iota - r_f = \sum_{\tau \in \mathcal{T}} \beta_{\iota,\tau} \cdot (\mu_{F_\tau} - r_f) ] Ideas with high beta to a particular intelligence type command a risk premium proportional to the market price of that type's systematic risk.

\[\begin{interpretation} This is a cognitive asset pricing model. Ideas requiring rare intelligence types (high $R_{\mathrm{kin}}$ in a population of sedentary scholars) command a premium: the "supply" of cognitive capital in that type is scarce, so the market-clearing return must be higher. Conversely, ideas loading on abundant types ($R_{\mathrm{ling}}$ in a literate population) have lower premiums. The compatibility tensor $\bK$ enters through the factor covariance structure: cross-type synergy means that factor returns are themselves correlated. \end{interpretation}\]

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Portfolio Rebalancing as Attention Dynamics¶

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Rebalancing--Replicator Equivalence

\tA\;

The continuous-time rebalancing dynamics of the idea portfolio are: \begin{keyeq} [ \dot{w}_i = w_i \left( r_i - \bar{r} \right) + w_i \sum_j \frac{\partial^2 g}{\partial w_i \partial w_j} (r_j - \bar{r}) ] \end{keyeq} where $\bar{r} = \sum_j w_j r_j$ is the portfolio return and $g = \E[\log W]$ is the Kelly growth rate. In the limit of small returns, the first term dominates, and the dynamics reduce to the replicator equation from Part III: [ \dot{w}_i = w_i (r_i - \bar{r}) ] Ideas with above-average returns grow in portfolio weight; ideas with below-average returns shrink.

Proof

The continuous-time Kelly criterion maximizes $g(\bw) = \E[\log(1 + \bw^\top \mathbf{r})]$. The gradient ascent dynamics are $\dot{w}_i = w_i \cdot \partial g / \partial w_i$ (multiplicative dynamics to preserve $\bw \in \Delta^{N-1}$). Computing: [ \frac{\partial g}{\partial w_i} = \E!\left[\frac{r_i}{1 + \bw^\top \mathbf{r}}\right] \approx \E[r_i] - \bw^\top \E[\mathbf{r}] \cdot \E[r_i] + O(r^2) = r_i - \bar{r} + O(r^2) ] The zeroth-order term gives the replicator equation. The second-order corrections involve the Hessian $\partial^2 g / \partial w_i \partial w_j$, which accounts for the covariance structure and prevents the dynamics from converging to a degenerate portfolio (all weight on a single idea).

\[\begin{interpretation} The replicator dynamics of Part III (attention allocation across intelligence types) and the portfolio rebalancing dynamics (attention allocation across ideas) share the same mathematical structure. Both are instances of *selection dynamics on the simplex*: ideas/types that perform above average grow; those below average shrink. The portfolio formalism adds the covariance-correction term, which acts as an endogenous diversification force absent from the pure replicator equation. This is the formal link between the attention simplex and the efficient frontier. \end{interpretation}\]

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Summary of Results¶

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\begin{portbox}[Thread 1: New Results] [nosep,leftmargin=1.5em] - Theorem ref:thm:eff-frontier (Tier A): Existence and parabolic form of the cognitive efficient frontier. - Theorem ref:thm:kelly (Tier B): Kelly criterion for growth-optimal idea allocation. - Theorem ref:thm:cog-efficiency (Tier B): IdeaRank as equilibrium portfolio under homogeneity (cognitive market efficiency). - Theorem ref:thm:rebalance-replicator (Tier A): Portfolio rebalancing $\equiv$ replicator dynamics on the idea simplex. - Proposition ref:prop:filter-adj (Tier A): Filter pipeline modifies portfolio problem through effective costs. - Definition ref:def:idea-cov (Tier B): Idea covariance matrix constructed from $\bK$ and graph structure. - Definition ref:def:sharpe (Tier B): Cognitive Sharpe ratio.

\end{portbox}

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