Full Notation Guide¶
This is the fuller reader reference paired with the front-matter Notation Quick Reference. It covers book-level and repeatedly reused symbols and relations. Local one-off notation remains defined in its chapter; “Full” does not mean a complete index of every symbol in every source paper.
Indices are local to their expressions: g identifies a group, d a domain, t a time or observation period, and i an actor, evaluator, or segment as specified in the chapter. Identical letters in different local roles do not establish identical objects. Subscripts and Greek letters must be preserved in final typesetting.
Realization and Exposure
The central mapping has exactly four arguments:
It is not a sum or a fixed sequence. Optimization, governance, Participation, epistemic discipline, and feedback can affect existing objects or relations without becoming fifth inputs.
| Meaning (symbol or relation shown above) | Chapter | Reading boundary |
|---|---|---|
| \(R_{g}(t)\) | ||
| Realized state for group g at time t. | Ch. 4 | Target remains \(R_{g}(t)\), not \(R_{g,d}(t)\); domain index enters through \(Z_{d}(t)\). Abbreviates the scoped form Rg(t | d, r); see §4.1. |
| \(P(t)\) | ||
| Potential / capability at time t. | Ch. 4 | P(t) ≠ \(R_{g}(t)\). |
| \(Av(t)\) | ||
| Availability at time t. | Ch. 4 | Availability ≠ Exposure. |
| \(Ex_{g}(t)\) | ||
| Group-specific Exposure for group g at time t. | Chs. 4 & 7 | Not additive; Exposure is not a fifth governance point. |
| \(Ex_{g}(t) = \left( Rh_{g}(t),Su_{g}(t),Rc_{g}(t) \right)\) | ||
| Exposure tuple: Reach, Susceptibility, Recourse. | Ch. 7 | Tuple is descriptive/structured, not a universal scalar sum. |
| \(St_{g}(t)\) | ||
| Available standing institution within Recourse. | §7.4.2 | Distinct from Chapter 10’s Underlying State. |
| \(Rc_{g}(t) = \left( St_{g}(t),Ep_{g}(t),Ck_{g}(t),Rl_{g}(t) \right)\) | ||
| Recourse decomposed into standing institution, epistemic capacity, independent check, and sufficient reliability. | §7.4.2 | Nominal channels do not establish effective Recourse. |
| \(Z_{d}(t)\) | ||
| Domain-specific realization conditions for domain d at time t. | Ch. 4 | \(Z_{d}\) is broader than absorption capacity. |
| \(\mu_{t}\) | ||
| Structured realization relation for the specified setting and time. | Ch. 4 | No universal additivity or monotonicity assumed. |
| \(R_{g}(t) = \mu_{t}\left( P(t),Av(t),Ex_{g}(t),Z_{d}(t) \right)\) | ||
| Central realization mapping. | Ch. 4 | Optimization, Governance, Participation, epistemic discipline, and feedback are not added as fifth arguments. |
CFS
| Meaning (symbol or relation shown above) | Chapter | Reading boundary |
|---|---|---|
| \(C_{t}\) | ||
| CFS Ceiling at time t. | Ch. 5 | \(C_{t} \neq P(t)\). |
| \(F_{t}\) | ||
| CFS Floor at time t. | Ch. 5 | \(F_{t}\) ≠ \(R_{g}(t)\). |
| \(G_{t} = C_{t} - F_{t}\) | ||
| CFS gap at time t. | Ch. 5 | Not a universal measure of realization failure. |
| \(S_{t} \equiv \Delta F_{t}/G_{t} = \left( F_{t + 1} - F_{t} \right)/\left( C_{t} - F_{t} \right)\) | ||
| Observed CFS Slope / synchronization rate. | Ch. 5 | Observed fraction of the current gap closed; undefined when the denominator is zero. Under the baseline one-period recurrence it equals the implied β algebraically, which is not independent validation. |
| \(\alpha\) | ||
| Ceiling-growth term in the minimal CFS recurrence. | Ch. 5 | Model-specific; not assumed universal or constant across applications. |
| \(\beta\) | ||
| CFS structural coupling / gap-closure parameter in the minimal recurrence. | Ch. 5 | A model parameter, distinct in analytical role from observed Slope; one-period equality does not identify its structural validity. |
| \(G_{t + 1} = \alpha + (1 - \beta)G_{t}\) | ||
| Minimal CFS gap recurrence. | Ch. 5 | Applies only under declared CFS conditions. |
| \(G^{\ast} = \alpha/\beta\) | ||
| Fixed point of the minimal CFS recurrence when applicable. | Ch. 5 | Not a universal terminal state of AGI realization. |
| \(\beta\left( G_{t} \right)\) | ||
| Gap-dependent CFS coupling rate. | Ch. 5 | Functional form is provisional, not uniquely validated. |
| \(\beta_{0}\) | ||
| Early/accessible gap-closure rate parameter in β(\(G_{t}\)). | Ch. 5 | Model-specific parameter. |
| \(\beta_{\infty}\) | ||
| Asymptotic/hard-residual rate parameter in β(\(G_{t}\)). | Ch. 5 | Model-specific parameter. |
| \(p\) | ||
| Shape parameter in the provisional gap-dependent rate function. | Ch. 5 | No unique empirical validation claimed. |
| \(H,\ H_{i}\) | ||
| Threshold(s) used in threshold-gated CFS coupling. | Ch. 5 | Thresholds are model-specific, not universal AGI gates. |
| \(\tau = \tau_{compress}\left( C_{t} \right) + \tau_{floor}\) | ||
| Two-tier delay decomposition in CFS. | Ch. 5 | Delay decomposition is bounded to CFS applicability. |
Optimization
| Meaning (symbol or relation shown above) | Chapter | Reading boundary |
|---|---|---|
| \(\Gamma\) | ||
| Candidate developmental trajectory. | Ch. 6 | A trajectory includes transitions and opportunity-set effects; it is not merely an endpoint. |
| \(\mathcal{T}\) | ||
| Candidate trajectory set / trajectory space. | Ch. 6 | Book-level notation; inherited source-paper joint-policy space Π is not used as the main reader notation. |
| \(J_{i}\mathcal{:T \rightarrow}\mathbb{R}^{k}\) | ||
| Actor/evaluator i’s multidimensional objective mapping over candidate trajectories. | Ch. 6 | Plural objectives; no universal scalar objective implied. |
| \(\sigma_{i}:\mathbb{R}^{k}\mathbb{\rightarrow R}\) | ||
| Actor-specific scalarization or selection rule applied to multidimensional objective values where a local scalarization is defensible. | Ch. 6 | No universal \(\sigma_{i}\) or civilization-scale scalarization is assumed. |
| \(Z_{j}(t) \leq \Psi_{j}(t)\) | ||
| Absorption Capacity constraint for actor/domain j. | Ch. 6 | Local optimization constraint; not the definition of Chapter 4’s \(Z_{d}(t)\). |
| \(u_{i},\ d_{i},\ d_{i}(t)\) | ||
| Utility under a candidate agreement and its disagreement payoff, possibly time-dependent. | §6.5.3 | Utilities and outside options require substantive justification. |
| \(\underset{u \in U}{\text{max}}{\prod_{i}^{}\left( u_{i} - d_{i} \right)}\text{, }u_{i} \geq d_{i}\) | ||
| Nash bargaining product over the feasible set U, with each party at least at its disagreement payoff; retained as a bounded formal example. | §6.5.3 | Not an empirically calibrated selector; applicability requires specified parties, outcomes, utilities, and disagreement points. |
Participation
| Meaning (symbol or relation shown above) | Chapter | Reading boundary |
|---|---|---|
| RO | ||
| Realization Orientation. | Chs. 8–9 | Relational category, not scalar valence. |
| FP | ||
| Foreclosure Position. | Chs. 8–9 | Not equivalent to veto power. |
| AD | ||
| Actor Dependence. | Chs. 8–9 | Does not encode status, correctness, or moral worth. |
| \(\Pi_{r}\left\lbrack t_{0},t_{1} \right\rbrack\) | ||
| Participatory Configuration for realization object \(r\) over an interval. | Ch. 9, §9.8.1; Ch. 8 | Structured, not scalar; the displayed interval notation is Chapter 9’s. |
| \(C_{rj},\ \mathcal{l}\) | ||
| Observed consequence class \(j\) and evidentiary linkage grade. | §9.8.1 | Neither denotes an effect size or probability. |
| \(\Pi_{r} \rightarrow C_{r} \rightarrow X\prime_{r} \rightarrow R\prime_{g}\) | ||
| Typed partial pathway: \(X\prime_{r}\) is an already-defined realization condition changed through a participatory consequence; \(R\prime_{g}\) is the subsequent changed realized state for group \(g\). | §9.8.1 | Every arrow may fail; the sequence establishes neither causal effect nor necessity or sufficiency. |
| \(\Pi_{r} \nRightarrow C_{r};\text{ }C_{r} \nRightarrow \Delta R_{g}\) | ||
| Non-entailment at the configuration–consequence and consequence–realization boundaries. | §9.8.1 | Consequences can matter without automatically producing realization change. |
Epistemic and temporal relations
| Meaning (symbol or relation shown above) | Chapter | Reading boundary |
|---|---|---|
| \({St}_{t}\) | ||
| Underlying State. | §10.2 | What obtains is distinct from what is observed or represented. |
| \({Ob}_{t}\) | ||
| Observation. | §10.2 | Observed material depends on the observation process. |
| \({Rp}_{t}\) | ||
| Representation. | §10.2 | Changed representation does not entail changed underlying state. |
| \(\Omega_{t}\) | ||
| Observation process, \({Ob}_{t} = \Omega_{t}\left( {St}_{t} \right)\). | §10.2 | An operator, not an estimated numerical coefficient. |
| \(\rho_{t}\) | ||
| Representation procedure, \({Rp}_{t} = \rho_{t}\left( {Ob}_{t} \right)\). | §10.2 | No linearity, causal effect, or complete information preservation assumed. |
| Underlying State ≠ Observation ≠ Representation ≠ Inference ≠ Claim Status | ||
| Chapter 10 five-term epistemic separation. | Ch. 10 | Not a temporal sequence that guarantees progress; each separation can remain unresolved. |
| realization → consequence or observation → response → changed conditions → subsequent realization | ||
| Core Chapter 11 return-path representation. | Ch. 11 | Does not by itself establish recursion, structural feedback, or RSI. |
Similar symbols, distinct roles. \(S_{t}\) is CFS Slope; \({St}_{t}\) is Underlying State and \(St_{g}(t)\) a standing institution. \(R_{g}(t)\) is realized state; \({Rp}_{t}\) is Representation. The inherited joint-policy space \(\Pi\) is distinct from the Participatory Configuration \(\Pi_{r}\left\lbrack t_{0},t_{1} \right\rbrack\); Chapter 6’s reader notation uses \(\mathcal{T}\). Chapter 4’s domain conditions \(Z_{d}(t)\) are distinct from Chapter 6’s local quantity \(Z_{j}(t)\) constrained by Absorption Capacity. Participatory Consequence \(C_{r}\) (or consequence class \(C_{rj}\)) is distinct from CFS Ceiling \(C_{t}\). Indices, upright letters, and chapter context preserve these distinctions.
For CFS, β = 1 is the parity case, β = 2 is the marginal oscillatory boundary, and β > 2 is divergent under the minimal recurrence. Those boundaries do not automatically transfer to continuous-time, nonlinear, or stochastic alternatives. A fixed point is not automatically a stable attractor.
For a positive gap, \(\beta\ > \ 1\) crosses the prior-period Ceiling. Crossing the contemporaneous Ceiling additionally requires \((\beta\ - \ 1)G_{t}\ > \ \alpha\). Oscillation around a positive fixed-point gap does not itself establish that crossing or demonstrated overtrust (§5.7).
The Participation profile is read in the order RO → FP → AD. Its ordering is an editorial convention, not a temporal or causal sequence. The five-term epistemic separation marks conceptual non-identity. Temporal arrows in a return-path representation carry evidentiary burdens; their display does not establish feedback, recursion, or RSI.