The equation library

Every recovered mathematical record has an individual page. Browse original equations, code expressions, indexed components, parameter definitions, and explicitly unresolved source slots.

2,910 recordsSource snapshot 2026-10-17

Counts refer to mathematical records, including implemented expressions and generated instances. They are not counts of independently validated theories. Exact occurrences and unresolved slots remain traceable. Read the complete 1–516 source index ↗ · Inspect the scan decisions ↗

2,910 records · page 10 of 59Search the source archive ↗

eq-1a940ec11c4757

Normalized minimum stochastic-interaction partition

For each nontrivial split, the numerator compares the two within-part conditional entropies with the whole-system conditional entropy. K is the smaller marginal entropy, J is their ratio, and S* minimizes that ratio. The benchmark saves both the dimensionless minimized ratio and the raw numerator…

Integration and coherencedocumented formula

\Phi_{\rm SI}(S)&=H(X_{t+1,S}\mid X_{t,S})+ H(X_{t+1,\bar S}\mid X_{t,\bar S})-H(X_{t+1}\mid X_t),\\ K(S)&=\min\{H(X_{t,S}),H(X_{t,\bar S})\},\qquad J(S)=\Phi_{\rm SI}(S)/K(S),\qquad S^*=\arg\min_S J(S).

eq-9011be227ccb43

Reciprocal variance score for the manuscript window

This score depends only on how much the Φ samples vary within the window. Constant samples always give R=1, whether their level is high or low, and permuting their time order leaves R unchanged. Since Φ lies in [0,1], population variance is at most 1/4 and this formula requires R≥0.8; the…

Integration and coherencedocumented formula

R_T = \frac{1}{1 + \text{Var}(\{\Phi_t\}_{t=T-w}^T)}

eq-373b25e4ffac64

Reported project aggregate score near 0.76

The source reports a φ_global value around 0.76 for its own aggregate-state score. This number measures a different construction from the reported PyPhi value around 0.006, so their magnitudes are not directly comparable. The value is a historical report tied to its implementation and state, rather…

Integration and coherencedocumented formula

• My φ_global = 0.76 (measures different property)

eq-19873775e8bf52

Reported raw information loss at the representative normalized optimum

The unnormalized stochastic-interaction numerator at the selected partition is approximately 2.1867917397 nats. It is saved separately from the dimensionless objective used to select the partition. Minimizing a ratio and reporting its raw numerator are different operations, so the two values should…

Integration and coherencedocumented formula

\Phi_{\rm SI}=2.1867917397

eq-1568af9a566fd8

Unity-score maximum condition

For Φ(x)=1/(1+|x−1|), the maximum value one occurs exactly when x=1. This follows because the absolute difference is nonnegative and equals zero only at that point. It specifies the score’s reference value, not an independently measured biological or computational threshold.

Integration and coherencedocumented formula

\Phi(x) = 1

eq-d858d8d35cc9ae

Unregularized inverse-discrepancy score

Taking the reciprocal of δ makes the score grow as positive discrepancy shrinks, but the expression is undefined at δ=0 and unbounded near zero. It differs from the bounded 1/(1+δ) score used elsewhere. A numerical implementation needs an explicit zero case, positive regularizer, or a restricted…

Integration and coherencedocumented formula

\Phi \left(\delta \right)=\frac{1}{\delta }.

eq-85bbcc32476e1f

Weighted composite score in the original manuscript

The score is a weighted sum of Φ, Ψ, and Ω with weights 0.4, 0.3, and 0.3. The weights sum to one, so if all three components lie in [0,1], the total also lies in that interval. The manuscript must specify how Ψ and Ω are measured and scaled before the expression becomes a reproducible measurement.

Integration and coherencedocumented formula

\text{CONSCIOUSNESS} = 0.4 \cdot \Phi + 0.3 \cdot \Psi + 0.3 \cdot \Omega

poster-eq-8648f3b74d21d3

Expected revenue from lead value and closing probability

The equation scales lead value L by the closing probability P_c. For a nonnegative L and a probability between zero and one, the product ranges from zero when a close is impossible to L when a close is certain. A probability has no monetary unit, so ER retains the value unit used for L. The poster…

MADDIE conversion framework / detailed postersource graphic; not an execution result

ER = L × P_c

poster-eq-bc843ff7115087

Lead value as a probability-weighted outcome sum

For each listed outcome i, multiply its value v_i by its probability p_i, then add those contributions from i=1 through n. When these are mutually exclusive outcomes covering the possibilities and their probabilities sum to one, the expression is the expected outcome value. Outcomes with larger…

MADDIE conversion framework / detailed postersource graphic; not an execution result

L = Σ_{i=1}^{n} v_i · p_i

poster-eq-55dd145604de56

Message score as a weighted sum of pain points

The Message component assigns a numerical quantity p_i to each pain point and multiplies it by an emotional weight w_i. Adding those products produces M, an additive score: each term contributes independently according to its weight, with no interaction term shown. The neighboring condition makes…

MADDIE conversion framework / detailed postersource graphic; not an execution result

M = Σ_{i=1}^{n} w_i · p_i

poster-eq-79572d386bfa11

Normalize the total emotional weight in the Message score

This equality constrains the emotional weights in the Message formula to add to one. It fixes their combined scale, so changing the allocation among pain points need not change the total assigned weight. If each weight is nonnegative, the weights can be read as shares of a whole and M becomes a…

MADDIE conversion framework / detailed postersource graphic; not an execution result

Σ_i w_i = 1

poster-eq-bc179aae1a78ca

The declared zero-to-one range of the desire score

The two inequalities require D_desire to lie in the closed interval from zero to one, including both endpoints. The adjacent component equation defines D_desire as emotional intensity E multiplied by perceived value V and identity alignment I. Constraining each of those three factors to [0,1] would…

MADDIE conversion framework / detailed postersource graphic; not an execution result

0 ≤ D_desire ≤ 1

eq-fd5a813d1eda60

Reported sides of a coarse PyPhi partition

The source places Input, Memory, Reason, and Integrate on one side and Meta on the other in a reported five-element bipartition. This identifies a cut of the coarse model. Interpreting it requires the exact transition model and state-dependent calculation; the partition names alone do not validate…

Partition optimizationdocumented formula

A = {Input, Memory, Reason, Integrate}

eq-1263815036a5d8

Adjacent-window gradient-norm reduction

The expression computes one minus the ratio of the later window mean to the earlier window mean, then converts to percent. With the supplied means it gives about 46.52465%. This is a reduction in the selected average gradient norm, not a percentage gain in capability or a replicated effect across…