01
The formula โ
and what it is derived from
COP is the fifth step in a derivation chain that begins
with the participation matrix and passes through
co-occurrence, value significance, and association strength
before arriving at a proper conditional probability.
Each step adds a layer of mathematical structure
that the previous step alone cannot provide.
PMkl
Participation Matrix โ which SC of order k participated in which SC of order l. The raw structural information of the composition.
COMk|l
Co-Occurrence Matrix โ COM = PM ร PMT. How many times each pair of SCs co-participated. Symmetric. Raw co-occurrence only.
VSM family
Value Significance Measures โ intrinsic, relational, novelty, associational novelty. The significance framework that elevates the derivation beyond co-occurrence counting.
ASMk|l
Association Strength Matrix โ f(com_ij, vsm_x_i, vsm_y_j). Asymmetric. Multiple variants. The directed graph of SC relationships.
COPk|l
Conditional Occurrence Probability โ normalised from ASM. A proper probability. Bayes-consistent. Substrate-independent. The basis for IC, DCEM, retrieval, generation, and navigation.
02
What COP produces โ
from one formula
Once computed from a body of knowledge, the COP matrix
is simultaneously a probability structure, an association matrix,
an embedding space, a retrieval index, a generative model,
and a navigation map. All analytically derived.
No training. No labels. No gradient descent.
01
Information Content
I(SC_i|SC_j) = โlog(COP(i|j)). IC of any composition computable analytically by chain rule. Closed-form formula for sentence IC.
02
Subject identification
DCEM_i = CEM1_i โ CEM2_i. Negative DCEM = the composition is about that SC. Analytically derived. ฮฃ_i DCEM_i = 0 โ proven theorem.
03
Knowledge retrieval
Query matrix formed from ASM vectors multiplied by PM โ association-weighted relevance of each document. Retrieval-Augmented Generation, analytically derived. Live since 2011.
04
Embedding vectors
Rows or columns of COPM are embedding vectors. Each SC's COP profile across all others encodes semantic similarity โ derived analytically, not trained.
05
State Navigation
COP drives transitions between states in the causal AI framework. Causal COP โ derived from the Shifted Participation Matrix โ is next-SC prediction. Priority: 2019.
06+
And many more
Knowledge graphs, novelty detection, consensus measurement, causal inference, genomics, image analysis, physical sensing. The full treatment is in the technical white paper.
Technical White Paper ยท Qualified parties
The full mathematical treatment
The complete derivation of COP โ the general form,
the elegant special case, the Bayes consistency theorem,
the CEM1/CEM2/DCEM analysis with proofs,
the IC chain rule and closed-form sentence formula,
and the full application set โ is available
in the technical white paper for qualified parties.
Investors in technical due diligence
Potential licensing and technology partners
Technical collaborators and researchers
Engineers evaluating the framework
Request the technical white paper โ
In one sentence
COP(i|j) is a probability derived from a body of knowledge โ
and from it, without approximation, the complete structure of
that knowledge becomes computable.