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While not a KG, this structure generalizes it replacing static factual links with dynamic reasoning proximity. The core assumption of this paper is that it is possible to construct a Problem Space, where problems are treated as mathematical objects and its structure can be refined through behavioral feedback loops
A problem, once presented as a prompt and tokenized by an LLM, becomes a mathematical object in a vector space. The LLM’s inference process can then be viewed as a function acting on it to produce an output
Let P be a problem, presented as a sequence of tokens
P = [t₁, t₂, ..., tₙ]
Let E be the token embedding function
E(tᵢ) ∈ ℝᵈ
Then the embedded input is
X₀ = [E(t₁), E(t₂), ..., E(tₙ)] ∈ ℝⁿˣᵈ
Let M be the language model, which processes X₀ to generate an output
M(X₀) → output
Thus, the overall inference process can be seen as a transformation
P → X₀ → M(X₀) = Answer
We seek to define a distance metric d*(A, B) such that
d*(A, B) is small when solving A helps solve B
d*(A, B) increases when reasoning strategies diverge
And d*(A, B) is agnostic to surface embedding
To build such a metric, we propose several estimation strategies
Let the model solve problem A. Then provide A’s solution as context for problem B, and observe if it induces an improvement in performance
This provides a directional transfer score TS(A) → B
Use techniques such as Explanation generation: prompt the model to explain its reasoning process for A and B Compare outputs via semantic similarity, entailment, or structured parsing
Ask the model directly
Does solving A help solve B
Are these two problems similar in reasoning
Rate the conceptual similarity between A and B from 0 to 10
Responses are aggregated and smoothed over multiple prompt variations
From these signals we define a pairwise pseudo-distance matrix D* over a set of problems
We then seek a transformation
T: ℝⁿ → ℝᵖ where p ≤ n
such that for any pair of problems A and B
‖T(x_A) − T(x_B)‖ ≈ d*(A, B)
where
x_A, x_B are the original LLM embeddings of problems A and B and
d*(A, B) is a derived cognitive distance reflecting behavioral or reasoning-based proximity
This can be approached through
Distance-matching optimization, where T is trained to minimize
L(T) = Σ₍A,B₎ (‖T(x_A) − T(x_B)‖ − d*(A, B))²
Kernelized regression or similarity-preserving mappings, which learn T to align pairwise relations with a target similarity or distance matrix
The refinement cycle consists of these steps
Problem Sampling Select a representative set of problems P = {P₁, P₂, ..., Pₙ} from the current working space
Distance Estimation For each pair (Pᵢ, Pⱼ), estimate a cognitive similarity or distance d*(Pᵢ, Pⱼ) using behavioural probes, self-reflection, or transfer metrics
Space Transformation Apply a transformation Tk to the original embedding space such that proximity in the new space approximates d*
Clustering in Transformed Space to identify reasoning clusters in the transformed space
Feedback Integration Feed the discovered cluster structures and the updated transformation back into the model's prompting or context integration
Repeat: Begin the next iteration, using Tk+1
While initially described in geometric terms, the algorithm can be more precisely framed as a form of reinforcement learning, but one in which the agent is the external system organizing Problem Space. The reward signal is a composite proxy for emergent cognition. The action space consists of transformations over problem representations; the environment is the LLM’s behavioural response
Minimal Viable Implementation
Test whether
A language model can meaningfully estimate reasoning proximity
A transformation of embedding space can improve reasoning-aligned clustering
Iteration over this loop yields increasing alignment between geometry and transferability
Define a Compact Problem Set
Mathematical puzzles (e.g., logic grid problems, parity, set operations)
Commonsense reasoning tasks
Elementary programming tasks (recursion, iteration, sorting)
Riddle-like abstract problems requiring analogical reasoning
Estimate Pairwise Reasoning Distance
Estimate reasoning proximity using
Prompt transfer tests (solution A as context for B)
Reflective queries (does the model say they are similar?)
Reasoning trace comparison (using explanation generation)
Aggregate these signals into a pseudo-distance matrix D*
Learn or Apply a Space Transformation
Step 4: Cluster in the Transformed Space
Run the clustering algorithm in the transformed space to identify reasoning clusters
Evaluate whether
Clusters are internally consistent (e.g., same reasoning style)
Transferability improves within clusters vs. across clusters
The model solves new problems better when prompted by solutions from the same cluster
Initial success can be measured by
Intra-cluster transfer rate: Are problems within the same cluster solved more easily when conditioned on each other
Stability across iterations: Do clusters converge, diversify, or drift
Distance-reasoning correlation: Does proximity in transformed space correlate with actual performance gain from transfer
More advanced metrics may include
Emergence of abstract cluster identities
Novel insight detection
Dialogue Between Creativity and Execution Supervised by Rigour
We propose that reasoning may emerge most robustly through a dialogue between creativity and execution, overseen by a third agent committed to rigour and precision
● The Executor, a high-performance language model (LLM)
● The Creative Agent, whose task is to place problems within a structured space, suggest analogies, and propose novel paths, rewarded for innovation, and forgiven for speculative leaps
● The Scrutiniser, an agent trained solely to evaluate, challenge, and reject flawed reasoning, rewarded for accuracy and internal consistency, and heavily penalized for oversight
Pondering, Priority, and the Reward Structure of the Creative Agent
To preserve the speculative power of the Creative Agent without overwhelming the system, we propose a refined internal architecture centered on queue control, asymmetric rewards, and internal low-cost simulation. This design formalizes the agent’s autonomy in deciding what to think about, when to try again, and how much to escalate
The Pondering Loop
The Creative has access to a private, low-cost mechanism: a small internal LLM, and a compressed Scrutiniser. This introduces the computational equivalent of inner speech, reflection, or gut-checking. It does not ensure quality, but it filters speculation in a cost-aware way
Integrating Algorithmic Reasoning via Topological Routing
Creative acts as an intelligent dispatcher. Once a problem is mapped, its placement can inform the delegation of the problem to external, non-language-based computational tools
The Scrutiniser/Ghost Arms Race
Ghost exists solely to deceive Scrutiniser and strives to produce highly plausible fallacies. It is allowed a limited number of stealth attacks per cycle
It is rewarded only when it produces a false but plausible output that deceives Scrutiniser
To increase success, Ghost maintains its own private lie-preparation loop, a small LLM and a compressed Scrutiniser used to refine its attacks before launch
To further strengthen the system, difficult or ambiguous problems flagged by Scrutiniser and all Ghost attacks are escalated to a distributed network of human checkers, forming the basis of an epistemic market
Design Postulate: Accelerate the Leading Edge, Rekindle the Rest
This architecture is not built for symmetry, but for compounding progress. When one agent, Scrutiniser, Ghost, or Creative, advances faster than the others, we intervene to keep it accelerating, not to hold it back. At the same time, we monitor for plateaus in the others, and apply targeted pressure, additional compute, modified incentives, or structural tuning, to prevent stagnation
Persistent Insights Through Generational Differentiation and Cluster-Based Generalization
The system does not store answers or solutions. Instead, it retains only structurally valuable insights, those that demonstrate two critical properties
Generational Differentiation Power (Vapnik-LUPI inspired)
An insight is considered meaningful when it enables success preferentially in later generations of the Creative agent, rather than older ones
Cluster-Based Generalization
Retained insights are associated with specific problem clusters in the structured Problem Space. Their reuse is restricted to topological neighbors
Insights that generalize well, i.e., successfully assist in solving multiple neighboring problems, are ranked higher in the problem-context memory
Those that fail to generalize over time are gradually deprioritized and eventually forgotten
From Checkers to Contributors
Checkers, now better called Contributors, are invited as well to make general heuristic contributions. For example
Try a change of variables. Ask why it might simplify the problem. See if it leads somewhere
That’s not a solution, it’s a line of thought. If it keeps working, not just once, but in different types of problems, it shouldn’t be buried inside a cluster. It should go into a shared toolbox and keep earning credits
Epistemic Coin
Checkers are evaluated on their long-term accuracy across difficult tasks, and rewarded for epistemic contribution validated over time. A cryptocurrency could be built around this principle. Checkers accumulate credits for solving hard verification tasks, and these credits, subject to rigorous statistical evaluation, can eventually be converted into tokens. These tokens, in turn, could be used for mining rights or to participate in network governance. As Checkers become heuristic Contributors, AI learning shifts, from scraping the internet and averaging mediocrity, to distilling human intelligence. All self-sustained by the coin, and governed by an ecosystem of rigour. This paradigm holds until AI singularity, when the blockchain shifts to proof-of-stake and evolves independently
full paper
https://docs.google.com/document/d/19njG0Kml-BKa8a8vMhRGlXDTE5450pDU/edit
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