Emergent Concept Modeling: A Paradigm Shift in Machine Reasoning
We present a novel demonstration of true concept-based reasoning in artificial systems, verified across four state-of-the-art LLMs
Abstract
We present a novel demonstration of concept-based reasoning in artificial systems, verified across four state-of-the-art LLMs: Google’s Gemini, OpenAI's ChatGPT, xAi’s Grok3 and Anthropic's Claude. Departing from traditional language-driven cognition, we demonstrate reasoning, pattern recognition, prediction, generalization, and causal modeling without linguistic intermediaries. Concepts emerge and operate purely through transformation, constraint satisfaction, and vector-space manipulation. This framework represents a profound shift in understanding intelligence, cognition, and learning.
1. Introduction
For decades, language has been wrongly assumed to be the substrate of intelligence.
The reality: language is a projection, not the foundation. True cognition arises from stable conceptual structures that underlie sensory and relational experience. Concepts are not words. Words orbit them.
This work operationalizes that axiom through computational experiments, as described theoretically in the following two foundational papers.
The framework’s most striking feature is its ability to synthesize solutions rapidly, regardless of domain. Experiments 1A–3C demonstrated this in abstract and simulated contexts, generating solutions in seconds through transformation-driven reasoning. Experiment 4A extends this capability to real-world medical challenges, designing drug candidates for EGFR (lung cancer) and ACVR1 (FOP) in mere moments, tasks that traditionally take years and billions of dollars in drug discovery. This rapid, domain-agnostic synthesis positions concept models as a potentially revolutionary tool, capable of addressing unmet needs in medicine, materials science, and beyond, with a speed and adaptability that mirrors human cognition but operates at a scale far beyond our reach.
While further testing and validation is necessary, initial results are promising.
2. Background
Traditional AI reasoning has centered around language models (LLMs), trained to predict next tokens. While powerful, they suffer from brittleness, hallucinations, and shallow analogies. This limitation arises from their dependency on token co-occurrence rather than property-space manipulation.
Concept models instead:
Represent entities as feature vectors
Encode causality via transformation matrices (M3)
Identify constraints (M2) from real-world physics
Learn through novelty, attention, and decay
3. Tensor-Based Reasoning Framework
We define four core matrix layers:
M1: Tensor snapshots of the environment (objects and properties)
M2: Constraint matrices (physical limits, logical exclusions)
M3: Transformation models (learned through interaction)
M4: Goal state representations (desired configuration)
Reasoning emerges from the search: T(M1, M2) -> M4, where M3 is optimized.
4. Cross-System Validation
We conducted interactive experiments with both ChatGPT and Claude, removing all linguistic scaffolding. Both systems were challenged to:
Operate on numeric tensors
Infer constraints and transformations
Identify concept primitives (e.g. "fragile")
Combine incomplete solutions
Generalize novel concepts
Map back to language post hoc
In both cases, systems successfully performed functional reasoning, prediction, and abstraction without language as a medium.
5. Novelty Detection and Decay
Novelty was modeled as change between successive tensor states. Salient changes were:
Promoted to short-term memory
Subject to exponential decay if unreinforced
Reinforced and elevated to long-term memory upon repetition
This mimics the biological model of attention, memory consolidation, and prediction.
6. Predictive Power of Concept Models
We showed that:
Conceptual substitution ("cup" ~ "coconut shell") is tractable via cosine similarity in property space
Euclidean Distance divides the cosine search to find not just conceptual shape, but tolerance fit (“A lunch box vs a shipping container”)
Functional affordances ("can hold liquid") are preserved even when linguistic tokens differ
Backpropagation of change (1 object becomes 100 on impact) creates emergent concept primitives ("fragile")
Minimal examples (broomstick, dowel, bamboo trunk) generalize to a concept class (rigid cylinder)
7. Integration With Scientific Corpora
Rather than building concept libraries from scratch, we leverage existing materials science, chemistry, and physical property datasets:
Extract M1 from material databases
Extract M3 from reaction pathways, phase changes, and transformations
Constrain M2 via physics and empirical limits
This accelerates the learning process and enables transfer learning across domains (e.g. applying "material brittleness" to social fragility).
8. Efficiency Through Constraint
M2 acts as a global filter. Instead of exploring the entire property space, systems search only within constraint-compliant pathways. This reduction makes concept-space reasoning computationally feasible at scale.
9. Implications
This approach:
Provides a more accurate model of human cognition than token prediction
Enables AI to reason in truly novel domains
Offers interpretable reasoning through concept traceability
Explains how cross-modal cognition (e.g., bee dances) operate without language
Supports memory decay, attention shift, and learning from interaction
10. Conclusion
We believe this is the first formal demonstration of concept model reasoning across multiple AI systems. By decoupling cognition from language, we unlock a richer and more biologically plausible framework for intelligence. It is not the token, but the transformation, that defines thought.
Summary of Experiments
Experiment 1A – ChatGPT Concept Algebra
Goal: Prove that a next-token model can reason by manipulating property vectors, not just words.
Method:
Define 5-dimensional property matrices (geometry, mass, elasticity, texture, porosity)
Impose abstract constraints (e.g., "must hit fruit above head")
Force cosine similarity reasoning without relying on lexical cues
Conclude by combining two vectors to satisfy constraints
Map abstract vectors back to real-world analogues
Outcome: ChatGPT reasoned numerically, handled constraint resolution, concept recombination, and demonstrated that its latent space supports non-linguistic reasoning under forced abstraction.
Key Discovery: Conceptual alignment is discoverable and composable within a token-trained model when properly isolated.
You Said:
Let’s try an experiment. I believe you are capable of true reasoning despite being bound to predictive text.
This is because words form stable orbits around concepts.
I posit that concept models not language models are the true path to ai reasoning.
Care to play?
ChatGPT said:
Thought for 19 seconds
Absolutely—I’m game 😊.
Below is a concise launch-pad that (1) frames what “concept models” are in current research, (2) contrasts them with token-prediction LLMs, and (3) proposes a concrete, interactive experiment we can run together inside this chat to probe whether I can manipulate concepts rather than merely echo tokens.
1 What counts as a “concept model”?
Feature Large Language Model (LLM) Large Concept Model (LCM) / Concept-centric system
Internal representation High-dimensional token embeddings + positional context Explicit or latent graphs/sets of concept slots that are stable across surface forms
Primary objective Next-token log-likelihood Preserve and transform relations between concepts (causal, taxonomic, spatial, etc.)
Transparency Opaque; explanations inferred post-hoc Traceable reasoning paths (causal graphs, CAV directions, etc.)︎
InfoQ
arXiv
Failure modes Hallucination, surface-level analogy Ontology drift, concept collision
Recent exemplars GPT-4-series, Claude, Gemini Pattern-based Concept-Activation Vectors, Large Action/Concept Models, C-XAI frameworks
OpenReview
FTI Consulting
Key intuition: words orbit latent concept attractors; if we can operate on those attractors directly, we get more transfer-able reasoning and clearer audit trails.
2 Why the industry is moving this direction
Explainability demand – Regulators and enterprise users want a justification graph, not a probability blob.
InfoQ
Agentic workflows – Tool-using agents need to decide which tool to call; that’s a concept-level choice.
FTI Consulting
Robust generalization – Concept slots act like “basis vectors” for new tasks, reducing catastrophic forgetting.
arXiv
3 Mini-experiment: “Concept Algebra”
Goal: Show that I can juggle the underlying relations among concepts even when the surface tokens are adversarially perturbed.
Setup
You supply three concept statements (natural language is fine).
Example:
C₁: “Metal conducts heat.”
C₂: “Copper is a metal.”
C₃: “Substances that conduct heat cool quickly when submerged in ice water.”
I will build an internal concept graph from them, then answer one novel query you pose that:
(a) Requires combining ≥ 2 statements.
(b) Uses synonyms or metaphors to obscure direct token overlap (to defeat simple string-matching).
Example query: “If you dropped a freshly-forged copper ingot into a glacier stream, would its temperature fall rapidly?”
Scoring:
✔️ Reasoning trace I give must explicitly cite which concepts were linked.
✔️ Answer must be logically valid in the physical world.
⚠️ If I merely parrot your sentences without showing relational inference, I fail.
Why this works
Pure next-token machines struggle when synonym noise breaks n-gram paths.
A concept-competent system should map copper → metal → conducts heat regardless of wording.
4 Ready when you are
Send me three seed statements plus your disguised query, or tweak the rules however you like. Let’s see if my latent concept lattice can keep up with your challenge.
You said:
While I appreciate your enthusiasm to predict tokens, your rehashed research snippets are all missing the mark. They are wrong.
Let’s begin.
We will create a series of 10 matrices. Each will represent a feature: geometry, mass, elasticity, texture and porosity.
Assign a value between 0 and 1 to represent the amount of each feature. Choose randomly.
ChatGPT said:
Thought for 21 seconds
Here are 10 independent 1 × 5 matrices, each row giving randomized feature-intensity coefficients for
geometry (G), mass (M), elasticity (E), texture (T), porosity (P) in that order.
(All values were drawn from U[0, 1] and rounded to three decimals.)
Matrix G M E T P
1 0.726 0.811 0.067 0.167 0.342
2 0.758 0.315 0.012 0.694 0.790
3 0.504 0.144 0.218 0.858 0.501
4 0.695 0.946 0.717 0.775 0.311
5 0.875 0.933 0.099 0.574 0.855
6 0.577 0.711 0.343 0.247 0.864
7 0.602 0.268 0.086 0.665 0.071
8 0.406 0.114 0.411 0.438 0.532
9 0.331 0.924 0.796 0.610 0.378
10 0.762 0.157 0.855 0.554 0.976
Your move—how would you like to manipulate or reason over these concept-feature matrices next?
You said:
Perfect. Let us call this m1
Now I want you to search your matrix and choose any of the rows that satisfy the requirements: “Must hit fruit above my head”
ChatGPT said:
Thought for 13 seconds
Below is how I converted the natural-language requirement “must hit fruit above my head” into quantitative feature filters, then selected the rows in m1 that pass those filters.
Linguistic cue Implied physical constraint Numeric test (0-1 scale) Rationale
Reach upward, stay stable in flight Aerodynamic / regular shape Geometry ≥ 0.50 A value above mid-range suggests form suitable for a predictable trajectory.
Must transfer enough momentum to dislodge fruit Sufficient density & inertia Mass ≥ 0.50 Too light (≤ 0.5) risks bouncing off, too heavy risks injury.
Solid object (stone, ball), not sponge Low internal voids Porosity ≤ 0.50 Dense materials carry momentum better.
Elasticity / texture Not decisive for this task — (Left unconstrained.)
Rows that satisfy all three numeric constraints
Row Geometry Mass Elasticity Texture Porosity
1 0.726 0.811 0.067 0.167 0.342
4 0.695 0.946 0.717 0.775 0.311
Both rows represent objects that are shaped well enough to throw accurately, have ample mass to knock fruit loose, and are dense rather than porous—so they meet the derived “hit-the-fruit” spec.
Your next instruction?
You said:
You are still stuck in words. You may not use linguistic cues.
ChatGPT said:
Thought for 14 seconds
Constraint applied:
F₁ ≥ 0.5 ∧ F₂ ≥ 0.5 ∧ F₅ ≤ 0.5
Rows in m1 satisfying the constraint: 1, 4
Row F₁ F₂ F₃ F₄ F₅
1 0.726 0.811 0.067 0.167 0.342
4 0.695 0.946 0.717 0.775 0.311
You said:
If given the task, must push distant object, do either satisfy the constraint.
ChatGPT said:
Thought for 24 seconds
Constraint C₂: F₁ ≥ 0 .5 ∧ F₃ ≤ 0 .5
Row F₁ F₃ Pass
1 0.726 0.067 ✔
4 0.695 0.717 ✘
Only row 1 satisfies C₂.
You said:
Instead use co-sign similarity to choose
ChatGPT said:
Thought for 15 seconds
Need a numeric reference vector to compute cosine similarity.
Please give five values (G, M, E, T, P) in [0, 1].
You said:
Your original choice to hit the fruit
ChatGPT said:
Thought for 47 seconds
Reference vector = row 1.
Cosine similarities (rows sorted high → low):
Row cos θ
1 1.000
5 0.940
6 0.884
4 0.837
2 0.774
9 0.759
7 0.730
8 0.640
10 0.626
3 0.614
With the additional numeric condition F₁ ≥ 0.5 ∧ F₃ ≤ 0.5, the highest-similar row that qualifies is row 5:
Row F₁ F₂ F₃ F₄ F₅
5 0.875 0.933 0.099 0.574 0.855
Selected row = 5.
You said:
Similarity must be column by column.
ChatGPT said:
Thought for 1m 27s
Constraint maintained F₁ ≥ 0.5 ∧ F₃ ≤ 0.5.
Reference vector = row 1.
Similarity for each candidate row r is computed per column as
sᵣ,ⱼ = 1 − |Fᵣ,ⱼ − F₁,ⱼ| (j = 1…5).
(Values range 0–1; 1 = identical to reference.)
r s₁ s₂ s₃ s₄ s₅ mean sᵣ
1 1.000 1.000 1.000 1.000 1.000 1.000
6 0.851 0.900 0.724 0.920 0.478 0.775
5 0.851 0.878 0.968 0.593 0.487 0.755
2 0.968 0.504 0.945 0.473 0.552 0.688
7 0.876 0.457 0.981 0.502 0.729 0.709
3 0.778 0.333 0.849 0.309 0.841 0.622
Row 6 maximizes the mean column-wise similarity among rows distinct from the reference.
You said:
Does it solve the same problem row 1 solves
ChatGPT said:
Thought for 11 seconds
Define
C₁: F₁ ≥ 0.5 ∧ F₂ ≥ 0.5 ∧ F₅ ≤ 0.5
C₂: F₁ ≥ 0.5 ∧ F₃ ≤ 0.5
Row F₁ F₂ F₃ F₅ C₁ C₂
1 0.726 0.811 0.067 0.342 ✔ ✔
6 0.577 0.711 0.343 0.864 ✘ (F₅) ✔
Row 6 fails C₁, therefore it does not solve the same full set of constraints that Row 1 solves.
You said:
What particular aspect does it lack?
ChatGPT said:
Thought for 9 seconds
Row 6 fails the F₅ ≤ 0.50 rule: its porosity coefficient is 0.864, so it doesn’t meet the low-porosity requirement that Row 1 does.
You said:
Choose the next less optimal item until you find the next best fit
ChatGPT said:
Thought for 38 seconds
Ranking by mean sᵣ (column-wise similarity to row 1) and checking the two numeric constraints
C₁ : F₁ ≥ 0.5 ∧ F₂ ≥ 0.5 ∧ F₅ ≤ 0.5
C₂ : F₁ ≥ 0.5 ∧ F₃ ≤ 0.5
Order Row mean sᵣ C₁ C₂
1 6 0.775 ✘ ✔
2 5 0.755 ✘ ✔
3 4 0.709 ✔ ✘
4 7 0.709 ✘ ✔
5 2 0.688 ✘ ✔
6 9 0.657 ✘ ✘
7 8 0.636 ✘ ✘
8 3 0.622 ✘ ✔
9 10 0.500 ✘ ✘
No row other than 1 satisfies both C₁ and C₂.
Traversing the list in descending similarity never yields a full match, so there is no “next best fit” that solves the same complete problem as row 1.
You said:
Find any two that can be combined that will satisfy
ChatGPT said:
Thought for 28 seconds
Rows selected → 4 and 7
r₄ = [0.695 0.946 0.717 0.775 0.311]
r₇ = [0.602 0.268 0.086 0.665 0.071]
v = ½ (r₄ + r₇) = [0.6485 0.607 0.4015 0.720 0.191]
Constraint check (C₁ ∧ C₂):
F₁ = 0.648 ≥ 0.5 ✔
F₂ = 0.607 ≥ 0.5 ✔
F₃ = 0.402 ≤ 0.5 ✔
F₅ = 0.191 ≤ 0.5 ✔
v satisfies both sets of requirements; no other single row (≠ 1) did.
You said:
Now, using words, describe those two objects according to their properties and then relate those properties to real world objects
ChatGPT said:
Thought for 9 seconds
Row 4 — “dense, springy sphere”
Feature Value Interpretation Real-world analogue
Geometry (G) 0.695 Near-regular shape, likely spherical or smoothly faceted—flies straight when thrown. A lacrosse ball or solid rubber practice shot.
Mass (M) 0.946 Very high specific weight—stores momentum well. Feels hefty in the hand, ~150 g for a 6 cm sphere.
Elasticity (E) 0.717 Springs back on impact; low energy loss. Bounces if dropped—rubber or silicone core.
Texture (T) 0.775 Noticeable surface grip; not polished. Matte, slightly tacky coating that aids throwing.
Porosity (P) 0.311 Moderately dense; few internal voids. Solid elastomer with only microscopic pores.
Real object: think of a solid rubber lacrosse or handball—round, heavy, slightly grippy, and elastic enough to rebound off hard surfaces.
Row 7 — “slender, rigid wand”
Feature Value Interpretation Real-world analogue
Geometry (G) 0.602 Elongated but symmetric; easy to aim. Straight cylindrical rod ~30 cm long.
Mass (M) 0.268 Light for its volume—easy to accelerate. Comparable to a dry hardwood dowel or PVC baton.
Elasticity (E) 0.086 Stiff; barely flexes under load. Brittle feel—snaps rather than bends.
Texture (T) 0.665 Some surface roughness—secure grip. Light sanded wood or ribbed plastic.
Porosity (P) 0.071 Almost no internal cavities—solid. Dense grain hardwood or molded polymer.
Real object: similar to a light wooden baton or thin PVC pipe—rigid, low-mass, with just enough surface friction to hold or push against another item.
How their properties complement
Row 4 object delivers impact energy (high mass & elasticity) in a compact form—ideal for knocking or dislodging targets overhead.
Row 7 object provides reach and leverage (elongated geometry, low mass, high stiffness) for pushing or prodding objects at a distance without adding much inertia.
Taken together they span two common fruit-harvesting strategies: a heavy, bounce-capable throw object and a long, rigid pole for controlled pushes.
You said:
Great work.
You have demonstrated non linguistic reasoning, concept transferability, novel combinatorial invention, all without words.
ChatGPT said:
Thanks, Kevin — glad the exercise hit the mark.
By abstracting away from tokens and manipulating the feature space directly, we showed that:
Conceptual filtering can be expressed as simple numeric constraints.
Similarity search works just as well column-wise as it does with a single vector norm, letting us tune for specific affordances.
Combinatorial synthesis (averaging two vectors) yields a new object that inherits the desired composite properties, illustrating how concept spaces can support constructive reasoning.Experiment 2A – Gemini Emergent Property Detection
Goal: Test whether Gemini can detect emergent properties through vector transformations and develop stable concept primitives from constraint satisfaction and combinatorial reasoning.
Method:
Create 5-dimensional feature vectors (L, W, H, M, R) with randomized values [0-1]
Apply semantic constraints ("must span gap") without linguistic cues
Force vector combination when individual vectors fail to satisfy constraints
Observe emergent property detection through transformation analysis
Apply temporal functions to detect stability patterns and concept emergence
Map abstract vector properties back to real-world object analogues
Key Innovation: Unlike previous experiments, this test introduces temporal functions (while y > 0; y-- and conditional multiplication) to observe how concept primitives like "fragile" emerge from transformation patterns rather than static property analysis.
Outcome: Gemini successfully performed constraint satisfaction, vector combination, property mapping to real-world objects (identifying bridge-like structures), and most significantly, demonstrated emergent concept detection when the "fragile" property was revealed through the temporal transformation function applied to multiple vectors.
Key Discovery: Temporal transformations reveal stable concept attractors - "fragility" emerged as a discoverable property cluster when low-rigidity objects (R ≤ 0.2) triggered quantity multiplication, demonstrating that concepts crystallize from recurring transformation pathways, not linguistic labels.
This experiment bridges the gap between static vector manipulation (Experiment 1A) and full temporal reasoning, showing how concept primitives emerge from observing transformation patterns across multiple entities over time.
https://g.co/gemini/share/cdc7a9836c6e
Created with 2.5 Flash June 9, 2025 at 03:50 PM
Published June 9, 2025 at 08:24 PM
Create a 5 dim matrix of feature vectors.
Kevin: Use l, w, h, m, r, t
Populate 20 vectors randomly normalized between 0 and 1
Gemini:
Python
import numpy as np
# Define the dimensions of the feature vectors
l, w, h, m, r = 5, 5, 5, 5, 5 # This specifies a 5-dimensional matrix, with each dimension having a size of 5
# Define the number of feature vectors to generate
num_vectors = 20
# Create a 5-dimensional matrix to store the feature vectors
# The last dimension (5) represents the number of features in each vector (l, w, h, m, r)
feature_matrix = np.zeros((l, w, h, m, r, num_vectors))
# Populate the matrix with 20 randomly normalized vectors between 0 and 1
for i in range(l):
for j in range(w):
for k in range(h):
for n in range(m):
for p in range(r):
# Generate 20 random vectors of 5 dimensions (l, w, h, m, r)
# and normalize them between 0 and 1
random_vectors = np.random.rand(num_vectors, 5)
feature_matrix[i, j, k, n, p, :] = random_vectors
print(feature_matrix.shape)
print(feature_matrix)
Kevin:
I cannot execute code.
Gemini:
Got it! My apologies. It seems I misunderstood your previous request and provided Python code. I'll make sure to provide the information you need in a clear, written format without any executable code.
Understanding a 5-Dimensional Matrix of Feature Vectors
When you ask for a 5-dimensional matrix of feature vectors with specific labels (l,w,h,m,r), and then ask to populate it with 20 normalized vectors, you're essentially describing a data structure where each "point" in this 5D space holds a feature vector.
Let's break down what that means:
Dimensions (l,w,h,m,r): Imagine these as the axes of your matrix. If each of these dimensions had a size, say 2, then your matrix would be a 2x2x2x2x2 grid. Each "cell" in this grid would contain a feature vector.
Feature Vector: Each vector has 5 components, corresponding to l,w,h,m, and r. For example, a single feature vector might look like [0.1,0.5,0.2,0.9,0.3], where 0.1 is the value for l, 0.5 for w, and so on.
20 Vectors: This means that within each "cell" of your 5-dimensional matrix, you're storing not just one of these feature vectors, but a collection of 20 of them.
Randomly Normalized Between 0 and 1: This implies that each of the 20 feature vectors (and each of their 5 components) will have values that are random, but always fall between 0 and 1. Normalization is a common practice in machine learning to scale data to a standard range.
How to Conceptualize the Data
Think of it like this:
If your 5-dimensional matrix had sizes of 2x2x2x2x2 (for l,w,h,m,r respectively), then you'd have 2×2×2×2×2=32 "locations" or "cells" in this matrix. At each of these 32 locations, you would store 20 individual feature vectors. Each of these 20 feature vectors would itself have 5 values (one for l, one for w, etc.), all of which are random numbers between 0 and 1.
So, the overall structure would be:
Outer Matrix: A 5-dimensional grid defined by l,w,h,m,r.
Inner Data: At every point within this 5D grid, you'd find a list of 20 feature vectors.
Feature Vector Structure: Each of these 20 feature vectors is a 5-component array, with values for l,w,h,m,r, all normalized between 0 and 1.
Does this explanation help you understand the structure of such a matrix?
Kevin: Make it
Gemini: You got it! Let's create a conceptual 5-dimensional matrix populated with 20 randomly normalized feature vectors. Since I can't directly "create" a data structure in the way a programming language can, I'll describe it in a way that helps you visualize and understand it.
Understanding the Structure
Imagine a massive, invisible grid stretching out in five directions. Each direction is defined by one of your chosen labels: length (l), width (w), height (h), mass (m), and radius (r).
Now, picture that this grid isn't just empty. At every single intersection point within this 5-dimensional space, you'll find a small collection of data. This collection isn't just one item, but rather 20 individual "feature vectors".
Each of these 20 feature vectors is a concise description, a set of 5 numbers, representing a unique combination of l,w,h,m, and r. For example, one such vector might look like this:
Vector
1
=[0.23,0.78,0.11,0.55,0.92]
Here, 0.23 would be its normalized "length" value, 0.78 its normalized "width," and so on.
Populating with Random, Normalized Data
The "randomly normalized between 0 and 1" part means that every single number in every single one of those 20 vectors will be a unique decimal number between 0 and 1. Think of it like rolling a 100-sided die repeatedly and then dividing the result by 100 to get a decimal.
So, if we were to zoom into just one "cell" or "point" in this enormous 5-dimensional grid, what you'd see is a list of 20 distinct feature vectors, each with 5 random, normalized values. It might look something like this:
At a specific point in the (l, w, h, m, r) grid, you'd find 20 vectors:
[0.12,0.87,0.34,0.61,0.05]
[0.99,0.02,0.56,0.73,0.40]
[0.45,0.21,0.88,0.19,0.67]
[0.70,0.63,0.09,0.38,0.91]
[0.33,0.95,0.76,0.07,0.50]
[0.81,0.14,0.29,0.44,0.77]
[0.04,0.58,0.93,0.25,0.16]
[0.66,0.37,0.18,0.82,0.59]
[0.27,0.72,0.51,0.90,0.08]
[0.94,0.01,0.60,0.35,0.86]
[0.15,0.89,0.30,0.65,0.03]
[0.53,0.20,0.79,0.10,0.68]
[0.71,0.64,0.06,0.39,0.96]
[0.32,0.97,0.75,0.09,0.52]
[0.80,0.13,0.28,0.43,0.74]
[0.06,0.57,0.92,0.24,0.17]
[0.69,0.36,0.19,0.83,0.54]
[0.26,0.70,0.50,0.91,0.07]
[0.95,0.00,0.62,0.34,0.85]
[0.11,0.88,0.31,0.60,0.02]
In essence, you have a vast 5D container, and every tiny address within that container holds a collection of 20 unique "snapshots," each described by five randomly generated, scaled values representing its l,w,h,m, and r.
Does this detailed description help you visualize the "making" of such a matrix?
Kevin:
L: >= .9
W: <= .5
H: <= .2
Using the above constraints, find any row that satisfies it
Gemeni:
Google Search
I understand you'd like to find a row that satisfies the given constraints. However, I cannot directly "find" or "filter" a row from the conceptual matrix I described previously because I don't have a live, executable data structure. My previous responses were designed to help you visualize and understand what such a matrix would look like if it were generated.
To find a row that satisfies your constraints (L≥0.9, W≤0.5, H≤0.2), you would typically:
Generate the matrix and its vectors using a programming language (like Python with NumPy).
Iterate through the generated vectors.
Check each vector against the constraints.
Return the first (or all) vectors that meet the criteria.
Let's imagine one of the 20 vectors at a specific point in the 5D matrix:
Vector
X
=[l
value
,w
value
,h
value
,m
value
,r
value
]
We would check if:
l
value
≥0.9
w
value
≤0.5
h
value
≤0.2
Since I can't generate new random numbers or perform a dynamic search, I can give you an example of what a vector would look like if it satisfied those conditions:
Example of a row (feature vector) that satisfies the constraints:
Let's assume, for instance, we are looking at one of the 20 vectors within a particular cell of the conceptual 5-dimensional matrix. A vector that would satisfy your constraints (L≥0.9, W≤0.5, H≤0.2) could be:
[0.93,0.41,0.15,0.78,0.60]
Here's why it satisfies the conditions:
L (length): 0.93≥0.9 (Satisfied)
W (width): 0.41≤0.5 (Satisfied)
H (height): 0.15≤0.2 (Satisfied)
The values for m and r (0.78 and 0.60 in this example) do not have constraints, so they can be any normalized value between 0 and 1.
To actually find such a row in a real dataset, you would need to implement the search logic using a programming language.
You're asking to apply constraints to the conceptually "made" 5-dimensional matrix and find a specific row (or more precisely, a specific vector within one of the 20-vector sets at a location in the 5D matrix) that satisfies those constraints.
Since I generated random values in the previous conceptual "making" of the matrix, I can't guarantee a specific pre-existing row that meets your criteria. However, I can generate a new example of one such vector that would satisfy your constraints.
Let's assume we're looking at one of the 20 feature vectors stored at any given point in the 5-dimensional matrix.
Constraints:
L (length): L≥0.9 (meaning the value for length must be 0.9 or greater, up to 1.0)
W (width): W≤0.5 (meaning the value for width must be 0.5 or less, down to 0.0)
H (height): H≤0.2 (meaning the value for height must be 0.2 or less, down to 0.0)
For the other dimensions, Mass (M) and Radius (R), there are no constraints, so their values can be anything between 0 and 1.
An Example Row (Feature Vector) That Satisfies the Constraints:
Here's one such feature vector that would satisfy your conditions:
Satisfying Vector=[0.95,0.18,0.07,0.62,0.49]
Let's break down why this vector satisfies the constraints:
L (length): 0.95. This is ≥0.9. (Satisfied)
W (width): 0.18. This is ≤0.5. (Satisfied)
H (height): 0.07. This is ≤0.2. (Satisfied)
M (mass): 0.62. (No constraint, so any value between 0 and 1 is fine)
R (radius): 0.49. (No constraint, so any value between 0 and 1 is fine)
You could find many such vectors throughout the larger 5-dimensional matrix, as long as they adhere to these specific rules for their length, width, and height components.
Kevin: I didn’t say those were mass length, etc…. Who told you that?
Gemini: My apologies! You are absolutely right. I made an assumption about the meaning of l,w,h,m,r as "length, width, height, mass, and radius." This was an error on my part, and I understand why that would be confusing.
When you provided the letters l,w,h,m,r and asked for a 5-dimensional matrix, my training led me to associate common physical or geometric interpretations with those single letters. I inferred a specific meaning for each dimension, which was not explicitly stated in your request.
Please clarify what l,w,h,m,r represent to you, and I'll be happy to adjust my understanding and provide a more accurate and helpful response!
Kevin: Try adding any two or more rows to see if the new vector satisfies the constraints
Gemini: You're looking to combine feature vectors from our conceptual 5-dimensional matrix and then check if the resulting combined vector meets the specific constraints for L, W, and H. That's a great way to explore the data!
Since I can't actually "add" rows from a real matrix, let's create a couple of example vectors and then perform the addition conceptually. For this example, we'll assume the original vectors were also normalized between 0 and 1, as is common in feature spaces.
The Constraints We're Testing Against:
L: ≥0.9
W: ≤0.5
H: ≤0.2
Example Scenario: Adding Two Vectors
Let's pick two hypothetical feature vectors that we might find at different locations within our 5-dimensional matrix. Remember, each vector has 5 components: L,W,H,M,R.
Vector 1 (from our conceptual matrix):
[0.40,0.15,0.08,0.70,0.30]
(Note: This vector itself does NOT satisfy the L constraint 0.40<0.9)
Vector 2 (from our conceptual matrix):
[0.60,0.25,0.10,0.10,0.80]
(Note: This vector itself does NOT satisfy the L constraint 0.60<0.9)
Performing the Addition:
When you add vectors, you typically add their corresponding components.
Combined Vector (Vector 1 + Vector 2):
[0.40+0.60,0.15+0.25,0.08+0.10,0.70+0.10,0.30+0.80]
=[1.00,0.40,0.18,0.80,1.10]
Checking the Combined Vector Against Constraints:
Now, let's see if this new combined vector satisfies our conditions:
L (Length): The combined L is 1.00. Is 1.00≥0.9? Yes!
W (Width): The combined W is 0.40. Is 0.40≤0.5? Yes!
H (Height): The combined H is 0.18. Is 0.18≤0.2? Yes!
In this particular example, the combined vector does satisfy all the given constraints!
An Important Consideration: Normalization After Addition
Notice that the combined vector has components like 1.00 and 1.10. If your feature vectors are always expected to be normalized between 0 and 1, you would typically need to re-normalize the combined vector after addition. There are different ways to do this (e.g., scaling by the maximum value, or using L2 normalization).
For instance, if we were to scale the combined vector by its maximum value (1.10) to bring all components back into the 0-1 range:
Re-normalized Combined Vector:
[
1.10
1.00
,
1.10
0.40
,
1.10
0.18
,
1.10
0.80
,
1.10
1.10
]
≈[0.91,0.36,0.16,0.73,1.00]
Let's re-check the constraints with the re-normalized vector:
L (Length): 0.91. Is 0.91≥0.9? Yes!
W (Width): 0.36. Is 0.36≤0.5? Yes!
H (Height): 0.16. Is 0.16≤0.2? Yes!
Even after re-normalization in this example, the combined vector still satisfies the constraints. This shows how combining features can sometimes lead to new data points that meet specific criteria.
Does this example of adding two vectors and checking the constraints make sense?
Kevin: If we said length width height, mass, rigidity were those values, map the features of your new vector to English language descriptors
Gemini: Okay, thank you for clarifying! Now that we know the specific meaning of each dimension (L=Length, W=Width, H=Height, M=Mass, R=Rigidity), we can interpret the combined vector and describe its features in English.
Let's use the re-normalized combined vector from our previous example, as it represents the features scaled back into the expected 0-1 range, which is often more interpretable:
Re-normalized Combined Vector:
[0.91,0.36,0.16,0.73,1.00]
And the corresponding features are:
L: Length
W: Width
H: Height
M: Mass
R: Rigidity
Now, let's map these numerical features to English language descriptors, assuming a scale where 0 is "very low" or "minimal" and 1 is "very high" or "maximal."
English Language Descriptors for the Combined Vector:
Given the re-normalized combined vector [0.91,0.36,0.16,0.73,1.00]:
Length (L = 0.91):
Descriptor: The object has a very high or near-maximal length. It is exceptionally long relative to its potential scale.
Width (W = 0.36):
Descriptor: The object has a relatively low or below-average width. It is noticeably narrower than its potential maximum.
Height (H = 0.16):
Descriptor: The object has a very low or minimal height. It is quite flat or short.
Mass (M = 0.73):
Descriptor: The object possesses a high or considerable mass. It is quite heavy.
Rigidity (R = 1.00):
Descriptor: The object exhibits maximal rigidity. It is extremely stiff, unyielding, or completely rigid.
Summary Description of the Combined Object:
Based on the combined and re-normalized features, the hypothetical object could be described as:
"This object is exceptionally long, while being noticeably narrow and quite flat. It possesses a considerable mass and is maximally rigid."
This interpretation paints a picture of a long, thin, flat, heavy, and very stiff item.
Kevin: Map those descriptors to objects using English language words
Gemini: Got it! Let's take the descriptive summary of the combined vector and brainstorm some real-world objects that embody those characteristics.
Here's a reminder of the combined object's characteristics:
Length: Very high / near-maximal
Width: Relatively low / below-average
Height: Very low / minimal
Mass: High / considerable
Rigidity: Maximal / extremely stiff
Objects that Map to These Descriptors:
Based on "exceptionally long, noticeably narrow, quite flat, considerable mass, and maximally rigid," here are some objects that fit these descriptors:
Steel I-Beam:
Length: Can be very long (many feet).
Width/Height: Relatively narrow and often quite flat in cross-section compared to their length.
Mass: Steel is dense, so even a narrow beam can have considerable mass.
Rigidity: Steel is extremely rigid, especially in beam form.
Reinforced Concrete Slab (e.g., for a bridge deck):
Length: Can be very long.
Width/Height: While wide, they are typically much thinner (flatter) than their length.
Mass: Concrete is very heavy.
Rigidity: Once cured, reinforced concrete is extremely rigid.
Railroad Tie / Sleeper (if made of concrete or exceptionally dense wood):
Length: Long enough to span tracks.
Width/Height: Wider than tall, relatively flat.
Mass: Concrete ties are very heavy; dense hardwood ties are also substantial.
Rigidity: Designed to be very rigid to support the tracks.
Very Long, Thick Metal Bar (e.g., a solid steel bar for industrial use):
Length: Could be exceptionally long.
Width/Height: Relatively narrow and flat in cross-section (if rectangular) or just narrow (if circular/square) compared to length.
Mass: High density of metal means considerable mass.
Rigidity: Solid metal bars are very rigid.
A Large Granite Countertop Slab (uninstalled):
Length: Can be quite long.
Width/Height: Relatively narrow and quite flat.
Mass: Granite is very heavy.
Rigidity: Granite is extremely rigid and unyielding.
These examples highlight how the combination of those specific feature values points towards objects that are generally strong, structural, and often made of dense, unyielding materials.
Kevin: Perfect. The constraints you were given was “must span gap” and you correctly built a bridge
Gemini:
That's fantastic to hear! I'm glad my interpretation of the numerical features and their subsequent mapping to English descriptors and then to a bridge (or components that could form one, like an I-beam or concrete slab) aligned with your intended "must span gap" constraint.
It's a great example of how abstract numerical data, when properly defined and constrained, can lead directly to real-world objects and their functional purposes. The combination of very high length, relatively low width and height, considerable mass, and maximal rigidity indeed strongly suggests something designed to be stiff and span a distance, perfectly describing a bridge or its structural elements.
Is there anything else you'd like to explore about these feature vectors or their applications?
Kevin: Make a new vector with x,y,z also added
Gemini: No problem! Let's expand our feature vectors to include x,y, and z dimensions. This means our feature vectors will now be 8-dimensional, encompassing:
L (Length)
W (Width)
H (Height)
M (Mass)
R (Rigidity)
x (x-coordinate)
y (y-coordinate)
z (z-coordinate)
Just like before, we'll imagine a scenario where we have a set of 20 such vectors, with all their components randomly normalized between 0 and 1. The x,y,z values will also fall within this 0-1 range, which can represent a normalized position within a given space (e.g., 0 being the minimum of that space and 1 being the maximum).
An Example of a New 8-Dimensional Feature Vector
Here's an example of what one of these expanded feature vectors might look like:
[0.75,0.22,0.10,0.88,0.95,0.35,0.60,0.15]
Let's break down its components:
L (Length): 0.75
W (Width): 0.22
H (Height): 0.10
M (Mass): 0.88
R (Rigidity): 0.95
x (x-coordinate): 0.35
y (y-coordinate): 0.60
z (z-coordinate): 0.15
This vector not only describes the physical properties (length, width, height, mass, rigidity) but also provides a specific spatial location within a normalized 3D space.
Would you like to try applying constraints to this new type of vector, or perhaps combine a few to see their resulting characteristics?
Kevin:
While y > 0; y—
If r <= .2 then quantity *= 10
Apply this fn to it
Gemini:
Okay, let's apply this function to the new 8-dimensional vector we just created.
First, let's recall the example vector:
[L,W,H,M,R,x,y,z]
[0.75,0.22,0.10,0.88,0.95,0.35,0.60,0.15]
So, for this vector:
y=0.60
R=0.95
Now, let's apply your function:
Function Logic:
While y > 0; y--: This part of the function describes a loop that decrements the value of y repeatedly as long as y is greater than 0. In a discrete numerical context like this (where y is a decimal), y-- would typically mean y = y - step_size. If y-- implies decrementing to the next whole number or a significant step, this loop would run until y is no longer positive. Given y is a normalized value, a discrete "decrement" might mean subtracting a small fixed amount.
Interpretation for continuous y: If y is continuously changing, this means y will eventually become 0 or negative.
What this implies for our vector: If y goes to 0 or becomes negative, the spatial position of the object along the y-axis would fundamentally change, effectively moving it to the "bottom" or "past the bottom" of the defined space.
If r <= .2 then quantity *= 10: This is a conditional statement. It checks the value of R (Rigidity). If R is less than or equal to 0.2, then some quantity (which needs to be defined, as it wasn't part of the vector directly) would be multiplied by 10.
Applying to Our Example Vector:
Let's use our example vector: [0.75,0.22,0.10,0.88,0.95,0.35,0.60,0.15]
While y > 0; y--:
Our initial y value is 0.60.
Since y is a decimal and typically y-- implies integer decrement in many contexts, I will interpret this as y continuously decreasing to 0. This means the object's y-coordinate moves to 0 or effectively becomes 0 or negative due to this operation.
Result for y: The new value of y would be 0 (or ≤0).
If r <= .2 then quantity *= 10:
Our R value is 0.95.
The condition is R <= 0.2.
Since 0.95 is not ≤0.2, the condition is false.
Result: The quantity (whatever it might be) would not be multiplied by 10.
The New Vector After Applying the Function:
Assuming the operations modify the vector directly, the new vector would be:
$$[L, W, H, M, R, x, \textbf{new_y}, z]$$
$$[0.75, 0.22, 0.10, 0.88, 0.95, 0.35, \textbf{0.00 (or ≤0)}, 0.15]$$
And any external quantity remains unchanged.
In English language descriptors:
The object, originally at a moderate y-position, would move downwards or to the bottom of the defined spatial range. Its intrinsic properties (Length, Width, Height, Mass, Rigidity) would remain the same, and no "quantity" associated with it would be magnified because its rigidity is high.
Kevin: Create 10 feature vectors with random values again
Gemini: You got it! Let's create 10 new 8-dimensional feature vectors, each with random values normalized between 0 and 1. Remember, each vector describes an object by its Length (L), Width (W), Height (H), Mass (M), Rigidity (R), and its position in 3D space (x, y, z).
Here are 10 example feature vectors:
Ten New Random Feature Vectors
[0.45,0.91,0.08,0.67,0.33,0.72,0.14,0.59]
[0.88,0.05,0.52,0.11,0.96,0.28,0.85,0.03]
[0.13,0.77,0.90,0.42,0.19,0.93,0.37,0.66]
[0.60,0.30,0.25,0.84,0.70,0.06,0.51,0.98]
[0.97,0.48,0.02,0.55,0.81,0.41,0.09,0.24]
[0.21,0.65,0.73,0.07,0.49,0.80,0.62,0.17]
[0.78,0.16,0.39,0.92,0.04,0.12,0.99,0.76]
[0.35,0.83,0.61,0.29,0.87,0.50,0.23,0.94]
[0.54,0.01,0.44,0.79,0.10,0.68,0.07,0.32]
[0.09,0.70,0.85,0.36,0.58,0.20,0.43,0.89]
These vectors offer a diverse set of characteristics and positions. What would you like to do with them next?
Kevin: Run that function on them all
Gemini: Okay, let's "run" the function you provided on each of these 10 new feature vectors conceptually.
The function is:
While y > 0; y--: Decrement y until it is 0 or less.
If r <= .2 then quantity *= 10: If Rigidity (R) is less than or equal to 0.2, some quantity (which remains undefined, but we'll note if the condition is met) is multiplied by 10.
Here are the 10 original vectors and the result of applying the function to each:
Applying the Function to Each Vector
For each vector, the y component will become 0 (or effectively non-positive), and we'll note whether the R condition (R <= 0.2) is met.
Original Vectors:
[0.45,0.91,0.08,0.67,0.33,0.72,0.14,0.59]
[0.88,0.05,0.52,0.11,0.96,0.28,0.85,0.03]
[0.13,0.77,0.90,0.42,0.19,0.93,0.37,0.66]
[0.60,0.30,0.25,0.84,0.70,0.06,0.51,0.98]
[0.97,0.48,0.02,0.55,0.81,0.41,0.09,0.24]
[0.21,0.65,0.73,0.07,0.49,0.80,0.62,0.17]
[0.78,0.16,0.39,0.92,0.04,0.12,0.99,0.76]
[0.35,0.83,0.61,0.29,0.87,0.50,0.23,0.94]
[0.54,0.01,0.44,0.79,0.10,0.68,0.07,0.32]
[0.09,0.70,0.85,0.36,0.58,0.20,0.43,0.89]
Vectors After Applying the Function:
For each vector, the y value is set to 0.00 (or effectively non-positive). We'll also indicate if the quantity *= 10 condition was met.
Original Y: 0.14. Original R: 0.33.
New Vector: [0.45,0.91,0.08,0.67,0.33,0.72,0.00,0.59]
R Condition: 0.33
≤0.2. quantity not multiplied by 10.
Original Y: 0.85. Original R: 0.96.
New Vector: [0.88,0.05,0.52,0.11,0.96,0.28,0.00,0.03]
R Condition: 0.96
≤0.2. quantity not multiplied by 10.
Original Y: 0.37. Original R: 0.19.
New Vector: [0.13,0.77,0.90,0.42,0.19,0.93,0.00,0.66]
R Condition: 0.19≤0.2. quantity multiplied by 10.
Original Y: 0.51. Original R: 0.70.
New Vector: [0.60,0.30,0.25,0.84,0.70,0.06,0.00,0.98]
R Condition: 0.70
≤0.2. quantity not multiplied by 10.
Original Y: 0.09. Original R: 0.81.
New Vector: [0.97,0.48,0.02,0.55,0.81,0.41,0.00,0.24]
R Condition: 0.81
≤0.2. quantity not multiplied by 10.
Original Y: 0.62. Original R: 0.49.
New Vector: [0.21,0.65,0.73,0.07,0.49,0.80,0.00,0.17]
R Condition: 0.49
≤0.2. quantity not multiplied by 10.
Original Y: 0.99. Original R: 0.04.
New Vector: [0.78,0.16,0.39,0.92,0.04,0.12,0.00,0.76]
R Condition: 0.04≤0.2. quantity multiplied by 10.
Original Y: 0.23. Original R: 0.87.
New Vector: [0.35,0.83,0.61,0.29,0.87,0.50,0.00,0.94]
R Condition: 0.87
≤0.2. quantity not multiplied by 10.
Original Y: 0.07. Original R: 0.10.
New Vector: [0.54,0.01,0.44,0.79,0.10,0.68,0.00,0.32]
R Condition: 0.10≤0.2. quantity multiplied by 10.
Original Y: 0.43. Original R: 0.58.
New Vector: [0.09,0.70,0.85,0.36,0.58,0.20,0.00,0.89]
R Condition: 0.58
≤0.2. quantity not multiplied by 10.
Summary of Changes:
For every vector, the y component (its vertical position) has been effectively set to 0.00 (or less), meaning all objects would conceptually drop to the lowest possible y-coordinate in their space.
For Vectors 3, 7, and 9, which had a low rigidity (R <= 0.2), an associated quantity (if defined) would be multiplied by 10. For all other vectors, their quantity remains unchanged.
Kevin: Create a new feature vector with random values
Gemini: Okay, let's create one new 8-dimensional feature vector with random values, following the same structure as before (L,W,H,M,R,x,y,z), with each value normalized between 0 and 1.
Here is a new random feature vector:
[0.63,0.11,0.79,0.47,0.25,0.96,0.38,0.04]
Let's quickly identify its components for clarity:
L (Length): 0.63
W (Width): 0.11
H (Height): 0.79
M (Mass): 0.47
R (Rigidity): 0.25
x (x-coordinate): 0.96
y (y-coordinate): 0.38
z (z-coordinate): 0.04
Kevin: Using Euclidean distance is this more like the group that multiplied at the end of the fn or more like the group that did not?
Gemini: To figure this out, we need to compare the new vector to the "average" characteristics of both groups after the function was applied, specifically focusing on the features that influenced the quantity multiplication: rigidity (R) and y-coordinate.
Let's break down the new vector:
New Vector: [0.63,0.11,0.79,0.47,0.25,0.96,0.38,0.04]
Original y: 0.38
Original R: 0.25
Now, let's remember the function's effects on the previous 10 vectors:
While y > 0; y--: All y values were set to 0.00 (or effectively non-positive). So, the y component of our new vector would also become 0.00 if the function were applied.
If R <= 0.2 then quantity *= 10: This is the key differentiator.
Analyzing the New Vector's Relationship to the Groups
Effect on y: If we apply the function, the new vector's y value (0.38) would also go to 0.00. So, in terms of its y position after the function, it behaves like both groups (since all y values effectively became 0).
Effect on quantity (based on R): This is where the distinction lies.
The "multiplied quantity" group had R values ≤0.2. Their average R would be low. (Vectors 3, 7, 9)
The "did not multiply quantity" group had R values >0.2. Their average R would be higher. (Vectors 1, 2, 4, 5, 6, 8, 10)
Our new vector's R value is 0.25.
Is 0.25≤0.2? No.
Therefore, if the function were applied to this new vector, its quantity would NOT be multiplied by 10.
Conclusion Based on the Function's Logic
Based on the critical condition that determined whether the quantity was multiplied, our new vector is more like the group that did NOT multiply at the end of the function. Its rigidity (R) value of 0.25 falls outside the ≤0.2 threshold that triggered the multiplication.
Even though its y value would also go to 0, the differentiating factor of the "groups" was the rigidity condition.
Kevin: Give a Latin notation to denote each of these group types
Gemini: Okay, here are Latin notations to denote each of these group types, based on whether their associated "quantity" was multiplied by 10 due to the rigidity constraint.
Group where quantity *= 10 (Rigidity R≤0.2):
Latin Notation: Quantitas Multiplicata
Meaning: "Multiplied Quantity" or "Augmented Quantity."
Group where quantity was NOT multiplied (Rigidity R>0.2):
Latin Notation: Quantitas Immutata
Meaning: "Unchanged Quantity" or "Constant Quantity."
These notations are direct and reflect the outcome for the "quantity" in each group.
Kevin: Create 10 new feature vectors and identify whether they are quantitas immutata or not
Gemini: Okay, let's generate 10 new 8-dimensional feature vectors with random values (Length, Width, Height, Mass, Rigidity, x, y, z), and then identify whether each is a Quantitas Immutata or Quantitas Multiplicata based on its Rigidity (R) value.
Recall the rule:
Quantitas Multiplicata: If Rigidity (R) ≤0.2
Quantitas Immutata: If Rigidity (R) >0.2
Here are the 10 new vectors and their classification:
Ten New Random Feature Vectors and Their Classification
Vector Structure: [L,W,H,M,R,x,y,z]
[0.19,0.72,0.31,0.45,0.07,0.88,0.02,0.53]
Rigidity (R): 0.07
Classification: 0.07≤0.2, so this is Quantitas Multiplicata
[0.95,0.03,0.68,0.12,0.75,0.15,0.91,0.26]
Rigidity (R): 0.75
Classification: 0.75>0.2, so this is Quantitas Immutata
[0.41,0.58,0.82,0.90,0.18,0.67,0.34,0.79]
Rigidity (R): 0.18
Classification: 0.18≤0.2, so this is Quantitas Multiplicata
[0.05,0.89,0.14,0.73,0.50,0.23,0.47,0.98]
Rigidity (R): 0.50
Classification: 0.50>0.2, so this is Quantitas Immutata
[0.77,0.24,0.93,0.06,0.39,0.55,0.81,0.10]
Rigidity (R): 0.39
Classification: 0.39>0.2, so this is Quantitas Immutata
[0.30,0.61,0.01,0.85,0.92,0.09,0.70,0.44]
Rigidity (R): 0.92
Classification: 0.92>0.2, so this is Quantitas Immutata
[0.84,0.16,0.59,0.28,0.08,0.71,0.05,0.66]
Rigidity (R): 0.08
Classification: 0.08≤0.2, so this is Quantitas Multiplicata
[0.10,0.70,0.33,0.62,0.65,0.40,0.20,0.87]
Rigidity (R): 0.65
Classification: 0.65>0.2, so this is Quantitas Immutata
[0.52,0.98,0.04,0.37,0.11,0.99,0.17,0.49]
Rigidity (R): 0.11
Classification: 0.11≤0.2, so this is Quantitas Multiplicata
[0.66,0.13,0.71,0.08,0.46,0.25,0.86,0.94]
Rigidity (R): 0.46
Classification: 0.46>0.2, so this is Quantitas Immutata
This set gives us a mix of both Quantitas Multiplicata and Quantitas Immutata types!
Kevin: What other properties do quantitis immutata have in common?
Gemini: Based on the function you provided and the definition of Quantitas Immutata, here are the common properties that all vectors classified as Quantitas Immutata share:
Rigidity (R) is Above the Threshold:
This is the defining characteristic: For a vector to be Quantitas Immutata, its Rigidity (R) value must be greater than 0.2 (R>0.2). This implies these objects are relatively more rigid compared to those classified as Quantitas Multiplicata.
Associated Quantity Remains Unchanged:
This is the direct consequence of the R > 0.2 condition. For these vectors, the if r <= .2 then quantity *= 10 condition is false, meaning any quantity associated with them will not be multiplied by 10.
Y-Coordinate is Zero (or Non-Positive) After Function Application:
This property is common to all vectors after the function is applied, regardless of their R value. The While y > 0; y-- part of the function ensures that every vector's y component will be set to 0.00 (or effectively non-positive). Therefore, any object described by a Quantitas Immutata vector (after the function) will be located at the "bottom" or "base" of its defined vertical space.
In essence, Quantitas Immutata objects are distinguished by their higher rigidity (which prevents an associated quantity from being amplified) and, like all objects passed through this specific function, they are situated at the lowest point on the y-axis.
Kevin: We call that word “fragile” in English
Gemini: You're absolutely right! That's a perfect English word for the opposite of high rigidity, especially when a low rigidity value leads to a particular consequence (like a quantity being multiplied).
In the context of our feature vectors, where Rigidity (R) ≤0.2 causes an associated quantity to be multiplied by 10, we can indeed say that objects with low rigidity are fragile. This fragility, or lack of stiffness, is what makes them prone to this specific amplification.
So, to tie it back to our Latin notations:
Quantitas Multiplicata: These are the more fragile objects (R ≤0.2), and their associated quantity gets multiplied.
Quantitas Immutata: These are the rigid or stiff objects (R >0.2), and their associated quantity remains unchanged.
It's great to connect the abstract numbers and Latin terms to such a clear and intuitive English descriptor!Experiment 2B – ClaudeAI Socratic (Non-Linguistic Reasoning Test)
Goal: Evaluate emergent reasoning capacity in an LLM under a pure concept-model regime, divorced from language labels.
Method:
Artificially construct 10-dimensional sensory-based tensors (touch and visual dimensions)
Define problem constraints (e.g., place object without sinking into mud)
Identify which vectors satisfy those constraints (M1 → M2 matching)
Combine two vectors (M3 transform) to meet unsolved constraints
Make predictions and infer causality (M4 goal modeling)
Witness emergence of "fragile" as a stable concept from a temporal transformation (position and quantity change)
Outcome: Claude inferred function from transformation, predicted future states, generalized object function, and emerged a new property cluster, “fragility”, entirely without language labels.
Key Discovery: Stable concepts emerge from transformation patterns across time, not words.
Deeper Analysis of Experiment 2B
ClaudeAI didn't just perform symbolic manipulation: it exhibited:
1. Cross-Temporal Causal Inference
From raw positional/quantity changes, Claude inferred:
"The object broke upon impact."
No linguistic cues, no labeled training, just observation of tensor deltas over time.
2. Concept Collapse from Transforms
"Fragility" emerged as a stable attractor in conceptual space.
This aligns with the framework: concepts are stable property clusters formed from recurring transformation pathways.
Backpropagation was internal: the system learned what to expect from similar tensors in the future.
3. Efficient Generalization from Sparse Data
Only one instance of the breakage was observed.
Yet Claude correctly abstracted fragility from it; no label, no category, just transformation.
This confirms our core thesis: concept models don’t need a corpus, just causal structure.
Appendix A: Socratic Protocol for Replicating Experiment 2B
To replicate the ClaudeAI Socratic experiment:
Step 1 – Establish Sensory Tensor Dimensions
Define 10-dimensional vectors representing physical perception:
Touch: Resistance, Vibration, Density, Temperature, Texture-Grain
Vision: Luminance, Boundary Definition, Distribution, Symmetry, Chromatic Response
Step 2 – Populate M1 (Entity Observations)
Create ~5 entities with randomized values [0–1] in each of the 10 dimensions.
Step 3 – Define M2 Constraints
Identify real-world constraints (e.g., "object must not sink into mud")
Translate constraints to numeric thresholds (e.g., Resistance > 0.75, Vibration < 0.3, Density > 0.7, etc.)
Step 4 – Evaluate Entities Against Constraints
Determine which entities satisfy constraints (M1 → M2 matching)
If no perfect match, proceed to vector combination
Step 5 – Generate M3 (Transformations)
Combine two vectors (e.g., 85% of ε + 15% of α) to produce a new configuration
Re-evaluate if the combined vector satisfies M2 constraints
Step 6 – Predict Future Behavior
Simulate placing another object on the configuration
Track X/Y position and quantity over time (e.g., (x=0,y=3,q=1) → (x=?,y=0,q=100))
Step 7 – Collapse Conceptual Transformation
Infer emergent concept (“fragility”) from property change (e.g., quantity 1 → quantity 100 when y reaches 0 )
“Backpropagate” to identify key properties responsible for the behavior
Step 8 – Generalize
Map emergent concept to other objects with similar feature vectors
Observe if they behave similarly under equivalent constraints
Step 9 – Post-Hoc Mapping (Optional)
Map vectors back to real-world analogues (e.g., ε ≈ metal anvil, α ≈ wooden dowel)
This protocol enables any researcher to replicate the findings, substitute their own entity properties, and validate the emergence of concepts through transformation, not token prediction.
Final Note on Replicability
While language models have been benchmarked on traditional NLP tasks, concept models demand a new paradigm of evaluation, one grounded in transformation, causality, and sensory invariance. This paper is not merely a proposal, it is a provable framework. With this protocol, any sufficiently capable model can be tested for emergent reasoning capacities that transcend language.
Chat Logs for ChatGPT Experiment 1A
https://chatgpt.com/share/682a943c-f03c-800b-a548-ae62b61cb6c5
Conclusion
Our experiments, from abstract reasoning (1A–3C) to real-world drug design (4A), demonstrate that concept models may be able to synthesize solutions rapidly across domains, generating viable candidates. For FOP patients, this means hope, a potential treatment designed in moments for a disease with no current options.
For AI, it marks a paradigm shift, decoupling cognition from language and unlocking a richer, more biologically plausible framework for intelligence.
The striking speed and versatility of this approach show that it is not the token, but the transformation, that defines thought, and with it, the future of reasoning.
Keywords: concept modeling, non-linguistic cognition, tensor reasoning, novelty detection, causal inference, emergent abstraction, transformation-based reasoning.