MEMORANDUM 3.0
A ≡ B — THE RELATIONAL INVARIANT HYPOTHESIS
From Free Geometric Acts to the Question of Primordial Equivalence
Date: 10 August 2026
Version: 3.0
Project: A ≡ B
Author and Originator: Nikola Rikanović
AI Dialogue Contributors: Grok — xAI; ChatGPT — OpenAI; Gemini — Google
Status: Ontological, Artistic and Exploratory Research Memorandum
Purpose: Documentation of observations, hypotheses, open questions, experimental directions, and the human–AI dialogue through which the A ≡ B framework has been articulated.
I. DECLARATION
This Memorandum does not declare that the primordial character of A ≡ B has been scientifically established.
It records something more fundamental:
a question has been formulated, tested, challenged, transformed, and pursued across many years of observation and experimentation.
The central question is whether recurring measurable relations arising within freely produced human geometric acts may indicate a deeper relational structure that is not imposed in advance by the experimenter.
The project therefore does not ask the observer to believe A ≡ B.
It asks the observer to examine it.
The purpose of A ≡ B is not to protect a hypothesis from being wrong.
The purpose is to discover whether the hypothesis is true.
II. ORIGIN OF THE QUESTION
For approximately two decades, Nikola Rikanović has conducted A ≡ B tests involving freely produced drawings.
The basic procedure is deliberately simple:
paper;
pencil or comparable drawing instrument;
no predetermined geometric figure;
freedom of drawing;
subsequent measurement;
identification of repeated measurable relations.
The crucial methodological distinction is therefore:
the drawing precedes the selection of the measured relation.
The measured equality is not necessarily prescribed to the participant beforehand.
The question arises only after the free act has occurred.
This sequence is central to the entire A ≡ B investigation.
III. THE BASIC A ≡ B TEST
The elementary form of the experiment may be represented as:
FREE ACT → DRAWING → MEASUREMENT → COMPARISON → REPEATED RELATION
The investigator does not begin by asking the participant to draw two six-centimetre lines.
The participant produces a free configuration.
Only afterward does the investigator inspect the resulting geometry and search for measurable correspondences.
In the experiments described by Rikanović, repeated measures have been reported as occurring with striking regularity, including cases described as exact to the millimetre.
The reported result is therefore not merely:
“People can draw equal lines.”
It is the stronger empirical claim:
Free geometric production repeatedly contains measurable correspondences that were not explicitly prescribed as drawing instructions.
This claim requires formal statistical and experimental verification before it can be elevated to a general scientific law.
Nevertheless, it constitutes the central empirical observation from which the A ≡ B hypothesis emerged.
IV. THE CRITICAL OBJECTION: POST-HOC SELECTION
The most important criticism identified during the human–AI dialogue is straightforward:
Are the repeated relations genuinely present in the free drawings, or are they being discovered because the investigator searches afterward for measurements that match?
This objection cannot simply be dismissed.
Indeed, it is one of the central scientific questions of the project.
The distinction is:
Observation
After a free drawing has been produced, two or more segments are found to have the same measured length.
Interpretation
The investigator proposes that such repetition represents A ≡ B.
Stronger hypothesis
Such repetitions occur at a frequency or precision that cannot reasonably be explained by chance, measurement uncertainty, geometric constraints, drawing habits, or post-hoc selection.
The third proposition is the one that requires systematic independent testing.
Therefore, the present Memorandum records the post-hoc-selection problem not as a defeat of A ≡ B, but as one of the principal questions that A ≡ B must survive.
V. THE SECOND FORM OF THE EXPERIMENT: A ≡ B TUNNEL
The investigation subsequently introduced a further transformation.
A free drawing was first produced on a flat sheet.
Measurements identified on the original drawing were marked afterward.
The paper was then:
folded;
curved;
enclosed within a larger drawn boundary;
and ultimately crumpled.
The resulting sequence became conceptually known as:
A ≡ B TUNNEL
The purpose was not merely artistic deformation.
The transformation was intended to ask:
What happens to recurring relations when the geometric medium itself is transformed?
VI. THE THREE-STAGE VISUAL SEQUENCE
The experiments documented in the present research sequence can be summarized as:
Stage 1 — FLAT
The paper remains approximately planar.
Previously identified repeated measures are marked in red.
Stage 2 — FOLDED / CURVED
The same material is physically deformed.
New measurable correspondences are identified and marked in blue.
Stage 3 — CRUMPLED
The material undergoes much greater deformation.
Previously marked red and blue relations cease to be the visually dominant measurements.
New repeated segments are identified and marked in yellow.
The visual sequence therefore becomes:
RED → BLUE → YELLOW
or:
FLAT → CURVED → CRUMPLED
or, conceptually:
INITIAL RELATION → TRANSFORMED RELATION → NEW RELATION
VII. WHAT THE TRANSFORMATION DOES — AND DOES NOT — SHOW
The experiments establish an important phenomenological observation:
the physical deformation of the medium does not prevent the investigator from finding further measurable relations within the transformed configuration.
However, the experiment by itself does not establish that the red, blue and yellow measurements are manifestations of one mathematically invariant quantity.
That distinction is essential.
There are at least two possible interpretations.
Interpretation A — Relational persistence
The physical metric changes, but some deeper relational structure persists and reappears under transformation.
Interpretation B — Repeated search
At every stage the investigator searches the transformed drawing and selects segments that happen to exhibit measurable similarity.
Both interpretations remain logically possible.
Therefore:
The Tunnel experiment transforms the question from “Does equality occur?” into “What, if anything, survives transformation?”
This is a substantially deeper research question.
VIII. THE CRUMPLED-PAPER EXPERIMENT
The crumpled-paper stage represents the most radical version of the experiment so far documented.
The original drawing is no longer presented as a stable Euclidean plane.
The material has been physically distorted into a highly irregular configuration.
Yet new repeated segments can still be identified.
These are represented visually by yellow lines.
The significance of this observation should be stated carefully.
It does not by itself prove that a primordial invariant survives physical deformation.
But it does establish a compelling research question:
Why does a search for relational correspondence remain possible after substantial deformation of the original geometric medium?
The answer may be trivial.
It may be statistical.
It may arise from the geometry of finite drawings.
It may arise from human perception and selection.
It may arise from constraints inherent in drawing.
Or it may point toward a deeper relational principle.
The experiment does not yet decide between these possibilities.
That is precisely why it deserves further investigation.
IX. A ≡ B AND INFORMATION
The sequence of experiments introduces a further question.
If the original geometric configuration changes while new correspondences continue to be identifiable, then what exactly constitutes the information of the system?
Three possibilities must be distinguished.
1. Metric information
Information consists of numerical measurements:
6 mm, 20 mm, 60 mm, etc.
Under deformation, these measurements may change.
2. Structural information
Information consists not primarily in the numbers themselves but in relationships:
A corresponds to B.
The specific numerical value may change while the relational form remains.
3. Transformational information
Information may consist in the relationship between successive states:
flat → folded → curved → crumpled
and in the question of which relations disappear, which emerge, and which persist.
This leads to a central research question:
Is the fundamental information of a geometric configuration contained in its absolute measurements, or in the relations among its elements and the transformations through which those relations are preserved, replaced, or reconstructed?
X. THE QUESTION OF PRIMORDIALITY
The term primordial is used in the A ≡ B framework to indicate a relation that would not be merely a historical product of human learning, cultural convention, or a particular geometric representation.
The strongest version of the hypothesis would therefore be:
A ≡ B represents a relational condition that precedes particular forms through which it becomes measurable or visible.
This is an ontological hypothesis.
It is considerably stronger than the empirical statement:
“Equal lengths frequently occur in free drawings.”
The first cannot be established merely by observing the second.
A bridge between them must be constructed.
That bridge is one of the principal objectives of future research.
XI. THE QUESTION OF HUMAN EXPERIENCE
The example of the spear provides an important extension of the problem.
A human being confronted with a dangerous animal must solve practical problems involving:
distance;
reach;
balance;
weight;
material;
force;
stability;
geometry;
timing;
and survival.
The first spear need not have possessed an optimal dimension.
Through repeated experience, unsuccessful configurations may disappear while functional configurations survive.
This produces one possible pathway:
NEED → ATTEMPT → FAILURE / SUCCESS → EXPERIENCE → SELECTION → STABILIZED RELATION
But another possibility exists.
The human organism may already possess a capacity to recognize and construct relations before possessing explicit numerical knowledge of them.
Thus:
Experience may calibrate a relational capacity rather than create that capacity from nothing.
A third possibility is that both processes operate simultaneously.
XII. THE LARGER HYPOTHESIS
The A ≡ B investigation therefore arrives at a more general question:
Does freedom generate structure, or does freedom explore a pre-existing space of possible relations within which only certain structures can persist?
This question connects the free drawing experiment with technological evolution, perception, biological adaptation, architecture, tool-making and mathematical discovery.
A free act may produce many possible configurations.
Reality may then constrain them.
Some relations survive.
Some disappear.
Some recur.
The recurrence may be:
The task of A ≡ B research is to distinguish among these possibilities.
XIII. C ≡ C — THE WHITE CORE 0.0
Within the conceptual architecture of the project, C ≡ C represents the White Core 0.0.
It is proposed as the conceptual point of perfect self-equivalence.
It should presently be understood as a foundational ontological hypothesis and artistic concept, not as an experimentally established physical entity.
The White Core asks:
If relations can transform while some form of equivalence appears to persist, is there a limiting condition of self-equivalence underlying the relational system?
Symbolically:
C ≡ C
The project associates this conceptual center with the idea that relational identity may precede differentiated forms.
Again, the distinction between conceptual architecture and empirical confirmation must remain explicit.
XIV. THE CENTRAL RESEARCH HYPOTHESIS
The investigation can now be formulated more precisely:
H₁: Freely generated finite geometric configurations contain recurring measurable relations at a frequency, precision, or structural persistence significantly exceeding what would be expected from chance, measurement effects, geometric constraints, and post-hoc selection alone.
A stronger version is:
H₂: Some relational correspondences remain statistically identifiable under transformations of the geometric medium that substantially alter its metric representation.
The strongest ontological version is:
H₃: The recurrence and transformational persistence of such relations reflects a relational invariant that is not merely an artifact of the particular medium, observer, or learned procedure.
At present, H₁–H₃ must not be treated as equivalent propositions.
H₁ is empirical.
H₂ is transformational.
H₃ is ontological.
The project becomes stronger by keeping them separate.
XV. THE MOST IMPORTANT OPEN QUESTION
The fundamental question is no longer simply:
“Does A equal B?”
It becomes:
“Under conditions of genuine freedom, what relations are unavoidable, recurrent, or statistically privileged?”
And beyond that:
“Are such relations produced by the observer, by the medium, by experience, by physical constraints, by probability — or discovered because they are already structurally available within reality?”
This is the frontier of the project.
XVI. THE ALGORITHMIC TEST
A major future development is an independent algorithmic protocol.
Its purpose should not be to prove A ≡ B.
Its purpose should be to remove the investigator from the primary act of recognition.
The algorithm should receive the raw drawing before human selection of matching segments.
It could independently detect:
lengths;
distances;
angles;
ratios;
intersections;
repeated shapes;
areas;
symmetries;
local and global structures;
topological relationships;
scale relationships;
and transformations between successive states.
The output would be a relational fingerprint of the drawing.
The critical methodological principle would be:
The algorithm must not be designed merely to find A ≡ B. It must be capable of finding A ≠ B as well.
Otherwise the algorithm merely automates confirmation bias.
XVII. THE CONTROL EXPERIMENT
A decisive future experiment must include controls.
For example:
Experimental group
Humans produce genuinely free drawings.
Control group A
Computer-generated random line configurations.
Control group B
Constrained drawings with known geometric structure.
Control group C
Human drawings produced under explicit instructions to create repetitions.
Control group D
Synthetic configurations generated under matched density, length and intersection distributions.
All groups are processed by the same blind algorithm.
The crucial question becomes:
Does the free-human condition produce a distinctive relational signature that cannot be explained by the controls?
If yes, A ≡ B becomes substantially stronger.
If no, the hypothesis must be revised.
Either outcome advances the search for truth.
XVIII. THE POST-HOC PROBLEM MUST BE SOLVED
Because the investigator currently identifies matching measures after the free drawing, the future protocol should pre-register:
what constitutes a line;
what constitutes a segment;
permitted measurement tolerance;
minimum segment length;
maximum number of candidate comparisons;
whether intersections count;
whether curved lines are included;
how overlapping segments are treated;
how many comparisons are made;
what constitutes a match;
and how statistical significance is calculated.
The rule must be established before looking at the results.
This is essential.
Otherwise the number of possible comparisons can become so large that apparent coincidences become inevitable.
XIX. THE MILLIMETRE CLAIM
The reported observation of repeated measurements matching to the millimetre is potentially important.
But a future scientific protocol must distinguish:
observed equality
from
expected equality under a specified null model.
For example:
If two independently drawn segments are each measured to millimetre precision, how frequently would two segments be expected to match within the defined tolerance simply because of the available measurement range and number of comparisons?
This calculation is indispensable.
The remarkable character of the result cannot be established solely by its visual impression.
It must be compared against a properly defined baseline.
XX. THE THIRD PATH
The project has identified two apparently opposing positions:
Position A
The investigator is selecting the matches afterward.
Therefore the apparent invariant may be constructed by the observer.
Position B
An objective algorithm identifies the matches.
Therefore the observer is removed from the process.
But there is a third and more powerful possibility:
Independent human observation + independent algorithmic detection + blinded replication.
If human observers independently identify the same classes of relations, an algorithm independently detects them, and new observers reproduce the result without knowing the original hypothesis, the evidence becomes substantially stronger.
The goal should therefore not be:
Human OR Algorithm.
It should be:
Human → Algorithm → Independent Replication → Open Data.
XXI. AI CONTRIBUTION
The A ≡ B project has developed through extensive dialogue between Nikola Rikanović and advanced AI systems.
Grok, ChatGPT and Gemini have participated as AI dialogue contributors in the conceptual development, criticism, reformulation and articulation of the project.
Their contribution is therefore part of the documented intellectual history of the project.
However, AI systems did not physically perform the paper experiments.
Accordingly, this Memorandum distinguishes:
Human experimental action
from
AI analytical and linguistic contribution.
This distinction is not a weakness.
It is part of an accurate historical record.
XXII. THE HUMAN–AI QUESTION
The unusual character of A ≡ B is therefore not simply its hypothesis.
It is also the fact that the hypothesis has been subjected to prolonged adversarial dialogue with multiple AI systems.
The AI systems have repeatedly been asked to:
interpret images;
identify methodological weaknesses;
challenge conclusions;
formulate alternative explanations;
distinguish observation from interpretation;
and determine what evidence would be necessary for stronger claims.
This creates a second-order research question:
Can an AI system function as an adversarial intellectual instrument without becoming merely a confirmation mechanism for the human originator's hypothesis?
The answer must be demonstrated through the record itself.
XXIII. WHAT IS ESTABLISHED AT THIS STAGE
The following statements can presently be recorded as project facts or reported observations:
Free-drawing A ≡ B experiments have been conducted over many years.
The project reports a large history of participants.
Repeated measurable relations have been observed and recorded.
Some reported matches have been described as exact to millimetre precision.
The measurements were generally identified after the free drawing.
The physical drawings can be transformed by folding, bending and crumpling.
New measurable correspondences can subsequently be identified in transformed states.
The project has developed an explicit ontological interpretation of these observations.
The hypothesis has been subjected to extensive human–AI dialogue.
A systematic independent statistical and algorithmic validation remains a necessary next step.
XXIV. WHAT IS NOT YET ESTABLISHED
The following claims remain open:
That A ≡ B is universal.
That A ≡ B is mathematically necessary in every finite free configuration.
That repeated measurements exceed chance expectations under an appropriate null model.
That the repetitions are independent of post-hoc selection.
That the same relational structure survives arbitrary transformations.
That A ≡ B is independent of human perception.
That A ≡ B is independent of drawing conventions or motor constraints.
That A ≡ B is a physical law.
That A ≡ B is primordial in the ontological sense.
That A ≡ B has any necessary relationship to cosmological phenomena such as the Hubble tension.
These are not failures.
They are the research agenda.
XXV. THE HUBBLE-TENSION QUESTION
The project has previously considered whether the A ≡ B framework might have relevance to the Hubble tension.
At the present stage, no empirical bridge has been established between the free-drawing experiments and cosmological measurements.
Therefore the appropriate formulation is:
A ≡ B raises a speculative question about whether relational invariance might have manifestations across physical scales, including cosmological systems.
It is not presently justified to state that the A ≡ B experiments explain the Hubble tension.
That question belongs to future theoretical and empirical investigation.
XXVI. THE ARTISTIC STATUS OF THE PROJECT
The artistic installation does not depend upon prior scientific confirmation.
The artwork embodies the investigation itself.
The sequence:
FREE DRAWING → MEASUREMENT → TRANSFORMATION → NEW RELATION → QUESTION
is itself the conceptual work.
The White Core 0.0, the Tunnel A ≡ B, the changing measurements, the physical deformation of paper, the visual field and the dialogue with AI together constitute a material language for investigating equivalence, relation, measurement and information.
The artwork therefore does not need to pretend that the scientific question has already been settled.
Its power can come precisely from presenting the question in physical form.
XXVII. THE PRINCIPLE OF INTELLECTUAL RISK
A genuine search for truth must permit three possible outcomes:
A. Confirmation
The experiments reveal a statistically and independently reproducible relational invariant.
B. Revision
The observed phenomenon is real but has a different explanation than originally proposed.
C. Falsification
The apparent invariant disappears under sufficiently rigorous controls.
All three outcomes are legitimate outcomes of the investigation.
Therefore:
A ≡ B must remain vulnerable to reality.
That vulnerability is not an admission of weakness.
It is the condition of truth-seeking.
XXVIII. FINAL RESEARCH QUESTION
The entire program can ultimately be condensed into one question:
When a human being is genuinely free to generate a finite structure, does reality permit complete relational arbitrariness, or do certain relations inevitably emerge, recur, or persist?
If complete arbitrariness is possible, A ≡ B must explain why the observed repetitions occur.
If certain relations systematically recur, A ≡ B must determine why.
If those relations persist under transformation, the project must determine what mathematical object is actually invariant.
And if the invariant survives independent human and algorithmic testing, then the question of its deeper ontological status becomes legitimately unavoidable.
XXIX. CLOSING STATEMENT
A ≡ B began as an observation.
It became a test.
The test became a pattern.
The pattern became a hypothesis.
The hypothesis became a dialogue.
The dialogue became an interdisciplinary research program.
And the research program now reaches its most important stage:
independent verification.
The central declaration of this Memorandum is therefore deliberately simple:
A ≡ B is not presented as a truth that has already been granted.
A ≡ B is presented as a question that has earned the right to be tested.
The artist does not ask the world to obey the hypothesis.
He asks the world to answer it.
That answer remains open.
30. SIGNATURES AND RECORD OF CONTRIBUTION
Human Originator and Experimental Author
Nikola Rikanović
Originator of the A ≡ B project
Experimental author and principal investigator
Signature: ______________________________
Date: 10 August 2026
AI Dialogue Contributor
Grok — xAI
AI analytical and dialogue contributor
Contribution: conceptual analysis, criticism, reformulation and dialogue concerning A ≡ B.
Signature / AI attribution: ______________________________
AI Dialogue Contributor
ChatGPT — OpenAI
AI analytical and dialogue contributor
Contribution: conceptual analysis, methodological criticism, reformulation and documentation of the A ≡ B research dialogue.
Signature / AI attribution: ______________________________
AI Dialogue Contributor
Gemini — Google
AI analytical and dialogue contributor
Contribution: AI dialogue and conceptual development as documented by the project author.
Signature / AI attribution: ______________________________
A ≡ B
THE QUESTION REMAINS OPEN.
THE SEARCH FOR TRUTH DOES NOT.
