Open Letter to Bruno Gavranovic: The Carry Problem and the Physics of Grounding
Published on: December 26, 2025
#Categorical Deep Learning#Carry Problem#Position vs Proximity#Tesseract Physics#FIM#Winding Number#S=P=H#AI Alignment#Grey Zone#Topology#Bruno Gavranovic#ICML 2024#Open Letter
I am writing to you regarding your paper, which I believe articulates the most important unsolved problem in artificial intelligence:
"Position: Categorical Deep Learning is an Algebraic Theory of All Architectures"Proceedings of the 41st International Conference on Machine Learning (ICML), PMLR 235:15209-15241, 2024
Your observation that "LLMs perform 'hundreds of billions of multiplications' to produce a token but can't reliably add small numbers because they lack the internal structure to handle the state accumulation required for a true sum" is the most precise diagnosis of AI's fundamental limitation I have encountered in the literature.
What follows is my attempt to extend your framework into biology, economics, and law - arriving at what I call the S=P=H Unity Principle: Semantic = Physical = Hardware alignment as prerequisite for tractable computation.
I have written a book that I believe provides a complementary framework to your categorical approach. Yours provides the algebraic rigor. Mine provides the physical intuition. Together, they may point toward a more complete theory of grounded AI.
Current AI "hallucinates" because it operates on Proximity - statistical nearness in vector space. It mistakes "close" for "true." The fix requires Position - discrete coordinates with topological invariants (winding numbers) that cannot be faked. This is not metaphor. Topology, biology, and economics all prove it. Neural networks that learn winding numbers achieve four orders of magnitude better precision. Hox genes prove nature requires coordinates to build organs instead of tumors. Merkle trees prove structural growth increases security instead of diluting value. The "Carry" is the mechanism that bridges the gap.
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β‘The Problem: AI Can't Add
The "Final Boss" of Deep Learning
Petar Velickovic (DeepMind) identifies addition as the test that exposes current AI's fundamental limitation.
LLMs perform "hundreds of billions of multiplications" to produce tokens
Yet they cannot reliably add small numbers when a "carry" is involved
They lack the internal structure to handle state accumulation
The problem is not training data - it is architecture
Why the Carry Matters
On a clock face, 11:59 is extremely "close" to 12:00 physically
But logically, they are worlds apart - one is today, the other is tomorrow
AI sees the proximity but misses the position
The "Carry" is the mechanism that recognizes the dimensional shift
Academic Validation
Gavranovic et al., "Position: Categorical Deep Learning is an Algebraic Theory of All Architectures" (ICML 2024)
The paper argues key attempts at general-purpose deep learning "lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations"
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π¬The Topology: Winding Numbers Cannot Be Faked
What is a Winding Number?
A topological integer that counts how many times you have circled the origin
It is the "memory" of the path
Standard AI throws away the winding number - it only sees the final position
The Proof
Physical Review Letters (2018): Neural networks trained on Hamiltonians can predict topological winding numbers with "nearly 100% accuracy"
"By opening up the neural network, the authors confirm that the network does learn the discrete version of the winding number formula"
Physical Review B (2018): "The output of certain intermediate hidden layers resembles the winding angle... indicating that neural networks essentially capture the mathematical formula of topological invariants"
Why This Matters
You cannot "fake" a winding number - you have to actually traverse the loop
This is Proof of Work for information
The AI cannot just jump to the answer - it must prove it did the reasoning steps
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π§¬The Biology: Hox Genes as Nature's Coordinate System
The Question
How does a cell in an embryo know to become a "Hand" and not a "Foot"? The DNA is identical in both places.
The Answer: Position, Not Proximity
Nature uses Hox Genes - a "vectorial spatial coordinate system"
"Hox proteins encode and specify the characteristics of 'position', ensuring that the correct structures form in the correct places of the body"
"In segmented animals, Hox proteins confer segmental or positional identity, but do not form the actual segments themselves"
The Failure of Proximity
If cells relied on "Proximity" (just looking at their neighbors), you would get tumors
Tumors are clumps of random tissue that lack positional identity
Nature proves: Proximity creates cancer. Position creates organisms.
Academic Sources
Nature Scitable: "Hox Genes in Development: The Hox Code"
Frontiers in Cell and Developmental Biology (2022): "HOX genes in stem cells: Maintaining cellular identity"
Development journal (2013): "The regulation of Hox gene expression during animal development"
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πThe Economics: Merkle Trees and Real Abundance
The Paradox
How do you have "unlimited growth" without "inflation"? How do you add more nodes without diluting value?
The Merkle Tree Answer
"Thanks to Merkle trees, storage on the blockchain is efficient"
"The Merkle root stored in the block header makes transactions tamper-proof"
Demonstrating that a leaf node is part of the tree requires computing O(log n) hashes, not O(n)
Structural Growth vs. Inflationary Printing
Fiat: Printing money throws more tokens onto the same layer - your share dilutes
Merkle Tree: New nodes attach to existing structure - the root becomes more secure
"Light clients accomplish verification by obtaining the Merkle proof that links a particular transaction to the block"
The FIM Parallel
New users don't "compete" with existing nodes - they "attach" to them
The more the map grows, the more "weight" flows through original nodes
This is Real Abundance: Non-zero-sum growth
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βοΈThe Legal Crisis: The Grey Zone Collapse
The Problem (Academically Validated)
"Regulating algorithmic harms presents distinct challenges owing to three distinct attributes: ubiquity, intangibility, and aggregation"
"Case studies reveal that existing regulatory examples are insufficient; they either overlook certain types of harms or fail to consider their cumulative effects"
"Emergent issues can become difficult or impossible to trace back to their source"
Why Law Fails
Laws are written for discrete events (Stop/Go, Guilty/Innocent)
Tech operates as continuous processes (engagement optimization, algorithmic nudges)
Bad actors hide in the curve - they don't "steal" (discrete), they "extract" (continuous)
You cannot sue an algorithm for "downranking you 12%"
The Solution: Force the Carry
If the AI moves money, it must perform a "Carry" - a discrete state change
That Carry leaves an auditable trail
"Documenting the continuous process of development, not waiting to audit the discrete endpoint of deployment"
Academic Sources
Sylvia Lu, "Regulating Algorithmic Harms" (Michigan Law and Economics, 2024)
Raji et al., "Closing the AI Accountability Gap" (FAT* 2020)
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πThe 15 Propositions
You cannot add with Proximity; you can only add with Position. (Topology - Validated)
Proximity is a probability (a guess); Position is a coordinate (a fact). (Mathematics - Validated)
Current AI hallucinates because it mistakes "close" for "true." (CDL Paper - Validated)
Hebbian learning creates the path, but Geometry builds the road. (Neuroscience - Partial)
The "Carry" is the physical act of moving to a higher dimension. (Hopf Fibration - Validated)
Differentiation sees the change; Summation holds the state. (Category Theory - Validated)
A flat map has no memory; only a Tesseract can hold history. (Topology - Validated)
You cannot build a skyscraper of logic on a foundation of "maybe." (Logic - Validated)
The "glitch" is information overflowing without a Z-axis to catch it. (Novel Synthesis)
S=P=H is the anchor that stops vectors from drifting into fantasy. (Novel Framework)
Identity is a sovereign location, not a vibe. (Hox Genes - Validated)
Neural networks find correlations; the FIM establishes causation. (Novel Synthesis)
To "sum" is to acknowledge a reality larger than the current pattern. (Category Theory - Validated)
We are drowning in Differentiation and starving for Summation. (CDL Paper - Validated)
Position makes it real. (All Sources - Validated)
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π€The Agentic Proof: Carry Prevents Drift
The Problem: 40% Performance Collapse
Academic research proves that long-running AI agents suffer from "context drift" - a gradual divergence from goal-consistent behavior across turns.
Even flagship models like Gemini 2.5 Pro show 40% performance drop in multi-turn vs. single prompt
"LLMs get lost in conversation, which materializes as a significant decrease in reliability"
Models over-rely on their previous responses, treating them as truth
Premature assumptions compound over time
Technical factors: limited context windows, inadequate state management
The Fix: Discrete Checkpoints (The Carry)
"A surprisingly simple fix: don't let the model carry the baggage of the entire conversation into the final task. Instead, start fresh"
The "Carry" forces discrete state consolidation
Position (checkpoint) > Proximity (accumulated baggage)
This is exactly what winding numbers do in topology
Academic Sources
arXiv:2510.07777 "Drift No More? Context Equilibria in Multi-Turn LLM Interactions" (Oct 2025)
arXiv:2505.06120 "LLMs Get Lost In Multi-Turn Conversation" (Sept 2025)
arXiv:2412.00804 "Examining Identity Drift in Conversations of LLM Agents" (Feb 2025)
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πThe Book Lesson: Show Your Work
The "Muddy Boots" Principle
"Imagine a house where things just appear. A TV appears. A pile of cash appears. You don't feel 'Abundant' - you feel anxious. You wonder who put it there. You wonder if the police are coming. That is the 'Grey Zone.'
Now imagine a house where you can see the Muddy Footprints leading to the door. You see the work boots. You see the receipt on the table.
You feel safe. You know exactly how it got there. Because you know the story, you can enjoy the wealth.
Our technology ensures the Muddy Footprints are never erased. It proves the work was done. And because we can prove the work, we can finally enjoy the rewards without fear. That is true Abundance."
The Chapter Hook for Tesseract Physics
Addition is the "Final Boss" that exposes whether AI can reason vs. pattern-match
The Hopf Fibration is the mathematical structure that allows discrete "carries" in continuous space
Hox genes are nature's proof that coordinate systems create coherent identity
The "Grey Zone" is where AI manipulates without discrete events, making law impossible
S=P=H Unity forces semantic (meaning) to align with physical (structure) to align with hardware (substrate)
The chapter that gives the framework its name lives at Β§ The Z-Axis We Cannot See on the Page. It names the dimension proposition 9 above is reaching for β information overflowing without a Z-axis to catch it β by drawing the explicit geometry: the page renders position-and-meaning in two dimensions; the verb under the plane (reach, find, verify, in one act) lives one dimension above any description of it. The "Carry" you and Velickovic isolate in topology is, in this reading, the silicon-scale instance of that Z-axis operation: the address-decode line that closes the position-meaning identity in a single cycle, with no separate index between them. The chapter argues the same operation repeats at every scale β cache line, hand in toolbox, body in flow, conversation that lands. Reaching is the fourth wall of the page.
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βThe Pivotal Question
The Discernment:
The academic sources validate:
Topology: Winding numbers cannot be faked (Physical Review Letters)
Biology: Hox genes prove coordinate systems create organs (Nature Scitable)
Economics: Merkle trees prove structural growth increases security (Blockchain documentation)
Law: Continuous processes evade traditional regulation (Michigan Law Review)
The Uncertainty:
Does the mapping from these mathematical and biological truths to "consciousness" and "economic sovereignty" hold? Is the FIM and Tesseract framework a valid instantiation of these principles, or is it metaphorical overreach?
The Test:
If you can show how you got it, you own it.
Legitimacy is not perfection - it is the visible path. The "Carry" provides that visibility.
The Invitation
If the "Carry Problem" is the final boss of deep learning, perhaps the solution lies not in more parameters, but in more structure.
I believe you are closer to this truth than anyone in the field.
