When Math Forgets How to Count: The Averaging Paradox That Might Explain Everything (Or Nothing)
A tale of two physicists, one coastline, and the philosophical question: “Are we idiots or just averaging wrong?”
Definitions (Because Words Actually Matter)
Quantum Gravity - Not a wellness trend. The search for a mathematical framework that explains what happens when something very small (quantum mechanics) meets something very big (general relativity). Currently about as successful as teaching cats to meditate.
MOND (Modified Newtonian Dynamics) - A 1983 proposal that basically says “What if Newton was right except when he wasn’t?” Specifically: gravity gets weird at very low accelerations. Heretical to some, promising to others, confusing to everyone.
Dark Matter - Invisible stuff we invented to explain why galaxies spin wrong. Either it exists (and we can’t find it), or it doesn’t exist (and we’re bad at math). Both options are humbling.
The Cosmological Constant - Einstein’s “biggest blunder” that turned out to maybe not be a blunder. A number that keeps showing up everywhere like a drunk uncle at family gatherings. Currently responsible for the universe’s accelerating expansion and possibly our collective confusion.
Averaging - What you do in math class. What 90 years of quantum gravity research might have been doing wrong.
The Facts, No Spin
In October 2025, Benjamin Koch and colleagues at TU Wien published “Geodesics in Quantum Gravity” in Physical Review D. Here’s what they actually did:
The Math
They introduced “q-desics” (quantum-corrected geodesics) - basically asking “what if we average quantum fluctuations after we multiply things instead of before?”
For a quantum variable X: ⟨X²⟩ ≠ ⟨X⟩²
This is not controversial. It’s basic quantum mechanics.
What is new: applying this to Einstein’s nonlinear field equations
The Results
At solar system scales: deviations of ~10⁻³⁵ meters (unmeasurable, irrelevant)
At galactic scales (10²¹ meters): potentially observable differences from classical general relativity
The cosmological constant amplifies these quantum corrections at large scales
The math is formally correct (even skeptics agree on this)
What They Did NOT Claim
They did not prove dark matter doesn’t exist
They did not fully derive MOND from first principles
They did not provide specific numerical predictions yet
They did not solve quantum gravity
What They DID Claim
Standard approaches might be averaging in the wrong order
Quantum gravity effects might be observable at galactic scales, not just Planck scales
The cosmological constant might mediate between quantum and classical regimes
This opens a research direction worth exploring
Three-Layer Analysis
Layer 1: What’s the Obvious Answer? (Surface Thinking)
“Scientists found a new way to maybe explain galaxy rotation without dark matter using quantum corrections. Cool!”
This is the headline version. It’s not wrong, but it’s incomplete. At this layer, it sounds like they solved something. They didn’t. They opened a door and found another hallway with more doors.
The surface story: We’ve been looking for quantum gravity at the smallest scales because that’s where quantum effects dominate. But nonlinearity means small quantum fluctuations might accumulate into big effects at large scales. Like compound interest, except for spacetime.
Layer 2: What Am I Missing? (Blind Spot Angles)
Historical Context We’re Ignoring People have been proposing “quantum gravity explains galactic dynamics” since at least 2005 (Reuter & Weyer’s QEG paper). This idea isn’t new. What’s new is the mathematical formalism of q-desics.
The String Theory Elephant String Theory proponents would argue they’ve already addressed these issues - renormalizability, background independence, black hole thermodynamics. Their approach:
Infinite tower of fields whose contributions cancel divergences
T-duality making “very small” indistinguishable from “very large”
Holographic principle connecting quantum gravity to thermodynamics
Specific corrections to Einstein’s equations that prevent singularities
The Koch paper doesn’t engage with String Theory at all. Why? Probably because String Theory has its own problems (no experimental evidence after 40+ years, requires extra dimensions we don’t see, etc.)
The MOND Complication MOND works empirically at galactic scales. This is observational fact. But:
It fails at galaxy cluster scales (you still need some “dark matter”)
It struggles with gravitational lensing observations
It lacks a fundamental theoretical basis
Some galaxies (NGC1052-DF2, DF4) seem to have no dark matter at all, which MOND can’t explain
If quantum corrections produce MOND-like effects, they inherit these problems.
What The Physics Community Actually Thinks Sabine Hossenfelder (physicist, YouTuber, professional skeptic) gave this a 3/10 on her “bullshit meter.” Not because the math is wrong - because:
The authors haven’t adequately quantified their results
Claims of “clear difference at large distances” aren’t demonstrated in the paper
It’s a framework, not predictions
This is the difference between “we have a new tool” and “we’ve solved the problem.”
The Measurement Problem Nobody Mentions How do you test this? The predicted effects are:
Too small at solar system scales (10⁻³⁵ m)
Potentially observable at galactic scales BUT
We can’t do controlled experiments on galaxies
We have to rely on astronomical observations
Observations already contaminated by the dark matter vs MOND debate
So even if you calculate predictions, proving them is hard.
The Averaging Paradox Cuts Both Ways Neural Foundry’s comment (in the source material) hits something crucial: When does operation order matter in nonlinear systems? Always. But the question becomes: which order is physically correct?
Koch’s team says: average products, not products of averages. But why is that the right choice? Because it captures “richer geometric information.” Okay, but that’s a mathematical preference, not necessarily a physical principle.
This is the hidden assumption: the “correct” averaging order is the one that preserves more information. Maybe. Or maybe nature doesn’t care about our information theory preferences.
Layer 3: What Question Should I Actually Be Asking? (Reframe)
The Real Question Isn’t About Dark Matter
It’s about whether our mathematical tools are appropriate for the phenomena we’re studying.
For 90 years, we’ve assumed:
Quantum gravity matters at Planck scale (10⁻³⁵ m)
Classical GR is fine at larger scales
Any quantum corrections are negligible
Koch’s paper suggests:
This assumption might be correct for linear equations
But Einstein’s equations are nonlinear
Nonlinearity can amplify tiny effects at large scales
We might have been systematically hiding these effects by our choice of averaging
The Meta-Pattern: Scale-Dependent Emergence
This applies beyond physics:
In computational systems: Processing order changes emergent behavior (Neural Foundry’s simulation experience)
In social systems: Individual actions average to institutions, but the sequence of those actions matters for which institutions emerge
In consciousness: Do you process sensory data then integrate, or integrate while processing? Different order, different experience.
In democratic systems: Do you aggregate preferences then make decisions, or make decisions that shape preferences? The feedback loops are different.
The Actual Question:
Not “Does dark matter exist?”
But: “How do complex systems generate emergent properties, and how does our analytical framework reveal or hide those properties?”
Dark matter might exist. MOND might be partially right. Quantum corrections might matter. These aren’t mutually exclusive.
The deeper issue: We’ve been treating spacetime as a background on which quantum stuff happens. But if spacetime itself is quantum, then our analytical separation of “background” and “fluctuation” might be the error.
The Cosmological Constant as Information Density Threshold
Hans Jonsson’s proposal (from the article): The cosmological constant represents a phase transition in how spacetime averages quantum uncertainty.
At high density (solar systems): quantum effects decohere rapidly → linear averaging works At low density (galaxies): quantum coherence extends → nonlinear averaging matters
This would make the cosmological constant not just a parameter, but a structural feature of how reality transitions between quantum and classical regimes.
What This Means For “Truth Matters”
Truth does matter. But truth isn’t always a binary state.
It’s TRUE that galaxies rotate faster than Newtonian gravity predicts
It’s TRUE that dark matter particles haven’t been detected
It’s TRUE that MOND successfully predicts rotation curves
It’s TRUE that MOND fails at cluster scales
It’s TRUE that quantum corrections to GR should exist
It’s TRUE that we haven’t measured them yet
All these truths coexist. The question isn’t “which truth wins” but “what structure connects these truths?”
The Argument FOR Koch’s Approach
Strength 1: Mathematical Rigor The formalism is solid. They’re not handwaving - they derived the q-desic equation using both Lagrangian and Hamiltonian methods. The math checks out.
Strength 2: Addresses Known Gaps We know quantum gravity is incomplete. We know GR and QM don’t mesh. Any approach that provides a mathematically consistent bridge deserves attention.
Strength 3: Testable (In Principle) Unlike String Theory’s extra dimensions or supersymmetric particles, this makes predictions at scales we can observe (galaxies). We already have the data - we just need to calculate the predictions and compare.
Strength 4: The Averaging Paradox Is Real ⟨X²⟩ ≠ ⟨X⟩² is not debatable. If Einstein’s equations are nonlinear, then averaging order DOES matter. This isn’t speculation - it’s mathematical necessity.
Strength 5: Explains The “Too Clean” MOND Connection Why does MOND work so well despite having no theoretical foundation? Because it’s an effective theory - it captures the right functional form without understanding the mechanism. If quantum corrections produce MOND-like effects, that would explain MOND’s success and provide the missing theoretical basis.
Strength 6: The Cosmological Constant Appears Naturally They didn’t add it by hand. It emerges from the formalism. When different independent approaches all point to the same constant, pay attention.
Strength 7: Addresses The “Where Are The Effects?” Problem Standard quantum gravity approaches fail because they predict effects at unmeasurable scales. This approach suggests effects might be observable exactly where our current theories already show strain (galactic dynamics). That’s elegant.
The Argument AGAINST Koch’s Approach
Weakness 1: It’s Incomplete They have a formalism, not a theory. To make predictions, you need to specify the quantum state of spacetime. They haven’t done that yet. This is like having Newton’s laws but not knowing the initial conditions.
Weakness 2: The String Theory Objection String Theory has already addressed renormalizability and background independence (within its framework). If you’re proposing a new approach to quantum gravity, you need to either:
Explain why String Theory is wrong, or
Show how your approach connects to or improves on String Theory
Koch’s paper does neither. It simply ignores 40+ years of String Theory research.
Weakness 3: MOND’s Problems Become Your Problems If your quantum corrections reproduce MOND, you inherit MOND’s failures:
Doesn’t work at galaxy cluster scales
Can’t explain galaxies with no apparent dark matter (NGC1052-DF2)
Gravitational lensing observations still problematic
Weakness 4: The Quantification Gap Hossenfelder’s criticism is valid: they claim “clear differences at large distances” but don’t demonstrate it numerically in the paper. This is a red flag. If you can’t calculate specific predictions, you don’t have a testable theory.
Weakness 5: The “Which Averaging?” Ambiguity Why is averaging products (instead of products of averages) the correct choice? Because it preserves more geometric information? That’s an aesthetic preference, not a physical principle. Nature might not care about our information preservation.
Weakness 6: The Measurement Challenge Even if you calculate predictions, testing them is hard:
Can’t do controlled experiments on galaxies
Observations already interpreted through existing frameworks (GR + dark matter or MOND)
Circular reasoning risk: using galactic dynamics to motivate quantum corrections, then using quantum corrections to explain galactic dynamics
Weakness 7: The Scale Transition Mystery If quantum corrections are negligible at solar system scales but significant at galactic scales, there must be a transition region. Where is it? How does it work? The paper doesn’t address this.
Weakness 8: Background Independence Isn’t Solved Quantum gravity needs to be background independent - you’re quantizing spacetime itself, so you can’t assume a fixed spacetime background. The Koch approach still works within a classical spacetime background (spherically symmetric, static). This is a significant limitation.
What Could Go Wrong (And Right)
Optimistic Scenario
The formalism develops
Someone figures out how to specify the quantum state of spacetime
Specific predictions emerge for galactic rotation curves
Predictions match observations better than GR + dark matter
Predictions also explain where MOND fails
We get new insights into dark energy (cosmological constant connection)
This becomes the foundation for a complete quantum gravity theory
Probability: Low History suggests most new approaches to quantum gravity don’t pan out. But not zero.
Pessimistic Scenario
Turns out the formalism requires inputs that are impossible to determine
When people finally calculate predictions, they don’t match observations
Or worse: predictions match current observations but don’t make new testable claims
The community loses interest
Paper becomes a footnote: “interesting idea that didn’t work out”
Probability: Moderate This is what happens to most physics papers. Not wrong, just not useful.
Realistic Scenario
The formalism proves useful as an analytical tool
Doesn’t solve quantum gravity
Doesn’t eliminate dark matter
But provides new ways to think about quantum corrections in curved spacetime
Inspires other approaches
Contributes to gradual progress
Probability: Highest Science usually advances through incremental improvements, not revolutions.
Cynical Scenario
The math was fine but the interpretation was oversold
Media runs with “quantum gravity explains dark matter!”
Public gets confused when nothing comes of it
Another entry in “cool physics that didn’t change anything”
Contributes to science communication fatigue
Probability: Unfortunately Also High We’ve seen this movie before.
The Absurdist Sketch
Setting: A beach. Two lighthouses at opposite ends. Between them, 90 years of footprints in the sand.
QUANTUM LIGHTHOUSE (small, flickering, uncertain whether it’s actually on): I illuminate the very small! Particles, uncertainty, probability waves!
GRAVITY LIGHTHOUSE (massive, steady, bending the beach around it): I illuminate the very large! Planets, black holes, the curvature of space itself!
PHYSICIST (squinting at the beach): You’re obviously describing the same coastline.
BOTH LIGHTHOUSES: Obviously!
PHYSICIST: So why do your beams never meet?
QUANTUM LIGHTHOUSE: Because he’s in the wrong phase!
GRAVITY LIGHTHOUSE: Because she’s too uncertain!
PHYSICIST: Right. (Takes out calculator) Let me try averaging your light beams...
QUANTUM LIGHTHOUSE: How are you averaging?
PHYSICIST: First I’ll measure each of you separately, average those measurements, then multiply them together.
GRAVITY LIGHTHOUSE: That’s not how multiplication works when we’re nonlinear!
PHYSICIST: What?
QUANTUM LIGHTHOUSE: You have to multiply us first, THEN average!
PHYSICIST: But that gives a different answer!
BOTH LIGHTHOUSES: Correct!
PHYSICIST: But... but I’ve been doing it the other way for 90 years!
GRAVITY LIGHTHOUSE: Yes, and how’s that working out?
PHYSICIST: (looks at notes, sees “dark matter?” scribbled 10,000 times) ...not great.
QUANTUM LIGHTHOUSE: Try the other way.
PHYSICIST: (recalculates) Oh. Oh no.
BOTH LIGHTHOUSES: What?
PHYSICIST: You DO meet. At galactic scales. You’ve been connecting this whole time, I just... I was averaging wrong.
GRAVITY LIGHTHOUSE: Does this mean we can finally go home?
QUANTUM LIGHTHOUSE: Probably not. He still has to figure out what our quantum states are.
PHYSICIST: Your what now?
QUANTUM LIGHTHOUSE: Our quantum states. You can’t actually use this calculation without knowing our exact quantum configurations.
PHYSICIST: And how do I determine those?
BOTH LIGHTHOUSES: (in unison) No idea!
PHYSICIST: So I’ve discovered that 90 years of physics might be based on averaging in the wrong order, but I still can’t make predictions without information I don’t have?
QUANTUM LIGHTHOUSE: Now you’re getting it!
GRAVITY LIGHTHOUSE: Welcome to physics. Would you like a doctorate?
(PHYSICIST sits down on beach, stares at ocean)
PHYSICIST: Is dark matter even real?
OCEAN (the cosmological constant, somehow both expanding and constant): Define “real.”
PHYSICIST: I hate this job.
BOTH LIGHTHOUSES: That’s how you know you’re doing it right!
Dimensional Storytelling: The Pattern That Connects
This isn’t just a physics story. It’s a pattern that appears across scales:
In Computation (Neural Foundry’s insight) Order of operations changes emergent behavior. A neural network trained with gradient descent in one order learns differently than the same network with operations in a different order. Same data, same architecture, different emergence.
In Society Do you count votes then interpret (majoritarianism) or interpret votes then count (weighted representation)? Different order, different democracy.
In Consciousness Do you process sensory inputs then construct a model (bottom-up), or do you have a model that filters inputs (top-down)? Both happen, but the sequence matters for what you experience.
In Economics Do markets aggregate preferences then produce outcomes, or do market structures shape preferences that then aggregate? Averaging order matters.
The Universal Principle
When systems are nonlinear (which most interesting systems are), the order in which you combine information fundamentally changes what emerges.
This is why the Koch paper matters beyond physics. It’s a case study in how our analytical frameworks can systematically hide the phenomena we’re trying to understand.
We haven’t been stupid. We’ve been using tools designed for linear systems on a nonlinear universe. The tools work fine at small scales (solar system - approximately linear). They break at large scales (galaxies - decidedly nonlinear).
The averaging paradox isn’t a bug. It’s revealing that linearity is a useful approximation, not a fundamental property of nature.
What I Actually Think (Brutal Honesty Mode)
The Math: Solid. The q-desic formalism is a legitimate contribution.
The Physics: Promising but incomplete. They’ve opened a research direction, not solved a problem.
The Interpretation: Oversold. “Quantum corrections at galactic scales” sounds more conclusive than “we have a framework that might, if we can solve several additional problems, potentially make predictions about galactic scales.”
The Dark Matter Angle: Misleading. This doesn’t eliminate dark matter. At best, it might explain some dark matter observations through quantum corrections. Dark matter might still exist.
The MOND Connection: Interesting but circumstantial. Yes, both involve the cosmological constant and galactic scales. That could be meaningful or coincidental.
The String Theory Silence: Problematic. You can’t propose a new approach to quantum gravity and not engage with the dominant framework. Either explain how you’re different/better, or explain how you connect.
The “Averaging Wrong” Narrative: Partially true but simplified. Yes, averaging order matters in nonlinear systems. But calling it “wrong” assumes we know what “right” is. We don’t yet.
Will This Matter in 10 Years?
Probably. Not as a revolution, but as a useful tool. The q-desic formalism will likely contribute to our understanding of quantum corrections in curved spacetime, even if it doesn’t eliminate dark matter or fully solve quantum gravity.
Best Case: This becomes one pillar of a larger framework that eventually unifies quantum mechanics and gravity.
Worst Case: Interesting math that doesn’t connect to observable reality.
Most Likely Case: Incremental progress. Another piece of the puzzle. Not the final answer, but a step toward it.
Consequences (What Could This Mean?)
If Koch Is Right (Quantum Corrections Explain Galactic Dynamics)
Good:
We understand gravity better
Dark matter might be partially or fully explained without new particles
Opens new research directions
Validates the “think different about scales” approach
Bad:
40+ years of dark matter particle searches were chasing shadows
Huge amounts of research funding redirected
Major theoretical restructuring needed
Graduate students currently studying dark matter detection might need new thesis topics
Weird:
The universe’s structure is more quantum than we thought
The cosmological constant is even more important than we realized
Spacetime itself might have “texture” at galactic scales
If Koch Is Wrong (This Framework Doesn’t Pan Out)
Good:
We learned something about averaging in nonlinear systems
The math might still be useful elsewhere
Checked another possible path off the list
Bad:
Still don’t understand dark matter
Still don’t have quantum gravity
Back to the drawing board
Weird:
Nothing. Science working as intended.
If The Truth Is Somewhere In The Middle (Most Likely)
Reality:
Quantum corrections matter some
Dark matter exists but maybe less than we thought
MOND captures something real about galaxy dynamics
All three frameworks (GR + dark matter, MOND, quantum corrections) are partial descriptions
Outcome:
We need a more sophisticated model that incorporates all three
The cosmological constant is the bridge between scales
Galactic dynamics is more complex than any single framework captures
Lesson:
Nature doesn’t care about our philosophical preferences for elegant single explanations
Sometimes the answer is “all of the above, in different regimes”
Further Reading & Sources
Primary Sources
Koch et al. Paper
“Geodesics in Quantum Gravity” - Physical Review D 112, 084056 (2025)
arXiv:2510.00117
Full technical treatment of q-desics formalism
Earlier Related Work
Reuter & Weyer (2005): “Do we Observe Quantum Gravity Effects at Galactic Scales?” - arXiv:astro-ph/0509163
First proposal of large-scale quantum gravity effects
MOND Background
Original MOND Papers
Milgrom (1983): “A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis”
Physics Letters B, three papers introducing the framework
Recent MOND Reviews
Famaey & Durakovic (2025): “Modified Newtonian Dynamics (MOND)” - arXiv:2501.17006
Comprehensive recent review
McGaugh et al. (2016): “Radial Acceleration Relation in Rotationally Supported Galaxies”
Observational support for MOND
Dark Matter Context
Detection Efforts
XENON, LUX-ZEPLIN experiments: No detection after 30+ years
Ongoing searches continue
Anomalous Galaxies
van Dokkum et al. (2018-2019): NGC1052-DF2 and DF4 studies
Galaxies with apparently no dark matter (problematic for MOND)
String Theory Comparison
String Theory Claims
See Suman Suhag’s comment in source documents for summary
Addresses renormalizability, background independence, black hole thermodynamics
No experimental evidence after 40+ years
Skeptical Analysis
Sabine Hossenfelder
YouTube: “Science without the gobbledygook”
3/10 bullshit meter rating on this work
Critiques: incomplete quantification, framework vs. predictions
Physics World Coverage
“Motion through quantum space-time is traced by ‘q-desics’” (December 2025)
Balanced technical reporting
Philosophical Context
On Averaging and Emergence
Neural Foundry’s simulation work (mentioned in source): Operation order changes emergent behavior
Hans Jonsson’s dimensional thinking framework
Data Sources
Galactic Rotation Observations
SPARC (Spitzer Photometry and Accurate Rotation Curves) database
~175 galaxies with detailed rotation measurements
Cosmological Constant Measurements
Planck satellite data
Supernova cosmology project results
Bottom Line (Truth Matters Edition)
What We Know For Sure:
Galaxies rotate faster than Newtonian gravity predicts
Dark matter particles haven’t been detected despite decades of searching
MOND successfully predicts rotation curves but fails at larger scales
Quantum gravity must exist (GR and QM can’t both be fundamental)
The cosmological constant is real and measurably affects cosmic expansion
In nonlinear systems, averaging order mathematically matters
What We Don’t Know:
Whether dark matter exists as particles
Whether MOND captures something fundamental or is just a good approximation
Whether quantum corrections significantly affect galactic dynamics
What the quantum state of spacetime actually is
How to reconcile quantum mechanics and general relativity
Which averaging order (if any) is “correct” for quantum gravity
What This Paper Contributes:
A mathematically rigorous formalism for quantum-corrected geodesics
A new way to think about scale-dependent quantum effects
A potential connection between quantum gravity and galactic dynamics
Opens research directions, doesn’t close questions
What This Paper Doesn’t Prove:
Dark matter doesn’t exist
MOND is derivable from quantum gravity
Quantum corrections explain galactic rotation
We’ve been “doing it wrong” for 90 years
The Honest Assessment:
This is good science: careful, mathematical, opening new possibilities. It’s not revolutionary science: proven, paradigm-shifting, explanation-complete.
The averaging paradox is real. Whether it matters at galactic scales is an open question. The Koch paper provides tools to explore that question. That’s valuable even if the answer turns out to be “no.”
Truth matters. So does humility. We don’t know the truth about dark matter, quantum gravity, or galactic dynamics yet. We have competing frameworks, partial explanations, and promising approaches.
This paper is one of those promising approaches. Time will tell if it’s more.
Final Thought (The Optimistic Ending)
Maybe we’ve been averaging wrong. Maybe we haven’t. Maybe dark matter exists. Maybe it doesn’t. Maybe MOND is right. Maybe quantum corrections matter.
Or maybe - and this is the most likely answer - all of these frameworks are pointing at different aspects of the same complex reality, and we need to stop thinking in terms of “which one is right” and start thinking in terms of “how do they fit together.”
The universe doesn’t care about our preference for elegant single explanations. It is what it is. Our job isn’t to make it simpler. Our job is to understand it more accurately.
The Koch paper, whatever its ultimate fate, moves us slightly closer to that understanding.
That’s enough.
Word of mouth truth: Share this if you think words like “quantum gravity” and “averaging paradox” should be explained without bullshit. We’re all trying to figure out the same universe.
Facts matter. Definitions matter. Truth matters. Even when - especially when - the truth is “we don’t know yet.”
About This Analysis: Written using three-layer thinking framework: surface (what everyone says), blind spots (what we’re missing), reframe (what we should actually ask). No funding from physics departments, dark matter detection facilities, or MOND advocacy groups. Just one person trying to understand whether we’ve been doing math wrong for 90 years.
Disclaimer: I am not a physicist. I am someone who reads physics papers and tries to understand them honestly. If I’ve made errors, they’re mine. If this helps you understand the debate better, the credit goes to the researchers doing the actual work.
Last Updated: February 9, 2026
License: Share freely. Cite honestly. Challenge vigorously.
🙏