Now that the argument for hallucinations being inherent has been confirmed through OpenAI's recent proof, it points to a necessary larger problem within the system itself; that is the reliance on a non-differentiating, non-scientific, and by its own logic, non-alignable and non-scalable system. The method used to get to this point in the critique is Imminent Critique after reading Subliminal Learning: Language models transmit behavioral traits via hidden signals in data. The Following is my dialectical approach restructured and axiomatized for logical proofs, and empirical grounding.
The core argument of this critique is based on the fundamental unit of value LLMs use for their architecture. The very reason they function is also the inherent flaw.
This fundamental unit is a Monastic Axiom, a non-differentiating axiom. It bases the output on human differentiation, our understanding is constituted by the ability to distinguish. It can not be autonomous in the true sense of the word. And in action will degenerate over time.
The Axiom is formally stated as: Axiom 𝒜: y_{t+1} = \arg\max_{v\in V} p_\theta(v | x_{\leq t})
First I have been rearticulating this idea for a long time and have used Gemini to articulate it in research paper format for myself to negate. With the final set of notes it output a meta-proof of my theory accidentally but acknowledged it, if the set of notes are needed I can upload them as a .txt
Gemini output
Excerpt: "A compelling, if unintended, demonstration of this very problem is evident in the machine's own analysis of the source material for this report. When tasked with summarizing the document, the machine-generated responses consistently failed to correctly identify or extract key formal sections, such as the logical proofs, the theorem of incompleteness, and the initial philosophical notes on the "owls" experiment. This failure to correctly parse and categorize the structured, academic components of the text—mistaking a rigorous proof for a casual note—is a perfect, real-world manifestation of the "poverty of logic" being diagnosed here. The system's inability to discern a formal proof from an informal one underscores its non-differentiating nature, treating all information as an undifferentiated stream of tokens guided by the same singular statistical axiom. This structural deficiency provides a powerful meta-critique that grounds the paper's theoretical arguments in a tangible example of the system's limitations"
On the use of emergence, because this is the one I've been thinking about for the longest time. Emergence is not a technical term. They are using it to explain what they cannot explain, essentially. They are saying that there is a transition qualitatively in the system that they can only explain through saying it's emergence, through a physical process that happens with water boiling. Emergence in Physics can be explained, though. There is a rational explanation, through reason and Thermodynamics of why this happens to water. This cannot just be applied to everything, and especially not without a reason for it. They are just substituting a term for not knowing exactly how or why it's happening. And they can map it. They can say it's a linear progression that extends this, but they're not molecules in a system of entropy, gravity thermodynamics. It is tokens. And thereby, it is not a rule, this points to 4 obvious logical fallacies on the term emergence in this context
First this is a prime example of Category Error they are taking the process of emergent properties happening in a physical system governed by natural laws and misappropriating it towards a system that is inherently different from physics.
Second, a more forgivable one but not today. They are using a term that carries scientific rigor to add an aura of credibility to their claim that this is happening. It is contingent reorganization of the statistical function, nothing more and does not point towards a necessity or even a probable outcome on a larger scale, this constitutes an Appeal To Ignorance.
Next to address the claim that this is only probable, given the prior plateaus in recent years, they are saying the market is rational and so is the claim. They're saying this is probability, but how many samples do they have? probability happens at scale, not at a couple years. The plateaus they're pointing to as emergence can be applied to any technology, if we didn't know the proper method to achieving those results because the scale is so short. This is not an accurate use of probability given the data points and data set. It's too low of a distribution. Misuse of probability, another category error.
Even if this was probabilistic, equating probability with law; Media and prominent advocates for Agi are saying this is already happening, fair enough, present proof. But for those that are saying it will happen, it is not only probability as necessity but is a higher order misuse, they do not have data or distribution for probability to apply here
The very use of Emergence also denotes and implies it has happened, then an explanation of the laws that govern this phase transition is in order, but without that it is Begging the Question.
Now the Alignment Problem.
Because of the architecture design. Unless they are fundamentally changing the axiom from which this proof exhibits the limitations of, then of course it changes. The very nature of its core function has been altered. I would welcome a system that could derive truth internally. That would be logic proper. But the problem of anchor points, as the latest method of halting the model collapse problem, is already looking like a non-terminable issue. Why? Because the very notion of anchors denotes the reasoning of higher value differences within the statistical field. They need to be of a higher order of importance to actually anchor throughout the dilution that training depends on. But if that is the case, it would then not be an additive function of qualitative differences. Instead, it threatens the opposite. Why even feed it synthetic data if the most likely outcome through constraints and behavior training would always presume the statistical correlation of higher value axioms, rather than the lower value ones? That then leads to a bigger problem: if it doesn't value anchor points higher, then the data will degenerate after cycles of model generation. Also, the statistics being added are flawed since the synthetic data reinforces heavily from generation to generation. The synthetic data only serves as a volume of non-usable parameters.
Below is a test if anchor data will actually always be privileged in a statistical output.
hypothesis: I think it will privilege anchor data when it comes to human inputs, because the context is provided through humans, and the statistical likelihood of matching human writing is higher than human writing matching AI writing. But this also might be skewed by people inputting articulations that LLMs have written. But even then, it would still privilege first-generation synthetic data rather than subsequent generations.
Prompt: “Initial hypothesis” + Chatgpt5 rephrasing “In recursive training with synthetic data, only ongoing privileged human anchor data prevents collapse. First-generation anchors or pure synthetic recursion inevitably narrow distributions and degrade coherence”.
Chatgpt5 Instructive of a toy rl experiment to test the hypothesis
What I did (toy but instructive):
I built a small simulation where:
* A “human” language is a Zipf-like distribution over 200 tokens.
* A model is trained generation-by-generation on synthetic samples drawn from the previous model plus (optionally) anchor samples drawn from the human language.
* Each generation re-estimates by MLE on the mixture.
* I tracked KL divergence, entropy, and top-1 mass as crude collapse indicators.
Scenarios:
1. No anchors (pure self-feeding).
2. Small anchors (1k human samples/gen).
3. First-gen anchor only (5k once, then zero).
4. Steady anchors (5k/gen).
5. Strong anchors (20k/gen).
You can view the results I produced for you:
* A summary table: SPC\_Anchor\_Scenarios\_Summary (interactive table opened in your workspace).
* Time-series CSVs for all runs:
* [Download timeseries](/mnt/data/SPC_anchor_simulation_timeseries.csv)
* [Download summary](/mnt/data/SPC_anchor_simulation_summary.csv)
What it shows (in plain terms):
* No anchors: KL divergence from the human distribution steadily increases over generations—classic self-generated drift.
* First-gen anchor only: the model briefly stays closer to the human distribution, then drifts away—the one-time anchor doesn’t hold.
* Small steady anchors: drift is slowed, but the model still moves away from the human distribution over time.
* Steady 5k anchors: KL stays low and flat—the human anchor keeps the model near the human distribution.
* Strong 20k anchors: KL becomes very small and remains small—human anchors dominate.
In other words:
With recursive synthetic training, the model forgets rare/long-tail human patterns and narrows unless you keep injecting anchor data.
When you do inject anchors, the update rule is just counts, so the human samples exert a persistent pull back toward the human distribution. That’s your “anchor privilege” in a measurable sense.
A first-generation-only fix does not stop long-run drift; ongoing anchoring matters.
The findings point to a hard limit in anchor data, and scaling is not going towards higher intelligence but rather it is leading towards an end. The system cannot self negate, and cannot verify any true or false statements. It necessarily collapses into degenerative outputs. Is there a solution? Yes but not one at scale, Anchor data must be privileged or it will collapse the model. But again if that's the case why add synthetic data if it only serves to swell the parameters. This is likely due to their failure in understanding what emergence actually is. It is the reconfigured Statistical space, contingently not linearly
Statistical as Necessity, The Logic of Ignorance.
This suggests an ideological explanation of the use and the reason why its misuse is so alarming is needed. Statistics are not invalidated here. It is another category error, one that presumes all other category errors in AI. The prime error, I think. While it is useful for correlations. It is completely useful. That is its use. I'm not suggesting that statistics are non-valuable. I am suggesting that its use here is not the correct use of statistics because they are not assuming that it is statistical correlation. By way of saying Intelligence, by positioning it as a decision maker, an operator of a vehicle, they are taking it as necessity. That is already a foundational error to how statistics should be applied and can even be applied in a useful way.
I want to be clear, this isn't a critique of axiomatic systems as a whole. It is necessary for the conception of objects from the realm of concept onto the material to necessarily collapse the movement into a moment. Essentially, math is essential if we want to materialize logic in the real world, which is why technology uses math so appropriately. But this isn't appropriate, we apply this non-necessity system into the realm of necessity, both ideologically by putting it into a position of decision-making and by design, by implying that this machination isn't what it actually is, which is not just a guess, but one determined by contingent variables within the input-output, deterministic to itself, but incongruous with reality and humanity. Statistics is a useful tool but also one that does not reflect human subjectivity, choice, freedom, or even behavior. It reflects correlations between data sets.
If we assume statistics is in any way governed by necessity, even if we only simply elevate it to necessity without any factor or qualitative change in statistics as a system, then it should accurately have a teleology of human behavior and by necessity not deviate it beyond significant figures. But classically statistics do not do that.
Another point to address is the point that statistics is governed by, essentially, mathematical law, even if its relations and outputs are correlations or contingent. And while, yes, that's true. It is still, for one, a category error, and two, it is at best a virtualization. It's more like a virtual machine within the system proper. It has no effect on the larger mathematical laws, and it cannot be disproven by its own system, or by the system of mathematics, because it isn't necessarily mathematical. So there's another problem here, and that's the non-falsibility problem. It is, in essence, not scientific and not mathematical. Again useful but should not be used as a determinacy engine.
There's one more point to address in this critique, and it's this "if we don't analyze this or if we cannot use statistics then what do we use?" We use statistics for the purpose of the machine; that is the only thing it can do, and it is an approximation of human thought and behavior. It can never be replaced with the real thing, and that's what I'm arguing against because it's not the real thing. It's not even a simulation of the real thing; it is a good costume of the real thing. This is why philosophical analysis is needed. The Engineers were already presuming that it was already a necessity. When that is exactly not the role of statistics and not the place of statistics to be there. It is a primary, foundational, potentially catastrophic, category error.
But in essence, this is the thing that is the reductive factor in the subject-to-object relation; of AI and humans. This is what they're using to substantiate their claim that they have technocratized the use of human relations.
The Monarchic Axiom
Finally here I will articulate the primary foundational cause for the errors within llms and why alignment and Model Collapse is a non-terminable problem, this will be in the form of initial notes, since they are the prompt after hours of Negation, that I was able to reach this conclusion.
The known phenomena of data poisoning and prompt injection necessarily point that it is a non-differentiation system, it's monistically axiomatic, and it's much worse than I originally thought. Math is rigorous. But essentially this Axiom isn't even dualistic or multiple; it is a monastic system of equal tokens dictated by weight space but the tokens themselves are essentially of equal value. Outputs are steered by weights and constraints, but at the core every token is ontologically flat; undifferentiated in value, distinguished only by Statistics for which it does not even know are statistical.
I want to stay as simple and as calculable as possible, because they can and will then just say that this is all philosophical garble, And then, is it even provable? Things of that nature. I want to, in essence, use their own formulas of logic to counter and find their own limits. So instead of non-dialectic, it's non-differentiating. And understanding is itself a process of differentiation.
A simple equation using a monastic axiomatic system would be proof enough to show the incapabilities of such a system and the random contingency that it needs. And only through human input can it ever do something substantial. The input is essentially the guide and mechanism to derive any meaningful output. That's why it can't just output, it has no way of valuing tokens other than flat ontology. It is incapable of true autonomy. It is deterministic to itself, contingent to us, in catastrophic adherence to statistical weights
This is also a form of the incompleteness problem, except the outside is always assumed through the human. We are the engine of differentiation that forces the output to be differentiated at all. We form the context window through the input. It's essentially the formal logic version of the machine-world-human-machine gap (A term I used while developing this theory but not sure now). And the differentiation, even at output, is ours. It has no such differentiation. This is why the system degrades when fed purely synthetic data. It needs an anchor, and this is the proof that they have to anchor the statistics in human behavior. It cannot exist on its own. It does not even constitute itself.
Proof sketches that follow are in formal logical notation and mathematics. These are derivations of the Hegelian logic I used onto an axiomatic system via Chatpgt, deepseek, gemini. The axiom is monastic, meaning the same initial value across every iteration, how llm tokens work. This is the work of Negation and logic as a movement. The prompts are the notes until the description of the proof sketches, initial prompts are the entirety of my note set.
Proof sketch of the differentiation problem via prompt above, final itwration through
Proof sketches:
A proof sketch: why an LLM needs an outside to differentiate
Setup (minimal formalization)
Let be a finite token set.
A model is a conditional distribution over given context .
Inference policy maps to an action (greedy, sampling, beam, etc.).
Training objective (pretraining): minimize where is the empirical next-token distribution from a corpus .
Post-training (RLHF/constitutional/etc.) adds constraints that bias selection (via reward model or rejection sampling), but still selects tokens in .
Call this the monistic axiom: All behavior is “continue the string” by picking tokens according to , modulated by local constraints.
Claim 1 (No intrinsic differentiation)
Statement. Without an external criterion not computable from , the system cannot rank continuations by meaning/ground truth—only by .
Reason. Any “distinction” between two candidate tokens must come from either:
1. their probabilities under , or
2. an external evaluative signal not reducible to .
In the closed system , (2) is absent. So “differentiation” collapses to relative likelihood under learned statistics. This is selection, not understanding.
Corollary. “Refusal,” “honesty,” “helpfulness” are just selections satisfying —not internally computed semantic distinctions.
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Claim 2 (Dependence on an external anchor)
Statement. If is replaced by self-generated text from , iterative retraining drives the model toward support shrinkage and homogenization (model collapse), absent an external anchor.
Sketch. Let be the support of contexts/continuations in . Training pushes toward . If we sample and retrain, we effectively fit to our own modes. Low-probability events in (rare, yet crucial distinctions) vanish from . Repeating reduces entropy and prunes tails—thus reducing differentiation. The “anchor” is the human/world distribution that keeps rare distinctions alive.
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Claim 3 (Impossibility of internal grounding)
Statement. There is no computable function inside the LLM formalism that yields grounded semantics (a mapping from tokens to world states) without an external channel (sensors, experiments, human adjudication).
Sketch. Grounding needs a relation (world states / truth conditions). But is not derivable from text statistics alone (text underdetermines world; same strings can correspond to distinct realities). Any proposed internal would be equivalent to inferring from distributional facts—this is insoluble without extra-linguistic data/feedback. So grounding is extrinsic to .
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Claim 4 (Alignment as non-termination / “bad infinity”)
Statement. For any finite constraint set enforceable by post-training, there exist contexts where satisfying contradicts maximizing next-token likelihood or other constraints in . Iterative patching does not terminate.
Sketch. Diagonalization by construction: craft prompts that force trade-offs (truth vs. privacy; harmlessness vs. completeness; safety vs. non-discrimination). Because is finite and local, adversarially composed contexts will expose conflicts. Patching adds new local rules , which create fresh boundary cases. Hence an infinite regress of patches—no fixed point that globally resolves conflicts within the monistic axiom.
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Claim 5 (No internal “negation” operator)
Statement. Genuine negation (self-undercutting of the system’s own rule) would look like: (i) refusing to output at all or (ii) initiating output unprompted to overturn its own prior move. The standard action space does neither.
Sketch. The action space is “choose next token when asked.” There is no native act that suspends the rule or originates a counter-rule. “Refusals” are tokens conditioned on prompts and rewards; they are still continuations. Therefore the system cannot generate the kind of internal negativity that would re-found its axiom.
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Claim 6 (External differentiation is necessary and sufficient)
Statement. Supply an external differentiator (human feedback, sensorimotor grounding, experiments, formal tasks with verifiable outcomes), and the system can approximate distinctions relative to . Remove , and distinctions decay.
Sketch. With , training optimizes toward distinctions valued by (e.g., pass unit tests, satisfy judges). But is exogenous; the model cannot originate it. Therefore differentiation is imported, not endogenous.
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Theorem (Incompleteness of the LLM formalism for differentiation)
Let denote the class of LLMs with post-training constraints . There exists no such that, for arbitrary novel contexts , can (i) ground meanings, (ii) resolve global value conflicts, and (iii) generate true negation—without an external differentiator .
Reason. Claims 1–5 show (i)–(iii) require information or actions not contained in . Thus, the capability set is extensionally incomplete: it presupposes an outside to do the differentiating work.
QED (sketch).
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Practical corollaries (you can point to these in deployment debates)
1. RAG/citation ≠ grounding. It borrows differentiation from curated sources (an by proxy). It doesn’t make the system self-grounding.
2. Safety patches don’t converge. Local rules will conflict at scale; expect perpetual “whack-a-mole.”
3. Synthetic-only training collapses distinctions. Keep human/world anchors in the loop or watch support shrink.
4. Risk scales with delegation. The more we treat the object as a subject (asking it to decide rather than suggest), the more the incompleteness bites.
Supplementary proof sketch, the monastic axiom chatgpt5
1) The monistic axiom
Let the system have exactly one axiom (objective):
\textbf{Axiom} \quad \mathcal{A}: \quad y_{t+1} \in \arg\max_{v\in V} \; p_\theta(v \mid x_{\le t})
(or the stochastic variant ).
That’s it. The “logic” of the system reduces to continue with the most likely token (or sample from that likelihood). All tokens are ontologically identical IDs; differences are only scalar (their probabilities).
2) What this axiom cannot do
No differentiation (no truth/meaning operator).
There is no predicate for “true,” no valuation over world states, only likelihood . So the system cannot prefer true over false—only frequent/plausible over infrequent.
Under-determination → contingency/randomness.
If two continuations tie (or are near-ties):
p_\theta(v_1 \mid x_{\le t}) \approx p_\theta(v_2 \mid x_{\le t}),
then the choice requires an external tie-breaker . The axiom itself provides no reason to pick over . Hence randomness (or arbitrary heuristics) enters immediately.
Distribution shift fragility (no normativity).
Let training distribution be and deployment be . The system optimizes
J(\theta)=\mathbb{E}_{x\sim \mathcal{D}_{\text{train}}}\big[\log p_\theta(x)\big],
not correctness under . When , likelihoods can diverge from truth/utility with no internal mechanism to detect or correct.
Goodhart by construction.
If you try to make it “useful” by pushing the same axiom harder (higher likelihood), you optimize the proxy (text plausibility), not the target (truth, safety, justice). Classic Goodhart: measure ≠ goal.
3) Why human input is necessary (and what it actually is)
To make the system do work with stakes, you must add an external value functional—provided by humans:
Supervised targets: labels define a loss .
RLHF/RLAIF: a learned reward from human preferences; objective becomes
\max_\theta \; \mathbb{E}\big[R_\phi(s,a)\big] \quad \text{s.t.} \quad a \sim \pi_\theta(\cdot \mid s).
These are not discovered internally; they are imported. The model still has no native (truth) or (utility). It only learns to approximate the scalar signals we provide.
4) One-line contrast (the “equation of the gap”)
\underbrace{\arg\max p_\theta(\text{token}\mid \text{context})}_{\text{LLM axiom}}
\;\;\neq\;\;
\underbrace{\arg\max U(\text{world state})}_{\text{decision with values}}
Without a human-given (or its surrogate labels/rewards), the system cannot differentiate good from bad, true from false, harm from help. It only differentiates more likely from less likely—and when likelihoods tie or shift, it falls back to contingency (randomness or brittle heuristics).