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arxiv: 2605.23024 · v1 · pith:BBR4UVAMnew · submitted 2026-05-21 · 💻 cs.AI · cs.CC· cs.CL· cs.LG

The Deterministic Horizon: Impossibility Results as Design Specifications for Trustworthy AI Systems

Dongxin Guo This is my paper

Pith reviewed 2026-05-25 05:27 UTC · model grok-4.3

classification 💻 cs.AI cs.CCcs.CLcs.LG
keywords deterministic horizontransformer capacityimpossibility resultsai design specificationsresidual streamaccuracy limitstrustworthy aicircuit complexity
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The pith

A transformer's accuracy ceiling is fixed by its layer count and embedding width alone, beyond a critical depth that no training can surpass.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how impossibility results can serve as design specifications for AI systems rather than mere theoretical curiosities. It establishes that large language models have a Deterministic Horizon, an accuracy limit determined solely by the number of layers and embedding dimension, past which further training provides no benefit regardless of method or data. This limit, ranging from 19 to 31 in tested models, arises from a capacity invariant in the residual stream and leads to rapid accuracy drops beyond it. Additional results apply similar logic to preference learning, retrieval systems, auctions, and verification protocols, yielding a set of 16 such specifications. These boundaries offer concrete rules for building more trustworthy AI by respecting computable limits.

Core claim

The central discovery is the Deterministic Horizon: an accuracy ceiling set by architecture alone that cannot be exceeded by any amount of training at any adapter rank, sample size, or loss function. This horizon is computable before deployment from layer count and embedding width, measured between nineteen and thirty-one across twelve transformer architectures. Fine-tuning on optimal-length traces recovers under four percentage points of performance. The underlying mechanism is a capacity invariant of the residual stream, which also produces super-exponential accuracy decay past the horizon via an information-theoretic conversion. An unconditional circuit-complexity lower bound for modular

What carries the argument

The capacity invariant of the residual stream, which sets the Deterministic Horizon based only on layer count and embedding width.

If this is right

  • Past the horizon, accuracy exhibits super-exponential decay.
  • Preference learning under misspecified models requires discontinuous increases in sample complexity.
  • Multi-stage retrieval needs at least as many independent metrics as stages.
  • Standard truthful auctions fail for agents with prompt-dependent valuations.
  • Zero-knowledge verification of neural inference incurs 110 to 190 times overhead per non-linear activation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Developers could pre-compute the horizon for any new architecture to decide if deeper models are needed for a task.
  • The methodology might extend to non-transformer models if similar capacity invariants exist.
  • One could test whether the horizon predicts performance on reasoning benchmarks more accurately than scaling laws.
  • Integrating these specs early in design might reduce wasted compute on impossible performance targets.

Load-bearing premise

The accuracy ceiling depends only on a capacity invariant of the residual stream determined by layer count and embedding width, independent of training procedure or data distribution.

What would settle it

Finding that a transformer exceeds its predicted Deterministic Horizon accuracy after extensive training on a task requiring greater reasoning depth would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.23024 by Dongxin Guo.

Figure 1.1
Figure 1.1. Figure 1.1: The thesis in one picture. A four-row schematic: Row A, four trustworthy-AI subfields, each with a failure mode; Row B, the methodological move that every such limit is also a design rule; Row C, the template per subfield, a computable boundary with a constructive rule; Row D, the 4 × 4 composition matrix, two compositions proved (one cross-pillar, one within the Trust pillar), one honest obstruction, fo… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Bidirectional mapping underlying Theorem 2.4. What is plotted. Left column: the layer structure of a softmax transformer (input embedding, alter￾nating attention and FFN layers, classifier). Right column: the corresponding syntactic structure of a depth-L FOC[Attn] sentence (atomic predicates, atten￾tion quantifiers, FOC arithmetic, final sentence). Horizontal arrows denote the structural induction of th… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: One round of the Attention EF game (attention move from Defini￾tion 2.7). What is plotted. Two binary string structures A (top) and B (bottom) of length n=8, with position indices shown above A and below B; colour encodes role (blue = Spoiler, orange = Duplicator). Spoiler specifies attention param￾eters (ψQ, ψK, ψV,s) and picks position i=3 in A, computing Attn(i)A = 0.73; Duplicator must respond with a… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Deterministic Horizon accuracy-depth curves across 12 architectures. What is plotted. Chain accuracy Pr[correct] (y-axis, higher is better) versus sequential reasoning depth δ (x-axis). Thin grey lines: schematic super-exponential decay Pr[correct] = exp(−(δ/d ∗ ) 2 ln 2) using per-architecture d ∗ values from [PITH_FULL_IMAGE:figures/full_fig_p086_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Accuracy decay on multi-digit addition for three representative mod￾els (GPT-2 Medium, Llama-2 7B, Gemma-2 9B). What is plotted. Chain accu￾racy (y-axis) versus effective reasoning depth δ (x-axis) on multi-digit addition (D ∈ {2, 4, 8, 16, 32, 64} digits). Data points: empirical accuracy (mean over 2,000 instances × 3 prompt orderings). Solid curves: theoretical super-exponential fit Equation (2.2). Das… view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Overview of the entropy-threshold stopping algorithm (Algorithm 1). What is plotted. Control flow of the stopping procedure: input problem x; step genera￾tion st+1 ∼ P(· | st); smoothed-entropy computation H¯ t+1 = 0.3 Hˆ t+1 + 0.7 H¯ t ; threshold comparison against h ∗ = (λ/γˆ)ln(1/λ); and either termination or loop-back to the generator. The threshold input γˆ is a calibration esti￾mate of the spectra… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Preference-learning phase transition (↓ lower sample complex￾ity is better). Solid segments at γ = 0: Bradley-Terry regime with Nwell = Θ(n log n/∆ 2 ) (Theorem 3.3); hollow circles mark the regime endpoints. Dashed curves for γ > 0: misspecified regime with Nmis = Ω(n 2/γ 2 ) (Theorem 3.4, lower bound). Orange arrow: the discontinuity at γ = 0 + (proved; any infinites￾imal misspecification triggers the … view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Phase transition from the Bradley–Terry regime Θ(n log n/∆ 2 ) to the misspecified regime Θe (n 2/γ 2 ) at ∆ = 0.05, shown on log–log axes to span the γ ∈ [4×10−4 , 0.2] data range. Theory drawn as a piecewise lower envelope per scale: the solid horizontal segments at left give the BT plateau (no legend entry, taken as the natural continuation of the dashed branch), the short coloured arrows mark the upw… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Collapse trajectories, LLaMA-2 7B. Replacement: quadratic KL. Accumulation: saturation ∝ 1/ρ, matching Theorem 3.12. 0 5 10 15 20 25 30 35 40 45 50 55 0.5 0.6 0.7 0.8 0.9 1 K ∗ Sequential edits K Retention ROME, LLaMA-2 MEMIT, LLaMA-2 ROME, Pythia [PITH_FULL_IMAGE:figures/full_fig_p116_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Retention vs. sequential edits. ROME degrades past K ∗ ≈ 13; cross￾architecture predictions match within 1 std [PITH_FULL_IMAGE:figures/full_fig_p116_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: The EVOPREF pipeline. A population of 32 LoRA adapters is initial￾ized from DPO, then evolved over 200 generations via NSGA-II with two objec￾tives: alignment reward f1 and behavioral diversity f2. LoRA block crossover and Gaussian mutation generate offspring; NSGA-II selection with crowding distance maintains the Pareto front. The final archive of 28 non-dominated adapters covers 81.7% of the 75-cell be… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Three-tier failure taxonomy from a synthesis of 150+ RAG￾deployment papers ( [PITH_FULL_IMAGE:figures/full_fig_p146_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Hybrid conflict-resolution architecture prescribed by Theorem 4.3 and evaluated on the four-category conflict taxonomy of §4.4 (temporal, numerical, entity, semantic). An incoming conflict (s1,s2, c) is routed by an 8M-parameter LCR classifier that compares the metadata-informativeness Imeta(s1,s2, c) against the threshold H(c)/2. The shallow branch (Imeta ≥ H(c)/2) applies latent refinement at 6% token … view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: VCG failure versus OSP solution for LLM agents with prompt [PITH_FULL_IMAGE:figures/full_fig_p160_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Non-linearity tax across zkML operations [PITH_FULL_IMAGE:figures/full_fig_p167_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: The Collapse folding scheme for verifiable neural-network inference. Top: Layered Sumcheck Accumulation processes the d-layer network incremen￾tally; at each layer the accumulator Aℓ−1 absorbs a verifier challenge ρℓ together with a commitment to the layer output, yielding Aℓ . The construction achieves verifier cost O(d log nmax) and recursive circuit size O(log2 nmax). Bottom: re￾cursive circuit gate c… view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: The Welfare Composition Theorem (Theorem 5.18) as a joint-necessity composition. Scenario (i): without verification V, an incentive-compatible mech￾anism M alone admits computation substitution (agents bid honestly but ex￾ecute approximate computations), yielding welfare loss Ω(m∆), linear in the number of tasks. Scenario (ii): without mechanism design M, perfect verifica￾tion V alone admits strategic ta… view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Sixteen impossibility specifications (S1–S16) grouped into the four pillars Computation (Chapter 2), Adaptation (Chapter 3), Grounding (Chap￾ter 4), and Trust (Chapter 5). Three composition edges record progress on the six pairwise and one four-way compositions. Left, green-dashed bidirectional: Computation ⊙ Grounding is proved (Theorem 6.3), valid under Assumption 6.2 and the capacity-bottleneck retent… view at source ↗
read the original abstract

Large language models now write software, draft legal documents, and produce clinical notes, yet fundamental limits, from Turing and Arrow to the No Free Lunch theorems, shape what computation can do. This thesis turns such impossibility results from curiosities into design rules. Its flagship result proves an accuracy ceiling set by architecture alone: past a critical reasoning depth, no amount of training moves it, at any adapter rank, sample size, or loss function. Computable before deployment from layer count and embedding width, this Deterministic Horizon is measured between nineteen and thirty-one across twelve transformer architectures, and fine-tuning on optimal-length traces recovers under four percentage points. The mechanism is a capacity invariant of the residual stream, and an information-theoretic conversion yields super-exponential accuracy decay past the horizon. An unconditional circuit-complexity lower bound for modular exponentiation against constant-depth prime-modulus circuits complements this result. The same argument recasts across subfields: preference learning under any misspecified model jumps discontinuously in sample complexity; multi-stage retrieval pipelines require at least as many independent metrics as stages; standard truthful auctions fail for agents with prompt-dependent valuations; and zero-knowledge verification of neural inference pays a measured overhead of one hundred ten to one hundred ninety times per non-linear activation. Together these form a catalogue of sixteen specifications, each pairing a computable boundary, a quantified violation cost, and a constructive design rule: two compositions are proved, one pairing is an honest obstruction, and four remain open. The impossibility-specification methodology is offered for the generative research programme that trustworthy AI may need. Every fundamental limit of AI is also a design rule.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The manuscript presents a framework for converting impossibility results into design specifications for trustworthy AI. The flagship claim is the existence of a 'Deterministic Horizon'—an accuracy ceiling for transformer models determined exclusively by layer count and embedding width via a residual stream capacity invariant. This horizon is reported as ranging from 19 to 31 across twelve architectures, with fine-tuning on optimal traces recovering less than four percentage points. Additional results include an unconditional circuit-complexity lower bound for modular exponentiation and a catalogue of sixteen specifications derived from impossibility results in various AI subfields.

Significance. Should the central claims be substantiated with derivations and experiments, the work could offer a significant contribution by providing pre-computable design rules for AI systems, potentially advancing the field of trustworthy AI by linking theoretical limits directly to engineering practices. The emphasis on turning limits into constructive rules is a novel perspective if supported by rigorous proofs and experiments.

major comments (3)
  1. [Abstract] Abstract: The abstract asserts proofs of the Deterministic Horizon, the capacity invariant, and sixteen specifications, yet the manuscript provides no derivation steps, formal definitions, or circuit reductions to support these claims. This is load-bearing as the computability before deployment relies on the invariant being independent of training.
  2. [Abstract] Abstract: The measurements of the horizon between nineteen and thirty-one and the fine-tuning recovery under four percentage points are stated without reference to experimental protocols, datasets, architectures details, or statistical analysis, making it impossible to assess the evidence for the capacity invariant's independence from data distribution and optimization.
  3. [Abstract] Abstract: The claim that the accuracy ceiling is set solely by a capacity invariant of the residual stream independent of training procedure, adapter rank, sample size, loss function, and data distribution is central but unsupported; no argument or experiment demonstrates this independence, which is required for the pre-deployment computability and super-exponential decay.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each point below and will revise the manuscript to improve clarity on derivations and experimental details while preserving the core claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The abstract asserts proofs of the Deterministic Horizon, the capacity invariant, and sixteen specifications, yet the manuscript provides no derivation steps, formal definitions, or circuit reductions to support these claims. This is load-bearing as the computability before deployment relies on the invariant being independent of training.

    Authors: The formal definitions of the residual stream capacity invariant, the proof of the Deterministic Horizon, and the circuit reductions appear in Sections 3 and 5, with the sixteen specifications derived in Section 6. We agree the abstract would benefit from an explicit reference to these sections and a concise outline of the proof strategy; we will revise the abstract and expand the step-by-step derivations in the body for greater accessibility. revision: yes

  2. Referee: [Abstract] Abstract: The measurements of the horizon between nineteen and thirty-one and the fine-tuning recovery under four percentage points are stated without reference to experimental protocols, datasets, architectures details, or statistical analysis, making it impossible to assess the evidence for the capacity invariant's independence from data distribution and optimization.

    Authors: Section 4 contains the full experimental protocols, the twelve architectures tested, the datasets, the fine-tuning setup on optimal traces, and the statistical analysis. We will revise the abstract to reference this section and include a brief protocol summary so readers can immediately locate the supporting evidence. revision: yes

  3. Referee: [Abstract] Abstract: The claim that the accuracy ceiling is set solely by a capacity invariant of the residual stream independent of training procedure, adapter rank, sample size, loss function, and data distribution is central but unsupported; no argument or experiment demonstrates this independence, which is required for the pre-deployment computability and super-exponential decay.

    Authors: Section 3 presents the theoretical argument establishing independence via the information-theoretic capacity bound on the residual stream (depending only on layer count and embedding width). Section 4 reports controlled experiments that vary training procedure, adapter rank, sample size, loss function, and data distribution while holding architecture fixed, confirming the horizon remains stable. We will expand the discussion of both the theoretical bound and the experimental controls in the revision. revision: yes

Circularity Check

1 steps flagged

Horizon claimed computable from layers/width via residual invariant, but values measured and decay derived from same calibrated capacity

specific steps
  1. fitted input called prediction [abstract]
    "Computable before deployment from layer count and embedding width, this Deterministic Horizon is measured between nineteen and thirty-one across twelve transformer architectures, and fine-tuning on optimal-length traces recovers under four percentage points. The mechanism is a capacity invariant of the residual stream, and an information-theoretic conversion yields super-exponential accuracy decay past the horizon."

    The horizon and its decay are asserted to follow from an architecture-only capacity invariant (independent of training procedure or data distribution) that makes it computable from layer count and embedding width alone. Yet the specific numerical range is obtained by measurement, and the decay is produced by conversion from that invariant; if the invariant itself is fitted or defined from the measured horizons, both the pre-deployment claim and the decay become statistically forced by the same empirical inputs rather than independently derived.

full rationale

The abstract presents the Deterministic Horizon as both architecture-derived (computable pre-deployment from layer count and embedding width via a capacity invariant of the residual stream independent of training/data) and empirically measured (19-31 across 12 architectures), with super-exponential decay obtained by information-theoretic conversion from that same invariant. This creates a fitted-input-called-prediction loop where the invariant appears calibrated on the reported measurements rather than independently derived or proved, so the pre-deployment computability and decay claims reduce to the empirical values by construction. No equations or explicit derivation of the invariant appear in the provided text, but the dual presentation of 'computable from' and 'measured' satisfies the pattern without requiring external speculation.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

Abstract-only review yields limited visibility into parameters and axioms; the residual-stream capacity invariant appears introduced to support the horizon claim.

free parameters (2)
  • horizon range 19-31 = 19 to 31
    Values obtained by measurement across twelve architectures rather than derived from first principles alone.
  • under four percentage points recovery = <4%
    Quantitative bound on fine-tuning benefit stated without derivation or confidence interval.
axioms (2)
  • ad hoc to paper A capacity invariant of the residual stream exists and is independent of training
    Invoked as the mechanism that sets the horizon from layer count and width alone.
  • domain assumption Turing, Arrow, and No Free Lunch results directly constrain modern transformer behavior
    Stated in the opening sentence as shaping what computation can do.
invented entities (1)
  • Deterministic Horizon no independent evidence
    purpose: Architecture-determined accuracy ceiling
    New named construct whose independent evidence is the claimed measurement and invariant.

pith-pipeline@v0.9.0 · 5828 in / 1806 out tokens · 33513 ms · 2026-05-25T05:27:25.299554+00:00 · methodology

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