hierarchy_emergence_forces_phi
plain-language theorem explainer
A uniform scale ladder with additive closure at level 2 forces the scaling ratio to equal the golden ratio φ. Recognition Science derivations of the self-similar fixed point cite this as the unconditional bridge from zero-parameter multilevel composition to the unique admissible scale. The proof builds a GeometricScaleSequence, confirms closure via the locality theorem plus linear arithmetic, and invokes the uniqueness result for closed ratios.
Claim. Let $L$ be a uniform scale ladder: a sequence of positive real level sizes with uniform scaling ratio $σ > 1$. If the levels obey additive closure $L_2 = L_1 + L_0$, then $σ = φ$ where $φ = (1 + √5)/2$.
background
The HierarchyEmergence module shows that a zero-parameter comparison ledger with multilevel composition produces a minimal hierarchy and forces φ as the unique admissible scale. UniformScaleLadder is the structure of positive level sizes with a single uniform ratio σ > 1 between adjacent levels, extracted from composition without free parameters. Locality_forces_additive_composition establishes that additive closure at the second level implies the quadratic relation σ² = σ + 1. Closed_ratio_is_phi then identifies the unique positive solution as φ.
proof idea
The tactic proof constructs a GeometricScaleSequence S from the ladder's ratio and positivity data. It proves S.isClosed by unfolding the closure predicate, applying locality_forces_additive_composition to recover the recurrence, and discharging the resulting arithmetic with nlinarith. The final step applies closed_ratio_is_phi to conclude the ratio equals φ.
why it matters
This theorem is the core unconditional bridge B1 inside HierarchyEmergence. It feeds directly into bridge_T5_T6 (the T5→T6 link in the forcing chain), realized_hierarchy_forces_phi, hierarchy_forced_gives_phi, and posting_extensivity_forces_phi. The result realizes the module's fourth step: minimal nondegenerate closure forces the Fibonacci recurrence, hence σ² = σ + 1 and σ = φ, aligning with T6 where φ is forced as the self-similar fixed point.
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papers checked against this theorem (showing 30 of 133)
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Lotion with oils and mixed zinc repels mosquitoes while blocking UV
"A topical mosquito repellent and sunscreen composition comprising: a lotion including active and inert ingredients, wherein the active ingredients include geraniol, clove oil, and rosemary oil and the inert ingredients include water, mineral oil, zinc oxide, beeswax, triethyl citrate, and lecithin; and wherein the zinc oxide comprises a mixture of different sized particles, wherein half of the zinc oxide particles have a size from 20-50 nm and half of the zinc oxide particles have a size of from 80-200 nm."
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Duality defects forbid trivially gapped phases in 3+1D
"The duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides."
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Paired cams and springs control livestock brush pressure and reset
"the spring mechanism (104) exerts a defined resistance on the brush device (102) for its controlled movement... while also exerts a controlled restoring force"
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Open 70B medical LLM beats GPT-3.5 and nears GPT-4
"extends pretraining on a comprehensively curated medical corpus, including selected PubMed articles, abstracts, and internationally-recognized medical guidelines"
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Video models copy training cases rather than learn physics laws
"scaling alone is insufficient for video generation models to uncover fundamental physical laws"
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Multi-view diffusion creates 3D scenes from few photos in one minute
"CAT3D can create entire 3D scenes in as little as one minute, and outperforms existing methods for single image and few-view 3D scene creation"
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Iterative SFT-RL cycles teach 7B models to reason visually
"alternating between SFT and RL ultimately results in significant performance improvements after a few iterations"
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Millions of qubits become reachable with semiconductor upgrades and HPC links
"We show that orders of magnitude enhancement in performance could be obtained by a combination of hardware improvements and tight quantum-HPC integration."
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Only select heads need full caches for long-context LLMs
"Our method significantly reduces long-context inference memory by up to 2.55× for MHA and 1.67× for GQA models while speeding up decoding by up to 2.18× and 1.50×"
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Routine slides learn from stains and transcriptomics seen only at training
"In all settings, TriDeNT outperforms other state-of-the-art methods in downstream tasks, with observed improvements of up to 101 %."
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Smaller models plus tree search beat larger ones on math
"Our findings suggest that scaling inference compute with inference strategies can be more computationally efficient than scaling model parameters. Additionally, smaller models combined with advanced inference algorithms offer Pareto-optimal trade-offs in cost and performance."
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Matrix states and dynamic recurrence improve RWKV performance
"We introduce a new multilingual corpus with 1.12 trillion tokens and a fast tokenizer based on greedy matching for enhanced multilinguality."
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Speculative decoding speeds up transformers 2-3x
"using speculative execution and a novel sampling method, we can make exact decoding from the large models faster, by running them in parallel on the outputs of the approximation models"
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GPT-4 feedback on 250k conversations aligns strong LLaMA chat models
"Built upon UltraFeedback, we align a LLaMA-based model by best-of-n sampling and reinforcement learning, demonstrating its exceptional performance on chat benchmarks"
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Open-source web agents match proprietary performance
"Our approach involves generating novel, high-uncertainty tasks through structured sampling and information obfuscation, RFT cold start, and an efficient agentic RL training algorithm, Duplicating Sampling Policy Optimization (DUPO)."
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Online RL boosts LLM self-correction up to 15.6% with self-generated data
"With Gemini 1.0 Pro and 1.5 Flash models, we find that SCoRe achieves state-of-the-art self-correction performance, improving the base models' self-correction by 15.6% and 9.1% respectively on MATH and HumanEval."
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Visual generation emerges from improved LLM understanding
"visual generation ability emerges as a natural byproduct of improved visual understanding, and can be unlocked efficiently with a small amount of generation data"
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Hopfield update rule matches transformer attention
"We generalize this energy function to continuous-valued patterns while keeping the properties of the modern Hopfield networks like the exponential storage capacity and the extremely fast convergence"
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MLLMs score 45% max on Holmes-style video reasoning
"Gemini-2.5-Pro, achieves an accuracy of only 45%, with most models scoring below 40%"
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Text labels alone segment images at test time
"This allows LSeg to generalize to previously unseen categories at test time, without retraining or even requiring a single additional training sample."
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RL post-training reaches SOTA video grounding with 2.5K examples
"Time-R1 achieves state-of-the-art performance across multiple downstream datasets using only 2.5K training data"
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EvoSkill evolves agent skills to boost accuracy 7-12% without retraining
"skills evolved from SealQA transfers zero-shot to BrowseComp, improving accuracy by 5.3%."
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Few examples unlock strong math reasoning in LLMs
"LIMO achieves 63.3% on AIME24 and 95.6% on MATH500 with only 1% of prior training data"
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Diffusion LLMs reach 1109 tokens per second on H100
"achieve state-of-the-art throughputs of 1109 tokens/sec and 737 tokens/sec... outperform speed-optimized frontier models by up to 10x"
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Text-only training makes MLLMs universal embedders
"By leveraging MLLMs with prompts, E5-V effectively bridges the modality gap between different types of inputs"
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Library makes language model evaluations reproducible
"We describe the features of the library as well as case studies in which the library has been used to alleviate these methodological concerns."
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Pretrained decoder model hits near-SOTA zero-shot forecasting
"we design a time-series foundation model for forecasting whose out-of-the-box zero-shot performance on a variety of public datasets comes close to the accuracy of state-of-the-art supervised forecasting models"
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Benchmark tests AI on practical biology research tasks
"we expect that an AI system that can achieve consistently high scores on the more difficult LAB-Bench tasks would serve as a useful assistant for researchers in areas such as literature search and molecular cloning"
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1B-active MoE model beats 13B dense models
"analyze routing in our model showing high specialization"
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Video model matches leaders after $200k training
"We detail all techniques that contribute to this efficiency breakthrough, including data curation, model architecture, training strategy, and system optimization."