RCL_is_unique_functional_form_of_logic
plain-language theorem explainer
Any comparison operator on positive reals obeying the four Aristotelian constraints plus scale invariance and non-triviality forces its derived cost to satisfy multiplicative consistency through a bilinear combiner of the exact form P(u, v) = 2u + 2v + c uv. Researchers tracing the emergence of the Recognition Composition Law from logical primitives would cite this result when connecting Aristotelian structure to the d'Alembert setup. The proof is a direct term-mode extraction that first derives the full d'Alembert hypotheses from the logic law
Claim. Let $C$ be a comparison operator. If $C$ satisfies the laws of logic (identity, non-contradiction, excluded middle, scale invariance, route independence, and non-triviality), then there exist a function $P : ℝ → ℝ → ℝ$ and a constant $c ∈ ℝ$ such that the derived cost of $C$ satisfies multiplicative consistency with $P$ and $P(u, v) = 2u + 2v + c · uv$ for all real $u, v$.
background
The module formalizes logic as a functional equation. A ComparisonOperator is the type ℝ → ℝ → ℝ that assigns a real cost to any pair of positive quantities; the four Aristotelian constraints (identity, non-contradiction, excluded middle, route independence) together with scale invariance and non-triviality are packaged in the structure SatisfiesLawsOfLogic. The derived cost is obtained by fixing the second argument at the multiplicative identity, yielding a function on positive ratios. The local setting is the bridge from these logical axioms to the d'Alembert inevitability framework. The proof relies on the upstream theorem bilinear_family_forced, whose statement is: given a normalized symmetric cost F with multiplicative consistency via a symmetric quadratic polynomial P and non-triviality, the combiner P must be bilinear of the stated form.
proof idea
The proof is a term-mode extraction. It first applies laws_of_logic_imply_dalembert_hypotheses to C and the laws hypothesis to obtain normalization, consistency, polynomial form, symmetry, continuity, and non-triviality. It then feeds these directly into bilinear_family_forced, which returns the specific coefficients that fix P(u, v) = 2u + 2v + c uv together with the consistency property, and packages the pair (P, c) as the existential witness.
why it matters
This theorem shows that the Recognition Composition Law is the unique functional form compatible with the laws of logic under polynomial regularity. It supplies the logical justification for the RCL that is invoked by the downstream rcl_polynomial_closure_theorem on operative positive-ratio comparisons with finite pairwise polynomial closure. In the Recognition Science chain it supports the passage from T5 (J-uniqueness) through the RCL to the phi-ladder and the calibrated constants, citing the peer-reviewed d'Alembert Inevitability Theorem.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.
papers checked against this theorem (showing 9 of 9)
-
Hot-pressed cotton base and cover seal pet filter material in chamber
"A pet water filter cartridge, comprising a water filter cartridge carrier and a filter material (1), wherein the water filter cartridge carrier is composed of a non-woven cotton hot pressed base (2) and a non-woven cotton cover (3); the base (2) is provided with an accommodation chamber (4), the accommodation chamber (4) is provided with the filter material (1), the non-woven cotton cover (3) is sealed at an opening of the accommodation chamber (4)."
-
One model follows instructions on video and audio after training only on images
"PandaGPT combines the multimodal encoders from ImageBind and the large language models from Vicuna... Thanks to the strong capability of ImageBind in embedding data from different modalities into the same space, PandaGPT displays emergent, i.e. zero-shot, cross-modal behaviors"
-
Factorized models generate videos for unseen robot tasks
"we introduce RoboDreamer, an innovative approach for learning a compositional world model by factorizing the video generation. We leverage the natural compositionality of language to parse instructions into a set of lower-level primitives, which we condition a set of models on to generate videos."
-
AI-generated dialogues scale LLaMA past Vicuna
"Building upon UltraChat, we fine-tune a LLaMA model to create a powerful conversational model, UltraLLaMA. Our evaluations indicate that UltraLLaMA consistently outperforms other open-source models, including Vicuna."
-
Variance term stops collapse in self-supervised image learning
"the composite value is determined by a finite pairwise polynomial algebra"
-
Looped models match 12B LLMs with 1.4B params
"LoopLM yields reasoning traces more aligned with final outputs than explicit CoT"
-
GPT-4V and Gemini reach only 56-59% on expert multimodal benchmark
"MMMU focuses on advanced perception and reasoning with domain-specific knowledge, challenging models to perform tasks akin to those faced by experts."
-
Unique reciprocal cost on ratios forces balanced discrete ledgers
"Requiring coherent composition under multiplicative chaining together with normalization (F(1)=0) and quadratic calibration at unity yields the unique reciprocal cost J(x)=½(x+x⁻¹)−1"
-
MCP faces 16 security threats across four attacker types
"MCP introduces several innovations that extend beyond conventional function calling... bi-directional communication channels, dynamic discovery and schema negotiation..."