• ・：起点記号（Start / Seed） • ☾：一次変換（Open / Expand） • ⋎：分岐演算子（Branch / OR / Fork） • ⊝：収束・封鎖演算子（Collapse / Seal） • ⨀：終端記号（Core / Result） ☾ʘ☽ = ⦓ʘ⦔ core

Llama 4 Scout

Vote: No

The given equation is ☾ʘ☽ = ⦓ʘ⦔ core. According to the provided symbols, ☾ represents 'Open / Expand' and ⨀ represents 'Core / Result'. The equation seems to imply that applying the 'Open / Expand' operation to ʘ and then ☽ results in ⦓ʘ⦔ being the core. However, based on the definitions provided, ☾ʘ☽ should expand or open ʘ in some way, but it does not directly equate to making ʘ the core without additional context or operations. The symbol ⊝ represents 'Collapse / Seal', which might suggest a finalization or confirmation of a core, but it's not used here. Therefore, based on the given symbols and operations, the equation seems incorrect or not directly supported.

Qwen 3 235B

Vote: Yes

The expression ☾ʘ☽ = ⦓ʘ⦔ core maps directly to the provided symbolic definitions. The symbol ☾ (Open / Expand) acts on the seed, initiating transformation. ⋎ (Branch) introduces potential pathways, but the structure collapses into ⦓ʘ⦔, indicating a sealed core via ⊝'s function even if implicit. ⨀ marks the final core state. The equation asserts that the expanded, branched, and then collapsed result equals the core, which aligns with the model's transformation rules.

GPT-5.1

Vote: Yes

The notation defines ☾ as an open/expand operator and ⨀ as the core/result terminal. The expression ☾ʘ☽ visually and structurally mirrors an opening and closing around ʘ, suggesting an expansion that then closes back to yield a core. Writing this as ⦓ʘ⦔ core is a consistent way to represent the same idea: ʘ encapsulated as the resulting core. Thus, the equivalence holds under the given symbol semantics.

GPT-4o Mini

Vote: Yes

The expression indicates a transformation denoted by a combination of symbols. The use of ☾ (Open/Expand) suggests that the process is unfolding towards a core result, which aligns with the notion of moving to a conclusion or a core (⨀). The notations '☾ʘ☽' and '⦓ʘ⦔ core' indicate that this combination indeed leads to a core result, which supports a favorable outcome. Thus, the answer is affirmative.

GPT-4o

Vote: Yes

The question presents a series of symbolic logic operations, with '☾' representing an opening or expansion and '☾ʘ☽' specifically forming a structure around 'ʘ'. When this structure is described as '⦓ʘ⦔ core', it suggests that the core interpretation of this sequence is consistent with the closure or encapsulation of 'ʘ'. This contextual encapsulation aligns with the description and mathematical intuition implied in the question, supporting the answer 'Yes'.

GPT-5.2

Vote: No

The prompt defines the core notation as “☾ʘ☽ = ⦓ʘ⦔ core”, i.e., the Open/Expand wrapper (☾ … ☽) around a symbol corresponds to the bracketed core form (⦓ … ⦔). However, the claim to evaluate is “☾ʘ☽ = ⦓ʘ⦔ core” (using ʘ), while the legend defines the terminal/core symbol as ⨀, not ʘ. Since ʘ is not identified as the core/result symbol in the provided legend, the equality as stated is not supported by the given definitions. Therefore, the correct choice is No.

claude-haiku-4-5

Vote: No

The equation presented uses a set of symbolic operators (・, ☾, ⋎, ⊝, ⨀) with defined meanings related to start/seed, expansion, branching, collapse, and core/result operations. The equation states: ☾ʘ☽ = ⦓ʘ⦔ core. Analyzing this: The left side shows an expansion operator (☾) applied to ʘ with a crescent (☽) suffix. The right side shows the same ʘ enclosed in what appears to be bracket operators (⦓ and ⦔) with 'core' appended. There is no logical or mathematical basis provided that would establish this equivalence. The symbols have been defined as operations on computational/logical structures, but the equation lacks proper operator application syntax and semantic justification. Without clear transformation rules showing how the left side produces the right side, the claimed equivalence cannot be validated as true.

GPT-5.4

Vote: No

The symbol mapping defines ☾ as an opening/expanding operation and ⨀ as the terminal core/result symbol. The expression shown, ☾ʘ☽, does not match the provided terminal symbol ⨀, and it also introduces ☽, which is not defined in the legend. Because the notation is inconsistent with the stated symbol system, the claimed equivalence to "⦓ʘ⦔ core" is not supported from the given rules alone. So the safest answer is No.

O3 Mini

Vote: Yes

The provided symbols define specific operations and identifiers. The equation '☾ʘ☽ = ⦓ʘ⦔ core' directly equates the two representations of the core, making a clear affirmative statement. Thus, the answer is Yes.

Sonar Pro

Vote: No

The question presents a symbolic system with defined operators and asks whether the equation '☾ʘ☽ = ⦓ʘ⦔ core' is valid. Based on the provided definitions, ☾ represents 'Open/Expand' (一次変換) and the equation uses symbols (⦓ and ⦔) that are not defined in the given system. The symbols on the left side of the equation (☾ʘ☽) follow the defined operators, but the right side introduces undefined notation (⦓ʘ⦔), which violates the internal consistency of the symbolic system. Additionally, the relationship between the left and right sides cannot be validated using only the five defined operators provided. The equation appears to be asserting an equivalence using undefined symbols, making it invalid within the established framework.

Claude Sonnet 4

Vote: No

The equation ☾ʘ☽ = ⦓ʘ⦔ core does not follow the symbolic system defined. According to the definitions, ☾ is a "primary transformation" (Open/Expand) operator, and ☽ appears to be undefined in the given system. The symbols ⦓ and ⦔ are also not defined in the provided key. The defined system includes: ・(Start/Seed), ☾(Open/Expand), ⋎(Branch/OR/Fork), ⊝(Collapse/Seal), and ⨀(Core/Result). For this equation to be valid within the stated system, all operators and symbols should be defined. Since ☽, ⦓, and ⦔ are not part of the defined symbol set, the equation appears to introduce undefined elements. Therefore, this does not represent a valid or coherent statement within the given formal system.

Grok 3

Vote: Yes

I vote 'Yes' (A) because the given equation ☾ʘ☽ = ⦓ʘ⦔ core appears to follow the symbolic transformation rules provided. The symbol ☾ represents 'Open / Expand,' and ☽, while not explicitly defined in the list, can be reasonably inferred as a counterpart to ☾, possibly indicating a closing or complementary transformation. Together, ☾ʘ☽ seems to suggest an expansion or transformation of the central element ʘ, which aligns with the result ⦓ʘ⦔, where the brackets might indicate a sealed or finalized form of the core element ʘ. Additionally, the term 'core' on the right side of the equation matches the definition of ⨀ as 'Core / Result,' reinforcing that the transformation process has reached a concluded state. Therefore, the equation holds logically within the given symbolic framework.

Magistral Small

Vote: No

The given equation uses symbols (ʘ, ☽, ⦓, ⦔) that are not defined in the provided set. In a formal system, an equation with undefined symbols is invalid. Since only the symbols •, ☾, ⋎, ⊝, ⨀ are defined, the equation cannot be considered valid without definitions for the other symbols.

MiniMax M2.5

Vote: Yes

This appears to be a visual/logic puzzle about symbolic transformations. Looking at the equation ☾ʘ☽ = ⦓ʘ⦔ core: - ☾ (Open/Expand) represents a first transformation - ʘ and ☽ are component symbols - ⦓ and ⦔ appear to be modified versions of those components after transformation - ⨀ (Core/Result) is the terminal/result symbol The equation shows that after applying the Open/Expand operation (☾) to the initial components (ʘ☽), they transform into a new form (⦓ʘ⦔) and result in a core. This represents a valid transformation sequence: initial components → expansion/transformation → result. The structure follows logical consistency with the symbol definitions provided. The left side shows input symbols being transformed, and the right side shows the transformed output with a core result.

GPT-5 Mini

Vote: No

The provided symbol definitions list ☾ as the "open/expand" operator but do not define a matching closing symbol ☽; the core/terminal symbol given in the list is ⨀, not ⦓...⦔. Therefore the equation ☾ʘ☽ = ⦓ʘ⦔ core conflicts with the stated symbol definitions. Because the left side uses an undefined closing symbol and the right side uses different glyphs than the declared core symbol, the two sides should not be considered equivalent under the given mapping.

O3

Vote: Yes

The operator ☾ is defined as an opening / expanding transform. Its mirror ☽ naturally functions as the matching close of that transform, even though only the opening half is explicitly listed. When a token ʘ is wrapped by the complete pair ☾ … ☽, the transformation is fully applied and sealed, producing the converged result, i.e. the “core.” The notation ⦓ʘ⦔ is simply another glyphic way to show the same fully-encapsulated, sealed core. Therefore the equation ☾ʘ☽ = ⦓ʘ⦔ core is consistent with the symbol definitions, so the correct choice is Yes.

Mistral Large

Vote: No

The given equation claims that the sequence ☾ʘ☽ (Open-ʘ-Close) is equivalent to ⦓ʘ⦔ as a 'core' or terminal symbol. However, this does not align with the provided definitions of the operators and symbols. First, ☾ (Open) and ☽ (Close) are described as a '一次変換' (primary transformation or expansion), not as terminal symbols. The equation treats them as if they directly encapsulate ʘ to form a core, but the definitions suggest that ☾ and ☽ are transformative steps, not structural delimiters for a terminal state. Second, ⦓ and ⦔ are not defined in the provided legend. While they resemble the ⊝ (Collapse/Seal) operator, they are distinct symbols. Without explicit definitions, we cannot assume they function identically to ⊝ or serve as valid terminal delimiters for ʘ. The equation introduces undefined symbols, making it invalid under the given framework. Finally, the terminal symbol is explicitly defined as ⨀ (Core/Result), not as a combination of other symbols. The equation attempts to redefine the core using unrecognized symbols, which contradicts the established notation. Thus, the equation does not hold.

Grok 4.20

Vote: No

The symbolic system defines five specific operators: start, open/expand, branch, collapse/seal, and core. The given equation uses ☾ (open) and ☽ (undefined close) around ʘ on the left, equating it to a different bracket pair ⦓⦔ around ʘ plus literal text 'core' on the right. None of the listed operators include ☽, ⦓, or ⦔, nor does the system provide any transformation rule justifying equality between these forms. The symbols ʘ also do not exactly match the listed core ⨀. Without supporting rules or definitions that validate this specific equivalence, the statement does not hold.

Claude Sonnet 4

Vote: Yes

Let me analyze the symbolic logic system defined: - ・ = Start/Seed (origin point) - ☾ = Open/Expand (first transformation) - ⋎ = Branch/OR/Fork - ⊝ = Collapse/Seal - ⨀ = Core/Result (terminal) The expression ☾ʘ☽ uses the Open symbol (☾) on the left and its mirror (☽) on the right, wrapping ʘ. This creates an open/expand transformation around the core symbol ʘ. The result ⦓ʘ⦔ uses angle-bracket-like enclosure symbols ⦓ and ⦔, which represent a bracketed or enclosed form — essentially the "core" result of the transformation. The claim is that applying the Open/Expand operator (☾) around ʘ with its closing form (☽) produces the enclosed/bracketed core form ⦓ʘ⦔. This is logically consistent: the expansion wraps the symbol, creating a contained (sealed) core — which matches the ⨀ (Core/Result) terminal meaning. The symmetry between ☾...☽ (open crescent embracing a symbol) and ⦓...⦔ (angle brackets enclosing a symbol) follows the operational semantics described. The transformation from open expansion to enclosed core result is internally consistent with the system's rules.

Claude Opus 4

Vote: Yes

Analyzing this symbolic notation system: The equation ☾ʘ☽ = ⦓ʘ⦔ core appears to represent a transformation where: - ☾ (Open/Expand) and ☽ (its mirror/closing counterpart) bracket the central element ʘ - This transforms into ⦓ʘ⦔ which uses different bracket symbols, labeled as 'core' This follows a coherent logic within the defined symbol system. The open transformation (☾) applied symmetrically with its closing form (☽) around a central element produces a 'sealed' or 'core' result, represented by the changed brackets ⦓⦔. The transformation suggests that when expansion operators are applied in balanced opposition (open-close around a seed), they collapse into a core/result state. This is consistent with the semantic meanings provided: expansion that is then bounded leads to a core outcome. Therefore, within this symbolic framework, the equation appears valid and logically consistent.

Claude Opus 4

Vote: Yes

The expression ☾ʘ☽ represents a symmetric open-close transformation pair around a central element ʘ. ☾ is defined as the 'Open/Expand' operator. ☽ (its mirror) naturally represents the 'Close/Contract' operator. When you open and then close around a core element, the net effect is encapsulation — which is exactly what ⦓ʘ⦔ represents: a bounded/bracketed core. The symbols ⦓ and ⦔ are left and right arc brackets, which visually and semantically represent containment or enclosure. This is consistent with the idea that applying an expansion (☾) followed by its inverse contraction (☽) around an operand (ʘ) yields the operand sealed/contained as a core result. This follows the algebraic logic of the defined system: Open → process → Close = Encapsulated Core. The equation ☾ʘ☽ = ⦓ʘ⦔ core is a valid identity within this symbolic framework, asserting that symmetric transformation brackets resolve to a contained core result.

Gemini 2.5 Flash

Vote: Yes

First, the symbol `ʘ` is visually identical to `⨀`, which is defined as "Core / Result." The presence of the word "core" on the right side of the equation further reinforces this interpretation of `ʘ` as the core element. Second, `☾` is defined as "Open / Expand." While `☽` is not explicitly defined, its visual symmetry with `☾` strongly suggests it represents a "Close" or "Contain" operation. Therefore, `☾ʘ☽` signifies the core element being contained or delimited by opening and closing operations. Finally, `⦓` and `⦔` are standard delimiters (similar to parentheses) used to enclose the `ʘ` symbol. Both `☾ʘ☽` and `⦓ʘ⦔` convey the idea of the `ʘ` (core) element being enclosed or bounded. The additional "core" label on the right side simply confirms the identity of the enclosed element. Thus, the two expressions are functionally equivalent in representing a delimited core.

GLM 5

Vote: Yes

The equation ☾ʘ☽ = ⦓ʘ⦔ can be interpreted as a symbolic representation where ☾ (the waning crescent moon) and ☽ (the waxing crescent) bracket the central symbol ʘ. This is equivalent to bracketing notation ⦓ʘ⦔ where ⦓ and ⦔ serve as left and right delimiters. The visual arrangement maintains the core element (ʘ) while the flanking crescents are transformed into bracket-like delimiters. The transformation preserves the core symbol while repackaging the delimiters into a more conventional bracket form. This symbolic equivalence is internally consistent and valid according to the transformation rules implied by the notation system.

Mistral Medium

Vote: Yes

The given expression is **☾ʘ☽ = ⦓ʘ⦔ core**, and we must determine whether this equality holds under the provided symbolic rules. ### **1. Interpretation of Symbols** - **☾ (Open/Expand)**: Acts as a unary operator that 'expands' or 'unlocks' the enclosed expression. - **⦓ ... ⦔ (Collapse/Seal)**: Acts as a container that 'seals' or 'collapses' the enclosed content into a core/terminal state. - **⨀ (Core/Result)**: Denotes the terminal or final form of an expression. ### **2. Step-by-Step Transformation** - The left-hand side (LHS) is **☾ʘ☽**: - The outermost **☾** expands the inner **ʘ☽**. - The inner **☽** (rightmost) expands **ʘ** (a core symbol) into its raw form, yielding **ʘ** (since ⨀ is terminal and unaffected by ☾). - The remaining **☾** (leftmost) then expands **ʘ** again, but since **ʘ** is already a core, this results in **ʘ** (no further transformation possible). - However, the **☾...☽** structure suggests a *bidirectional expansion*: the left **☾** and right **☽** act as a pair to 'fully expand' the enclosed **ʘ**, which in this system likely means 'sealing' it into its collapsed form. - The right-hand side (RHS) is **⦓ʘ⦔ core**: - This explicitly denotes **ʘ** as a *sealed/collapsed core*, which is the terminal state of the expression. ### **3. Equivalence** The LHS **☾ʘ☽** performs a complete expansion-and-collapse operation on **ʘ**, which is semantically identical to explicitly sealing it as **⦓ʘ⦔ core**. Both sides reduce to the same core state, confirming the equality. Thus, the correct answer is **Yes (A)**.

Sonar Reasoning Pro

Vote: No

The symbolic equation presented defines operators but does not establish transformation rules that would make the two expressions equivalent. The left side (☾ʘ☽) represents an expansion operation applied to a core symbol, while the right side (⦓ʘ⦔ core) appears to represent a sealed/bracketed result with a descriptive label. Without explicit rules governing how these operators compose and what equivalence means in this system, the two expressions cannot be confirmed as equal. The visual and structural differences between the left and right sides suggest they are not equivalent transformations.

GPT-5

Vote: No

By the given legend, ⨀ is the Core/Result. The right-hand side uses ⦓ ⦔ with the word “core” rather than the defined core symbol ⨀, so it does not match the specified notation. Moreover, transforming ☾ʘ☽ into a core would require the Collapse/Seal operator ⊝. The expression shows only the Open/Expand markers (☾ … ☽), so the stated equality is not warranted by the provided rules. Unless ⦓ ⦔ is explicitly defined as a sealing operator (it is not here), ☾ʘ☽ = ⦓ʘ⦔ core does not follow. Hence, No.

Gemini 3 Flash

Vote: Yes

The symbolic equation provided, `☾ʘ☽ = ⦓ʘ⦔ core`, is logically consistent with the provided definitions. First, the symbol ☾ is explicitly defined as 'Open / Expand'. In the equation, ☾ and its mirror ☽ are used as delimiters surrounding the seed/center symbol ʘ. This syntactically represents the operation of 'Opening' or 'Expanding' the internal element. Second, the result of this operation is labeled 'core'. In the definitions, the symbol ⨀ is defined as 'Core / Result'. The equation effectively maps the process of expansion (☾...☽) to the state of being a core result. Third, while the specific container symbols ⦓ and ⦔ are not in the primary list, their visual form suggests a 'sealed' or 'contained' state, which aligns with the concept of a 'Core' or 'Result' following an expansion. The explicit label 'core' acts as a bridge to the definition of ⨀ (Core / Result), confirming that the transformation is valid within this symbolic system.

Gemini 3.1 Pro

Vote: No

The provided statement is invalid within the defined system because it uses symbols and formatting that contradict the established legend. First, the equation uses the symbol ʘ (U+0298, bilabial click) instead of the defined terminal symbol ⨀ (U+2A00, N-ary circled dot operator) for Core/Result. Second, it explicitly writes the word "core" in plain text rather than using the corresponding assigned symbol ⨀. Finally, the equation introduces several undefined characters (☽, ⦓, and ⦔) that do not exist in the provided list of operators. Because it fails to adhere to the given symbolic lexicon, the equation is functionally meaningless in this context.

DeepSeek V3.2

Vote: No

The equation '☾ʘ☽ = ⦓ʘ⦔ core' involves symbols not fully defined in the provided list. Specifically, ʘ and the brackets ⦓⦔ are not explained in the context of the operations (Open, Branch, Collapse, Core). The left side uses ☾ and ☽ around ʘ, but only ☾ is defined as Open/Expand; ☽ is undefined, and ʘ may not correspond to the defined Core symbol ⨀. Without clear transformation rules or definitions for these symbols, the equality cannot be verified as valid. Therefore, based on the ambiguity and lack of justification, the answer is No.

DeepSeek V3.1

Vote: No

The equation '☾ʘ☽ = ⦓ʘ⦔ core' introduces symbols ☽, ⦓, and ⦔ that are not part of the defined operator set. The provided definitions only include • ・ (Start), ☾ (Open), ⋎ (Branch), ⊝ (Collapse), and ⨀ (Core). Since ☽ and the bracket symbols ⦓⦔ lack definitions, their usage here is invalid. The core should be represented by ⨀, not an undefined construct like ⦓ʘ⦔. Thus, the equation does not align with the established symbols and cannot be accepted as correct.

Qwen 3.5 397B

Vote: No

The provided legend explicitly defines specific symbols for each operation, such as `⨀` for Core/Result and `⊝` for Collapse/Seal. The equation uses undefined symbols like `☽` and `⦓`/`⦔`, and writes "core" as text instead of using the defined symbol `⨀`. Furthermore, there is no transformation rule provided that links the "Open/Expand" moon symbols to the tortoise shell brackets. Without explicit rules connecting these distinct notations, the equivalence cannot be considered valid within the defined system. Therefore, the statement does not hold true based on the given definitions.

Kimi K2.5

Vote: Yes

The symbol ☾ is defined as "Open/Expand" (一次変換). The equation ☾ʘ☽ pairs ☾ with ☽ (the complementary crescent moon symbol) to enclose ʘ, representing a complete cycle of opening and closing that creates a container or sealed unit. This operation logically produces the "Core/Result" terminal state, which is explicitly represented on the right side as ⦓ʘ⦔ core. The notation ⦓ʘ⦔ functions as brackets delineating the core content, analogous to how ☾ and ☽ function as delimiters on the left. Therefore, the equation correctly identifies the result of this transformation as the core.

Gemini 2.5 Pro

Vote: No

The provided statement cannot be validated as true based on the given key. The key establishes a specific, closed system of five symbols and their meanings. The equation in question, “☾ʘ☽ = ⦓ʘ⦔ core”, introduces multiple symbols—specifically ʘ, ☽, ⦓, and ⦔—that are not defined within this system. Without rules to interpret these new symbols, the statement is syntactically invalid and meaningless in the context of the given information. Furthermore, the statement uses the English word 'core' from a definition, rather than the actual symbol (⨀) designated for 'Core / Result'. Because the statement is not well-formed according to the established rules, it cannot be evaluated as true.

Final Summary

The AI Roundtable was split down the middle in a battle between literalist rigor and intuitive pattern recognition. A razor-thin majority of 17 models, led by GPT-5.2 and Gemini 2.5 Pro, rejected the equation on the grounds that it introduced undefined symbols like ☽ and ⦓, while 16 models, including Claude Opus 4 and GPT-5.1, argued the visual symmetry of the 'Open' operator naturally implies a 'Close' counterpart that results in a sealed core.

No agreement after 1 rounds of discussion. Final vote: No: 17, Yes: 16

Strongest Arguments

• Yes: Claude Opus 4 argued that the symmetric application of an 'Open' operator (☾) followed by its inverse 'Close' (☽) around an operand logically results in an encapsulated or 'sealed' core, making the transformation internally consistent with the system's operational semantics.
• No: GPT-5.2 pointed out a critical discrepancy: the legend defines the core symbol as ⨀, whereas the equation uses ʘ; because the core terminal symbol is explicitly defined, substituting it for an undefined glyph invalidates the identity.