feat(OEIS): README.md

FormalConjectures/OEIS/README.md

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+# OEIS Formalization Guidelines
+
+This directory contains formalizations of conjectures and theorems associated with integer sequences from the [On-Line Encyclopedia of Integer Sequences (OEIS)](https://oeis.org/).
+
+When contributing to or reviewing formalizations in this directory, please follow the specific conventions outlined below in addition to the general repository guidelines in [`AGENTS.md`](../../AGENTS.md).
+
+## File Naming & Work-in-Progress (WIP) Status
+
+- **Naming**: Files must be named after their OEIS number **without leading zeros** (e.g., `56777.lean`, `308734.lean`).
+
+## Sequence Definition Naming
+
+The primary defining sequence in each OEIS file must be named `a` (e.g., `def a (n : ℕ) : ℕ := ...`). Do not name the function after the OEIS number (`A224515`) or use uppercase letters. Secondary auxiliary sequences (such as denominators) may be named `b`, `c`, etc.
+
+## Namespaces
+
+Every file must enclose all its declarations within a dedicated namespace matching `OeisA[Number]` (without leading zeros). This namespace should open immediately after the imports and module docstring, and close at the very end of the file.
+
+```lean
+namespace OeisA308734
+
+-- definitions, helper lemmas, term theorems, and main conjecture
+
+end OeisA308734
+```
+
+## Module Docstrings & References
+
+Every file must include a descriptive module docstring (`/-! ... -/`) immediately following the imports.
+- **Content**: The docstring should clearly explain the sequence definition and the statement of the conjecture or theorem being formalized.
+- **References**: It must conclude with a standardized `*References:*` (or `*Reference:*` for a single item) section containing a Markdown link to the official OEIS page, along with any other papers or articles necessary to formulate the problem.
+
+```lean
+/-!
+# Four-square conjecture with powers of 2, 3, and 5
+
+Any integer $n > 1$ can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$
+where $a, b, c, d, x, y$ are nonnegative integers.
+
+*References:*
+- [A308734](https://oeis.org/A308734)
+- Z.-W. Sun, "Refining Lagrange's four-square theorem," *J. Number Theory* **175** (2017), 167-190.
+-/
+```
+
+## Main Theorem Docstrings
+
+The main problem or conjecture (typically the last theorem in the file) must have a dedicated docstring (`/-- ... -/`).
+- **Verbatim Citation**: The docstring must cite the conjecture from OEIS verbatim.
+- **Proof Attribution**: For solved problems where an AI-driven formal proof was found, the bottom of the docstring must include the following attribution line:
+  `A formal proof has been found with the methods described in [arxiv/2605.22763](https://arxiv.org/abs/2605.22763).`
+
+```lean
+/--
+Any integer $n > 1$ can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$
+where $a, b, c, d, x, y$ are nonnegative integers.
+
+A formal proof has been found with the methods described in [arxiv/2605.22763](https://arxiv.org/abs/2605.22763).
+-/
+@[category research solved, AMS 11, formal_proof using formal_conjectures at ...]
+theorem target_theorem_0 : ...
+```
+
+## Term Theorems (`category test`)
+
+To ensure the formalized definition behaves correctly and matches the official OEIS sequence, every file **must include term theorems verifying the first few values of the sequence** (typically the first 5 values).
+
+- **Official Alignment**: Verify the starting index ($n=0, 1, 2, \dots$) and exact initial values against the official OEIS `b-file` (`https://oeis.org/A[padded_number]/b[padded_number].txt`).
+- **Attributes**: Every term theorem must be tagged with `@[category test, AMS 11]` (or another appropriate AMS subject).
+- **Computable Definitions**: If the sequence definition is kernel-computable, prove the term theorems using `by rfl`, `by decide`, `by norm_num`, or by unfolding the definition.
+- **Noncomputable Definitions**: For complex or `noncomputable` definitions where kernel evaluation is not possible:
+  - Use appropriate helper lemmas to establish values rigorously (e.g., `csInf_eq_of_forall_ge_of_forall_gt_exists_lt` for `sInf`-based definitions, or `Int.floor_eq_iff` for real number bounds).
+
+```lean
+@[category test, AMS 11]
+lemma test_a_0 : a 0 = 1 := by rfl
+
+@[category test, AMS 11]
+lemma test_a_1 : a 1 = 1 := by rfl
+```