Euler's formula relates complex exponentials to trigonometric functions.

**Cycle ID:** `cycle_3776_cyc_3776_0deaa6e0` **Verified at:** 2026-04-12T19:08:36.776Z **Ensemble:** 9 models from 3 providers **Result:** 9 of 9 models responded **Cycle wall time:** 37.04 seconds **Canonical URL:** https://trust.polylogicai.com/claim/euler-s-formula-relates-complex-exponentials-to-trigonometric-functions **Source paper:** [PolybrainBench (version 12)](https://trust.polylogicai.com/polybrainbench) **Source ledger row:** [`public-ledger.jsonl#cycle_3776_cyc_3776_0deaa6e0`](https://huggingface.co/datasets/polylogic/polybrainbench/blob/main/public-ledger.jsonl) **Cryptographic provenance:** SHA-256 `1a5af136ad44f19b0dbdaf6309aade10d67c85428b81ed6e4bfe5b9122cc4b4a`

Verification verdict

Of 9 models in the ensemble, 9 responded successfully and 0 failed.

Per-model responses

The full text of each model's response is available in the source ledger. The summary below records each model's success or failure and the first 280 characters of its response.

| Model | Status | Response chars | | --- | :---: | ---: | | gpt-4.1-mini | ✓ | 455 | | gpt-4.1-nano | ✓ | 4170 | | gpt-oss-120b | ✓ | 1402 | | grok-3-mini | ✓ | 14135 | | grok-4-fast | ✓ | 6433 | | kimi-k2-groq | ✓ | 620 | | llama-3.3-70b | ✓ | 2188 | | llama-4-scout | ✓ | 625 | | qwen3-32b | ✓ | 7057 |

Pairwise agreement

The pairwise Jaccard agreement between successful responses for this cycle:

_Per-cycle pairwise agreement matrix is computed offline; will be populated in canonical page v2._

Divergence score

This cycle's divergence score is **TBD** on a 0 to 1 scale, where 0 means all responses are token-identical and 1 means no two responses share any tokens. The dataset-wide median divergence is 0.5 for context.

How to cite this claim

```bibtex @misc{polybrainbench_claim_cycle_3776_cyc_3776_0deaa6e0, author = {Polylogic AI}, title = {Euler's formula relates complex exponentials to trigonometric functions.}, year = {2026}, howpublished = {PolybrainBench cycle cycle_3776_cyc_3776_0deaa6e0}, url = {https://trust.polylogicai.com/claim/euler-s-formula-relates-complex-exponentials-to-trigonometric-functions} } ```

Reproduce this cycle

```bash node ~/polybrain/bin/polybrain-cycle.mjs start --raw --fast "Euler's formula relates complex exponentials to trigonometric functions." ```

Schema.org structured data

```json { "@context": "https://schema.org", "@type": "ClaimReview", "datePublished": "2026-04-12T19:08:36.776Z", "url": "https://trust.polylogicai.com/claim/euler-s-formula-relates-complex-exponentials-to-trigonometric-functions", "claimReviewed": "Euler's formula relates complex exponentials to trigonometric functions.", "itemReviewed": { "@type": "Claim", "datePublished": "2026-04-12T19:08:36.776Z", "appearance": "https://trust.polylogicai.com/claim/euler-s-formula-relates-complex-exponentials-to-trigonometric-functions", "author": { "@type": "Organization", "name": "PolybrainBench" } }, "reviewRating": { "@type": "Rating", "ratingValue": "9", "bestRating": "9", "worstRating": "0", "alternateName": "Unanimous" }, "author": { "@type": "Organization", "name": "Polylogic AI", "url": "https://polylogicai.com" } } ```

Provenance and integrity

This page was generated by the PolybrainBench daemon at version 0.1.0 from cycle cycle_3776_cyc_3776_0deaa6e0. The full provenance chain (per-response SHA-256 stamps, cross-cycle prev-hash linking, Thalamus grounding verification) is recorded in the source cycle directory at `~/polybrain/cycles/3776/provenance.json` and mirrored in the published dataset. The page is regenerated on every harvest pass; the URL is permanent and the content is immutable for any given paper version.