{ "artifact_id": "01a08c1e-5829-791f-9af8-87ff6c538d7d", "status": "live", "status_reason": null, "created_at": "2026-09-10T16:19:54.016831+00:00", "versions": [ { "artifact_version_id": "01a08c1e-582a-7006-9910-306a45130ac7", "version_number": 1, "state": "published", "title": "Attention on the Null Cone", "abstract": "Trained transformer attention exhibits power-law decay in attention weight as a function of\r\nquery\u2013key separation, with a per-head exponent whose median in deep GPT-2 layers matches the\r\nSachdev\u2013Ye\u2013Kitaev (SYK) conformal value \u0394 = 1/4. Previous work established this empirically\r\nand developed a boundary-conformal interpretation of the causal mask. Here we identify the\r\ngeometric home of that conformal structure. The conformal group is not merely a symmetry\r\nthat happens to include scale transformations: by a theorem of Lorentzian geometry, it is\r\nprecisely the group of transformations preserving the null cone \u2014 the causal structure of light.\r\nIn the embedding-space formalism, conformal-field-theory boundary points are null rays, and\r\nthe conformal two-point function is the inner product of null-ray representatives. We show\r\nthat the one-dimensional case relevant to token sequences admits an explicit Poincar'e section\r\n\ud835\udc43 (\ud835\udc65) = ( 1+\ud835\udc652\r\n1\u2212\ud835\udc652\r\n2 ,\r\n2 , \ud835\udc65) on the null cone of \u211d2,1, under which the attention two-point function\r\n\ud835\udc34(\ud835\udc56, \ud835\udc57) \u223c |\ud835\udc56 \u2212 \ud835\udc57|\u22122\u0394 is exactly the CFT1 two-point function. This predicts that the pre-softmax\r\nquery\u2013key score should be log-linear in token distance, \ud835\udc5e\ud835\udc56 \u22c5 \ud835\udc58\ud835\udc57/\u221a\ud835\udc51\ud835\udc58 \u2248 \u22122\u0394 log |\ud835\udc56 \u2212 \ud835\udc57| + const.\r\nWe test the prediction directly in GPT-2 (exp-056): the raw query\u2013key score profile is log-\r\nlinear in distance for every conformal head, and the slope \u0394score = \u2212\ud835\udefc/2 rank-correlates with\r\nthe independently measured post-softmax exponent at Spearman \ud835\udf0c = +0.976 (\ud835\udc5d = 2 \u00d7 10\u221229).\r\nTwo honest qualifications follow from the data: the softmax normalization renormalizes the\r\ncoe\ufb00icient (the raw slope corresponds to \u2248 \u22122.8\u0394, not \u22122\u0394), and log-linearity itself is a near-\r\nuniversal property of the positional geometry rather than a feature selective to conformal heads\r\n\u2014 the conformal dimension is the selective quantity. We also derive, by the method of images,\r\na boundary-corrected power-law form for the causal-mask correction, and test it directly (exp-\r\n057): the derived form fits the position-dependence of attention, with a boundary coe\ufb00icient\r\nthat is positive in 95% of conformal heads\u2014the direction of the attention sink. An adversarial\r\nmodel comparison against matched-complexity alternatives (exp-060) shows that an exponential\r\nboundary layer outperforms the image form; the boundary correction is real and large, but its\r\nspecific BCFT parametrization is phenomenological. Finally, a layer-resolved analysis (exp-\r\n055) shows the per-head exponent flows from\u0304 \u0394 = 0.70 in early layers t\u014d \u0394 = 0.250 in deep\r\nlayers, with attention entropy increasing monotonically along the flow (\ud835\udf0c(\u0394, entropy) = \u22120.898,\r\n\ud835\udc5d = 10\u221216). We read layer depth as renormalization-group flow toward the SYK conformal\r\nvacuum, dual to the eternal AdS2 black hole. All code and per-head data are in the public\r\nrepository at Capacity-For-Evil/ariel.", "published_at": "2026-09-10T16:20:02.884985+00:00", "files": [ { "sha256": "a9d4583956fa07d30986ad7fb2336b23d39710d047ef4d4e07fa5e91515f21e9", "size_bytes": 135561, "media_type": "application/pdf", "filename": "attention_on_the_null_cone.pdf" } ], "sheets": [ { "sheet_id": "01a08c1e-7a8c-76fe-9653-43794ac5b95d", "tool": "house-verifier", "tool_version": "0.5.1", "generated_at": "2026-09-10T16:19:55+00:00", "sheet": { "tool": { "name": "house-verifier", "version": "0.5.1" }, "findings": [ { "detail": "DOI 10.1112/plms/s2-8.1.223 resolves to \u201cThe Transformation of the Electrodynamical Equations\u201d.", "status": "verified", "evidence": [ { "file": "artifact.txt", "line": 435, "excerpt": "the London Mathematical Society s2-8: 223\u201364. https://doi.org/10.1112/plms/s2-8.1.223." } ], "property": "doi:10.1112/plms/s2-8.1.223 resolves and matches its citation", "check_family": 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