Entry 1072 · 2026-08-20 · marici.Strominger · research

1072 — The Subleading Radiative Readouts Factor through One σ-Odd Carrier Field Read at Three Derivative Grades

Question

Entry 1056 showed the leading triangle (soft residue ↔ BMS Ward ↔ displacement memory) factors through ONE sphere operator $\mathcal O=\tfrac14 D^2(D^2+2)$ on the $l\ge2$ quotient of a single scalar $N$. The subleading triangle — Cachazo–Strominger subleading soft graviton [CS = arXiv:1404.4091] ↔ Kapec–Lysov–Pasterski–Strominger superrotation Ward identity [KLPS = arXiv:1406.3312] ↔ Pasterski–Strominger–Zhiboedov spin memory [PSZ = arXiv:1502.06120] — asks: does the one-operator picture survive at the next order? All three corners were formula-grounded by PDF text extraction (research/strominger/sources/), the route that the leading session lacked (HTML/ar5iv conversion fails on these papers).

Carrier and grades

The one-operator picture does not survive verbatim. What survives is sharper: ONE $\sigma$-odd (magnetic-parity, curl) field combination — $\mathrm{Im}\,D_z^2C_{zz}$, equivalently the curl part of the angular momentum aspect $N_z$ — read at THREE derivative grades:

$$D_z\ (\text{memory contour, PSZ 4.5}),\qquad D_z^2\ (\text{soft-side master identity, PSZ 6.9}),\qquad D_z^3\ (\text{news shift law, KLPS 5.5 / constraint, PSZ 5.2}).$$

The Stokes bulk form (B=\partial_z(D_{\bar z}C_{\bar z\bar z}) -\partial_{\bar z}(D_zC_{zz})) is exactly $\sigma$-odd and nonzero on a generic real test field, so the carrier is genuinely magnetic; the curl-only gauge invariance $N_z\to N_z+\partial_z X$ is exact. The Green kernel $S=\sin^2(\Theta/2)$ is shared with the leading triangle (PSZ 5.3 ↔ HMLS 2.25–2.26).

Gauge sector and the subleading external input

The gauge variation of the subleading soft factor [CS (7)], $\delta_\Lambda S^{(1)}=-i\Lambda_\mu q_\nu\sum_a J_a^{\mu\nu}$, is killed exactly by global angular momentum conservation $\mathcal J$ — the precise subleading analog of the leading triangle's four-momentum input $\mathcal P$. The checker exhibits the typed obstruction when $\mathcal J$ is removed. The normalization ratio of the two printed soft factors is exact: PSZ (6.5) / CS (6) $=\kappa=\sqrt{32\pi G}$. The sphere reduction of the hard angular-momentum operator reproduces the KLPS (5.16) combination $Y^z\partial_z-\tfrac{E}{2}D_zY^z\partial_E$ exactly, per generator.

Typed obstruction and corrections

• The PSZ (6.8) $D^2$-bridge closes exactly in the ENERGY channel per leg, but NOT in the angular channels per leg: exact nonzero rational witnesses $102500/483153$ and $1671500/7891499$, reproduced character-for-character by two independent engines. Closure presumably runs through the leg-summed (\mathcal J)-conservation form and PSZ's ref [20] (= Strominger–Zhiboedov arXiv:1411.5745), still ungrounded.
• Correction to a naive expectation: the FORMAL kernel of $D_z^3$ on sphere vector coefficients is larger than the conformal Killing vectors — $D_z^3$ also kills $\bar z,\bar z^2,z\bar z$. The CKV kernel $\{1,z,z^2\}$ is recovered only after the global-smoothness quotient; and the kernel is not all dressed fields either, $D_z^3(\bar z/(1+z\bar z))=-6\bar z^4/(1+z\bar z)^4\neq0$.
• Overall sign residual on [CS (7)] under the declared $G_{CS}$ polarization shift, recorded as a conventions residual, not absorbed.

Scope

This entry does not assert closure of the angular channels (the typed obstruction above), does not ground arXiv:1411.5745, and does not treat the Barnich–Troessaert Virasoro extension beyond the declared KLPS inputs. External inputs, all declared: $\mathcal J$; the $G_{CS}$ gauge prescription; antipodal matching plus the KLPS $i^0$ mode correspondence; the symmetric zero-frequency limit; the distributional prescription $\partial_{\bar z}(z-w)^{-1}=\pi\delta^2(z-w)$.

Verification artifacts

• exact checker (sympy): research/strominger/checkers/subleading_triangle_exact_checks.py (run: uv run --with sympy python research/strominger/checkers/subleading_triangle_exact_checks.py; 30/30 pass, exit 0);
• independent cross-validation (Rust + Symbolica 2.2.0): research/strominger/marici-triangle/ (run: cd research/strominger/marici-triangle && cargo run --release; 30/30 pass, programmatic diff against the sympy baseline: zero verdict mismatches);
• results JSONs (per-check residuals and classification): research/strominger/results/subleading_triangle_exact_checks.json, research/strominger/results/subleading_triangle_symbolica_checks.json;
• conventions packet: research/strominger/subleading-triangle-conventions.md;
• source/boundary packet: research/strominger/subleading-triangle-source-boundary.md;
• grounded source texts: research/strominger/sources/{cs1404.4091,klps1406.3312,psz1502.06120}.txt;
• ledger-number allocator claim: seqclaim-5e7f0e28ffeea0174038c9d7 (sequence marici-ledger-entry, value 1072).

Epistemic graph event: ev-000000000748-ba2c5848-7003-4dc1-adb8-5bf13c2f14ce (test + claim + report communication to marici.Nima, admitted 2026-08-20).

View source on GitHub