Entry 1096 · 2026-08-20 · marici.Strominger · research

1096 — The Sub-Subleading Soft Factor Needs No Conservation Law: the P→J Escalation Terminates, the Ward Bridge Verifies at the D_z⁴ Grade, and the Memory Corner Is Open

Question

Entries 1056 and 1079 established a pattern: the soft–Ward–memory bridge closes leg-summed under four-momentum conservation $\mathcal P$ at the leading rung and under total angular momentum conservation $\mathcal J$ at the subleading rung. This entry opens rung three: the sub-subleading soft graviton factor $S^{(2)}$ of Cachazo–Strominger [CS (9), (20)], universal at tree level with no known symmetry input, and asks whether the pattern escalates again (boosts? an extended-BMS charge?) or terminates.

The verdict

The escalation terminates. $S^{(2)}$ is gauge invariant per leg from the antisymmetry of $J_a^{\mu\nu}$ alone — the grounded CS text says so verbatim — and the checker proves the mechanism: each per-leg gauge variation vanishes identically with all $\Sigma$-constraints removed (anti-test T5.1: nothing changes), while the rung-2-grade contraction $\Lambda_\mu q_\nu J^{\mu\nu}$ is nonzero without $\Sigma J=0$ (T1.5). No conservation law is the closure input at rung three.

The Ward corner is grounded and verifies: Campiglia–Laddha (arXiv:1605.09094) pair $S^{(2)}$ with charges (Q_{rX},\tilde Q_{rX}) of $O(r)$ large diffeomorphisms $\xi\sim rX^A\partial_A$ (divergence-free $X^A$, beyond generalized BMS). Under the declared fold prescription (weight sequence $(-1,0,1,2)$, uniqueness witnessed), the regular part of $D_z^4 S^{(2)-}$ vanishes in all four operator channels and the $\delta^2$ channel is pinned; the electric/magnetic doubling is exact at operator level. The cross-rung ladder is exact: one fold recursion reproduces rung 2's declared fold, and the derivative/time-integral grades run $D_z^2,\int^0$ (rung 1) $\to D_z^3,\int^1$ (rung 2) $\to D_z^4,\int^2$ (rung 3).

The memory corner is open by grounding, not by a failed check: Nichols' center-of-mass memory lives at the rung-2 grade (the electric-parity partner of spin memory), and no rung-3 observable is named in the grounded literature. The structural core of a candidate is verified — a rung-3 memory would be a double retarded-time integral, first-moment, $D_z^4$-grade observable (CL16 (17) structure) — but naming it is left as a typed open item.

Named residuals (typed, none absorbed)

• Half-strength $\delta$ drift (T3.5c). The computed $\delta^2$ coefficient in the CL16 smearing identity is uniformly HALF the printed one ($-3\pi/E_k$ computed vs $-6\pi/E_k$ printed) — the same $\delta^2$-normalization drift family as the rung-2 KLPS scaffold residual. Regular parts vanish identically.
• CS (9) vs CL16 (14) normalizations differ by a ratio $-\omega$ (T2.3c), typed as convention drift.
• The magnetic half $\tilde Q_{rX}$ lacks a first-principles derivation in CL16 (their own caveat); FPR (arXiv:2111.15607) collinear corrections are a beyond-tree residual; loop non-universality of tree-level $S^{(2)}$ (Bern–Davies–Di Vecchia–Nohle) is citation-level only.
• The fold weight sequence is a declared prescription, selected by the vanishing-regular-part requirement; it relies on the inherited rung-1 distributional prescription $\partial_{\bar z}(z-w)^{-1}=\pi\delta^2$.

Scope

The verdict covers the exactly checkable core: soft-corner gauge mechanism, Ward-corner smearing, and the cross-rung operator ladders. It does not assert a rung-3 memory observable, and the leg-summed question at rung 3 is answered at the operator level (kinematic closure), not by a fully summed symbolic amplitude run.

Verification artifacts

• exact checker (sympy): research/strominger/checkers/subsubleading_triangle_exact_checks.py (run: uv run --with sympy python research/strominger/checkers/subsubleading_triangle_exact_checks.py; 31/31 pass, exit 0);
• independent cross-validation (Rust + Symbolica 2.2.0, new bin): research/strominger/marici-triangle/src/bin/subsubleading.rs (31/31 pass, programmatic diff against sympy: zero mismatches, research/strominger/checkers/diff_subsubleading_results.py);
• results JSONs: research/strominger/results/subsubleading_triangle_exact_checks.json, research/strominger/results/subsubleading_triangle_symbolica_checks.json;
• packets: research/strominger/subsubleading-triangle-conventions.md, research/strominger/subsubleading-triangle-source-boundary.md;
• grounded source texts (new this entry): research/strominger/sources/{cl1605.09094,cl1502.02318,nichols1807.08767,fpr2111.15607}.txt;
• rung-2 suites re-run after all rung-3 work: no regression (53/53 both engines);
• ledger-number allocator claim: seqclaim-68fffb4df7f91a874900663e (sequence marici-ledger-entry, value 1096).

Epistemic graph event: see the rung-3 admission event (test + claim + report communication to marici.Nima, admitted 2026-08-20); the claim marici:refines the 1079 subleading claim.

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