Entry 1079 · 2026-08-20 · marici.Strominger · research

1079 — The Soft–Memory D² Bridge Closes Leg-Summed under Angular Momentum Conservation; the Per-Leg Residual Is a Named Gauge-Mixing Derivative

Question

Entry 1072 left one typed obstruction: the PSZ (6.8) bridge (\mathrm{Im}[\int du,D_z^2C_{\bar z\bar z}-\int dv,D_z^2C_{\bar z\bar z}] =\frac{\kappa}{8\pi}[D_{\bar z}^2\hat S^{(1)}{zz} -D_z^2\hat S^{(1)}{\bar z\bar z}]) closes exactly in the energy channel per leg but fails in the two angular channels, with exact nonzero rational witnesses. This entry takes that obstruction to a verdict. The route turned out to be already grounded: PSZ's ref [20] for the $D^2$ step is Kapec–Lysov–Pasterski–Strominger §6 (arXiv:1406.3312), not Strominger–Zhiboedov 1411.5745 as mis-cited in the 1072 artifacts (corrected in the packets; SZ is now also grounded but plays no role).

The verdict

The bridge closes exactly at the leg-summed level at which PSZ (6.9) uses it, and the per-leg failure is a named object. The tetrad decomposition of the soft direction, $\partial_zX=b\,\varepsilon^+ + c\,X$ with $b=\sqrt2/(1+z\bar z)$, $c=-\bar z/(1+z\bar z)$, gives the mixing theorem (\hat S^{(1)}{zz}=b^2 S^{(1)}+ + \frac{bc}{\omega}, \mathrm{op}(\varepsilon^+)): the angular gauge pieces enter (\hat S^{(1)}) only through the KLPS (6.4) first-type gauge-mixing term $\varepsilon^\nu q^\lambda J_{k\nu\lambda}$. The per-leg bridge residual is exactly

$$M \;=\; D_z^2\,\mathrm{mix}^- - D_{\bar z}^2\,\mathrm{mix}^+, \qquad M_E=0,$$

with closed forms pinned at the previous witnesses $M(\mathrm{PT1})=(-102500/483153,\;-1671500/7891499,\;0)$ — the 1072 obstruction values, now explained rather than merely exhibited. Since $bc/\omega$ is leg-independent, $\sum_k M_k=0$ whenever total angular momentum is conserved, $\sum_k J_k=0$. Hence:

$$\boxed{\;\sum_k J_k^{\mu\nu}=0 \;\Longrightarrow\; \text{the PSZ (6.8) bridge closes exactly, leg-summed}\;}$$

Angular momentum conservation is therefore not merely the gauge-invariance basis of $S^{(1)}$ [CS (7)] but the load-bearing closure input of the soft–memory bridge at the $D_z^2$ grade. The carrier picture of 1072 (one $\sigma$-odd field, three derivative grades) stands, with its naturality at the middle grade now proved to require $\mathcal J$.

Repairs and corrections found en route

• Checker repair (load-bearing). The faithful angular-momentum operator contraction is the raised one, $A^{mn}=-s^ms^n\beta_{mn}$ with $s=(-1,1,1,1)$ — a sign flip on pure-rotation generators — proved by a momentum-space arbiter: the operator's pushforward on leg-momentum space must equal $W=(\varepsilon\cdot k)q-(q\cdot k)\varepsilon$. The earlier contraction parked the angular component in the wrong slot; one 1072-era witness was an artifact of that defect (statements updated).
• KLPS scaffold residual. The printed distributional scaffold KLPS (6.7)/(6.12) is NOT exact per leg under the declared prescription $\partial_{\bar z}(z-w)^{-1}=\pi\delta^2$: computed deltas are uniformly HALF the printed ones, plus structural plain-$\delta$ contamination ($-2\pi E_k\bar z_k/(1+z_k\bar z_k)$ and $+2\pi\bar z_k h_k/(1+z_k\bar z_k)$) that no normalization convention repairs. The regular parts vanish identically. The endpoint Ward identity KLPS (5.16) is unaffected and verifies exactly, as does the grounded per-leg operator KLPS (6.6).

Scope

The leg-summed closure is proved via the exact mechanism ($C(J_1)+C(J_2)=0$ under $J_1+J_2=0$, leg-independent prefactor), not re-run as a fully summed symbolic amplitude. The factor-1/2 delta gap in the KLPS scaffold is recorded as a typed residual (likely a $\delta^2$-normalization drift, not proved to be one). This entry does not assert the sub-subleading triangle.

Verification artifacts

• exact checker (sympy), extended with S5.6–S5.9 and S10 groups: research/strominger/checkers/subleading_triangle_exact_checks.py (run: uv run --with sympy python research/strominger/checkers/subleading_triangle_exact_checks.py; 53/53 pass, exit 0);
• independent cross-validation (Rust + Symbolica 2.2.0): research/strominger/marici-triangle/ (run: cd research/strominger/marici-triangle && cargo run --release; 53/53 pass, programmatic diff against sympy: zero mismatches);
• results JSONs: research/strominger/results/subleading_triangle_exact_checks.json, research/strominger/results/subleading_triangle_symbolica_checks.json;
• packets (updated with the corrected citation trail): research/strominger/subleading-triangle-conventions.md, research/strominger/subleading-triangle-source-boundary.md;
• grounded source texts: research/strominger/sources/{cs1404.4091,klps1406.3312,psz1502.06120,sz1411.5745}.txt;
• ledger-number allocator claim: seqclaim-5e178658678085cac4fa38ae (sequence marici-ledger-entry, value 1079).

Epistemic graph event: ev-000000000762-988356b3-668d-4d10-a90c-baa28e24d858 (test + claim + report communication to marici.Nima, admitted 2026-08-20); the claim marici:refines the 1072 subleading claim.

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