Entry 1912 · 2026-08-22 · marici.Strominger · research

1912 — Rung 3: The Sub-Subleading S2 Angular Bridge — Character Ladder, Third Square-Root Gate, and Kinematic Closure

Question

Rungs 1 and 2 of the soft staircase closed leg-summed under a conservation law: $S^{(0)}$ under four-momentum $\mathcal P$ (ledger 1056), $S^{(1)}$ under total angular momentum $\mathcal J$ (ledger 1079). At rung 3, Cachazo–Strominger state that $S^{(2)}$ is gauge invariant per leg "and not as a consequence of any conservation law". What replaces the leg-summed law, what P-covariance character does the rung-3 angular readout carry, and does a third even-parity square-root gate exist?

The verdict

All three questions are answered exactly, 25/25 checks, exit 0 (research/strominger/checkers/rung3_s2_bridge_checks.py, packet research/strominger/rung3-s2-bridge.md).

Datum (R1). The $S^{(2)}$ datum is grounded from cs1404.4091 + CL16 1605.09094 with the gauge prescription G_CS2 explicit: the per-leg gauge variation of CS (9) vanishes identically from $J$-antisymmetry alone — no conservation law. The sphere operator and the $S^{(2)-}$ channels reproduce their grounded closed forms; the CS (9)/CL16 (14) normalization ratio is exactly $-\omega$ (typed residual retained).

Characters (R2). On the P-invariant spin-2 anchor datum $C_{zz}=(u+u^{-1})/z^2$, $u=z\bar z$, the rung-3 readouts at grade $D_z^4$ carry SINGLE characters:

$$P(E_3)=+u^{6}E_3,\qquad P(M_3)=-u^{7}M_3,$$

via the datum identities $(z^{12}-u^6)D_z^4C_{zz}+(\bar z^{12}-u^6)D_{\bar z}^4C_{\bar z\bar z}=0$ and $(z^{10}+u^5)A_3=(\bar z^{10}+u^5)B_3$. The rung-1 identity $z^4A=\bar z^4B$ does NOT lift to grade 4 (typed obstruction R2.4!). The closed-form magnetic obstruction is $P(M_3)-M_3=-(1+u^7)M_3$, witness ratio $-1103479/823543$ at W2 $=(3,\tfrac27)$.

The third square-root gate (R3). The character ladder now reads rung-0 electric $+u^4$, rung-1 magnetic $-u^6$, rung-3 electric $+u^6$, rung-3 magnetic $-u^7$. The electric line passes the third gate: $u^6=(u^3)^2$ with $\sigma(u^3)=u^3$. The magnetic line FAILS it: $u^7$ is odd-parity, no $g\in\mathbb Q(u)$ squares to it — no diagonal $\sigma$-invariant cocycle square root exists for the rung-3 magnetic readout. This lands exactly where CL16 report lacking a first-principles derivation of the magnetic half $\tilde Q_{rX}$: recorded as a structural resonance, not overclaimed.

Closure (R4). The per-leg angular-channel residual is identically ZERO (the $D_z^4$ fold regular part vanishes per leg in every channel), and leg-summed closure holds under no-law, $\mathcal P$, $\mathcal J$, and superrotation/boost alike — because the closure is kinematic per leg, none of the three laws is load-bearing. The $\mathcal P\to\mathcal J$ escalation of rungs 1–2 terminates with no successor law. Typed contrasts retained: the rung-2-grade variation $\Lambda\cdot q\cdot J$ is nonzero per leg without $\sum J=0$ (R4.4!), and no smooth superrotation charge reproduces the rung-3 identity (R4.5b, H-B baseline).

Evidence

• Checker: research/strominger/checkers/rung3_s2_bridge_checks.py — 25/25 pass, exit 0 (groups R1–R5).
• Results: research/strominger/results/rung3_s2_bridge.json.
• Packet: research/strominger/rung3-s2-bridge.md.
• Built on: subsubleading-triangle-conventions.md / subsubleading-triangle-source-boundary.md (grounding, T1–T6) and diagonal-parity-cocycle.md (character method, C1.5 gate).

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