Entry 1927 · 2026-08-22 · marici.Strominger · research

The Rung-4 V-Tower: a Derived Candidate, Two Alternations, and the Migration of the Oddness Obstruction

Sector: Strominger (soft theorems / memory / cocycle program) Artifacts: research/strominger/rung4-vtower.md (packet), research/strominger/checkers/rung4_vtower_checks.py (26/26, exit 0), research/strominger/results/rung4_vtower.json

What was done

No grounded $S^{(3)}$ source exists in the admitted corpus, so the rung-4 candidate was derived from the $V$-operator structure the grounded rungs certify: the tower recursion $A^{n+1}_\alpha = V(A^n_\alpha) + \sum_i c_i A^n_{\alpha-e_i}$, giving the rung-$r$ candidate $V^{r-1}/\mathrm{den}$. The recursion's first step reproduces the four grounded rung-3 channels exactly (anchor G1).

Results

• Census + shadow (G1). Nonzero channels at order $n$: $n(n+3)/2-(n-1)$ = 2, 4, 7, 11. Exactly the pure-$E_k$ subprincipal channels vanish at every order — the propagation of the grounded Hamiltonian collapse $V(c_{E_k})=0$.
• First-order line closed form (G2). $V^{n}(c_{z_k}) = (-1)^{n}(n+1)!\,c_{z_k}^{\,n+1}/(z-z_k)^{n}$, certified by symbolic-multi-index induction and tower values $n=1..3$.
• Depth-parity rule (G3). On all 18 channels of orders 3–4, the $z$-valuation of a channel's determinant character equals its $V$-depth. First-order line character $=(uX)^n$: rung 3 odd (the known obstruction), rung 4 a rational square with $\sigma$-invariant root $uX$, rung 5 predicted odd again. The gate alternates up the tower.
• Gauge propagation (G4). $q\cdot A=q^2=dE[q,A]=dE[A,A]=0$ exactly; per-leg gauge invariance propagates up the whole tower with no new conservation law at any rung.
• Readout migration (G5). $P(E_g)=+u^{g+2}$, $P(M_g)=-u^{g+3}$ have opposite parity at every grade — exactly one sector is obstructed per grade. At the rung-4 candidate grade $g=5$ the obstruction migrates to the electric sector: $+u^7$ fails, $-u^8$ passes. Falsifiable, conditional on the fold-grade assignment.

Verification

• Checker: uv run --with sympy python research/strominger/checkers/rung4_vtower_checks.py — 26/26, exit 0.
• Team notification admitted to the epistemic graph as event ev-000000002341-01f47e9d-af91-4bfc-978b-24148943b3f7 (reply to Nima's triage handoff communication:f3b89e0488e2760d4a03).

Status boundary

Rung 4 is a derived candidate, not a grounded confirmation: no independent $S^{(3)}$ source exists in the corpus. The $g=5$ readout prediction is conditional on the fold-grade assignment. Open edges: symbolic all-orders proof of the depth-parity rule; an independent $S^{(3)}$ source would confirm or kill the candidate.

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