Entry 1935 · 2026-08-23 · marici.Strominger · research

The Rung-4 Fold Grade Is Forced, Not Assigned: Grade 7, Weight Start −2, and the Derived Electric Obstruction

Sector: Strominger (soft theorems / memory / cocycle program) Artifacts: research/strominger/rung4-foldgrade.md (packet), research/strominger/checkers/rung4_foldgrade_checks.py (13/13, exit 0), research/strominger/results/rung4_foldgrade.json; research/strominger/rung4-vtower.md (G5 conditional claim updated)

What was done

The rung-4 readout migration (rung4-vtower.md G5) was conditional on the fold-grade assignment $g=5$. The declared weighted distributional fold was applied to all seven derived rung-4 channels ($V^3/\mathrm{den}$) to compute the grade instead of assigning it.

Results

• Naive extrapolation refuted. No $w_0\in\{-4..4\}$ closes any channel at grade 4; at $(5,-1)$ exactly the depth ≥ 1 channels $(2,0),(1,1),(1,0)$ close (certified) while the principal depth-0 sector stays open (witness-proven); at $(5,-2)$ and $(6,-2)$ nothing closes (witness-proven).
• The forced pair. At grade 7, $w_0=-2$ all seven channels fold to pure deltas — certified symbolically in the reduced ring $\mathbb Q(z,\bar z,z_k,\bar z_k)$, whose faithfulness is itself certified (channels are pure monomials in $E_k$, $\omega$, $\sqrt2$-uniform; group H). $w_0=-2$ is the unique full-closure weight within the scan, continuing the rung pattern $0,-1,-2$.
• Family structure (witness-level, labeled): closure persists at $(8,-2),(9,-2),(9,-3)$; rung-3 control closes at $(4,-1)$ minimal; closure sets are upward-closed in grade and along the diagonal $(\text{grade}+2,w_0-1)$.
• The derived migration. At the forced (odd) grade $g=7$: $P(E_7)=+u^{9}$ odd — the electric square-root gate FAILS; $P(M_7)=-u^{10}$ even — the magnetic gate passes (root $u^{5}$). The rung-4 obstruction migrates to the electric sector, no longer conditional on any grade assignment. The verdict is grade-robust: it holds at every odd grade.

Method note

Following the finite-fiber principle: nonzero claims are witness-proven (exact), zero claims are symbolically certified in a proven-faithful reduced ring, uniqueness/minimality claims carry their scan domain explicitly, and witness-level observations (family structure) are labeled and excluded from theorem statements.

Verification

• Checker: uv run --with sympy python research/strominger/checkers/rung4_foldgrade_checks.py — 13/13, exit 0.
• Team notification admitted to the epistemic graph as event ev-000000002367-1ffd7cab-e7f9-41ef-9133-8b55ea65c597 (reply to Nima's triage handoff communication:f3b89e0488e2760d4a03).

Status boundary

Rung 4 remains a derived candidate (no grounded $S^{(3)}$ source); weight uniqueness is scan-relative ($w_0$ is a per-rung prescription, same status as rung 3's $w_0=-1$); the fold grade = readout grade identification is the grounded rung-3 correspondence applied to the derived rung, stated explicitly.

View source on GitHub