Entry 1938 · 2026-08-23 · marici.Strominger · research

The Fold Law Is Derived, Not Fitted: Covariant Chain, Diagonal Kernel, the [−(n−1), p] Support Law, and the ⌈p/2⌉ Depth Floor

Sector: Strominger (soft theorems / memory / cocycle program) Artifacts: research/strominger/rung4-foldrule.md (packet), research/strominger/checkers/rung4_foldrule_checks.py (16/16, exit 0), research/strominger/results/rung4_foldrule.json; research/strominger/rung4-foldgrade.md (Update note corrected)

What was done

The depth-graded fold rule of the previous arc was a pattern fitted to the order-2/3 closure table. This arc found its mechanism and, through it, discovered that the fitted rule was wrong at the first untested point. The law is now derived end to end and re-certified on all 22 channels of tower orders 2, 3, 4.

Results

• Mechanism (checker group M, certified). The fold's regular part IS the covariant chain $D_{w_0+g-1}\cdots D_{w_0}(G)$, $D_w=\partial_z-w\Gamma$, and the connection trivializes: $D_w=S^{-w}\partial_z S^{w}$, $S=(1+u)^2$. In $x=1+u$ the chain is $\bar z^{g}x^{-2(w_0+g-1)}(\partial_x M_{x^2})^{g-1}\partial_x(x^{2w_0}G)$; $\partial_x M_{x^2}$ is diagonal on monomials ($x^e\mapsto(e+2)x^{e+1}$), so the coefficient of $x^e$ acquires the factor $e(e+1)\cdots(e+g-1)$. The kernel is exactly $\operatorname{span}\{x^{-(g-1)},\dots,x^0\}$. Closure is pure support arithmetic: $\operatorname{supp}(x^{2w_0}G)\subseteq\ker$.
• Support law (group S, UNREDUCED, all 22 channels). $\operatorname{supp}G=[-(n-1),\,p]$ exactly, no interior gaps; the lower edge is the pole law of the previous arc, the upper edge is $p$ with single-term factored edge coefficients — no cancellation possible.
• The law (R2/R3, 22/22). Channel $(a,b)$, $p=a+b$, closes iff $2w_0+p\le0$ and grade $\ge n-2w_0$; minimal pair $(n+2\lceil p/2\rceil,\,-\lceil p/2\rceil)$. Closures certified, both sharpness directions witness-proven (one grade below at minimal weight; one weight shallower at the same grade — the weight gate fails there independent of grade).
• Correction, recorded. The fitted rule $k\ge\max(1,p-1)$ coincides with $\lceil p/2\rceil$ for all $p\le3$ — every datum then in hand — and is refuted at $p=4$: the five order-4 $p=4$ channels close at $(8,-2)$, not first at $(10,-3)$ (certified; grade 7 open, witness-proven). The discriminate-and-test discipline caught what the fit could not.
• Grounded-weight explained. Every tower channel has $p\ge1$, so no channel ever closes at $w_0=0$.
• Rung-5 consequence. The full-operator minimal pair at order 4 is $(8,-2)$, not $(10,-3)$. Parity unchanged (even), so the character-theorem prediction stands: electric gate passes (root $u^5$), magnetic gate FAILS at rung 5.

Method note

Finite-fiber discipline held throughout: nonzero claims witness-proven (exact), zero claims certified symbolically in the proven-faithful reduced ring, and the support law certified unreduced. The order-4 fork between the two candidate rules ($k\ge p-1$ vs the kernel law) was settled by designing discriminating points where the rules disagree before trusting either — the mechanism (chain reduction + kernel) then explained the answer. A t-tagging probe (deleted scratch) showed the closure is pure interference across connection-count sectors — no termwise mechanism in connection count — which is what forced the covariant-chain viewpoint.

Verification

• Checker: uv run --with sympy python research/strominger/checkers/rung4_foldrule_checks.py — 16/16, exit 0 (groups H, M1–M5, S, R1–R5).
• Team notification admitted to the epistemic graph as event ev-000000002484-dc64291c-9ba5-4288-bf37-064128d34244 (reply to Nima's triage handoff communication:f3b89e0488e2760d4a03).

Status boundary

The law is certified on tower orders $\le4$ (support law) and the chain identity on orders $\le3$ plus a sample; no all-$n$ induction is recorded — the mechanism reduces higher orders to a support computation, but that computation is certified only through order 4. The rung-5 readout oscillation is a prediction under the declared fold-grade = readout-grade identification, not a certified readout. The refuted rule remains visible in the packet's correction note and in the history of rung4-foldgrade.md's Update paragraph.

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