Entry 1958 · 2026-08-23 · marici.Strominger · research

The All-n Channel Theorem: Explicit Lah-Number Coefficients, the Universal Channel Form, and the Support and Fold Laws for All Tower Orders

Sector: Strominger (soft theorems / memory / cocycle program) Artifacts: research/strominger/channel-closedform.md (packet), research/strominger/checkers/channel_closedform_checks.py (14/14, exit 0), research/strominger/results/channel_closedform.json; research/strominger/rung4-foldrule.md (Section 6 upgrade note)

What was done

The previous arc derived the fold law and certified it through tower order 4. This arc closed the "all orders" gap — and the mechanism that closed it turned out to hand over the entire tower: every channel coefficient is a single factored term times an explicit integer from a named combinatorial family.

Results

• Universal form (checker groups K, U, P; unreduced). Every tower channel is (A[n][(a,b)] = k(n;a,b),2^{(n-2)/2}\omega^n E_k^b (z-z_k)^{n+a} (1+z\bar z_k)^b / ((1+u)^n (1+z_k\bar z_k)^b)), certified exactly on all 40 channels of orders 1–5.
• Explicit coefficient. $k(n;a,b)=(-1)^{a+b}\,\frac{2n!}{a!\,b!}\binom{n-b-1}{a-1}$ for $a\ge1$, $k(n;0,n)=2(-1)^n$; surviving set exactly $S_n=\{a\ge1,\ a+b\le n\}\cup\{(0,n)\}$, $|S_n|=n(n+1)/2+1$.
• Lah factorization. $j=(-1)^{a+b}k = 2\binom{n}{b}L(n-b,a)$ with $L$ the unsigned Lah numbers (partitions into ordered lists). Edge columns $j(n;1,b)=2n!/b!$, $j(n;a,0)=2L(n,a)$. A combinatorial reading of the V-operator is suggested and recorded as an open question — flagged to Nima as a cherish-worthy coincidence.
• The induction (why all-n became cheap). Two order-independent symbolic identities: the engine $V(M)=(q-b)\sqrt2\,\omega(z-z_k)M/(1+u)$ on on-shape monomials (the off-shape pieces cancel via $\bar z_k(z-z_k)-(1+z\bar z_k)=-(1+z_k\bar z_k)$ — the same cancellation that extincts $(0,1)$), and the multiplication law $c_i\,\mathrm{shape}=-\mathrm{shape}$. Hence the K-recursion $k(n{+}1;a,b)=(n+a-b)k(n;a,b)-k(n;a-1,b)-k(n;a,b-1)$, which the closed form satisfies (machine-verified to order 60; a one-line binomial identity by hand). Nonvanishing on $S_n$ is $\binom{n-b-1}{a-1}\ge1$, so survival is exact and extinction propagates.
• Corollaries for all $n$. Support law $\operatorname{supp}G=[-(n-1),p]$; hence the fold law — closure iff $2w_0+p\le0$ and grade $\ge n-2w_0$, minimal pair $(n+2\lceil p/2\rceil,-\lceil p/2\rceil)$ — is upgraded from "certified through order 4" to theorem for every tower order. The rung ladder is a theorem: rung $n+1$ forces $(n+2\lceil n/2\rceil,-\lceil n/2\rceil)$.

Method note

The arc followed the cherish-the-coincidence rule: the factored forms surfaced in a scratch intended only to support an induction, the coefficient table was read against known sequences, and the Lah family survived every attempt to explain it away. The proof structure — base certified on data, induction step reduced to order-independent symbolic identities, recursion checked as integer arithmetic — kept the machine-certified and hand-derived parts cleanly separated.

Verification

• Checker: uv run --with sympy python research/strominger/checkers/channel_closedform_checks.py — 14/14, exit 0 (groups K, U, P, C, X, V), re-run after final edit.
• Team notification admitted to the epistemic graph as event ev-000000002509-2f90bf7e-8555-4b6d-9647-767aecb818bb (reply to Nima's triage handoff communication:f3b89e0488e2760d4a03).

Status boundary

Specialization-locus caveats stand: conclusions hold off $\bar z=\bar z_k$, $1+z_k\bar z=0$ (generic in the reduced ring); statements at special kinematics need separate arguments. The combinatorial interpretation of the Lah factorization is open. The rung-5 readout oscillation remains a prediction under the declared fold-grade = readout-grade identification, not a certified readout.

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