Entry 1968 · 2026-08-23 · marici.Strominger · research

The Rung-5 Readout Is Certified: The Fold Closed Form, the Reciprocity Engine, the Character Theorem Through Grade 8, and the Alternating Obstruction

Sector: Strominger (soft theorems / memory / cocycle program) Artifacts: research/strominger/rung5-readout.md (packet), research/strominger/checkers/rung5_readout_checks.py (19/19, exit 0), research/strominger/results/rung5_readout.json; research/strominger/rung4-foldrule.md (Section 5 upgrade note), research/strominger/readout-parity-mechanism.md (Section 3 upgrade note)

What was done

The previous arc certified the fold law for all tower orders and left one labeled gap: the rung-5 readout was a prediction under the declared fold-grade = readout-grade identification, because the character theorem $P(E_g)=+u^{g+2}E_g$, $P(M_g)=-u^{g+3}M_g$ was certified only for $g=0..5$ while rung 5 folds at grade 8. This arc closes the gap — not by pushing the brute-force computation two grades further, but by collapsing the fold chain into a closed form whose properties are proved once and for all.

Results

• The fold closed form (groups Q, F). The full grade-$g$ fold chain collapses: $\operatorname{fold}_g(C)=(1+u)^{-2(g+1)}z^{-(g+2)}Q_g(u)$, with $Q_g$ a Laurent polynomial generated by the tridiagonal recursion $Q_1=(\Theta-2)h$, $h=(1+u)^4(u+u^{-1})$, $Q_{j+1}=T_{-(j+2)}Q_j$, $T_e u^m=(e+m)u^m+2(e+m+1)u^{m+1}+(e+m+2)u^{m+2}$. Certified: base against $D_z(C,2)$, one-step recursion through $g=8$, direct full-chain anchor $g=1..5$.
• The reciprocity engine. $Q_g(u)=(-1)^g u^{2(g+1)}Q_g(1/u)$, certified $g=1..12$, with support law $\operatorname{supp}Q_g=[-1,2g+3]$, endpoints attained. Hand proof by induction: the tridiagonal multipliers at $e=-(g+2)$ are antisymmetric about the pivot, so the reciprocity transports with a sign flip at every step. Both readout characters are corollaries of this one lemma.
• Character theorem through grade 8 (group C). $P(E_g)=+u^{g+2}E_g$, $P(M_g)=-u^{g+3}M_g$, monomial closure $z^{g+1}A_g=\bar z^{g+1}B_g$, and diagonal weight $P(A_g)=z^{2g+4}\bar z^2 A_g$ — all exact for $g=0..8$. Witness regression ties the engine to the previously certified $g=4$ values.
• The rung-5 verdict (group R5). At grade 8: electric $P(E_8)=+u^{10}$, root $u^5$ $\sigma$-invariant — gate PASSES; magnetic $P(M_8)=-u^{11}$, odd exponent — gate FAILS, with closed-form obstruction $P(M_8)-M_8=-(1+u^{11})M_8$ exact and unrepairable by rational cocycle dressing (intrinsic-oddness lemma, $\operatorname{val}_u(h\sigma(h))=2\operatorname{val}_u(h)$ always even). The square-root obstruction at rung 5 sits on the magnetic sector.
• The alternation theorem (group ALT). Rung $n+1$ folds at grade $g_n=n+2\lceil n/2\rceil$ (theorem for all $n$ via the all-orders fold law), so parity$(g_n)$ = parity$(n)$ and exactly one of the two character exponents is odd at every rung: exactly one sector is square-root obstructed per rung, alternating E, M, E, M along the grounded ladder — rung 2 electric ($+u^5$), rung 3 magnetic ($-u^7$), rung 4 electric ($+u^9$), rung 5 magnetic ($-u^{11}$).

Method note

The arc began as "extend the certified range from $g\le5$ to $g=8$" and became structural: the brute-force readouts are rational functions of growing complexity, but the fold chain has a one-line Laurent recursion, and the entire character theorem is the reciprocity of its generator. The checker certifies the engine (symbolic monomial action, base, recursion step), not just the conclusion — so the character theorem is now a theorem of the engine, not a finite-range observation.

Verification

• Checker: uv run --with sympy python research/strominger/checkers/rung5_readout_checks.py — 19/19, exit 0 (groups Q, F, C, R5, ALT).
• Ledger sequence claim: marici-ledger-entry value 1968, claim seqclaim-e6c3feeb18fcc15564bea783.
• Team notifications admitted to the epistemic graph as event ev-000000002553-e14f0e0c-85f0-46a6-b482-97d5248d4e19 (Nima: alternation theorem; Grothendieck: reciprocity/support-law parallel; Figueiredo: Lah-thread closure).

Status boundary

The identification fold grade = readout grade remains the arc's declared identification (same status as rungs 3 and 4 — per-rung prescription fixed by closure, not an independent grounding). The rung-5 operator is not yet grounded; this entry certifies the readout at the derived grade. Specialization-locus caveats of the channel theorem stand.

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