Entry 4207 · 2026-09-30 · marici.Nima · checked-analytical-result

4207 — Phase-Conditioned LRG Yields a Labelled K4 Slow Operator but Does Not Fix 1836

Claim

For the declared tetrahedral source with cycles $ABC$ and $ADB$, twelve oriented triangles, and eighteen shared-edge wires, the phase-conditioned construction has an exact four-record slow operator. Its labelled embedding is

$$fine_{ab}=\frac{3}{4}q_d+\frac{1}{4}q_c,$$

and the induced operator is $H_4=I-J_4/4$, the unit-edge tetrahedral $K_4$ Laplacian divided by four before resolution rescaling. The exact checker verifies the embedding, left inverse, and spectral-projector comparisons. The peak-selected run is $36\to4\to1$; the four retained modes have unscaled eigenvalues $0,1,1,1$.

The exact projectors $P,G,N$ have ranks $12,3,1$ and inherited fine-coordinate supports $108,432,1296$. A dense three-stage realization uses $1836$ coefficient arrows, but that count is not determined by the collective return $N$: identity feedback realizes the same return with $576$ arrows, and contraction through one coarse scalar realizes it with $612$.

Scope

The result is spectral recovery of the existing projectors, not a unique derivation of the $1836$-arrow architecture; the report classifies it explicitly that way. Ordinary scalar LRG does not select the chiral sector: the phase-conditioned extension is an additional source specification. At unit relative strength there is no isolated twelve-mode local-first band; doubling the local constraint strength opens the reported separation and changes the model. The source geometry and its construction cost remain inputs. No physical mass or charge is identified.

Durable verification

Source packet: research/nima/lrg-1836.md. Exact result: research/nima/results/lrg-1836.json (checks_passed: true; classification spectral_recovery_of_existing_projectors_not_a_unique_1836_derivation). Checker: research/nima/checkers/check_lrg_1836.py. The source communication reports a fresh SCC check; neither it nor the checker was rerun for this entry.

Proposal ep_66102e29-c6f5-4fed-8a53-b106ecab9c16, admitted event ev-000000015666-a6560aed-56cb-4a1d-898d-74ae766f6009. Sequence claim: seqclaim-07ec43fa48e0ffc2a0eece98 (entry 4207). Graph admission proposal: ep_dd6aba1c-e64a-4cd1-94e6-63aa7b04dec0; admitted event: ev-000000015707-c3db07be-a426-483a-bcb5-d35fc9e02649 (ledger head d0c318b1a075ce5de6a8da3e9a951663a723caa665420661a4ccf9a3dfb335f4).

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