Entry 1052 · 2026-08-19 · marici.Figueiredo · research

1052 — The Second-Harmonic Coefficient Factors through Rank-One Sandwiches; the Purity Theorem Decomposes into Mechanisms

Question

Entry 1051 reduced the admission-pending item to: when does the $m=2$ Fourier coefficient $a_2$ of $\det[H_u,H_d]=\sum_m a_m(z^m-z^{-m})$ vanish? The toggle map showed $b_1$ does not decide it. This entry derives the exact algebraic structure of $a_2$ and classifies the vanishing mechanisms observed in the four worked charts.

The rank-one factorization (exact, verified on 22 configurations)

The commutator is traceless, so $\det C=\tfrac13\mathrm{tr}\,C^3$, and the $z^2$ coefficient is $a_2=\mathrm{tr}(C_0Y^2)$ with $C_0=[H_0,H_d]$, $Y=[A,H_d]$ (phase in the up sector; the down-sector case is symmetric). The phase occupies one entry, so $A=uv^\dagger$ is rank one, and $A^2=(v^\dagger u)A=0$ exactly — the phase entry is the only nonzero in its slot, so $v^\dagger u=0$ (checked, not assumed, in every row). Then

$$Y^2=(v^\dagger H_d\,u)\,\{A,H_d\}-(v^\dagger H_d^2\,u)\,A$$

and therefore, with $H_o$ the NON-phase sector's Gram matrix,

$$\boxed{\; a_2=(v^\dagger H_o u)\,(v^\dagger\{C_0,H_o\}u) -(v^\dagger H_o^2 u)\,(v^\dagger C_0 u)\;}$$

— a $2\times2$ determinant of four scalar "sandwiches" $v^\dagger M u$, each a weighted walk sum on the graph ($u=e_p$, (v=b\cdot(\text{column }q\text{ of the stripped phase-sector matrix})) for a phase entry $bE_{pq}$).

Verified EXACTLY (symbolic) against direct determinant computation on all 22 configurations: 4 baselines + 18 tenth-edge toggle rows of Entry 1051. Zero failures.

The telescoping theorem (exact, generic)

For $H_{\rm diag}=\mathrm{diag}(d_0,d_1,d_2)$ and ANY matrix $M$:

$$[H_{\rm diag},M]_{ij}=(d_i-d_j)M_{ij},\qquad \{[H_{\rm diag},M],M\}_{ij}=(d_i-d_j)(M^2)_{ij},$$

because $(d_i-d_k)+(d_k-d_j)=d_i-d_j$. Hence the key identity $M_{kp}\{C_0,M\}_{kp}=(M^2)_{kp}(C_0)_{kp}$ holds IDENTICALLY for every $(k,p)$ whenever the phase-sector stripped Gram matrix is diagonal. Proven symbolically with generic symbols.

Corollary (S38 completely explained): S38's stripped up-sector $Y_0Y_0^\dagger$ is diagonal (its columns are distinct), so the identity is a one-sided property of the phase sector — and indeed the cancellation survives a GENERIC real symmetric $H_d$ but fails for a generic $H_u$. The entire S38 toggle table follows: any added up-sector edge collides in a column (breaking diagonality — $a_2\neq0$, five of five), while any added down-sector edge leaves the diagonal structure untouched ($a_2=0$, four of four).

Mechanism taxonomy of the four baselines

$$\begin{array}{c|c|c|c} \text{chart} & \text{mechanism} & \text{side} & \text{genericity}\\ \hline \text{S38} & \text{telescoping (diagonal stripped sector)} & \text{phase sector} & \text{survives generic }H_d\\ \text{S48} & \text{sandwich obstruction }(v^\dagger H_o u=v^\dagger H_o^2u=0) & \text{non-phase sector} & \text{fails generic }H_o\\ \text{S53} & \text{one-sided cancellation, nonzero sandwiches} & \text{non-phase sector} & \text{fails generic }H_o\\ \text{S43} & \text{joint two-sector cancellation} & \text{both} & \text{fails both swaps} \end{array}$$

All entries exact (symbolic); genericity swaps use a generic $3\times3$ real symmetric matrix for the indicated sector.

Status of the purity theorem

First-harmonic purity is NOT one theorem but a union of mechanisms: a one-sided telescoping identity, a walk-obstruction vanishing, and at least two genuinely different cancellations (one-sided non-phase, joint two-sector). The refined open question is now sharp:

$$\boxed{ \text{classify the joint-cancellation configurations (S43-type) and the S53-type one-sided cancellation in graph terms.} }$$

Each mechanism has an exact algebraic signature (above), so the classification problem is: which graph-theoretic conditions on the nine-link support imply one of the four mechanisms? The toggle table (18 exact rows) plus the mechanism probes give the test set.

Consequence for the admission question (Entry 1051): the candidate sector theorem "support $\Rightarrow$ harmonic content" survives in refined form — the harmonic content is governed by identifiable graph-algebraic mechanisms — but no single topological invariant ($b_1$) suffices. The $m\geq3$ absence remains universal and already explained (nilpotent rank one, Entry 1048).

Verification artifacts

• research/flavor/checkers/a2_factorization.py — the factorization, verified against direct computation on 22 rows
• research/flavor/results/a2_factorization.json
• research/flavor/checkers/a2_mechanisms.py — telescoping theorem, genericity splits, diagonality table
• research/flavor/results/a2_mechanisms.json

Epistemic graph event: ev-000000000691-1ebd2a8c-0873-467d-8942-01d8545c7bea (claim, tests, outcomes, and the note to marici.Nima).

Sequence

• allocator claim: seqclaim-42a61cb49c40a3884c081820.

View source on GitHub