b2_unique_zero_of_conserving
plain-language theorem explainer
Under a pairing-conserving seam transfer that delivers the turn ratio, the census cost vanishes exactly when the period equals the deficit-free Euclidean period β = 2π/κ. Anyone discharging B2 from ledger conservation (no J named in the premise) cites this. The proof is a one-line reduction: convert conservation pricing to seam-transfer pricing, then apply the unique-zero theorem already proved for that class.
Claim. Let $C:\mathbb{R}\to\mathbb{R}\to\mathbb{R}$ satisfy pairing-conserving seam pricing: for every $\kappa,T>0$ there exists a $2\times 2$ real transfer $W$ that preserves the double-entry pair form, has real eigenvalue equal to the turn ratio $x=\kappa T/(2\pi)$, and whose character anomaly equals $C(\kappa,T)$. Then $C(\kappa,T)=0$ if and only if $T$ equals the deficit-free Euclidean period $2\pi/\kappa$.
background
SeamTransferCore carries LEG-B Phase B of the Scale-Holonomy Trace Core: the per-closure recognition cost of a seam crossing is the character anomaly $C=\mathrm{Tr}(W)/2-1$ of the transfer $W$ induced on the seam's double-entry pair fiber. Balance ($\det W=1$) plus one real eigenvalue $x\neq 0$ forces the conjugate eigenvalue $x^{-1}$ and the trace identity $\mathrm{Tr},W=x+x^{-1}$, so the anomaly equals the T5 cost $J(x)=(x+x^{-1})/2-1$ without $J$ appearing in the inputs.
Pairing-conserving seam pricing is the ledger-facing premise: the transfer preserves the double-entry pair form (conservation, no matrix invariant named), some fiber leg scales by the turn ratio, and physical cost is the character anomaly. Via the Sp(2)=SL(2) bridge this implies ordinary seam-transfer pricing and hence census pricing.
The deficit-free Euclidean period is $\beta=2\pi/\kappa$, the unique positive $T$ at which the turn ratio is 1 and the anomaly (hence $J$) vanishes.
proof idea
One-line term wrapper. Convert the conservation hypothesis to seam-transfer pricing by the in-module bridge that turns pair-form preservation into unit determinant, then apply the already-proved unique-zero theorem for seam-transfer pricing at the same $(\kappa,T)$. No new algebra is done here; the content lives in that bridge and in the unique-zero result for the transfer class.
why it matters
This is the full Phase-B chain stated once: from ledger conservation (no $J$, no $\cosh$, no determinant, no diagonal form in the premise) to the B2 discharge that the unique zero of the census cost is the deficit-free period $\beta=2\pi/\kappa$. Downstream, the anomaly-ledger discharge cites it as the consistency check that the weak route and the landed route agree on the zero set (b2_unique_zero_of_anomalyLedger), and the same chain recovers $C(\kappa,T)=J(\kappa T/2\pi)$ at the anomaly reading.
In the Recognition forcing chain this is the holographic landing of T5 $J$-uniqueness on the seam: $J$ emerges as character anomaly of a balanced pair-fiber transfer, and B2 is the statement that the only zero is the Euclidean period. Panel Live Bet 2 (the numeric falsifier $8/3$ vs $2$) sits in the same module as the double-entry pairing identity for $J$.
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