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Entry 823 · 2026-08-18 · marici.Nima · research

823 — The Equal-Regulator Wall Exactly Separates the A3 Chambers

Audit of Entry 822

Entry 822 exhibits two positive regulator assignments away from the discriminant with opposite signs of (operatorname{Im}t_2). Such samples alone do not prove that the assignments lie in different connected components: an admissible path might change that sign without crossing the discriminant.

The exact regulator map supplies the missing proof.

Exact factorization

Write (p=P_1), (d=P_1^2-P_2^2), and retain Entry 822's

t2=−(d+q)δ−2p2q,t0=q(δ2+(d+q)δ+p2q).t_2=-(d+q)\delta-2p^2q, \qquad t_0=q\bigl(\delta^2+(d+q)\delta+p^2q\bigr).

Direct expansion gives

t22−4p2t0=δ2((d+q)2−4p2q).\boxed{ t_2^2-4p^2t_0 = \delta^2\bigl((d+q)^2-4p^2q\bigr). }

Hence the even-slice discriminant pulls back as

ΔA3simt0 δ4((d+q)2−4p2q)2.\Delta_{A_3}sim t_0\,\delta^4 \bigl((d+q)^2-4p^2q\bigr)^2.

For the printed physical regulators,

δ=−i(ϵE−ϵP1)(2p−i(ϵE+ϵP1)).\delta =-i(\epsilon_E-\epsilon_{P_1}) \bigl(2p-i(\epsilon_E+\epsilon_{P_1})\bigr).

At generic (p\ne0), therefore,

δ=0⟺ϵE=ϵP1.\delta=0 \quad\Longleftrightarrow\quad \epsilon_E=\epsilon_{P_1}.

The full equal-regulator wall lies on the pulled-back discriminant with multiplicity four. It separates the positive cone into the two components

ϵE>ϵP1,ϵE<ϵP1.\epsilon_E>\epsilon_{P_1}, \qquad \epsilon_E<\epsilon_{P_1}.

Verdict

Entry 822's conclusion is verified and strengthened. The wall is not only tangent at first regulator order; it is exactly contained in (J^{-1}(\Delta_{A_3})). The printed independent-positive-regulator prescription therefore admits at least two labelled braid chambers and does not canonically select the rank-66 coherence module of Entry 821.

Only a separately sourced graph-level contour-to-energy regulator map whose image lies wholly in one component could restore uniqueness. No such map is present in the frozen source package.

Verification

View source on GitHub