responseDiff_rev
plain-language theorem explainer
On the seated generator-1 edge of the Freudenthal patch, the cfgA-minus-cfgB history-response difference flips sign when the two endpoints are swapped. Anyone proving that this difference lies outside the metric edge-current image needs the sign flip to match strain-current symmetry. The proof is a three-line algebraic reduction: unfold the difference, apply the forward and reverse history-response evaluations, then ring.
Claim. Let $A$ and $B$ be the two fixed Loom configurations (master-key open vs master-key lock), and let $(p,q)$ be the seated generator-1 endpoints on the Fin-16 patch. Then the edge-current response difference satisfies $\Delta R(A,B;q,p)=-\Delta R(A,B;p,q)$, where $\Delta R(c_1,c_2;i,j)$ is the difference of history responses of $c_1$ and $c_2$ on the ordered pair $(i,j)$.
background
The module freezes claims G2/G3 of the Order-Sensitive Gravity proposition: a depth-two commutator reading of a Loom Config is seated as an antisymmetric Fin-16 edge current on the generator-(0,2) edge at record-time false, with no metric $H$ and no $\mu$-coordinate table.
The seating seatGen1 is the ordered pair of Fin-16 vertices obtained by seating Loom generator 1 at record-time false. The history response of a config $c$ on an ordered pair $(i,j)$ is the antisymmetric edge current built from the depth-two fingerprint amplitude of $c$. Upstream, the forward evaluation identifies the response on $(p,q)$ with the real fingerprint of the depth-two reading, while the reverse evaluation identifies the response on $(q,p)$ with the negative of that same fingerprint.
The response difference $\Delta R(c_1,c_2)$ is the pointwise difference of the two history responses. The concrete configs $A$ and $B$ are the certificate pair whose depth-two readings differ by a single integer entry in the fourth block (the open-vs-lock master-key distinction).
proof idea
Term-mode proof by simplification and ring. Unfold the definition of the response difference on both sides. Rewrite the reverse-edge history responses of $A$ and of $B$ via the reverse evaluation lemma (each becomes minus the fingerprint), and rewrite the forward-edge responses via the forward evaluation lemma (each becomes the fingerprint). The resulting identity is an elementary ring equality of the form $(f_A-f_B)=-(-(f_A-f_B))$.
why it matters
Parent use is the exclusion theorem that the cfgA/cfgB response difference is not in the metric edge-current image. That argument assumes a metric strain current $H$ equal to the difference, evaluates at both endpoint orders, and invokes strain-current symmetry $H(i,j)=-H(j,i)$; the present sign-flip supplies the matching antisymmetry on the history-response side, so the two evaluations force a contradiction with the nonzero fingerprint gap between $A$ and $B$.
In the Recognition gravity stack this is a local algebraic step inside the order-sensitive history response on the Freudenthal patch: it converts the antisymmetric seating of the depth-two reading into a concrete obstruction against interpreting that reading as a linearized flat-patch metric perturbation. It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law; it is gravity-side analysis supporting the frozen G2/G3 claims.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.