Prove
Lemma 3 ([The marked body model]). The two-stage grope body used in has a…Proposition 2 ([Geometric reduction]). For the integers m,p and the finite list…Proposition
Write D_ x,j for the single cap on the a -side of S_j , and use as f_j its proper truncation on ∂ν(S_j) . Its product framing is the framing required for compression of the bottom. Let U_j be the upper punctured torus. Write D_ y,j for its chosen surgery cap and D_ z,j for the other cap. Compress U_j along D_ y,j . Cutting U_j along its y -curve gives a pair of pants; fill its two new boundary components by opposite parallel copies of D_ y,j . The result is a disk C_j=U_j^ cut ∪D_ y,j ^ + ∪(-D_ y,j ^ - ) with boundary a parallel of b_j . Its collar is on the outgoing side of the upper attachment. The unused D_ z,j is not part of C_j : it will instead give a null-homotopy in E identifying the group-ring labels of the two surgery-cap sheets. Take the normal-circle torus over a parallel of b_j near ∂ν(S_j) , and compress it with two parallels of C_j . Here is a collar model fixing the relevant separations. Use coordinates (s,t,r,θ) , where s is along b_j , t is transverse to it on S_j , and (r,θ) are polar coordinates normal to S_j . The torus is t=0, r=ρ . The a -cap collar and the upper attachment depart at distinct angles θ_A,θ_B . Remove the thin compression band at θ_B , keep the annulus containing θ_A , and place the interiors of the two compression disks outside radius ρ . Thus g_j=R_j^ cut ∪C_j^+∪(-C_j^-). The retained annulus meets the a_j -collar in one transverse point. Orient and base it so that this point contributes 1 . All other body pieces, approach tracks and collars are in the separated neighbourhoods of Lemma 3 ([The marked body model]). In particular no collar point occurs for f_i,g_j with i≠j . Figure shows the normal-disk cross-section of this compression. Restoring the s -coordinate turns the retained arc into the annulus and the deleted arc into the compression band. We count all remaining intersections. They occur in cap--cap plumbing charts, because the parent bodies and the collar pieces just described are disjoint away from their prescribed attachments. Each occurrence of a D_ y,j sheet in has two indices: ε∈+1,-1 specifies the copy of C_j , and η∈+1,-1 specifies the cap sheet inside that copy. Its orientation coefficient is εη ; the two indices are shown in Figure. At a plumbing chart with coordinates (u,v)∈D^2×D^2 , these sheets are small parallel coordinate disks v=v_ ε,η ; the opposite core sheets are u=u_ ε',η' . For any fixed opposing sheet, their local intersection signs differ by the factor η . Fix ε . The two paths to the η -sheets have the same stem from the rim torus to that copy of the upper body. Their difference is, up to this common conjugation, the upper basis loop z_j . That loop bounds D_ z,j , together with its tip-annulus collar, in E . Indeed those upper caps were disjoint from all the bottom tubes; intersections with other caps do not remove them from E . An infinitesimal normal turn around the upper body also bounds a normal fibre disk in E , since that body was not removed. It follows that the two intersection labels in π_1E agree. The paired contributions are therefore εη\, h+ε(-η)\,h=0 ℤ[π_1E]. The potential winding around the bottom normal circle distinguishes the outer ε -copies. We never compare those copies in this cancellation: ε is held fixed throughout. This accounts for every cap contribution to λ(f_i,g_j) . For g_i,g_j , fix a sheet of g_i and the outer index of g_j , and cancel the inner index of g_j by the same argument. It also applies when all parallel sheets at a plumbing point are counted simultaneously. Hence the off-diagonal pairings of the g_j vanish. The diagonal and framing admit a direct geometric check. The disk C_j uses just the one core type D_ y,j , which has no self-plumbing. That cap and its parallels are embedded and miss the parent body off their attachments. Thus C_j , and then the sphere in , can each be individually embedded. Mutual intersections with other g_i do not affect this assertion. The normal frame of the rim torus is (∂_t,∂_r) . Along the outgoing radial compression collar the cap normal frame is (∂_t,∂_θ) . Rounding the surgery corner rotates the second vector from ∂_r to ∂_θ , while the first remains fixed. The product framing matched in Lemma 3 ([The marked body model]) extends this ordered pair. The earlier compression is the same framed operation, so its outgoing b_j -annulus has exactly the required inherited framing. Consequently g_j has trivial normal bundle. Its normal push-off is disjoint from it, proving the diagonal λ(g_j,g_j)=0 ; its embedded representative also gives μ(g_j)=0 . Combining this with the distinguished collar point proves all the stated equalities. Small perturbations make the finite collection generic while preserving the calculation and the framings.