Prove
Lemma 22 ([Northern comparison]). There are fixed c≥0 and b_0≥1 with the…Lemma 23 ([Compatibility of reference expectations]). Let two fluxes q,q' use…Proposition
The proof keeps O unchanged while descending through common northern slots. Only the small changes caused by each transport are estimated inductively. This distinction prevents an increasing radius from being assigned repeatedly to the original observable. We first describe the centered one-step identities under the northern condition of Lemma 22 ([Northern comparison]). For an E step put t=q-3 , so t^0=q^0-3 , and write R_ q,E (O) for its uncentered polar remainder in Lemma 22 ([Northern comparison]). The filled E step at q^0 , together with compatibility of the northern references, gives the exact scalar identity ref_q(O)-ref_t(O) =ref_ t^0 ( D_ q^0,3 (O)+R_ q^0,E (O)). Thus the parent-to-child reference change is determined by a filled step. Define R_ q,E '(O), =R_ q,E (O) -ref_ t^0 (R_ q^0,E (O))I,; s_q(O), =ref_t( D_ q,3 (O)) -ref_ t^0 ( D_ q^0,3 (O)). The references in the second line use the same filled state. Lemma 23 ([Compatibility of reference expectations]), first on common northern slots and then on rapid tails by norm approximation, therefore gives s_q(O)=ref_ t^0 ( D_ q,3 (O)- D_ q^0,3 (O)) +O(q^ -D ). The bounded-flux comparison in Lemma 22 ([Northern comparison]) supplies the additional factor q^ -1 . Substituting these scalar identities into the branch comparison yields J_ q,E ^*(O-ref_q(O))J_ q,E = , (O-ref_t(O)) +( D_ q,3 (O) -ref_t( D_ q,3 (O))); , +s_q(O)I+R_ q,E '(O),; |s_q(O)|≤, Cr^c u/q^3+C_Dq^ -D , \|R_ q,E '(O)\|≤C_Dq^ -D . Here and below the symbols denote compressions to the relevant child zero space. When h=0 the comparison is its own filled calibration, so all centered contributions vanish exactly. For an F step, t=q-1 . The nearest smaller filled fluxes are either identical or differ by three. In the latter case apply the filled E comparison just used. Together with , this proves J_ q,F ^*(O-ref_q(O))J_ q,F =O-ref_t(O)+S_ q,F (O), \|S_ q,F (O)\|≤Cr^cu/q^2+C_Dq^ -D . The off-diagonal blocks have norm C_Dq^ -D . An F step already records a flag, so its entire error will be charged to that flag. Here is a uniform closure argument, including the parameter choices. Let c,b_0 be as in Lemma 22 ([Northern comparison]). Choose b>b_0,c , then choose B>ab,c large enough to absorb the fixed polynomial costs of the local calculus. A constant M will be fixed after B . Set τ=Mr^b u. If q≤τ and h>0 , every flag has b'≤q , so W_m≥(2Mr^b)^ -a I. The norm bound \|O-ref_q(O)\|≤2 proves in this region with constant 2(2M)^a . When h=0 the centered compression is zero. Increasing M includes all stopping cases in this argument. Fix a finite Q , and let K_Q be the least constant in for fluxes at most Q , with the chosen B , all number sectors, and all allowed frames. This number is finite: for h>0 use W_m≥(1+Q)^ -a , while the filled compression is exactly centered. We prove K_Q≤C_M+12K_Q with C_M independent of Q . Suppose q>τ . Expand the unchanged O down the branching tree, using and , until the current flux is at most τ . The original northern slots, center, and radius are unchanged through this part of the descent. The terminal comparisons have just been bounded. In an E child, decompose D_ q,3 (O) into its core and dyadic ball shells, all centered at m . Equation, with a tail order larger than B+2 , gives ∑_νr_ν^B\| D_ q,3 (O)_ν\| ≤C(B)r^ B+c u/q^2. Applying the definition of K_Q to each smaller-flux shell, and using linearity of the reference, bounds its centered contribution by K_QC(B)r^ B+c u/q^2 times the remaining history weight in that child. Along any word, the flux strictly decreases; thus ∑_ q'>τ (u)/((q')^2)≤C(u)/(τ) =(C)/(Mr^b). Consequently all induction-dependent contributions are bounded by K_QC(B)M r^ B+c-b W_m. Choose M so large that this is at most 12 K_Qr^BW_m . This choice is possible because b>c and r≥2 . We verify that the other errors have a bound independent of K_Q . At a node with positive remaining hole number, its identity is bounded by the sum of the identities assigned to its remaining flags. A scalar electron error may therefore be charged to every such flag. For a flag at ℓ , the total charge from ancestors in is at most Cr^cu τ,1+ℓ^2 . The F error in occurs at its own flag and obeys the same bound, up to a fixed constant. Write w_ℓ=1,(u/(1+ℓ))^a . Since a<2 and τ=Mr^bu , r^cu τ,1+ℓ^2 ≤C M^ a-2 r^ c+b(a-2) w_ℓ ≤C_M r^B w_ℓ. The q^ -D remainders, including the off-diagonal blocks, obey the same estimate after choosing D>a+2 . For a self-adjoint off-diagonal block, its norm bounds the full block matrix above and below by that norm times the identity at its node, so the same charge applies. Nodes with h=0 contribute nothing, by the exact centered filled identity. Summation of these diagonal bounds along words proves . It follows that K_Q≤2C_M for every Q , proving the uniform claim. Finally, the assertion for A=∑_νA_ν follows by linearity of the reference and by summing the form inequalities.