Weak partial-evaluation spectral expansions satisfy full P(z) compatibility

Choose $u=exp(s)$, and let ${\mu }_s[w_3](w_1,w_2)=⟨e^{-s{L}^{\mathrm{\prime }}(0)}{w}_4^{\mathrm{\prime }},Y_1(w_1,a+ub)Y_2(w_2,ub)e^{sL(0)}{w}_3⟩\mathrm{.}$ A fixed small disk $\mathrm{\mid }s\mathrm{\mid }⟨ϵ$ satisfies |a+ub|>|ub|>|a| for every s, with consistent local logarithms. This disk depends on a,b, not on inner tests. The vectors exp(-sL'(0))w4' and exp(sL(0))w3 span finite dimensional spaces: each original vector has finite generalized-weight support, and L(0) is locally finite on each of these finitely many spaces. Thus the scalar product is analytic in s on the same geometric disk for all inner tests. The L(0) bracket for each intertwiner and its L(-1) derivative give an exact differential identity. Writing G for the unscaled product, its L(0) identity is $G(L^{\mathrm{\prime }}(0)w_4^{\mathrm{\prime }};w_1,w_2,w_3)=G(w_4^{\mathrm{\prime }};L(0)w_1,w_2,w_3)+G(w_4^{\mathrm{\prime }};w_1,L(0)w_2,w_3)+G(w_4^{\mathrm{\prime }};w_1,w_2,L(0)w_3)+{\zeta }_1{\mathrm{\partial }}_{1}G+{\zeta }_2{\mathrm{\partial }}_{2}G\mathrm{.}$ Differentiate mu_s, including the two coordinate derivatives ub(partial_1+ partial_2). The outer and third-state L(0) terms cancel by the displayed identity. Since zeta1=a+ub and zeta2=ub, the remainder is $\frac{d}{ds}{\mu }_s(w_1,w_2)=-{\mu }_s((L(0)+aL(-1))w_1,w_2)-{\mu }_s(w_1,L(0)w_2)=-{\mu }_s(B(w_1\otimes w_2))\mathrm{.}$ Every formal input derivative here is a finite algebraic state combination. The scalar derivative identities hold analytically because the radial scalar series are normally convergent on smaller logarithmic coordinate rectangles. There is a finite logarithmic bound uniform in the intermediate weight: on a weight-specified intertwiner coefficient the L(0) bracket says that derivative in the log variable is $N_{out}$ acting on its output minus the two nilpotent input actions. These three commuting operations have nilpotent sum of order at most $K_{out}+K_{in1}+K_{in2}-2$, by the multinomial formula (a term of larger total degree contains one exponent at least its corresponding K). Thus its log polynomial degree is at most $K_{out}+K_{in1}+K_{in2}-3$. The fixed modules in the two-operator chain give one finite bound on both logarithmic degrees, independent of the intermediate weight. For doubly homogeneous external vectors, its real exponent pair is affine in that intermediate real weight; general external vectors add only finitely many such families. One can see that uniformity without a discreteness assumption: expand the finitely many logarithmic powers, interpolate them at finitely many imaginary shifts, and bound each real exponent at the two real endpoints of a slightly larger logarithmic rectangle. The resulting endpoint sums are finite by absolute convergence. Polynomial factors from any fixed derivative are absorbed by exp(-delta|h|). This is precisely the locally proved interpolation estimate in lem-cft-real-exponential-polynomial-uniqueness, applied successively to the two logarithmic coordinates. Finite outer exponentials preserve it. Induction now gives ${\mu }_s^{(k)}{\mid }_{s=0}=(-1{)}^k\mu [B^k(w_1\otimes w_2)]\mathrm{.}$ Let lambda_(h,j)=(D-h)^j lambda_h. The same real-exponential interpolation lemma converts the assumed GROUPED absolute convergence into locally uniform absolute convergence of the double family $F(s)={\sum }_h{e}^{-hs}{\sum }_{j=0}^N\frac{(-s{)}^j}{j!}{\lambda }_{h,j}[w_3]\mathrm{.}$ Its derivatives are legitimate termwise. At zero, its kth derivative equals $(-1{)}^k{\sum }_h(D^k{\lambda }_h)(w1,w2)$. By the transpose formula for D this is $(-1{)}^k{\sum }_h{\lambda }_h(B^k(w1\otimes w2))$, equal to the previous derivative of mu_s by the original weak expansion applied to that finite test vector. Analytic Taylor uniqueness proves $\boxed{{\displaystyle {\mu }_s[w_3]=\sum _h{e}^{-hs}\sum _{j=0}^N\frac{(-s{)}^j}{j!}{\lambda }_{h,j}[w_3]\mathrm{.}}}$ The equality holds on one neighborhood independent of the inner tests. The coefficient family is unique by the same real-exponential lemma, so lambda_h is linear in the exterior vectors whenever all expressions exist: compare the expansion for a sum of two exterior vectors with the sum of their expansions on the intersection neighborhood. In particular all finite exterior linear combinations and nilpotent components used below are well defined. There is also one FINITE auxiliary-degree support for ALL lambda_h with these fixed exterior data, not just a separate finite support for each h. Decompose w4' and w3 into their finitely many auxiliary degrees. Product grade compatibility makes mu_s vanish on every inner homogeneous test outside the corresponding finite set of total degrees, for every s. Real-exponential uniqueness applied to each such test makes every lambda_(h,j) vanish there too. This proves the uniform auxiliary support used in the integer-string truncation argument below. It is not inferred by taking a potentially infinite union of the individual supports.

Fix a homogeneous state v in V of integer weight d. The opposite-coordinate state has a FINITE expansion $A(x)v=e^{xL(1)}(-x^{-2}{)}^{L(0)}v={\sum }_m{x}^m{v}_m,\mathrm{wt}(v_m)+m=-d\mathrm{.}$ Finiteness holds for a strongly graded Möbius algebra as well: L(1) preserves the auxiliary degree and lowers integer weight, and that fixed degree is lower truncated. No conformal vector or CFT-type grading is used here. In the last-input Ward term the conjugated field is $u^{-L(0)}{Y}_3(A(x)v,y)u^{L(0)}{w}_3=Y_3(u^{-L(0)}A(x)v,u^{-1}y)w_3\mathrm{.}$ Put $y=x{0}^{-1}$. Its summand with mode index l-1 is $u^{l-\mathrm{wt}(v_m)}{x}_0^l{x}^m{w}_{l,m},w_{l,m}=(v_m{)}_{l-1}{w}_3\mathrm{.}$ For each of the finitely many m, module lower truncation bounds l above by one integer L. The residue in y selects only $l\ge 1$. Thus only the FIXED FINITE exterior family $w_{l,m},1\le l\le L$, occurs, independent of every final x power. For these finitely many states choose the intersection of their deformation neighborhoods and the maximum of their finite nilpotence bounds. These choices remain independent of inner tests and of the x power. The exterior delta kernel is $K_{ub}(x,y)=x\delta \text{\hspace{0.17em}⁣}\left(\frac{-ub+y}{x^{-1}}\right)={\sum }_{r\in \mathbb{Z}}{\sum }_{q\ge 0}\left(\genfrac{}{}{0px}{}{r}{q}\right)(-ub{)}^{r-q}{y}^q{x}^{r+1},$ with expansion in nonnegative powers of y. Multiplication by $x{0}^l=y^{-l}$ and taking Res_y requires $q=l-1$. For TOTAL power $x^p,r=p-m-1$. The residue coefficient is therefore $\left(\genfrac{}{}{0px}{}{p-m-1}{l-1}\right)(-ub{)}^{p-m-l}\mathrm{.}$ Multiplying by the conjugation factor above gives $u^{p-m-wt(v_m)}=u^{p+d}$. All l,m dependence CANCELS from that exponent. Put $C_{p,l,m}=\left(\genfrac{}{}{0px}{}{p-m-1}{l-1}\right)(-b{)}^{p-m-l}\mathrm{.}$ Substitute the actual spectral deformation from part 1 of this proof, and reindex its real exponent nu by $\nu =h+p+d$. The coefficient of $x^p$ in the exterior term is $H_s(x){\mathrm{\mid }}_{x^p}={\sum }_h{e}^{-hs}{\sum }_j\frac{(-s{)}^j}{j!}{H}_{h,j,p},$ where $\boxed{{\displaystyle H_{h,j,p}=\sum _{1\le l\le L,m}{C}_{p,l,m}(D-(h+p+d){)}^j{\lambda }_{h+p+d}[w_{l,m}]\mathrm{.}}}$ These H coefficients are INDEPENDENT OF s. Omitting the u power in the kernel would leave parameter-dependent alleged expansion coefficients and invalidate coefficient uniqueness. Each sum above is absolutely convergent, being a finite linear combination of the assumed double expansions after a real-index shift.

Write $g_{h,j}(x)=Y_{P(a)}^{\mathrm{\prime }}(v,x){\lambda }_{h,j}[w_3],H_{h,j}(x)={\sum }_p{H}_{h,j,p}{x}^p\mathrm{.}$ For fixed h, every term lambda_(h+p+d)[w_(l,m)] belongs to $W_{all}[w_{l,m}]$ in the fixed integer string h+d+Z. Its auxiliary support lies in a finite set: first decompose the exterior states and w4' into their finitely many auxiliary components. Degree compatibility fixes the degree of the resulting functional from these exterior degrees and the degree of the product, independently of p. The all-terms space is lower truncated in each such degree/string. There are finitely many exterior states and degrees. Hence there is an integer $P_h$ such that $H_{h,j,p}=0(p<P_h),$ simultaneously for all inner tests and all finitely many j. Nilpotent powers preserve the weight and auxiliary degree, so they obey the same bound. This uses the generated space containing ALL expansion terms for each exterior state; it does not follow from a separate grading restriction on each single lambda_h. It uses no finite-coset or dual-C1 assertion. The deformed product Ward residue supplies ${\sum }_h{e}^{-hs}{\sum }_j\frac{(-s{)}^j}{j!}(g_{h,j}(x)+H_{h,j}(x))=R_s(x)\mathrm{.}$ Here $R_s$ is the full output-dual field, with the Möbius-coordinate substitution $x⟼{\varphi }_{ub}(x)=\frac{x}{1+ubx}\mathrm{.}$ More explicitly, with B(x)=A(x), the transformed output state is $C_{ub}(x)v=B(x^{-1}+ub)B(x)v=e^{-ub(1+ubx)L(1)}(1+ubx{)}^{-2L(0)}v\mathrm{.}$ $C_{ub}(x)v={\sum }_{j\ge 0}\frac{(-ub{)}^j}{j!}(1+ubx{)}^{j-2d}L(1{)}^{j}v\mathrm{.}$ Each sum is finite in j, and its scalar coefficients are power series in x with no negative powers. $R_s$ is evaluation of the actual field Y4'(C_(ub)(x)v,phi_(ub)(x)) on exp(-sL0')w4' through the same product. The formal derivation of this identity is the actual two-vertex Jacobi calculation, not a recurrence assumption. Its output term is obtained by moving the current across I2 then I1; the two occurrences on the M edge cancel. The remaining residues are the two inner residues defining Y'P(a), the exterior residue K(ub) computed above, and the output residue $R_s$. The finite rational-action and shifted-output identities are F2, Proof 2.2 and 3.2, and its coefficient delta-fusion estimate is Proof 5.1. F4, Proof 1.1–4.1, proves the actual product and iterate Ward premise, including the intermediate-completion majorants. The additional delta convolutions will be justified by the absolute-majorant argument following the truncation calculation. There is ONE lower Laurent cutoff p0 for $R_s$ as a functional, independent of s near zero and of the inner tests. To verify this assertion, the current state sent to the output has the finite-state expression $(-x_0^2{)}^{L(0)}{e}^{-x_{0}L(1)}A(x)v,x_0={\varphi }_{ub}(x)\mathrm{.}$ It is a finite linear combination of fixed states with Laurent-rational x coefficients having one finite pole order. The vectors exp(-sL'(0))w4' span a finite space. Each corresponding actual output-dual field is lower truncated on each vector in a finite basis of that space. Take the maximum of these finitely many cutoffs and the finite state-coefficient pole orders. Substitution by phi, whose x order is one, preserves this common lower cutoff. The maps I1,I2 and all inner tests are then merely evaluations of these output-dual vectors; they cannot introduce a lower x power. Consequently for $p<p0$ the real-exponential series with coefficients $[g_{h,j}+H_{h,j}]{\mathrm{\mid }}_{x^p}$ is identically zero in s. Each coefficient series is absolutely convergent: the g coefficients are obtained from the expansion of mu_s by the finite input-mode evaluations in the definition of Y'; the H coefficients were treated in part 2 of this proof. The REAL-exponent uniqueness lemma therefore gives $[g_{h,j}(x)+H_{h,j}(x)]{\mathrm{\mid }}_{x^p}=0(p<p_0)\mathrm{.}$ All equations are equations of functionals, as the same neighborhoods and p0 work for every inner test. Let f_(h,j)=g_(h,j)+H_(h,j). Every f_(h,j) has the COMMON lower cutoff p0; every H_(h,j) has the fixed-h cutoff $P_h$. Subtracting proves that g_(h,j) has lower cutoff $min(p0,P_h)$, again as a functional. In particular the required P(a)-lower truncation for lambda_h is now proved.

Let ${\mathrm{\Delta }}_a(x,q)=q^{-1}\delta \text{\hspace{0.17em}⁣}\left(\frac{x^{-1}-a}{q}\right)\mathrm{.}$ For a fixed coefficient $x^A{q}^B$, put r=-B-1 and $P=A-B-1$. In its product with a Laurent series $T={\sum }_p{T}_p{x}^p$ the coefficient is ${\sum }_{k\ge 0}\left(\genfrac{}{}{0px}{}{r}{k}\right)(-a{)}^k{T}_{P-k}\mathrm{.}$ If $r\ge 0$ the sum terminates. If $r<0$, the bound $\mid \left(\genfrac{}{}{0px}{}{r}{k}\right)\mid =\left(\genfrac{}{}{0px}{}{k-r-1}{-r-1}\right)\le C_r(1+k{)}^{-r-1}$ follows by writing the fixed-degree factorial product explicitly. For fixed $l\ge 1,m$, the other binomial factor binom(P-k-m-1,l-1) is a polynomial of degree l-1 in k, and hence bounded by C_(P,l,m)(1+k)^(l-1). For the exterior coefficients in part 2 of this proof the SUM over h of absolute exponential coefficients, after reindexing $\nu =h+P-k+d$, is bounded by a finite constant times $\mathrm{\mid }u{\mathrm{\mid }}^{P-k+d}{\sum }_{\nu ,j}\mathrm{\mid }{u}^{-\nu }\mathrm{\mid }\frac{\mathrm{\mid }s{\mathrm{\mid }}^j}{j!}\mid {\lambda }_{\nu ,j}[w_{l,m}](w_1,w_2)\mid \mathrm{.}$ The latter sum is finite and locally uniformly bounded on compact s sets by the real-exponential interpolation lemma. The remaining scalar factor from C_(P-k,l,m) has magnitude |b|^(P-k-m-l). Thus the FULL absolute sum over k,h,j is bounded by a finite sum, over the fixed exterior family, of $C{\sum }_{k\ge 0}(1+k{)}^M{\mid \frac{a}{ub}\mid }^k{\sum }_{\nu ,j}\mathrm{\mid }{u}^{-\nu }\mathrm{\mid }\frac{\mathrm{\mid }s{\mathrm{\mid }}^j}{j!}\mid {\lambda }_{\nu ,j}[w_{l,m}](w_1,w_2)\mid \mathrm{.}$ Shrink the s neighborhood so that $\mathrm{\mid }a\mathrm{/}(ub)\mathrm{\mid }\le \rho <1$. The geometric-polynomial sum is finite, uniformly on compact s sets: applying (rho d/drho)^M to 1/(1-rho), or multiplying finitely many elementary geometric sums, proves this directly. This establishes absolute coefficient convergence of Delta H, including the intermediate weight index and the binomial index together. It is not an assertion merely about a previously grouped scalar sum. For Delta f each coefficient uses only finitely many k, uniformly in h,j, because f_(h,j) has the common lower cutoff p0. Its weight sums are therefore absolutely convergent finite combinations of coefficient sums already proved. Since $g=f-H$, the coefficient sums of Delta g are differences of two absolutely summable families. Hence they are absolutely summable, and ${\mathrm{\Delta }}_a{\sum }_{h,j}{e}^{-hs}\frac{(-s{)}^j}{j!}{g}_{h,j}={\sum }_{h,j}{e}^{-hs}\frac{(-s{)}^j}{j!}{\mathrm{\Delta }}_a{g}_{h,j}\mathrm{.}$ For each fixed h,j the product Delta g is also its usual formal product: the just-proved lower truncation makes its coefficient sums finite. Every exchange of the k and weight sums is now justified by the explicit absolute majorant. For an arbitrary real indexing family absolute summability first reduces its nonzero evaluation support to countable support; finite exterior families and countably many k preserve that reduction. No countability of the entire category's weights is assumed.

Here is the actual scalar derivation of the partial identity, to avoid assuming it from a theorem about partial evaluations. The companion rational Ward proof constructs the scalar rational current Q(T) with local leaf expansions equal to the actual current actions. Translate the pair origin by ub. Its poles now lie at a,0,-ub. In the raw current variable y, the finite two-point residue formula for $Y_P^{\mathrm{\prime }o}(a)$ identifies its scalar series with the Laurent expansion of Q(y+ub) on |a|<|y|<|ub|. To check the expansion explicitly, split Q into its finite principal parts and Laurent polynomial: the input at zero gives its inside expansion, and the positive input modes at a add the difference between the outside and inside expansions of the principal part at a. The exterior pole at -ub remains in its Taylor expansion throughout this annulus. These operations involve only finite principal-part families. The exterior Taylor series is absolutely convergent there by the ordinary binomial formula. The scalar Jacobi delta relation for the two local input expansions and this middle-annulus expansion can now be verified term by term in that partial-fraction decomposition. For poles at a or 0 and Laurent monomials it is the usual integer-binomial delta identity, obtained by expanding (y-a)^n in its two regions. For an exterior pole term (y+ub)^(-l), expand in y/(ub); shifting from the local origin zero to a gives binomial coefficients bounded by a fixed polynomial times |a/(ub)|^k. Thus its coefficient convolution converges absolutely; either successive expansion gives the same rational Taylor coefficients because |a|<|ub|. Differentiating the geometric identity for $l=1$ supplies every finite l, with the same polynomial-times-geometric bounds. This is also the scalar delta-fusion proof in the local formal-action supplier part 5 of this proof. Inverting $y=x^{-1}$ and applying the FINITE state transformation A(x)v gives exactly the current-definition ordinary transported identity below. Its additional weight-index exchanges are the bounds proved in part 4 of this proof, not an assumed convergence of an untruncated field. Therefore the two-vertex Jacobi calculation gives the deformed partial identity ${\tau }_{P(a)}({\mathrm{\Delta }}_a(x,q)Y_t(v,x)){\mu }_s={\mathrm{\Delta }}_a(x,q)Y_{P(a)}^{\mathrm{\prime }}(v,x){\mu }_s\mathrm{.}$ On the left, every fixed coefficient is a FINITE sum of evaluations of mu_s on input-mode vectors: A(x)v has finitely many terms and input module lower truncation removes the otherwise nonnegative mode tail. Thus its weight series may be brought outside coefficientwise. On the right, that interchange is exactly part 4 of this proof, with the full $g=f-H$ decomposition; it cannot be justified by weak convergence at $s=0$ alone. Consequently the identity becomes ${\sum }_{h,j}{e}^{-hs}\frac{(-s{)}^j}{j!}{\tau }_{P(a)}({\mathrm{\Delta }}_a{Y}_t(v,x)){\lambda }_{h,j}={\sum }_{h,j}{e}^{-hs}\frac{(-s{)}^j}{j!}{\mathrm{\Delta }}_a{Y}_{P(a)}^{\mathrm{\prime }}(v,x){\lambda }_{h,j},$ with absolute convergence of each scalar coefficient on both sides. All its expansion coefficients are independent of s. Real-exponential uniqueness, applied to each fixed formal monomial and each fixed test pair, now gives $\boxed{{\displaystyle {\tau }_{P(a)}({\mathrm{\Delta }}_a{Y}_t(v,x)){\lambda }_h={\mathrm{\Delta }}_a{Y}_{P(a)}^{\mathrm{\prime }}(v,x){\lambda }_h\mathrm{.}}}$ The truncation assertion was proved in part 3 of this proof. These two assertions are precisely full P(a)-compatibility, using the current definition's delta and transported-field convention. They hold for every homogeneous v and hence for all v by linearity. The local lem-cft-strong-local-grading-and-cyclic-module-admission supplier then uses the EXPLICIT injective grading/mode-preserving ambient embeddings from the expansion condition to place the algebraic sum of generated coefficient spaces in an actual doubly graded generalized module. The all-terms grading supplies finite dimensional components and string lower truncation. This use of an assumed ambient embedding is not a deduction of category membership from the grading conditions alone; the analytic proof here supplies full COMP, a separate obligation from vertex Jacobi.

Start now with the actual ITERATE $G(a,b;w_4^{\mathrm{\prime }},w_1,w_2,w_3)=⟨w_4^{\mathrm{\prime }},{\mathcal{Y}}^1({\mathcal{Y}}^2(w_1,a)w_2,b)w_3⟩,$ and put $c=a+b$. Assume |c|>|b|>|a|>0 as before. Fix w1 and write nuw1=G(a,b;w4',w1,w2,w3). Use the corrected P^(1)(b) expansion and its all-terms local-grading restriction for these two inner arguments. Set D1=L'P(b)(0), transposed by $B_1=(L(0)+bL(-1))\otimes 1+1\otimes L(0)$ on W2 tensor W3. The required deformation is NOT obtained by keeping the old iterate separation a fixed. It is ${\nu }_s[w_1](w_2,w_3)=⟨u^{-L^{\mathrm{\prime }}(0)}{w}_4^{\mathrm{\prime }},{\mathcal{Y}}^1({\mathcal{Y}}^2(u^{L(0)}{w}_1,uc-b)w_2,b)w_3⟩,u=e^s\mathrm{.}$ There is a common small neighborhood of $s=0$ where |b|>|uc-b|>0 and |uc|>|b|, so this iterate has its actual convergent germ there. The L(-1) bracket of the INNER intertwiner gives the exact test identity $G(w_1,L(-1)w_2,w_3)=({\mathrm{\partial }}_b-{\mathrm{\partial }}_a)G(w_1,w_2,w_3)\mathrm{.}$ Indeed commuting L(-1) past the inner intertwiner produces its derivative in a, whereas L(-1) on its output produces the outer derivative in b. The iterate L(0) identity has coordinate term a partial_a+b partial_b. Differentiating nu_s and canceling the first-input and output L(0) terms therefore gives ${\nu }_s^{\mathrm{\prime }}=-G_s(L(0)w_2,w_3)-G_s(w_2,L(0)w_3)-b({\mathrm{\partial }}_b-{\mathrm{\partial }}_a)G_s=-{\nu }_s\circ B_1\mathrm{.}$ The same Taylor/interpolation proof of part 1 of this proof, now with exterior vector w1, proves the ACTUAL equality ${\nu }_s={\sum }_h{e}^{-hs}{\sum }_j\frac{(-s{)}^j}{j!}(D_1-h{)}^j{\eta }_h[w_1],$ where eta_h are its assumed P^(1) partial coefficients. This proof requires no transformation of generalized weights or auxiliary gradings by an unproved skew-symmetry identification. The inner pair W2,W3 sits at positions b and 0; the exterior first input sits at +uc. Thus the exterior kernel, in its local coordinate y, is $K_{uc}^{+}(x,y)=x\delta \text{\hspace{0.17em}⁣}\left(\frac{uc+y}{x^{-1}}\right)\mathrm{.}$ For $A(x)v={\sum }_m{x}^m{v}_m$, the same conjugation moves the first-input current past $u^{L(0)}$ and gives factor $u^{l-wt(v_m)}$, with exterior states $(v_m{)}_{l-1}w1$. The residue selects $1\le l\le L$, and its $x^p$ coefficient is binom(p-m-1,l-1)(uc)^(p-m-l). Its FULL u power is again p+d. Consequently the reverse correction has the EXACT s-independent coefficients $H_{h,j,p}^{(1)}={\sum }_{1\le l\le L,m}\left(\genfrac{}{}{0px}{}{p-m-1}{l-1}\right)c^{p-m-l}(D_1-(h+p+d){)}^j{\eta }_{h+p+d}[(v_m{)}_{l-1}{w}_1]\mathrm{.}$ This also fixes the sign convention: the exterior position is +uc, whereas it was -ub after the forward pair-origin translation. The reverse inner pair origin is already the global origin, so its full output term is the actual field Y4'(v,x) applied to $u^{-L^{\mathrm{\prime }}(0)}w{4}^{\mathrm{\prime }}$; it requires NO coordinate translation phi. It has the common output-dual lower cutoff directly. The two-vertex iterate Jacobi, with the first-input current retained as the exterior residue, gives g^(1)+H^(1)=R^(1) and the corresponding partial Delta_b identity. These are the same residue calculation with the explicit locations (external uc; inner b,0), not an assertion of unknown path transport. For each fixed h, the all-terms grading of the finitely many exterior states now places eta(h+p+d) in a fixed h+d+Z string and finitely many auxiliary degrees. Its lower truncation makes H^(1) vanish at sufficiently negative p. The common output cutoff and REAL-exponent uniqueness then make f^(1)=g^(1)+H^(1) have a common lower cutoff, exactly as in part 3 of this proof. Subtracting H^(1) proves lower truncation of each transported field on eta_h as a functional. Finally multiply by Delta_b. At $x^A{q}^B$ its binomial index k still uses $p=P-k$ and the same two fixed-degree binomial polynomials. Its full absolute bound is now the explicit convergent series $C{\sum }_{k\ge 0}(1+k{)}^M{\mid \frac{b}{uc}\mid }^k{\sum }_{\nu ,j}\mathrm{\mid }{u}^{-\nu }\mathrm{\mid }\frac{\mathrm{\mid }s{\mathrm{\mid }}^j}{j!}\mid (D_1-\nu {)}^j{\eta }_{\nu }[(v_m{)}_{l-1}{w}_1](w_2,w_3)\mid \mathrm{.}$ Shrink s so $\mathrm{\mid }b\mathrm{/}(uc)\mathrm{\mid }<\rho <1$. This justifies all coefficient exchanges and real-exponent extraction for the reverse partial Delta_b identity. Its $j=0$ coefficient proves full P(b)-compatibility of every eta_h. This completes the reverse analytic passage without any finite-coset premise.