Braided monoidal structure from HLZ analytic hypotheses

Fix the canonical restricted-dual models of the $P(z)$-products. F2 gives them at every nonzero $z$ with the fixed endpoint logarithms, from the one-point existence in A1. F6 realizes each as $S_z^{\mathrm{\prime }}$, where $S_z$ is the sum of images of transposed maps into the two-input dual. To check its category membership, universality identifies $S_z$ with the image of the canonical map's transpose from an object of $\mathcal{C}$; ambient-image closure puts that image in $\mathcal{C}$, and contragredient closure puts its dual there. Thus no undeclared globally grading-restricted or CFT-type theorem is used. F6 also proves that the homogeneous components of each universal tensor value span its representing object. The bifunctor on morphisms is the unique one obtained by applying universality to the two input maps and the universal map; identity and composition follow from uniqueness.

The triple clause of A3, with the third intertwiner a module action and its input the vacuum, supplies convergence and continuation for binary products, since $Y_W(\mathbf{1},z)$ is the identity. For an iterate let $\mathcal{U}={\mathrm{\Omega }}_{-r-1}({\mathcal{Y}}^1)$ be the inverse skew transform of its outer operator, so ${\mathcal{Y}}^1={\mathrm{\Omega }}_r(\mathcal{U})$. F9 gives, coefficientwise with the matching odd-$\pi i$ branch, $⟨f,{\mathcal{Y}}^1({\mathcal{Y}}^2(w_1,z_0)w_2,z_2)w_3⟩=⟨e^{z_2{L}^{\mathrm{\prime }}(1)}f,\mathcal{U}(w_3,e^{(2r+1)\pi i}{z}_2){\mathcal{Y}}^2(w_1,z_0)w_2⟩\mathrm{.}$ The exterior exponential is a finite sum: $L^{\mathrm{\prime }}(1)$ lowers weight in each of the finitely many fixed auxiliary-degree integer strings supporting $f$. The expression on the right is an actual binary product in its radial region $\mathrm{\mid }{z}_2\mathrm{\mid }>\mathrm{\mid }{z}_0\mathrm{\mid }>0$. Its intermediate weight projections are the original iterate's projections, by the coefficientwise skew identity before summation. Thus its finite sum of absolute product bounds proves absolute iterate convergence on the entire required region, with no sector extension. The same identities with finitely many leaf or exterior nilpotent components and the real-exponential normality estimates in F7 give analytic branches and legitimate derivatives. Continuation fixes the matching branches on common overlaps.

Take the unit object $V$ from A2. F9 gives $P(z)$-maps with types $(W;V,W)$ and $(W;W,V)$ from the module action and its skew transform. Universality gives $l_z:V{⊠}_{z}W\to W$ and $r_z:W{⊠}_{z}V\to W$. In the first case the universal intertwiner on $\mathbf{1}$ is constant in its sewing coordinate by the derivative identity, so $k_L(w)=\mathbf{1}{⊠}_{z}w$ is an actual vector, not only a completion. F9 gives $Y_{V⊠W}(v,x)k_L(w)=\mathcal{Y}(v,x)w$, so $k_L$ is a module map and $l_z{k}_L=1$; binary component spanning gives $k_L{l}_z=1$. For the right unit F9 gives $\mathcal{Y}(w,x)\mathbf{1}=e^{xL(-1)}{k}_R(w)$ with an actual constant coefficient $k_R(w)$, and comparison with the skew module action gives $r_z{k}_R=1$ and $k_R{r}_z=1$ on spanning components. Thus both maps are isomorphisms. Their naturality follows directly from their pairings, and $l=l_1,r=r_1$ are the desired unitors.

F9 also gives a $P(z)$-map to $W_2{⊠}_{-z}{W}_1$ by $w_1,w_2\mapsto e^{zL(-1)}(w_2{⊠}_{-z}{w}_1)$, with the odd-$\pi i$ logarithm chosen to match the fixed endpoint evaluation. It gives a natural map $R_z:W_1{⊠}_z{W}_2\to W_2{⊠}_{-z}{W}_1$. Use the inverse odd-$\pi i$ branch in the reversed skew transformation; the two translation factors cancel and F9's inverse skew identity makes the two composites fix universal values. Binary spanning proves invertibility. Define $c_{X,Y}=T_{{\gamma }_1^{-}}{R}_1$, where ${\gamma }_1^{-}$ is the path from $-1$ to1 in the closed upper half-plane avoiding0, with the backwards logarithmic lift prescribed by F2. This is the HLZ upper-half-plane braiding convention; fixed endpoint logarithms, rather than a new unrecorded branch at $-1$, determine its evaluation. Naturality and invertibility follow from those of $R_z$ and transport.

Apply F7 to each corrected partial expansion in A3. The finite rational Ward identity comes from the actual two intertwiner Jacobi identities, not an assumed conformal-block theorem. Its precise scale reindex and the all-terms grading restriction give one functional lower cutoff; geometric-binomial majorants justify multiplication by the full compatibility delta before real spectral uniqueness. Thus every term is fully compatible in both directions. Its given ambient cyclic realization and the local grading lemma give an actual strongly graded cyclic module $E$ injected in $X\in \mathcal{C}$. Full compatibility propagates from its generating term to $E$, as follows. In opposite coordinates let $\tau (v,f)$ be F7's finite rational-current anti-action on the two-input dual, and let $\sigma (v,f)={\mathrm{Res}}_t({\iota }_{\mathrm{\infty }}f)(t)Y_E^o(v,t)$ be the actual module's anti-action. The latter residue is finite, since $f$ has finitely many positive powers at infinity and the opposite field is upper truncated. Compatibility on the generating term says $\tau (v,f)\lambda =\sigma (v,f)\lambda$ for every rational $f$ with poles at0 and$z$: the delta coefficients give the principal parts at$z$, and Laurent monomials give the rest by partial fractions. The two anti-actions agree identically on Laurent generators by the definition of the transported field. Their common rational-current commutator is the finite sum of state products and derivatives of $f$. If they agree on a vector for every rational current, commuting one Laurent generator past that current proves they agree on the next generated vector; induction on word length proves equality on $E$. The same Möbius commutators handle the three additional generators. Thus every vector of $E$ is fully compatible, with its lower truncation supplied by its actual strong grading. The embedding $E\hookrightarrow X$ has a restricted transpose $X^{\mathrm{\prime }}\twoheadrightarrow E^{\mathrm{\prime }}$: every finite-component functional extends to the corresponding finite-dimensional component of $X$, proving surjectivity. Strong duality makes $E^{\mathrm{\prime }}$ an actual generalized module; ambient-image closure puts it in $\mathcal{C}$. No membership of an arbitrary ungenerated ambient subspace is inferred. Transpose the fully compatible inclusion $E\hookrightarrow (W_1\otimes W_2{)}^{\ast }$ to its $P(z)$-map with target $E^{\mathrm{\prime }}\in \mathcal{C}$. Consequently every term belongs to the actual image sum $S_z$. All hypotheses of F7's spectral factorization theorem are now satisfied. It supplies the outer $P(z_2)$-map for a product and the outer $P(z_1)$-map for an iterate, with full Jacobi and all three Möbius identities. This is the actual local product-iterate comparison; neither formal universality nor path transport alone is being used to create it.

For $\mathrm{\mid }{z}_1\mathrm{\mid }>\mathrm{\mid }{z}_2\mathrm{\mid }>\mathrm{\mid }{z}_0\mathrm{\mid }>0$, let $L=(W_1{⊠}_{z_0}{W}_2){⊠}_{z_2}{W}_3$ and $R=W_1{⊠}_{z_1}(W_2{⊠}_{z_2}{W}_3)$. Apply the product factorization of step 3.1 to the two universal maps defining $R$; outer universality gives a module morphism $A_{z_1,z_2}^{-1}:L\to R$ sending the nested iterate universal value to the product universal value. Apply reverse factorization to the universal iterate defining $L$; universality gives $A_{z_1,z_2}:R\to L$ with the reverse characterization. Both compositions fix the relevant nested universal values. Binary convergence from step 2.1 and generic real-weight spanning in F8 make their homogeneous components spanning in $L$ and $R$, so both compositions are identities. Naturality follows from the unique factorization, checked on those same spanning values. This constructs the associativity isomorphism itself.

Set $⊠={⊠}_1$. To define the usual left-to-right associator ${\alpha }_{X,Y,Z}:(X⊠Y)⊠Z\to X⊠(Y⊠Z)$, use geometric parameters $(z_1,z_2,z_0)=(3,2,1)$: transport the outer product on the left from1 to2, apply $A_{3,2}^{-1}$, then transport its inner product from2 to1 and its outer product from3 to1 along positive-real segments. The inner left product already has parameter1. F2 naturality makes these transports commute with the input maps. Different positive geometric parameters give the same morphism after those transports: on the common radial overlap step 4.1 preserves the scalar analytic germ, and continuation through the connected positive chamber preserves equality; the positive paths have unique fixed logarithmic lifts. Thus this definition is compatible with every local comparison used below, not just the chosen numbers3,2,1.

We prove the pentagon on actual four-leaf universal values. Choose positions $(z_1,z_2,z_3,0)=(7,6,4,0)$, so $z_{12}=1,z_{23}=2,z_{13}=3$. All five regions in F8's tree-expansion lemma hold: $7>6>4$, $4>2$ and $7>4+2$, $4>3>2$, $2>1$ and $4>2+1$, and $6>1+4$. By that lemma their double projection sums converge absolutely, and every adjacent binary comparison remains its actual analytic equality when its third input is a completed inner universal value. Following either route around the geometric pentagon therefore gives the same fully reassociated universal value, with the same leaf positions and continued branches. The homogeneous components of the source tree span by F8, so the two composite morphisms are equal. Transporting all edge lengths to1 along positive-real segments gives the exact pentagon for $\alpha$ in F4: the local comparison/transport squares commute by step 5.1. This proves the pentagon without invoking an external coherence theorem.

The triangle follows from the vacuum and translation, with an explicit geometric check. For the triple $X,V,Y$ use positions3,2,0. Its left tree is $(x{⊠}_1\mathbf{1}){⊠}_{2}y$. First applying the right unit to its inner factor gives $e^{L(-1)}x$ in the outer $P(2)$ map. The derivative identity makes this exactly the continued value of the outer universal intertwiner with input $x$ at position3. The other route first applies $A_{3,2}^{-1}$ to get $x{⊠}_3(\mathbf{1}{⊠}_{2}y)$, then the left unit gives the same value at position3. The exponential input substitution is legitimate after pairing with the actual restricted dual: its translation identity is the convergent Taylor identity on a small overlap, and then analytic continuation; no unrestricted vector-space exponential is used. Nested spanning proves equality of morphisms. Positive transport to1 therefore gives $(1_X⊠l_Y){\alpha }_{X,V,Y}=r_X⊠1_Y$, exactly F4's triangle.

We next explain rigorously why the maps just constructed are the maps of their insertion paths. The local associativity map in step 4.1 preserves the binary scalar germ by construction; step 6.1's local-move lemma extends this to a spectator intertwiner with all absolute sums controlled. F9's skew identity is exactly interchange of the two labeled insertions followed by translation of the origin, and F2 is the stated logarithmic path continuation. On a small coordinate neighborhood these characterizations determine the maps on spanning universal components. Along any finite path cover use finitely many overlapping coordinate neighborhoods and analytic uniqueness to compose them. Endpoint-fixed homotopic paths give the same continued scalar germs: subdivide a homotopy square into sufficiently small rectangles landing in such neighborhoods; continuation round each small rectangle is identity by local uniqueness, and the interior edges cancel. Spanning then gives equality of the corresponding module maps. This proves the needed path rule for the two- and three-insertion configurations from local analytic continuation itself, rather than assuming that binary logarithm transport already knows multivariable coherence.

The two hexagons are block-exchange path equalities for three labeled inputs. We first fix the orientation from step 2.3, rather than inferring it from the word upper. At two normalized positions, a representative from $(p_X,p_Y)=(1,0)$ to $(0,1)$ is $p_X(t)=\frac{1+e^{-\pi it}}{2},p_Y(t)=\frac{1-e^{-\pi it}}{2},0\le t\le 1.$ The relative coordinate $p_Y-p_X=-e^{-\pi it}$ traverses the upper semicircle from $-1$ to $1$, exactly ${\gamma }_1^{-}$; the first label $X$ therefore passes below the second label $Y$. Translation of the common origin does not change these relative coordinates. For $c_{X,Y⊠Z}$, the block path passes $X$ below both $Y,Z$; the successive path passes it below $Y$ and then below $Z$. Choose both representatives, after harmless small spectator displacements, with $\mathrm{Im}{p}_X<\mathrm{Im}{p}_Y<\mathrm{Im}{p}_Z$ at every interior time. For $c_{X⊠Y,Z}$, both representatives pass $X,Y$ below $Z$ and can use the same strict height order. Such representatives are obtained by lifting the labels to three distinct heights, moving their real coordinates to the required distinct final positions while retaining those heights, then lowering them. The lifts and lowering keep distinct real coordinates until the heights separate or after the real coordinates have reached their distinct endpoints. Dividing the real-coordinate motions into the successive exchanges gives the individual-braiding representative; doing them together gives the block representative. Any auxiliary displacement of a spectator pair whose real order is unchanged is contractible without collision, so it introduces no spectator braid. For fixed endpoints, the pointwise linear interpolation of the two representatives stays in the convex region of this strict imaginary order at every interior time, and hence gives an endpoint-fixed collision-free homotopy. If a leaf is used as the coordinate origin, translate the entire homotopy by its position; distinctness and relative heights are preserved. Step 7.1 equates the associated module maps. Reading the three binary regroupings along each path gives precisely ${\alpha }_{Y,Z,X}{c}_{X,Y⊠Z}{\alpha }_{X,Y,Z}=(1_Y⊠c_{X,Z}){\alpha }_{Y,X,Z}(c_{X,Y}⊠1_Z)$ and ${\alpha }_{Z,X,Y}^{-1}{c}_{X⊠Y,Z}{\alpha }_{X,Y,Z}^{-1}=(c_{X,Z}⊠1_Y){\alpha }_{X,Z,Y}^{-1}(1_X⊠c_{Y,Z})\mathrm{.}$ Composition is read right to left. These are the exact two equations in F4. The relative-coordinate check above makes every crossing agree with the upper-half-plane convention in step 2.3; no unjustified cancellation of a double braiding is used.

Finally $\mathcal{C}$ is additive: fullness makes each hom-set a complex vector space with bilinear composition, and A1 supplies zero and finite direct sums. The $P(1)$ bifunctor is additive in each input, because a map from a direct sum of inputs to a completed output is a pair of $P(1)$-maps and universality represents this pair by the direct sum of the two representing products; uniqueness identifies the comparison maps and proves preservation of sums of morphisms. The preceding constructions and coherence checks give the natural associator, unitors, braiding and all required coherence identities, so F4's definition is satisfied. If $\mathcal{C}$ is additionally abelian this is an abelian braided monoidal category, by that same definition. No step constructs evaluation/coevaluation maps or their triangle identities, so no rigidity conclusion has been added. This proves the entire preserved Statement by local constructions.