Knot cobordism maps and functoriality

By the assumed functoriality of $F$, gluing cobordisms is sent to composition of maps and disjoint union is sent to tensor product. Thus the only additional relations to check before factoring through the quotient are the three local relations in [F2].

A sphere is evaluated by $ϵ\eta$, and $ϵ(\eta (1))=ϵ(1)=0$. A torus is evaluated by $ϵm\mathrm{\Delta }\eta$; on the generator $1$, $\mathrm{\Delta }(1)=1\otimes X+X\otimes 1$, then $m\mathrm{\Delta }(1)=2X$, and $ϵ(2X)=2$. These are exactly the S and T relations, including when glued into any surrounding cobordism because the TQFT respects composition and tensor product.

In the four-tube relation, each side consists locally of two ways to attach one tube. Applying the TQFT to the local four-disk input $1$, either side gives

This follows by applying $\mathrm{\Delta }(1)=1\otimes X+X\otimes 1$ at each tube attachment and $ϵ(1)=0,ϵ(X)=1$ at the two cut disks. The equality on the local input implies equality as maps: tensoring with identity maps and composing with the maps for the unchanged part of the surface preserves equality. Thus the four-tube relation is respected.

Every generating local relation of ${\mathsf{C}\mathsf{o}\mathsf{b}}_{BN}$ is respected by steps 1.2 and 1.3, while the Frobenius identities respect the isotopy and gluing presentation by step 1.1. Therefore the TQFT factors through the quotient defining ${\mathsf{C}\mathsf{o}\mathsf{b}}_{BN}$, as asserted.

Treat rank zero directly. The target and all empty product tori are one point. The fixed-map operator is the zero map from the zero section space to the zero forcing space; it is onto of index zero. The three-marked source has no conformal parameter or automorphism. Hence its unique map is a regular isolated triangle with all marked degrees zero, in its assigned constant-class Spin-c extension. With another prescribed label or a nonzero degree vector the relevant set is empty. A fixed index-one triangle or universal index-minus-one triangle cannot occur. A matching one-parameter family is exactly its compact parameter interval, with its two identical endpoint counts; there are no nonconstant strips, spheres or boundary disks. Thus its differential is zero and its triangle and parameter-boundary identities hold directly. No strip translation is divided out and no empty tuple is claimed to supply a constituent surface point.

Now suppose that $m>0$ and apply [F1] with the marked points and one witness in every original region protected. The resulting representative has no triple torus intersection and has the same followed pair generators. [F2]'s actual collar extension carries the compression disks and Morse data, identifies the actual Spin-c lift and capped classes, and preserves every old strict witness coefficient under the swept-annulus lattice isomorphism. Thus the new triple has the same required strict admissibility and Spin-c indexing. Its polygon spaces represent exactly the transported literal continuous classes, not classes modulo a sphere or fundamental-group action. A constant triangle would lie in all three tori, so none occurs. This explicitly permitted auxiliary curve choice is necessary: at a constant triangle the almost-complex parameter derivative vanishes because $du=0$, so no J-only universal argument can be substituted for it.

Apply [F3] to this actual triple and its restricted pair sectors. The geometric strict-domain theorem first supplies a finite lower strip index and finite nonnegative domains in every upper-index band. Its actual auxiliary all-index regular path and bubble budget then produce the open dense good genuine-collar end data with no nonconstant index at most zero and index-one regularity. Hence the ensuing polygon limits have the required regular strip levels; a claim of index-one regularity alone would not justify that conclusion. The source-local Hilbert perturbations give actual fixed-data and one-parameter universal transversality at every nonconstant triangle, preserving the protected divisors, tori and exact product-angle end charts. There are no constant triangles by step1.2.

A triangle has three marked ends and no conformal modulus or translation action. Its fixed count index is therefore $\mu$, and a one-parameter universal count has index $\mu +1$. The domain bounds in [F2] and actual domain-area inequality in [F3] give one energy bound for each required low-dimensional set, across all its literal classes and all nonnegative degree vectors. Every limiting nonconstant strip has positive index. Every nonconstant disk bubble contributes at least two, every sphere at least $2k$, and every regular triangle vertex has nonnegative count dimension. The unconditional determinant index budget of [F3] thus excludes all bubbles in count dimensions zero and one. The only broken one-dimensional ends have one rigid triangle and one rigid strip. The same calculation with the family parameter counted ONCE gives rigid universal index-minus-one triangles and the stated universal index-zero ends. This retains $k=1$ and general intermediate $Y$; it does not borrow a knot-in-$S^3$ disk theorem outside its scope.

The actual product-chart, correction and exhaustion proof in [F3] supplies exactly one collar for every such compatible broken pair. Its global tail estimate promotes compact-core convergence to the true Sobolev chart, so a rigid total set is compact discrete and finite under Countable Choice. Its explicit Hausdorff collar compactification of a one-dimensional space is a compact one-manifold with finite boundary. Degree vectors and the actual total Spin-c structure are carried by each neck smoothing; restricting either these labels or an actual class gives the corresponding open-and-closed part. Thus neither multiple classes with the same domain nor the Spin-c extension orbit creates a missing or an extra end.

Count the boundary of this actual compact one-manifold over ${\mathbb{F}}_2$. Writing $F$ for the bilinear triangle count, the strip breaks give $dF+F(d\otimes 1)+F(1\otimes d)=0$. The marked vectors add at each actual gluing, so the component weights multiply, with every $w,z$ exponent retained. For a fixed-end one-parameter family its universal one-manifold also has the two parameter endpoints; the universal index-minus-one counts $H$ give $F_1+F_0=dH+H(d\otimes 1)+H(1\otimes d)$. These are finite weighted equations, also after specializing every $w$ color to $U$ and every $z$ color to $V$. If the intermediate input $\mathrm{\Theta }$ is an actual cycle, substituting it removes the $F(1\otimes d)$ term and gives the required one-input chain map and homotopy. Together with the direct point model in step1.1 and the raw convention [F4], this proves exactly the analytic identities. No existence or cycle property of a distinguished top input, elementary cobordism-map construction, or movie-invariance theorem has been assumed to obtain them.

Use the actual smooth traced realizations of the two planar movies, with their prescribed endpoint realizations and collars. These lie in the standard spatial chart $S^3\setminus \{\mathrm{\infty }\}$; by [F1] and the hypothesis, their oriented traces have the specified spherical isotopy relative to those collars. Apply [F2]. The comparison is a finite sequence of Euclidean isotopies and isolated sweep-around events, with the same parametrized boundary. At the intermediate cuts choose generic movies fixed on the retained collars; the same finite-jet construction in [F3] applies with these cuts protected.

On each Euclidean isotopy interval, [F3] replaces its comparison by finitely many of the fifteen local moves, distant interchanges and planar isotopies. All hidden-coordinate height orders and orientations are preserved. At each missing-arc passage retain the sweep-around supplied by [F2]; it is not silently replaced by a Euclidean move. The two boundary movies of each interval match those at the adjacent pieces, because the construction fixes the chosen cuts and collars.

Concatenate these finite lists. There are finitely many intervals and passages, so the total list is finite. Every retained frame is a regular frame of a generic traced surface, and every elementary replacement has its stated oriented local trace with the same outside and boundary data. Consequently all intermediate movies have the prescribed parametrized knots and collars. The total sequence starts and ends at the given movies and proves the claimed spherical comparison.

Choice scope. Fix the actual compact smooth cobordism, decoration, Spin-c structure and, for a comparison, the two actual decompositions. A compact smooth input has finite atlas/handle data. A Morse decomposition has finitely many critical points; a generic Cerf comparison has finitely many exceptional times (Zemke Lemma 11.18, p.80, and Theorem 11.5, pp.74 and 84–85). Choosing a finite list of charts, gradient-like fields, band neighborhoods, puncturing balls and standard auxiliary classes uses finite choice only. No choice of one decomposition for every cobordism is required: existence and pairwise comparison are quantified separately for arbitrary given inputs.

The dividing surface and boundary data meet the source category in [F1]: basepoints alternate, the dividing set separates the two regions, every three-manifold component meets the link, and every four-manifold component meets the divided surface. Closed dividing curves are allowed by its properly embedded one-manifold definition. The two-color assignment in [F3] is compatible on every elementary piece, including multicomponent and multiply based intermediate links. Its curvature is zero, so the imported curved maps are genuine maps of complexes over ${\mathbb{F}}_2[U,V]$.

Use the elementary maps of [F1] for the chosen parametrized Kirby decomposition, with [F2]'s actual auxiliary cycles and the fixed restricted Spin-c labels. Their finite polynomial counts, chain-map property and compatibility with the natural Heegaard transition maps are supplied by that external construction. Thus their composition is the displayed representative. If a component has an empty incoming or outgoing end, apply [F1]'s standard puncturing prescription before composing, not an undefined intermediate empty-link Heegaard diagram. No graph-TQFT reduction or hat-only construction is substituted for this two-variable map.

For that finite list of diagrams, the homotopy and periodic-domain data are countably indexed: use the finite-cell models of the compact manifolds and finite products of circles, and integer words in their finite attaching data; Spin-c extensions form a subset of a finitely generated integral cohomology torsor. Integer degrees, multiplicities, polygon types and neck lengths in $\mathbb{N}$ are countable indices as well. The count theorem [F4] supplies coherent perturbations relative to faces and fixed ends, rather than making independent choices at every real parameter value. Model counts and the finite move identities of [F1] use such zero- and one-dimensional polygon families; the elementary top classes in [F2] are algebraic classes on specified standard diagrams. A cycle representative, if needed, may be selected as the first representative in the explicit enumeration of finite polynomial chains over the countable ring. Nonempty single choices and finite choices require no additional axiom; countably many choices are covered by the assumed Countable Choice.

The geometric generic functions and comparison families can be chosen in the separable smooth spaces on those finitely many compact charts, with their countable derivative seminorms. Their residual conditions are indexed by countable jet charts/exhaustions, the finite singularity types and the countable curve classes; the analytic regularity choices are already furnished by [F4]. For the geometric Baire choices one may recursively select the first basic ball whose closure lies in the preceding ball and in the next dense open set, with diameters tending to zero. Completeness supplies the limit, so this step requires no choice over an uncountable set of functions. Source model relations are imported as finite geometric/count identities, with these actual CC-regular counts; their use does not require a well-ordering of all diagrams or a full-Choice selection of a global transitive system. Consequently the external proof specializes to the declared CC setting. This is a bounded choice inventory of its use here; it is not a claim that the publication itself discusses foundational optimality.

The distinguished class notation in the formula never selects a canonical cycle on every diagram: choose a representative on the actual standard auxiliary pair and carry it by the transition maps. Its boundary ambiguity is exactly the homotopy in [F2]. The local analytic triangle theorem supplies its own weighted bilinear identity, but the existence and correctness of the required elementary inputs come from [F1]–[F2]. Independence of all chosen decomposition data and the Spin-c extension sum are established in the separate naturality lemma by the external Theorems A and B.

Send the w-regions and z-regions of the given surface to the two colors $U,V$. The source ring is now exactly ${\mathbb{F}}_2[U,V]$ and its variable-power filtration is exactly the specified two-coordinate filtration. The curvature vanishes by the two-color identity. Hence [F1] applies to the actual maps of [F3] as genuine filtered equivariant chain maps, not merely as hat homology maps or uncolored counts.

For different endpoint diagrams first use [F1]'s canonical transition chain-homotopy equivalences to compare maps on the same chosen endpoint representatives. Theorem A then gives a filtered equivariant chain homotopy for any two admitted decompositions and regular auxiliary choices. The source's fourteen Cerf moves include the band, basepoint and quasi-stabilization relations; §12.5 additionally handles components missing one or both ends. Boundary-fixed ambient isotopy preserves the given decorations and induces precisely the required diffeomorphism of decorated cobordisms, so it is covered by the same theorem.

The matching boundary restrictions make the target sector of the first map the source sector of the second, after the natural intermediate-diagram identification. Apply [F2] to this typed composition and the two specified structures ${\mathfrak{s}}_1,{\mathfrak{s}}_2$. Restriction of each summand to the two pieces is exactly the condition displayed in the Statement; no unmarked or total-Spin-c map has replaced these maps. The two-color specialization respects the composition and homotopies, giving the displayed formula over ${\mathbb{F}}_2[U,V]$.

Finally $H^2(S^3\times [0,1];\mathbb{Z})=0$ and the product is spin. Its nonempty Spin-c torsor therefore consists of one element. If the chosen structures on the pieces restrict from this unique product structure, the extension set has that one element; otherwise it is empty. In the compatible case the law reduces to ordinary composition up to filtered equivariant chain homotopy. The empty-sum convention does not define a composition between different intermediate sectors. The argument imports the established naturality/composition theorems, not an unproved assertion that the local triangle identity checks all movies.

Inverse clips. For MM1–MM5, a Reidemeister move is followed by its selected inverse, possibly with a planar change of frame between them. The equations in [F2] make the composite homotopic to the identity. All reflections and compatible strand orientations use the corresponding specified local coordinate identification in [F1]; the cochain and internal shifts agree at both ends. Thus the five inverse clips satisfy their formal movie relation before any closed evaluation.

Simple base frames for the five circular clips. Each of MM6–MM10 consists solely of Reidemeister moves, so it gives a degree-zero homotopy automorphism of its starting tangle complex by [F2]. The following explicit starting frames of the models in [F7] are simple. For MM7 choose the frame before the small cusp loop: it is a single bent arc, hence a pairing. For MM8 choose the frame before the cusp crosses the third strand: its two arcs have one transverse crossing and no small loop; peeling this sole boundary crossing leaves a pairing. For MM9 choose the frame before the cubic double-curve turn: it likewise has two arcs with one crossing, which peels off at the boundary. The MM6 starting frame is two parallel arcs crossed by a third transverse arc. Its two crossings peel in order from an end of the transverse arc, leaving three disjoint arcs. More explicitly, the outermost crossing has two adjacent free legs in the disk boundary; cut a small collar rectangle containing that crossing and those legs, leaving one boundary-crossing addition to the smaller tangle. After removing it the other crossing has the same property. For MM10 the initial frame is the four small affine strand segments resolving the projected quadruple point, with six pairwise crossings. Choose a linear height not perpendicular to any segment and place upper and lower cuts beyond all six crossings. Each segment is a graph over that height. Order the six crossing heights after a small planar perturbation; the highest crossing is between adjacent outgoing strands, so its two free legs give a boundary-crossing collar rectangle. Remove it and repeat down the ordered list. This peels all six crossings and leaves a pairing, for every retained over/under order. These cuts merely parametrize the local disk and straighten its outside arcs; no crossing change is used. Thus the boundary-crossing equivalences in [F4] and [F2], beginning with [F3], prove simplicity of all five base frames. Starting the same circular clip at a different frame conjugates its endomorphism by a product of equivalences, so [F4] also gives simplicity there.

The five Morse clips. For MM11 use the explicit disk cancellation. For MM12 and MM13 use the two columns $D_a-T,D_b-T$ and the integral identity $D_a+D_b=2T$; the upward clips have their separately checked cap/saddle maps. For MM14 use its two circle-birth columns, with the reverse-order RII cube sign retained in both. For MM15 use the residual saddle column $(\alpha ,\beta {)}^T$, or its actual target-retract row in the upward version, with the middle cube permutation and last-vertex sign. These are exactly the five proved calculations of [F5]. Hence the two formal maps agree up to a single sign for every nonreversible clip as well.

Circular clips. The degree-zero homotopy automorphism of each MM6–MM10 base frame is therefore homotopic to $\epsilon 1$, where $\epsilon =1$ or $-1$. This is one scalar in the formal endomorphism group of the whole local complex, not a separate sign on its states. Changing a base frame conjugates that equation by equivalences and preserves the same assertion. A relation presented as two alternative paths is equivalent to its circular clip: compose one path with the selected inverse of the other; its homotopy inverse exists by [F2]. Multiplying the circular equation by the other path therefore makes the two path maps equal up to this one sign. This proves all five circular relations without assuming that arbitrary maps between two homology groups agree up to sign.

Outside complexes and distant events. Insert each formal equation or homotopy into the fixed planar outside by [F4]. For two distant elementary events, decompose their disjoint support disks and the outside as one planar operation. Their maps have cochain degree zero; a Morse map can have nonzero quantum degree, but the totalization signs use cochain degree alone. Acting in the first disk and then the second therefore gives the same matrix as acting in the opposite order. For an adjacent crossing transposition, conjugating the raw cube differential of [F7] by the vertex sign $(-1{)}^{v(a)v(b)}$ changes its edge sign by $(-1{)}^{v(b)}$ in coordinate $a$, by $(-1{)}^{v(a)}$ in coordinate $b$, and by $1$ in every other coordinate. These are precisely the differences between the two crossing orders. Transporting the fixed normalization shifts gives the crossing-order identification in [F1]; use it consistently for both paths, also when a Reidemeister event changes the crossing set. In particular it holds for the two distant surface-height critical events in the statement. Now all objects are closed resolutions. Apply [F6] entry by entry; its additivity, gluing and relation factorization preserve each formal chain-map equality and homotopy. The images are precisely the integral maps fixed in [F1], including empty resolutions and the identity on unaffected circles.

Pass to homology in step 3.1. A chain homotopy becomes equality, and the one scalar $\epsilon$ becomes multiplication by that same overall sign on the full integral homology group. The construction covered MM1–MM5, MM6–MM10 and MM11–MM15, all their stated variants, and the distant interchanges. Thus every relation asserted in the statement preserves the selected map up to one sign. No assertion about spherical sweep-around follows from these Euclidean local computations.

A local integral comparison category. Use oriented blue facets of label $1$ and red facets of label $2$; two blue facets meet a red facet along each seam, with the usual merge orientation. Decorations on blue facets are powers of a dot. One can define their closed evaluation by cutting at the circular seams and applying the two Frobenius algebras $B=\mathbb{Z}[X]\mathrm{/}(X^2)$, ${ϵ}_B(1)=0$, ${ϵ}_B(X)=1$, ${\mathrm{\Delta }}_B(1)=1\otimes X+X\otimes 1$, ${\mathrm{\Delta }}_B(X)=X\otimes X$, and $R=\mathbb{Z}$ with counit $-1$ and comultiplication $-1\otimes 1$. Contract a seam by ${ϵ}_B(x\bar{y})$, where the positive blue side is selected by right-hand cyclic order and $\bar{X}=-X$. Cut-independence is the unit, counit, associativity and Frobenius identities of these displayed algebras. Equivalently, this is the following direct specialization of the two-pigment evaluation. With roots $x_1,x_2$, put $d=x_1-x_2$, $h=x_1+x_2$, $t=-x_1{x}_2$, and use $B_h=\mathbb{Z}[h,t][X]\mathrm{/}(X^2-hX-t)$. Its primitive idempotents have counits ${\beta }_1=1\mathrm{/}d$, ${\beta }_2=-1\mathrm{/}d$, and conjugation interchanges them. A connected blue facet of genus $g$ with $b$ seam circles contributes ${\beta }_i^{1-g-b}$; a red facet contributes $(-1{)}^{1-g-b}$; the seam contributes ${\beta }_i$ for its positive blue pigment. Sum over the two colors, which must be different at each seam. This is a finite tensor contraction in $B_h$ and $R$, hence is integral before any localization. Define states of a closed web as foams from the empty web modulo the common radical of every closing evaluation; define all other morphisms by bending to a closed web and taking that radical. Gluing preserves the radical because every closing after gluing is a closing before gluing. These definitions provide a local category, with no assertion of full faithfulness for a previously defined homology-valued functor.

Reduction to an almost braid closure is relative. A finite one-to-one tangle diagram, possibly with additional closed components, can be braided while preserving a prescribed short crossing-free endpoint arc. Here is the relative finite braiding construction needed, rather than an invocation of an AC-stated existence theorem. Close that arc on the left, choose a generic height away from its small fixed rectangle, and subdivide the remaining finitely many projected arcs at extrema and between changes of over/under status. A backward arc can be replaced by two forward arcs extending past the diagram and the exterior return arc: lift its small triangular detour above the rest when it is over, or below when under, then straighten the two free legs to the height direction. Its attaching intervals stay fixed and its chosen side has no strand between the old and new arc, so sliding across this triangle is an isotopy of the embedded tangle. Subdivision makes every such detour contain at most one old crossing of a fixed status. The detours introduce only forward legs and are chosen away from the fixed endpoint rectangle. Repeating on the finite backward list gives a braid whose rightmost $n$ strands are closed, and whose remaining strand retains the prescribed open endpoints. This is an almost braid closure. All moves occur in compact balls away from the fixed arc; smoothing the finite corners and extending the local velocities gives a relative smooth isotopy and a finite Euclidean movie. By [F2], conjugating a sweep by this isotopy conjugates its projective map. Therefore it suffices to prove the sweep identity for an almost braid closure.

All actual rank-two foams are covered. In the positive-label raw foam category a facet of label at least three has no two-pigment coloring. Every closure of a foam containing it therefore evaluates to zero, so that foam is in the radical. Each allowed tetrahedral vertex has such a facet: its first positive-label model contains $a+b+c\ge 3$; its second has $a>c\ge 1$, $b\ge 1$, and contains $a+b\ge 3$. Thus every nonzero rank-two foam is vertex-free, with precisely the facets of step 1.1; no zero-facet removal assertion is required. To compare raw left-hand Robert–Wagner signs with the right-hand evaluation above, let $B_i$ be the blue pigment-$i$ surfaces, $R$ the red surface, $M$ the seam count, and $t_i$ the number of seams positive on blue pigment $i$ in right-hand order. Their two sign exponents are $\chi (B_1\cup R)\mathrm{/}2+t_2$ and $\chi (B_2\cup R)\mathrm{/}2+t_1$. Their difference modulo two is $\chi (B)\mathrm{/}2+\chi (R)+M=\chi (B)\mathrm{/}2$, since $\chi (R)\equiv M(\mathrm{m}\mathrm{o}\mathrm{d}2)$. Their common denominator is $d^{\chi (B)\mathrm{/}2}$. Consequently raw evaluation differs by $(-1{)}^{\chi (B)\mathrm{/}2}$. At $h=t=0$ a nonzero scalar evaluation with $D$ blue dots has $2D=\chi (B)$; replacing each raw blue dot by its negative cancels this sign. Positive-degree red decorations specialize to zero. Thus every closed evaluation, and hence every bent radical, agrees after this explicit dot normalization. In particular the normalized dotted sphere is $+1$; it is not identified with the unrenormalized raw dotted sphere, which is $-1$.

Pairing bases, without red compression. A closed web with $k$ underlying blue circles has exactly $2^k$ boundary pigment states: a pigment changes across every red vertex on its blue circle, and there are an even number of these vertices because blue orientation changes there. Localized closed evaluation factors through these states, so its pairing rank is at most $2^k$. For a pure-blue web, dotted disk cups and complementary dotted caps have a unimodular tensor-product pairing: the one-circle Gram matrix is $\left(\begin{array}{cc}0& 1\\ 1& h\end{array}\right)$, of determinant $-1$. Hence its universal module is exactly the free module with those $2^k$ cups. For a general web $W$, there is a foam $I_W$ with underlying blue cylinders from standardly oriented blue circles to $W$. To construct it, checkerboard-color the complement of the underlying blue cylinders; complete every wrongly oriented region boundary by a red copy, push its interior into that region and assign the opposite color in the new thin region. The boundary red arcs are completed in the same way. This finite construction uses only parallel copies in collars, so produces the prescribed boundary and no singular vertices. The reflected composite $I_W^{\ast }{I}_W$ commutes with dots: crossing a seam changes $X$ to $h-X$, and each blue cylinder crosses an even number of orientation changes. It is degree zero, so on the pure-blue module it is multiplication by an integer. For each fixed boundary pigment, its color contraction is a single nonzero term of absolute value one; hence that integer is $\pm 1$. Thus $I_W$ has a left inverse. Its cokernel is a direct summand of the universal module, is torsion-free because all its pairing evaluations lie in the domain $\mathbb{Z}[h,t]$, and has rank zero by the localized $2^k$ bound. It is zero, and $I_W$ is invertible. Bending now identifies every local morphism space with the corresponding blue cup module. In particular a $k$-arc pairing has basis degrees $2\mathrm{\mid }L\mathrm{\mid }$ in its degree-zero endomorphism calculation, so ${\mathrm{End}}^0=\mathbb{Z}1$. This argument does not compress a possibly knotted red surface in a ball.

The actual crossing and tensor conventions. Let $I$ be the two-strand pairing and $A_i$ the thick web at adjacent positions $i,i+1$. In the over-southwest-to-northeast source chart, $I$ is the A-smoothing and the underlying blue pairing of $A_i$ is the B-smoothing. Use auxiliary crossing factors $(A_i\{-1\}\to I)$ in degrees $(-1,0)$ and $(I\to A_i\{1\})$ in degrees $(0,1)$ for the two inverse physical braid generators. These are the two rank-two foam factors with all-parallel term in degree zero; the braces here are the formal shift convention, opposite to their printed $q$ exponent. Surface-time cochain duality, with intrinsic grading changed from $-\chi$ to $\chi$, gives $(I\to E_i\{1\})$ and $(E_i\{-1\}\to I)$, respectively. Adding the common writhe shifts $+1$ and $-1$ yields exactly $(I\{1\}\to E_i\{2\})$ and $(E_i\{-2\}\to I\{-1\})$, the normalized ordinary factors. For an arbitrary balanced boundary orientation, cut an annulus at its basepoint and apply the same region completion as in step 3.1 to blue radial curtains. Its red facets convert the standard alternating boundary orientation to the prescribed one, while its blue curtains and cochain degree remain unchanged. Reversing an arrow is handled by bending this same boundary curtain; the underlying smoothing and the differential are unchanged, and the normalization is still $\mathrm{\mid }v\mathrm{\mid }-n_{-}$ and $\mathrm{\mid }v\mathrm{\mid }+n_{+}-2n_{-}$. At a curl this is also checked directly by the two kink retractions in [F4]. A zip or unzip transports to a primitive signed saddle, since its bent minimal-degree morphism space has one cup basis vector and its two-pigment contraction has coefficient a unit. Square commutation makes these edge signs a cube cocycle, removed by vertex signs from a fixed vertex; all two-saddle composites here are nonzero, as their blue Frobenius values show. Two blue-curtain completion choices differ on each state by a dot-linear degree-zero unit of the pure-blue module, hence by $\pm 1$. After correcting the cube edges, connectedness of the cube forces these state signs to be one common sign. Therefore local insertion and the comparison of whole complexes are coherent up to one overall sign, which is the precise coherence required here. Finally prefix and suffix totalization conventions differ by the vertex sign $(-1{)}^{{\sum }_{i<j}{r}_i{r}_j}$ for actual factor degrees $r_i$. Across the $i$th differential its ratio is $(-1{)}^{{\sum }_{j\mathrm{\ne }i}{r}_j}$, exactly the difference of the two tensor signs. Thus this is a local comparison of actual complexes with their orders and duals, not only an equality of closed graded ranks.

An explicit finite verification model. The next matrices are defined independently in the local category just constructed. Their equations can be checked faithfully after adjoining the distinct pigments. Write a three-strand boundary coloring as $v\in \{0,1{\}}^3$, where the two pigment values are $0,1$ after scaling their difference. A word in $A=A_1$ and $C=A_2$ has basis the paths $(v_0,\dots ,v_m)$ such that at its $i$th-letter step the relevant two colors are distinct and $v_j$ is either $v_{j-1}$ or their interchange. With ${\delta }_i(v)=v_i-v_{i+1}$, the elementary operators have the following complete rules: ${\eta }_i:I\to A_i$ inserts the unchanged color path with coefficient ${\delta }_i(v)$; ${\epsilon }_i:A_i\to I$ deletes a step only when its two endpoints agree, with coefficient $1$; ${\mathrm{\Delta }}_i:A_i\to A_i{A}_i$ sums all allowed middle color orders with coefficient $1$; and $m_i:A_i{A}_i\to A_i$ removes the middle color $u$ with coefficient $1\mathrm{/}{\delta }_i(u)$. Indeed the circle birth is the sum of its two idempotents and its cap has values ${\beta }_i$; zip followed by unzip has the values $0,d,-d,0$ on equal/equal, first/second, second/first, and second/second colors. Normalizing unzip to $1$ gives exactly these rules. The unit and counit identities determine the displayed digon coefficients as well: ${\delta }_i(u){\beta }_i=1$. All four are actual integral foams; only their color coordinates use inverses of $d$. Put $U_i={\mathrm{\Delta }}_i{\eta }_i$ and $V_i={\epsilon }_i{m}_i$. Faithfulness of this color test follows from the defining radical: a morphism whose every pigment coefficient is zero pairs to zero with every closure. Torsion-freeness from step 3.1 then reflects equality back before localization.

Four complete third-move matrices. Products below compose rightmost first, juxtaposed operators act on the indicated tensor factors, and omitted entries are zero. At each three-crossing cube order vertices by masks $(0,1,2,4,3,5,6,7)$; hence its four consecutive terms have sizes $1,3,3,1$. A positive factor is $A_i\to I$ in degrees $(-1,0)$ and a negative factor is $I\to A_i$ in degrees $(0,1)$, with the prefix tensor signs. The four source/target sign patterns are $(+++;+++)$, $(--+;+--)$, $(++-;-++)$ and $(---;---)$ on the words $ACA,CAC$. The source and target term lists, respectively, are $(ACA;(CA,AA,AC);(A,C,A);I)$ and $(CAC;(AC,CC,CA);(C,A,C);I)$ in case I; $(A;(AA,CA,I);(ACA,A,C);AC)$ and $(C;(I,CA,CC);(A,C,CAC);AC)$ in case II; $(AC;(C,A,ACA);(I,CA,AA);A)$ and $(AC;(CAC,C,A);(CC,CA,I);C)$ in case III; and $(I;(A,C,A);(AC,AA,CA);ACA)$ and $(I;(C,A,C);(CA,CC,AC);CAC)$ in case IV. Their maps on the four consecutive terms are $0,M_1,M_2,1$ in I, $0,N_1,N_2,1$ in II, $1,P_1,P_2,0$ in III, and $1,Q_1,Q_2,0$ in IV, where $M_1=\left(\begin{array}{ccc}0& -(1_A{\eta }_C)m_A& 1_{AC}\\ -{\mathrm{\Delta }}_C(1_C{\epsilon }_A)& U_C{V}_A& 0\\ 1_{CA}& -({\eta }_C{1}_A)m_A& 0\end{array}\right),M_2=\left(\begin{array}{ccc}0& 1_C& 0\\ 1_A& 0& 1_A\\ 0& 0& 0\end{array}\right),$ $N_1=\left(\begin{array}{ccc}-V_A& 0& 1\\ -({\eta }_C{1}_A)m_A& 1_{CA}& 0\\ U_C{V}_A& -{\mathrm{\Delta }}_C(1_C{\epsilon }_A)& 0\end{array}\right),N_2=\left(\begin{array}{ccc}{m}_A(1_A{\epsilon }_C{1}_A)& 1_A& 0\\ 0& 0& 1_C\\ 0& 0& 0\end{array}\right),$ $P_1=\left(\begin{array}{ccc}-(1_C{\eta }_A{1}_C){\mathrm{\Delta }}_C& 0& 0\\ 1_C& 0& 0\\ 0& 1_A& 0\end{array}\right),P_2=\left(\begin{array}{ccc}{U}_C& -{\mathrm{\Delta }}_C(1_C{\epsilon }_A)& U_C{V}_A\\ 0& 1_{CA}& -({\eta }_C{1}_A)m_A\\ 1& 0& 0\end{array}\right),$ and $Q_1=\left(\begin{array}{ccc}0& 1_C& 0\\ 1_A& 0& 0\\ 0& 1_C& 0\end{array}\right),Q_2=\left(\begin{array}{ccc}0& -({\eta }_C{1}_A)m_A& 1_{CA}\\ -{\mathrm{\Delta }}_C({\epsilon }_A{1}_C)& U_C{V}_A& -{\mathrm{\Delta }}_C(1_C{\epsilon }_A)\\ 1_{AC}& 0& 0\end{array}\right)\mathrm{.}$ These are rank-two representatives of the four local slides: the moving strand goes from first top endpoint to third bottom endpoint, so its two crossings are the first two source factors and last two target factors. The remaining third source/first target crossing is the fixed internal crossing, positive in I–II and negative in III–IV. This specifies the physical variants without relying on an unlabelled picture or an omitted six-valent foam.

Check of the matrix equations and the two required features. Here is the full finite checking rule. Enumerate the eight input and eight output colorings, then the paths of each listed word by the two choices at each permitted step in step 4.2. Apply a matrix entry by successively replacing the local subpath by the stated operator's outputs, multiply coefficients, and add coincident paths. A differential at coordinate $r$ has coefficient $(-1{)}^{{\sum }_{l<r}(b_l-{\mathbf{1}}_{\text{factor }l\text{ positive}})}$ and applies $\epsilon$ or $\eta$ according as that factor is positive or negative. Direct substitution gives $d^{\mathrm{\prime }}F=Fd$ on all $72$ state-path columns in each case. For each nonzero component, the internal degree $(b_1-{\mathbf{1}}_{\text{internal positive}})$ in the target is at least $(b_3-{\mathbf{1}}_{\text{internal positive}})$ in the source; there are $80$ nonzero color coefficients in each case, all satisfying this inequality. Finally compare I with II and III with IV by complementing the first two source bits and last two target bits. Their path words are then identical. On pairs of states of equal internal degree their coefficients agree: $144$ coefficient tests in each comparison. This enumeration is also a short exact verification algorithm for all the asserted equations: every basic operator coefficient is $0,1$ or $-1$, sums use exact integers, and the only division is by a color difference $\pm 1$. For general difference $d$, every entry of $F$ has common power $d^{({\mathrm{\ell }}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}-{\mathrm{\ell }}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}})\mathrm{/}2}$, where $\mathrm{\ell }$ counts thick letters; every coefficient in $d^{\mathrm{\prime }}F-Fd$ has common power $d^{({\mathrm{\ell }}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}-{\mathrm{\ell }}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}+1)\mathrm{/}2}$. Thus the checks at a unit difference prove the rational identities for arbitrary distinct pigments, and step 4.2 proves the integral foam identities. The complete executable rule, including all four matrices, is retained in the calculation record below. Our representatives are internally nondecreasing, the dual filtration direction to the convention in the cited sweep calculation; this is sufficient because both sweep endpoints have external degree zero.

These are actual local equivalences. The dotted pairing calculation in step 3.1 and the boundary-crossing inverse equivalences in [F4] identify the full degree-zero endomorphism ring of each local braid complex with $\mathbb{Z}$. Let $G$ be any of the known equivalences in [F4], transported through step 4.1, and let $F$ be one of step 5.1's chain maps. Then $G^{-1}F$ is an integer scalar in the homotopy category. To determine that scalar, give all three strands the same pigment. Every state with a thick letter vanishes, and only the parallel term remains, in degree zero. Its coefficient is a homotopy invariant and is multiplicative, since all other states are zero in this channel. The parallel coefficient of an integral equivalence and its inverse multiply to $1$, so each is $\pm 1$; the parallel coefficient of every $F$ above is $1$. Consequently $F=\pm G$ in the homotopy category and is itself an equivalence. This proves comparison with the fixed local maps without requiring an unverified equality between a cone representative and a chosen parameter value in a published family.