Elliptic geometric copy Homs and parallel Floer products

Finite genuine Hamiltonian copies. Fix ell_0(x,y)=x+sqrt(2)y, nonzero on every rational parent direction. In a flat cotangent annulus of each parent, with x modulo 2π and $\omega =dy$ wedge dx, use exact graphs $y=a$ cos x+b sin x with primitive $H=a$ sin x−b cos x. Their coefficient local systems are moved by the Hamiltonian isotopy. Choose each parameter vector in an arbitrarily small open disk so that its graph tangent has the same nonzero ell_0 sign as its own parent. This requires no margin uniform over rational directions. A finite chart chooses distinct parameters avoiding pairwise tangencies, triple concurrence and collinear same-parent parameter triples. Later charts retain old curves literally; the genericity and smooth-corner proofs below justify adjoining new curves even when their intersections occur on an old curved edge. For each increasing word use fixed distinct curve labels and the actual nonconstant holomorphic polygons with those boundary curves. All boundary splittings retain these same curves. Formal scalar identities will be added only in the directed category, before localization; repeated geometric polygons and clean constant limits are not presumed.

Smooth transverse corners and immutable charts. Use the actual smooth transverse half-strip end theorem in lem-gts-transverse-strip-local-action-neck-and-end-estimates, Statement and proof 4.1–5.1. In the holomorphic source coordinate $z=exp(-\pi (s+it))$, it gives, for some β>0, u(z)−p=O(|z|^β), u'(z)=O(|z|^(β−1)), up to both physical boundaries; all strip derivatives obey the corresponding exponential bounds. Replace β by a smaller number in (0,1). The following is a linear boundary-phase correction, not a simultaneous holomorphic straightening of the target arcs. Let θ+ and θ− be the limiting tangent phases, taken modulo π. Along the two source half-edges let φ be the difference between the actual tangent phase at u(x) and its limiting phase. The smooth target tangent and the preceding estimates make φ Hölder, continuous across zero, and φ(0)=0. The chosen strip coordinate lies in the lower half-disk; use the branch $-\pi <\arg z<0$. Extend φ with compact support on a slightly longer real interval, and put g(z)=−(1/π) integral φ(t)/(t−z) dt, a(z)=exp(g(z)). For $z=x-iy$ with $y>0$, $\mathrm{Im}g(x-iy)={\pi }^{-1}{\int }_{\mathbb{R}}y\varphi (t)\mathrm{/}((t-x{)}^2+y^2)dt$, the positive Poisson extension of φ. Thus the imaginary boundary value of g is φ. The Cauchy integral extends continuously at zero: subtract φ(0), split the integral at 2|z|, and bound the near part by C|z|^β and the far difference by C|z|^β(1+|log|z||). Thus a has a finite nonzero limit there. The same bounds with a smaller Hölder exponent give its boundary continuity and exponential strip-derivative bounds. Select δ in (0,1) so that e^(iθ+)z^(δ−1) has the two limiting derivative lines on the two half-edges. Then f(z)=u'(z)/(e^(iθ+) a(z) z^(δ−1)) has real boundary values on both half-edges. Schwarz reflection makes f holomorphic on the punctured disk. Its growth is O(|z|^(β−δ)); since β−δ>−1, no negative Laurent power is possible. Hence $f=z^k(c+O(z)),k\ge 0$ and c≠0. Integration yields u(z)−p=c' z^(δ+k)(1+o(1)). In particular $k=0$ is a local homeomorphism to the convex curvilinear wedge with angle πδ. The same nonvanishing holomorphic factor corrects the boundary phase of every linearized section ξ. Therefore the substitution ξ=du·v in the existing F2 proof has exactly its former compact divisor orders: corner contribution k, interior contribution twice the branch order, and edge contribution the branch order. The factor a changes no winding or divisor. On the strip it converges exponentially to a nonzero constant, so multiplication by it and its inverse preserves the actual weighted Fredholm realizations. F2's compact-line cokernel calculation, domain variations and transverse inverse/gluing argument consequently apply to smooth transverse corners as well. No geometric holomorphic chart straightens both arcs. The no-three-concurrence and essential-circle bubble exclusions remain unchanged. This proves the precise local extension needed here. Existing copies need never be changed when later curves are added. Their later intersections may lie on curved edge interiors. The common-height embedding/count argument then applies with the preceding leading convex wedge in place of the exact affine corner expansion. The statement and original narrow analytic carrier F2 need not be rewritten; this is an additional argument in the present geometric construction using the explicitly named smooth-end supplier.

1. Engineer one common monotone height Fix once ell_0(x,y)=x+sqrt(2)y. It is nonzero on EVERY rational parent direction (p,q), since $p+sqrt(2)q\mathrm{\ne }0$. Choose each Hamiltonian graph copy C1-small relative to its own strictly positive ell_0 tangent margin. Thus every finite collection of copies has a common positive margin, without an angle bound uniform over all rational slopes. For a fixed finite collection choose ell in a sufficiently small neighborhood of ell_0, and keep |ell(tangent)| bounded away from zero on EVERY lifted copy curve. Each copy is then a globally ell-monotone embedded line in the universal cover: its parametrization along the parent coordinate has derivative whose ell-value has one fixed nonzero sign. The geometric copy constructions are generic/no-three-concurrence, with their smooth transverse corners covered by step 2.1; the strictly positive tangent margin is open and is preserved by those smaller changes. Choose ell further generically so that no two distinct lifted arrangement vertices have the same ell-value. The forbidden directions are countably many lines, since the finite arrangement has finitely many physical vertices and countably many deck lifts. Removing that countable set from the preceding open direction set leaves a choice. This genericity is used separately for each finite arrangement, not asserted uniformly over all branes in the whole category.

2. A rigid polygon's lifted map is globally injective Let u be an index-zero nonconstant polygon with N cyclically ordered boundary edges, every edge assigned a DISTINCT copy curve. Lift it to $U:D\to R^2$. By the analytic supplier, U is a local holomorphic diffeomorphism at every interior point, has no branch on an open edge, and extends at each corner with the minimal exponent delta in (0,1). Let $H=ell$ composed with U. H has no interior critical point: dU is an invertible real-linear map and ell is nonzero. On every open boundary edge, U is an immersion into its assigned copy curve; its orientation cannot change without a boundary branch. Since ell is nonzero on that curve tangent, H is strictly monotone along that whole edge, even if that edge winds around its circle many times. All critical height values can therefore occur only at the N corners. They are distinct by step 3.1 for the realized distinct corner lifts. With distinct copy labels and no triple concurrence, corner lifts in an $N\ge 3$ word are distinct. For a bigon, coincident corner lifts would force a nonconstant monotone edge to have identical endpoint heights, which is impossible. Between two consecutive corner heights, the level foliation is an actual product, not an assumed Reeb-graph regularity theorem. On a compact slab with no corner, choose locally a smooth vector field V with dH(V)=1. Near a boundary edge choose the vector field tangent to that edge and divide by its nonzero H derivative. In the interior use grad(H)/|grad(H)|². A partition of unity preserves the equality dH(V)=1, so its finite-time flow identifies the slab with a regular level times its height interval. The regular level is a compact one-manifold. It has no circle component: such a circle bounds a subdisk in D, and the maximum/minimum principle for harmonic H with constant boundary value would make H constant on that subdisk, contradicting $dH\mathrm{\ne }0$. Thus it is a disjoint union of arcs. At a corner the exact local expansion identifies U with a convex wedge, of angle strictly less than pi, to leading order with a nonzero coefficient. The height foliation has precisely these local possibilities: • If both bounding rays increase H away from the vertex, a single short level arc is born (a cap). • If both rays decrease H, a single short level arc dies. • If the rays have opposite H signs, an endpoint merely passes from one boundary edge to the next; the level continues regularly. There is NO saddle/merge/split alternative. To verify this directly, intersect a convex wedge with $ell=t$: it is empty followed by one interval, one interval followed by empty, or one interval whose endpoint changes ray. The leading corner expansion from step 2.1 is a local homeomorphism to the corresponding curvilinear wedge and preserves this topology on sufficiently small neighborhoods. In particular this conclusion would fail for an excess/reentrant corner; its exclusion is the essential index-zero input. Collapse every connected regular height arc to a point, using the above product structures and the local corner caps. This constructs a finite graph explicitly: every regular vertex has degree two and every extremal cap has degree one; no other vertices occur. It is connected since D is connected. It has extrema because D is compact. A finite connected graph with all degrees at most two and an endpoint is an interval. Height is strictly monotone on that interval; hence EACH regular height has exactly ONE level arc. Along that arc U takes values on the target line $ell=t$. Its tangential derivative is nonzero, since dU is invertible, so its coordinate along that line is strictly monotone. Thus U is injective on every regular level. If two distinct interior points had the same image at a corner height, their two local inverse branches would persist for a nearby regular target height and contradict that injectivity. A boundary/interior or two-boundary collision is excluded by the same open-wedge argument and the distinct corner-height choice. Consequently U is an embedding of the closed polygonal disk. In fact its image is ell-convex: every regular target line $ell=t$ meets it in one interval. This is stronger than the general curved-edge analytic supplier. It uses both its convex minimal corners and the specially engineered COMMON global monotone height. Unbranchedness alone would not justify the conclusion.

Engineered exact graph family. Use an annular lift with coordinate x modulo 2π and symplectic primitive y dx. Fix a finite set of distinct real parameters t in a small interval, and ε>0. Set $H_t(x)=ϵ(t\sin x-t^2\cos x)$, $h_t(x)=H_t^{\mathrm{\prime }}(x)=ϵ(t\cos x+t^2\sin x)$, $L_t=\{y=h_t(x)\}$. All graphs are exact Hamiltonian copies of the same original circle. For $r<s$ write α_rs=π/2+arctan(r+s), β_rs=α_rs+π, ℓ_rs=ε(s−r)√(1+(r+s)²). The only crossings of $L_r,L_s$ occur at α_rs,β_rs modulo 2π. The action $A_{rs}=H_s-H_r$ has maximum +ℓ_rs at α_rs and minimum −ℓ_rs at β_rs. If the directed order of the objects reverses r,s, negate A; thus its minimum switches from β to α. These facts concern actual periodic graph functions, not limiting Morse trajectories. No corner modification is made in the immutable family. The smooth-corner construction of step 2.1 supplies the precise extension of the narrow affine-germ analytic interface. Thus graph functions, crossing coordinates, primitive values and exact area formulas remain literal at every later chart. Each graph is globally x-monotone. Thus the already checked common-height embedding applies with height x, without any angle gap uniform over future probes or slopes.

3. Uniqueness for a prescribed lifted boundary word Suppose two rigid polygons have the same oriented lifted edge paths and corners. step 4.1 makes each an embedding. For a point off their common boundary, the holomorphic argument principle says the number of its preimages is the winding number of that oriented boundary. Since each map is injective, that winding number is either zero or one, and determines its image exactly. The two images are therefore equal. Composing one conformal inverse with the other gives a biholomorphism of their disk domains preserving every marked corner, which is precisely the equivalence relation in polygon moduli. Thus there is AT MOST ONE rigid map per lifted boundary word. No Riemann-mapping existence theorem or immersed-filling uniqueness is assumed in this count bound.

Actual two-cell differential. For a directed pair i,j, let $c_{ij}$ denote its degree-zero minimum of $A_{ij}$ and $e_{ij}$ its degree-one maximum, with coefficients $Hom(V_i,V_j)$. These two generators are in the source's usual lifted gradings; the graph isotopies have zero Maslov winding. A rigid bigon is embedded by the fixed-copy common-height argument. Each edge is an x-monotone graph segment, and its two endpoints are successive crossings: if it passed another crossing, its two boundary edges would meet there. Thus the two bounded lens regions between successive roots are the only possible bigons. Conversely they are positively oriented simple domains with convex corners.

3. Literal complex coefficients: separate the annular case For a polygon all of whose parent circles coincide, adjacent intersections force the same integer normal lift, so its entire lifted boundary is in ONE thin band. The harmonic maximum principle traps its interior there. On the resulting annulus $\omega =d(ydx)$, and each graph has the periodic primitive $H_i$. Stokes gives Area(u)=sum_corners (H_out−H_in)(corner), 0<=Area(u)<=2N max_i ||H_i||infinity for a fixed N-corner word. Local-system transports on that word are finite matrices. Area-cutoff compactness and the zero-dimensional regularity theorem therefore give finitely many rigid polygons. There is no infinite high-energy tail at all in this case. A degenerate straight reference polygon's zero area is NOT used to bound its lift diameter.

Positive finite-face chains count actual mixed-parent maps. For the fixed N DISTINCT copy labels, let Gamma be their finite transverse arrangement and R1,...,RF its connected complementary faces, of positive areas $a_j$. Faces need not be disks. For a point of $R_j$, the preimage of a polygon is a compact discrete subset away from its boundary and hence finite; by local covering continuation its nonnegative integer multiplicity $n_j$ is constant on that connected face. Change of variables gives $Area(u)={\sum }_j{a}_j{n}_j$. Stokes against smooth test one-forms gives $partial(u_{\ast }[D])=partial({\sum }_j{n}_j[R_j])$; at an arrangement segment its coefficient is the difference of neighboring face multiplicities. The supplied corner exponents make the cut-off Stokes limits at corners valid. Each copy occurs ONCE as a word edge and different copies have no common open arc. Thus the segment coefficients on that copy cannot hide a cancellation between two occurrences of the same curve. With endpoint vertices and orientation fixed they determine its entire path and winding; fixing the first corner lift then determines all other lifts. The common-height embedding and uniqueness in steps 4.1 and 5.2 make this at most one ACTUAL rigid map per chain and finite endpoint/orientation pattern. Therefore # {rigid maps: $\mathrm{Area}\le E\}\le C{\prod }_j(1+\lfloor E\mathrm{/}{a}_j\rfloor )$. Testing any one open arrangement segment on a copy also bounds its edge winding by $C(1+{\sum }_j{n}_j)$: each complete turn traverses it once, and there is at most one final partial turn. Monotonicity of the lifted edge excludes backtracking. This is an actual mapcount/winding estimate, not just a count of area classes, and applies when parent circles repeat or are parallel. The constants belong to this fixed finite chart; no $a_{min}\to 0$ or uniform-in-slope assertion is made.

Actual graph-domain parametrizations. Let P be any bounded simple graph bigon or triangle used here, with smooth edges and convex transverse corners. Its interior is a topological disk. Apply the already available Riemann mapping theorem to obtain f:P→D. Properness gives |f(z)|→1 at its boundary. For a small circular crosscut γ_r cutting off the local cap at p∈∂P, put L(r)=length(f(γ_r)). Cauchy–Schwarz gives L(r)^2≤2πr integral_(γ_r)|f'|², hence integral L(r)^2 dr/r≤2π Area(D). There are radii $r_n\to 0$ with $L(r_n)\to 0$. Finite length gives endpoint limits of these image crosscuts; properness puts them on ∂D. Their diameters tend to zero. Each cuts off, on the side excluding a fixed interior image point, the small circular cap adjacent to its two nearby endpoints: the crosscut lies in its small endpoint ball, whose complementary disk is connected and contains the fixed point for all sufficiently small crosscuts. The source cap at p is exactly that side, by crosscut separation in the graph disk. Nested source caps therefore have images of diameter tending to zero. This proves continuous extension of f at every boundary point, including corners. A boundary fiber is connected: otherwise choose one point with a different image in each intervening boundary arc and join those points by an interior crosscut. Its image is a crosscut with distinct endpoints; the two source boundary arcs must map to the two different closed circular arcs, whose intersection consists only of the endpoints, contradicting the proposed common value. A nontrivial connected fiber contains an open smooth boundary arc on which f is constant. Extend f minus that constant by zero across this arc. This is continuous and holomorphic on both sides; cutting test triangles along the smooth arc and cancelling the common integrals proves Morera's condition. It is therefore holomorphic across the arc and identically zero by the identity theorem, a contradiction. Thus the extension is injective; compactness and interior surjectivity give boundary surjectivity, so it is a homeomorphism. Its inverse u:D→P is consequently continuous on the closure. Conformal change of variables gives integral_D |du|²=2Area(P)<∞, so u is W1,2 also up to each open boundary arc. Flatten just that single smooth target arc by an ordinary smooth chart and flatten the source circle by a Möbius chart. The transformed complex structure is smooth, the real boundary is totally real, and the weak chain rule follows by mollification. The exact continuous-weak boundary regularity supplier in [F5] now makes u smooth up to the open edge; continuity and finite Sobolev norm were established before invoking it. At corners the smooth-corner phase calculation of step 2.1 gives exponent δ+k. Injectivity and the actual convex target tangent wedge force $k=0$: a larger exponent covers sectors outside that wedge or overlaps itself. The same local phase reflection at a smooth edge excludes a zero derivative, since an order greater than one covers a half-neighborhood more than once or leaves its target half-neighborhood. Thus u is a genuine unbranched polygon with minimal convex corners, not merely an interior Riemann map. The extended F2 calculation proves regularity; the checked boundary winding argument proves uniqueness. This construction uses no new import, clean limit or parameterized triangle compactness assertion.

Absolute complex sums and their finite jets. For any integer m, positive or negative, a monodromy $M=\lambda$ I+J with |lambda|=1 and $J^r=0$ satisfies $M^m={\lambda }^m{\sum }_{l=0}^{r-1}binom(m,l){\lambda }^{-l}{J}^l$. The generalized binomial coefficients are polynomials in m, so $\mathrm{\mid }\mathrm{\mid }{M}^m\mathrm{\mid }\mathrm{\mid }\le C(1+\mathrm{\mid }m\mathrm{\mid }{)}^{r-1}$. Partial transports are bounded on the fixed compact edge charts. The winding estimate proves that a fixed polygon coefficient has norm at most $C(1+\sum n_j{)}^s$; any prescribed finite holonomy/nilpotent jet only increases s. Since $c=2piIm(\tau )>0$, the FULL mixed-parent absolute sum is bounded by C sum_(n in $N^F)(1+\sum n_j{)}^{s}exp(-c$ sum $a_j{n}_j$). Bound its polynomial by $C{\prod }_j(1+n_j{)}^s$ to obtain a finite product of convergent geometric-moment series. This is literal complex convergence at $\rho =\tau$, for arbitrary fixed finite arity and arbitrary finite nilpotent order. No formal specialization, associated-graded acyclicity or copy-coincidence limit is used. For each arity there are finitely many breaking types and finite intersection bases; their sums are finite products of these absolute sums. Thus the actual area-cutoff compactification/collar identities of [F2] pass to unrestricted sums, with all rearrangements justified by absolute convergence. Finite subdivisions and common chart refinements have their own positive face areas, so the same argument applies to their operations as well. There is no need for a uniform analytic estimate over infinite arities, all slopes or coinciding copies.

The elementary conformal-domain argument in step 7.1 supplies their maps, and F2 supplies regularity and uniqueness. Their orientation signs are opposite, as follows by the oriented boundary of the one-dimensional strip translation/fixed-end determinant convention; equivalently choose the two-cell orientation so the two interval endpoints have opposite signs. Parallel transport along the two lenses differs by one turn around the circle. The exact area of a polygon is $A_{out}-\mathrm{\Sigma }{A}_{in}$. Replacing an intersection generator z by $exp(2\pi i\tau A_{ij}(z))z$ therefore removes every action coefficient from every fixed-word operation: the scalar factors telescope in the displayed area formula. These are nonzero complex changes of basis, not a Novikov specialization or a limiting operation. With a common parent-circle basepoint and transported coefficient trivializations, the complex is consequently $Hom(V_i,V_j)--(D_{ij}-1)-\to Hom(V_i,V_j)$, where $D_{ij}(T)=M_{j}T{M}_i^{-1}$. The minimum generator with identity coefficient is closed whenever $V_i=V_j$ with its transported local system. The same argument works for arbitrary finite Jordan monodromy; no semisimplicity is used.

All degree-compatible continuation triangles. Take three distinct parameter values $r<s<t$. Put $a={\alpha }_{rs},b={\alpha }_{rt},c={\alpha }_{st},A=a+\pi ,B=b+\pi ,C=c+\pi$. Then $a<b<c<A<B<C<a+2\pi$. Each directed pair has exactly the two indicated roots. A lifted three-edge boundary is simple if and only if the two successive x-gaps after ordering its three vertices are less than π. Indeed the boundary edges on each graph are the intervals between its two vertex coordinates. For the pair corresponding to the least vertex, the overlap of its two edge intervals runs from that least vertex to the middle one; its next crossing occurs exactly π later. The same assertion, with direction reversed, holds at the greatest vertex. The pair at the middle vertex has interval overlap only at that vertex. Thus an x-gap at least π creates an additional boundary intersection, while two gaps less than π exclude every additional intersection. Each graph edge is itself embedded. Equality does not occur because all six displayed coordinates are distinct. For a degree-compatible product, the root set is one of exactly six patterns: all α; all β; {a,b,C}; {a,B,C}; {A,b,c}; {A,B,c}. The other two choices {a,B,c} and {A,b,C} are precisely the two degree-incompatible root patterns. In each of the six allowed patterns there is exactly one circular x-gap greater than π. Cutting immediately after that gap yields the unique lifted simple boundary, modulo a common deck translation. Every other choice of lifts leaves one of the two successive gaps greater than π and hence fails the preceding boundary test. This enumerates actual boundary paths and windings, not only area classes. For completeness the six parameter orders of three degree-zero continuation arrows give the following patterns, written in unordered-pair order rs,rt,st: r→s→t: (β,β,β). r→t→s: (β,β,α). s→r→t: (α,β,β). s→t→r: (α,α,β). t→r→s: (β,α,α). t→s→r: (α,α,α). The same six sets arise from the degree-compatible products (0,1→1) and (1,0→1). Substitution in $h_j^{\mathrm{\prime }}-h_i^{\mathrm{\prime }}$ gives a negative slope difference at α and a positive one at β when $i<j$. With the boundary order of the given product, the unique simple boundary has the positive sector at every corner. At the two extreme-x corners its two inward rays have the same x sign and enclose the small sector; at the middle corner the displayed root order and slope-difference sign give the same convex sector. Thus all three interior angles are strictly less than π. This is a direct corner check; no implicit regularity at a constant triangle is invoked. Write $v_u=\epsilon (u,u^2)$, so ${\mathrm{\ell }}_{ij}=\mathrm{\mid }{v}_j-v_i\mathrm{\mid }$ for unordered i,j. The signed area of a degree-compatible boundary is respectively ℓ_ij+ℓ_jk−ℓ_ik for (0,0→0), ℓ_ik+ℓ_ij−ℓ_jk for (0,1→1), ℓ_ik−ℓ_ij+ℓ_jk for (1,0→1). Every quantity is strictly positive: three distinct points on the parabola u↦(u,u²) are not collinear, so all three strict triangle inequalities hold. Therefore the boundary orientation is the holomorphic one. Step 7.1 provides its actual map; F2 makes it regular. Its index is zero because its corners are minimal and it has no branches. The checked embedding/uniqueness result and the complete lifted-boundary enumeration show there is exactly one rigid triangle for each of these products, and no further winding contributions. This works for any order of the distinct parameters, not only a monotone parameter path.

Coefficients and both cohomological units. After the exact action gauge, each enumerated triangle has scalar coefficient ±1. Its local-system operation is the literal boundary parallel-transport composition. If one input is the continuation identity between two copies of the same V, the V transports on the two adjacent edges telescope. Comparing the input and output coefficient fibers by fixed parent-circle basepoint paths leaves either identity or a simultaneous full winding: $T\mapsto D^{n}T,n\in Z$, where $D(T)=M_{V}T{M}_W^{-1}$ for the probe W. There is no independent one-sided monodromy factor: the projected triangle boundary is contractible, so its two total parent-circle paths have the same winding. The integer can depend on the chosen lifts and basepoint trivializations. It need not be set to zero globally. The remaining signs can be normalized without assuming an existing unit. First use trivial rank-one coefficients and a fixed auxiliary copy b preceding a finite family in the directed object order. Each minimum/minimum product is nonzero by step 8.2. Normalize $c_{bi}$ arbitrarily and normalize $c_{ij}$ uniquely so $m2(c_{ij},c_{bi})=c_{bj}$. The distinct-label A-infinity arity-three identity implies associativity on cohomology. Applying it to b,i,j,k gives $m2(c_{jk},c_{ij})=c_{ik}$, since multiplication by $c_{bi}$ is a nonzero map between the one-dimensional degree-zero spaces. This fixes all minimum-product signs coherently. Passing to finite upper charts with the same b preserves the normalization. Changing b gives a unique scalar gauge, so no new analytic compatibility is needed. The normalization is independent of local-system coefficients, since orientation signs depend only on the underlying graphs and brane lines. Orient each maximum relative to its minimum so the actual two-bigon differential is D−1. For a continuation acting on a probe, the distinct-label A-infinity differential identity says its two degree maps form a chain map. Each degree map is one signed transport $D^n$ by step 8.2. Test this identity with one-dimensional probe monodromy λ on the unit circle, λ≠1. Equality $(D-1){\sigma }_0{D}^{n_0}={\sigma }_1{D}^{n_1}(D-1)$ for every such λ forces σ_0=σ_1 and $n_0=n_1$: these are Laurent monomials agreeing on infinitely many λ. The minimum sign is already +1 by the preceding normalization. The same scalar orientation calculation therefore fixes the maximum sign to +1 for every finite local system; its transports are the same geometric words. Thus multiplication by continuation is, in the chosen two-cell identifications, $D^n$ in both degrees, and in particular induces identity on ker(D−1), coker(D−1). For n≥1 the explicit homotopy identity is $D^n-1=(D-1)(1+D+\mathrm{.}\mathrm{.}\mathrm{.}+D^{n-1})$; for $n<0$ use the corresponding negative finite Laurent sum. Hence $D^n$ is chain homotopic to identity on the two-cell complex: put that Laurent sum in the unique degree-minus-one component. This is valid for arbitrary matrices and nilpotent parts. Both left and right continuation actions are therefore quasi-isomorphisms on every same-position same-parent probe. In particular the normalized degree-zero continuation cocycles satisfy actual cohomological transitivity for any three distinct copies, with the true geometric m2 products; no intended geometric map has been replaced by an abstract unit.

Different parallel fluxes. For completeness a future parallel probe need not be disjoint from an old bent graph. Lift two parallel Hamiltonian graph copies to the same cylinder. Their height difference is δ+F'(x), with nonzero mean δ (the original positions are distinct modulo the normal period). Arrange finite transverse intersections; for the sufficiently thin copies used here at most one normal deck translate can intersect. If there are no intersections the complex is zero. Otherwise its crossings alternate minima and maxima of G(x)=δx+F(x), and the only rigid bigons are the lenses between neighboring crossings, by the same embedded-boundary argument as step 8.1. The graph-domain construction in step 7.1 gives each such lens exactly once. Their coefficients are invertible transport/action factors. Choose basepoint paths and eliminate successively a neighboring minimum/maximum pair along one of these invertible differential entries. Gaussian cancellation is an explicit elementary chain contraction: solve that maximum coefficient for the chosen minimum, substitute in its other neighboring differential, and remove the resulting contractible two-term summand. The new entry is the product of the two neighboring transports with the inverse of the eliminated entry, with its differential sign. Repeating around the circle reduces the finite complex to one copy of Hom(V,W) in each degree, with differential 1−exp(2πiτ ∫_0^{2π}δ dx) D. The scalar is the full-period action/flux weight; intermediate exact F values telescope. If a different normal lift is used replace δ by δ+m, which remains nonzero for distinct positions. Since Im τ>0, the scalar has modulus different from one. Every eigenvalue of D has modulus one, including all finite nilpotent blocks. The displayed operator is therefore invertible. The complex is acyclic, and every continuation action between such parallel-position complexes is automatically a quasi-isomorphism. This argument handles close future parallel probes without a global separation bound.

Finite generic families and immutable extensions. In a flat parent coordinate (x mod 2π,y), choose genuine exact graph copies L_v: y=a cos x+b sin x, v=(a,b), with |v| as small as needed for that parent's common-height tangent margin. Their primitives are $H_v=a$ sin x−b cos x. Different local systems on the same physical parent receive different copy parameters. A new parameter can be arbitrarily small while avoiding all pairwise tangencies with finitely many old curves: at each source x the two parameter variations cos x,sin x span the normal direction. For completeness use a local graph $y=g(x)$ for an old curve at a possible tangency. The equations $F=g(x)-a$ cos x−b sin $x=0$ and $F_x=0$ have parameter Jacobian [[−cos x,−sin x],[sin x,−cos x]], of determinant one. Their zero set in (x,a,b) is therefore a one-dimensional smooth manifold. Its projection to the parameter plane has measure zero by the elementary Lipschitz-chart covering argument below. An old curve with vertical tangent is automatically transverse to the new graph. A countable chart cover handles all points and normal lifts. Thus arbitrarily small parameters outside the exceptional set exist, with no Sard theorem assumed. Every finite set of old triple points excludes finitely many affine equations in v. Choose new parameters also off each line through two old same-parent parameters. Thus every comparable finite chain has distinct transverse curves and no triple concurrence; old curves and their operations are retained. The same-parent triangle calculation is not restricted to a special parameter ordering: for three noncollinear vectors $v_i,v_j,v_k$ the differences obey $d_{ik}=d_{ij}+d_{jk}$. The minimum critical direction of $H_k-H_i$ lies between the two input minimum directions along their minor circular arc. The three minimum directions therefore have one complementary gap greater than π; the other two gaps are less than π. This is the precise interlacing used by step 8.2. Its unique positive triangle has area $\mathrm{\mid }{d}_{ij}\mathrm{\mid }+\mathrm{\mid }{d}_{jk}\mathrm{\mid }-\mathrm{\mid }{d}_{ik}\mathrm{\mid }$. The action/transport gauge gives [m2(c_jk,c_ij)]=[c_ik]. All c carry the identity local-system coefficient. For auxiliary isotopy subdivision one can use a strict convex parabolic arc of parameter vectors through any two endpoints. Tiny C3 perturbations of this entire compact family preserve the root interlacing uniformly, including as adjacent parameters coalesce: divide each graph difference by its parameter difference. In a parabolic coordinate the reference quotient is cos x+(s+t)sin x. Its simple root has both parameter derivatives 1/(1+(s+t)^2)>0, uniformly on the compact interval. A sufficiently small C3 perturbation preserves simplicity and those strict derivatives. Thus the triangle identity remains valid for every finite subdivision, however fine. A generic auxiliary path relative a finite list of fixed probes can be chosen without assuming a transversality theorem. In local graph coordinates write F(t,x) for the probe/path intersection equation. Add a finite-dimensional, arbitrarily C3-small trigonometric-polynomial perturbation of the family that vanishes with its first t derivative at the fixed endpoints. Finitely many such modes span the required $(F,F_x,F_{xx})$ and $(F,F_x,F_t)$ jets on the compact interior cylinder; the zero-mean constraint is automatic for nonconstant Fourier modes. The two bad zero sets each have full rank three in all variables and hence, by inverse charts, dimension p−1 if p is the number of parameters. Their projections to $R^p$ have measure zero: on each compact Lipschitz chart, cover a bounded (p−1)-dimensional set by O(η^(−p+1)) cubes of side η; its image has p-volume O(η). A countable exhaustion suffices. Select an arbitrarily small parameter outside those projections. Then every tangency has $F_t\mathrm{\ne }0$ and $F_{xx}\mathrm{\ne }0$. The equations $F=F_x=0$ are regular isolated points in the compact path cylinder and hence finite. The same argument using independent spatial jets at two distinct points avoids two tangencies at one time if desired. Endpoints already transverse, so a fixed endpoint neighborhood contains no bad event.

All nonparallel probes, including tangencies. There is no smallness condition relative to all future rational angle gaps. Fix just the actual finite list of probes needed in a finite diagram. Let X be a fixed probe whose primitive homology direction is not parallel to the parent of L. The action is single-valued on each component of paths from X to L. Indeed a loop of such paths has boundary winding n[X]=m[L] in π1(T²)=Z²; independence of the two primitive directions forces $n=m=0$. Its remaining sphere class is zero. Its symplectic area is therefore zero. The Hamiltonian isotopy identifies these components for nearby copies. There are finitely many critical points and finitely many relevant components; choose action normalizations by the same isotopy. First consider a sufficiently small path step with no tangency. Intersections p of X with L correspond to p' of X with L'. The triangle/Stokes identity for multiplication by the continuation cocycle is Area = A(p)−A'(q')+κ, where κ and A'(p')−A(p) tend uniformly to zero with the isotopy step. This follows directly by lifting the edges, integrating y dx, and inserting the exact primitive difference on the two graph edges. Consequently a nonzero triangle coefficient can only go in one action direction, except possibly inside a cluster of equal limiting action values. Inside such a cluster, distinct intersection labels cannot be connected by triangles of arbitrarily small area. The X edge of a triangle from p to q' lies on one of finitely many fixed lifted X arcs: the closing homology equation fixes its X winding, since [X] and [L] are independent. If p≠q, its limiting arc contains a point a fixed positive distance from L. A small patch at that point has physical boundary only on X. The required uniform local area bound follows directly from the already proved embedding in step 4.1. Choose an interior point $a_0$ of that fixed lifted X arc away from its limiting endpoints, and a small closed disk $B_r(a_0)$ disjoint from L. For all sufficiently small path steps it is also disjoint from L', and the actual X edge contains $a_0$. Make r small enough that the fixed smooth embedded X arc cuts the disk into two connected components $U_{+}$ and $U_{-}$, each of strictly positive area. The embedded polygon image $\mathrm{\Omega }$ has no boundary in this disk except on X: its other boundary edges lie on L and L'. At the ordinary X-edge point $a_0$, unbranched boundary regularity puts a one-sided neighborhood inside $\mathrm{\Omega }$. In each component $U_{\pm }$, membership in $\mathrm{\Omega }$ is both open and closed relative to that component, since $\mathrm{\partial }\mathrm{\Omega }\cap B_r(a_0)\subset X$. Therefore at least one whole component lies in $\mathrm{\Omega }$. Change of variables for the embedded holomorphic map gives $\mathrm{Area}(u)\ge \min \{\mathrm{Area}(U_{+}),\mathrm{Area}(U_{-})\}>0$. The disk and this positive bound depend on the fixed limiting X arc, not on the separation of L and L'. There are only finitely many relevant lifted arcs, so take the minimum of their positive bounds. The identical argument applies to a rigid strip, which is an embedded two-edge polygon under the same embedding result. This proves the fixed-patch energy cost used here and at the tangency below from the actual surface geometry. This contradicts Area→0. Thus equal-action off-diagonal coefficients vanish for a sufficiently small step. The diagonal coefficient is the unique thin curvilinear triangle, one for each intersection. Its region is explicitly the strip between the two close graphs on the appropriate arc from their degree-zero critical point to p, closed by the short X arc. The sign of their difference selects exactly one of the two arcs. Extra periods would include a negative-winding region. Its existence, boundary regularity and minimal convex exponents are precisely the directly constructed graph-domain maps of step 7.1. The embedding/winding argument gives uniqueness. Its coefficient is nonzero and its local-system transport invertible. Hence multiplication is a finite triangular matrix with invertible diagonal blocks, and is an actual chain isomorphism. The identical argument applies to left multiplication. At a generic quadratic tangency, take the path step sufficiently small about that single event. Two nearby generators of adjacent degrees are born or die. Their one tiny positive bigon is the entire local differential between them, with nonzero invertible weighted transport. It is unique by the same positive-region/embedding argument. Filter the mapping cone of the continuation chain map by the finite set of limiting action values, grouping equal values. In every cluster away from the tangency, the associated graded is a direct sum of cones of the preceding diagonal invertible maps. At the tangency cluster it additionally has the two-generator complex with its invertible bigon differential. There are no other same-cluster terms: a triangle or strip connecting a different limiting intersection has an X edge leaving the fixed small neighborhood and thus has the positive energy cost just proved, whereas its area tends to zero by the action formula. All other terms strictly lower the filtration. Each associated-graded complex is acyclic. A finite filtration and the elementary long exact sequence therefore make the full mapping cone acyclic. This proves quasi-isomorphy across a birth or death, without a clean-limit inverse or a purported uniform transverse spectral gap. Take the finite subdivision furnished by step 11.1 and compose these quasi-isomorphisms. The A-infinity associativity relation, on distinct physical copy labels, identifies its cohomological action with multiplication by the product of the continuation cocycles. The same-parent triangle identity makes that product $c_{AB}$. Thus the actual marked $c_{AB}$ induces a quasi-isomorphism against the chosen probe. This uses actual polygon maps and identities, and works for every finite required list of nonparallel probes.

Parallel-probe contraction. A fixed bent copy need not remain disjoint from a future parallel parent with a nearby, different flux. Its complex is nevertheless acyclic, as follows. Choose a normal lift of the two graphical circles. Their difference is f(x)=a+H_j'(x)−H_i'(x), a≠0, where a is their flux difference for this lift. If there are no zeros the complex is zero. Otherwise its 2r transverse zeros alternate. Positive winding and the embedding result show that strips are exactly the adjacent positive-lobe bigons. A nonadjacent arc includes a negative lobe; an additional full period includes every sign and is likewise impossible. Every adjacent positive lobe has one conformal parametrization with the prescribed ends. The direct graph-domain construction in step 7.1 gives each positive lobe its actual unique smooth finite-energy map, without a deformation or a clean limit. The degrees alternate, giving r copies of the coefficient space in each of two consecutive degrees. Its differential is a cyclic two-diagonal matrix with invertible weighted parallel-transport entries. After diagonal transport/action changes of basis, eliminate r−1 rows. The only remaining block is 1−νD, D(T)=Hol(F) T Hol(E)^(−1), ν=exp(2πiτa) or the simultaneous inverse depending on edge convention. To verify the coefficient, multiply the r successive adjacent ratios: the local transports telescope to D and the signed lobe-area difference is integral f dx, namely the normal period times a. This is precisely the stated area exponent after the parent's period normalization. Opposite boundary orientation supplies the minus sign; this is the same two-cell orientation convention as the zero-flux differential D−1. Every eigenvalue of D has modulus one, including arbitrary finite Jordan parts. Since |ν|≠1, 1−νD is invertible. The whole differential is consequently invertible. Distinct relevant normal lifts give separate finite path components; for different flux no a+n is zero, so the calculation applies to each. Any continuation map between these acyclic complexes is a quasi-isomorphism. Same-flux probes, including different finite local systems, are exactly the same-parent case of step 9.1.

Ordered immutable polygon charts. An index records a finite original-object set, a finite downward-closed ancestry, and fresh copy curves against every ancestor curve; keep old curves literally. The generic choice in the preceding construction supplies such fresh curves, even when their intersections occur on old curved edges, because the smooth-corner extension already applies. Two incomparable records have a common upper record by taking their finite histories and making fresh choices; incomparable old curves need not be mutually generic, since they never occur together in an increasing word. Let $A_{ord}$ have objects $(I,X)$, actual Floer complexes for $I<J$, only formal scalar identities at equality, and zero backward Homs. On strictly increasing words use the actual polygon operations. The preceding absolute convergence permits their area-cutoff identities to pass to the complex sums. Adjoin strict units: binary multiplication by the formal identity is identity, and every higher unit-containing operation is zero. In a unit-containing bar equation the adjacent binary unit contractions cancel, so these are actual unital A-infinity identities. Mark the closed degree-zero continuation cocycles $c_{IJ}(X)$ carrying the identity coefficient. Their pre- and post-composition actions on every forward Hom where defined are quasi-isomorphisms by the same-parent, nonparallel and different-flux calculations. This is established before localization, using only geometric operations.

The dg envelope is constructed, not assumed. A right module assigns graded spaces to the objects of $A_{ord}$ and a degree-one coderivation on its tensor product with the reduced suspended bar coalgebra; its square is zero. A module morphism is a comodule map, with differential $d\varphi =b_N\varphi -(-1{)}^{\mathrm{\mid }\varphi \mathrm{\mid }}\varphi b_M$ and ordinary composition. Expansion gives $d^2=0$ and the graded Leibniz rule. The representable $h_X$ has entries ${\mathrm{Hom}}_{A_{ord}}(Y,X)$ and module operations given by the original polygon operations. The all-arity Yoneda map inserts the external word into these operations. Evaluation of a closed module map $h_X\to M$ at the strict unit has an inverse obtained by inserting the resulting element in the module operations. The homotopy between their composite and identity inserts that unit at the terminal empty place of the augmented bar word: contractions away from the inserted unit cancel in pairs, the adjacent binary contraction is identity, and higher unit-containing operations vanish. There is one insertion on each word, so this proves the Hom quasi-isomorphism without an infinite sum. Denote the full dg category of representables by D and adjoin finite cones. Thus actual forward Floer complexes map quasi-isomorphically to the corresponding dg Homs, and their binary actions agree on cohomology by the arity-two functor equation. In particular the marked continuation actions are still quasi-isomorphisms on those forward dg Homs.

Actual quotient inverses and acyclic forward coefficients. Adjoin for every marked $c:A\to B$ its cone $C_c=B\oplus A[1]$, then a degree-minus-one endomorphism ${ϵ}_c$ with $d{ϵ}_c=1$. The resulting dg quotient Q has Homs that are algebraic direct sums of finite strings through these cones. The differential is the internal differential plus replacement of an epsilon by identity, with the graded Leibniz signs. In the cone order $(B,A[1])$, the bottom-left entry v of ${ϵ}_c$ is an underlying closed degree-zero map $B\to A$. Its diagonal entries give degree-minus-one maps $h_B,-h_A$, with $cv+d{h}_B=1_B$ and $vc+d{h}_A=1_A$. Hence c is invertible in the actual quotient cohomology. Now fix finitely many marked cones and a copy $Y_l$ later than all their endpoints and a chosen source X. The complex $V_c={\mathrm{Hom}}_D(C_c,Y_l)$ is the shifted cone of precomposition by c between two forward dg Homs. It is acyclic by the independently proved geometric continuation quasi-isomorphism, without a coherent target. For each degree split $V_c^n=Z_c^n\oplus T_c^n$; acyclicity makes $d:T_c^n\to Z_c^{n+1}$ bijective. Its inverse on cycles and zero on T defines a square-zero contraction $h_c$ with $d{h}_c+h_{c}d=1$. Full AC permits these actual vector-space complements.

Finite-string contraction. Restrict temporarily to quotient strings using one finite cone set S and ending at the chosen later target $Y_l$. Its length-zero part is $W_l={\mathrm{Hom}}_D(X,Y_l)$. Write a positive-length string with its leftmost factor first as $a_r{ϵ}_{C_r}{a}_{r-1}\cdots {ϵ}_{C_1}{a}_0$, where $a_r\in V_{C_r}$. Write $d=d_0+\mathrm{\Delta }$, with $d_0$ preserving epsilon length and $\mathrm{\Delta }$ lowering it. On positive-length strings let h apply $h_{C_r}$ to the leftmost factor, and put h zero on length zero. With i inclusion and p projection onto $W_l$, tensor signs give $d_{0}h+h{d}_0=1-ip$, $h^2=hi=ph=0$ and $pi=1$: cross terms cancel since h has degree minus one. Define $P=p{\sum }_{k\ge 0}(-\mathrm{\Delta }h{)}^k$, $I=i$ and $H=h{\sum }_{k\ge 0}(-\mathrm{\Delta }h{)}^k$. Every sum is finite on each string because $\mathrm{\Delta }$ strictly lowers its length. Expanding the finite geometric inverses and cancelling successive terms proves $PI=1$, $Pd=d_{W_l}P$ and $dH+Hd=1-IP$. Since $\mathrm{\Delta }i=0$, the induced differential on $W_l$ is its original one. This is a contraction of the full quotient differential, rather than a claim based only on an associated graded object or a filtered nerve.

All localized classes have forward representatives. For $I<J$, the forward inclusion ${\mathrm{Hom}}_D(X_I,Y_J)\to Q(X_I,Y_J)$ is a quasi-isomorphism. To prove surjectivity, a closed quotient element q has finite string support. Choose l later than its source, target and every cone endpoint in that support. Postcompose with the marked $s:Y_J\to Y_l$. The finite contraction gives $sq-IP(sq)=dH(sq)$, with $P(sq)$ a forward cocycle. Forward multiplication by s is a quasi-isomorphism, so choose a closed forward a and a forward homotopy between sa and $P(sq)$. The displayed relation makes $sq$ cohomologous to sa; the explicit inverse of s then makes q cohomologous to a. For injectivity, if a forward cocycle a bounds a quotient primitive, choose l later than the primitive's finite support. Apply the contraction to obtain a forward primitive of sa. Forward continuation quasi-isomorphy implies a was already a forward boundary. This proves the assertion. For arbitrary chosen representatives X,Y of original objects, first replace Y by an equivalent later copy so that the pair is strictly forward; the same argument applies. Thus self-Homs in Q are computed by two distinct genuine copies of the same original circle, not by the formal scalar endomorphism at a single ordered label. The all-arity map from an increasing polygon chart is quasi-isomorphic on strictly forward Homs, and localization supplies the missing self/parallel Homs. No quasi-isomorphism from a chart's formal scalar self-Hom to the final two-degree self-Hom is asserted. Choose one representative per original object and call the resulting full dg category G. Its actual quotient composition gives the required full geometric Homs and binary products. For a geometric pair of immutable curves in the reverse index order, first postcompose geometrically with the target's later continuation copy. This is a quasi-isomorphism by the continuation calculation and now has strictly forward target, so the established forward map identifies its Hom with the same G-Hom. The actual arity-three identities on distinct generic labels make these transports compatible with binary products. In particular opposite geometric pairs and cyclic rotations of a three-curve diagram can be used before transport to G; they are not discarded merely because one displayed order is not increasing. Fresh auxiliary copies avoid every curve in that finite diagram, so no operations on incompatible incomparable histories are required.

Parallel binary products and their trace pairing. The two-copy calculation now computes the actual G-Hom: its degree-zero group is $\ker (D_{ij}-1)$ and its degree-one group is $\mathrm{coker}(D_{ij}-1)$. A degree-zero class is precisely a flat intertwiner. For three copies with the same parent, the degree-compatible triangle enumeration gives one actual triangle for minimum/minimum to minimum, minimum/maximum to maximum and maximum/minimum to maximum, and no degree-two output. The scalar orientations were normalized independently by the auxiliary-copy argument; boundary transports give coefficient composition, possibly with a simultaneous full winding $D^m$. On kernels and coinvariants $D^m$ induces identity, because $D^m-1$ is $(D-1)$ times a finite Laurent polynomial for every integer m. A flat intertwiner may be moved along its incident boundary edge, since it intertwines the monodromies. Consequently the three induced products are respectively composition of intertwiners and its left/right action on coinvariants; two degree-one classes multiply to zero. The trace pairing between opposite coefficient spaces descends to the kernel/coinvariant groups: the trace identity makes the image of $D-1$ the annihilator of the opposite invariant kernel. Finite-dimensional linear algebra makes this descended pairing perfect. Fix the maximum orientation dual to the normalized minimum, so that its value is the ordinary coefficient trace. This is the same geometric pairing obtained by rotating the enumerated triangles: boundary parallel transports cancel around their contractible boundary and cyclic matrix trace retains the suspended rotation sign. Distinct parallel fluxes give zero cohomology by the actual different-flux contraction, so introduce no uncomputed binary product.

Mixed products, including the actual transverse actions. Compute a finite required diagram in sufficiently small copies relative to its fixed nonparallel angles; every transverse Hom is then concentrated in its original single Maslov degree. General immutable charts are identified with it by the already proved continuation actions, including their tangency mapping-cone calculation, so this choice does not assume a global angle margin. For a flat intertwiner T, replace the identity coefficient of a continuation minimum by T. In every polygon coefficient, T can be slid along the parent boundary transport because $M_{j}T=T{M}_i$. Its product with any transverse morphism is therefore the product with the identity-coefficient continuation followed by ordinary coefficient composition. The identity-coefficient continuation is the actual invertible marked arrow; under the forward-Hom identification this proves the claimed action in G, rather than declaring the desired action to be its definition. A parallel degree-one class composed with a transverse class vanishes: the transverse source and output have the same unique Maslov degree, whereas the proposed product raises it by one. The remaining case consists of two transverse morphisms whose product has a parallel endpoint. Pair that product with an arbitrary opposite degree-zero flat intertwiner, rotate its actual three-boundary triangle, and use the just-proved coefficient action. Coherent brane caps give the suspended Koszul rotation sign, and cyclic trace of the boundary coefficient composition is unchanged. The pairing is compatible with continuation: rotating a continuation triangle identifies its left action with the trace-adjoint right action, and their marked inverses cancel. The perfect kernel/coinvariant pairing therefore determines the product uniquely. It is exactly the trace-dual composition asserted in the Statement. This checks every binary case with parallel objects, using actual triangles and the actual localized Homs. It proves neither a theta dictionary nor a mirror equivalence.

Compatibility and conclusion. Different copy records have a common upper record. Actual A-infinity arity-three identities identify the two binary transports after refinement, and the independently normalized continuation cocycles compose transitively on cohomology. The finite-string identification therefore preserves binary composition and is independent of the chosen representatives. Literal straight generic labels can be retained in a finite history and placed in strictly increasing records; their original polygon words are unchanged, so their all-arity forward maps retain those actual operations. A finite cover, equipped with the pulled area form and local systems, lifts every disk uniquely from one chosen vertex. It preserves area, determinant-cap orientations and boundary transports; summing over its finitely many lifted components preserves all formulas, including the two-cell differential and the trace-dual binary products. Shifts and finite direct sums are the corresponding graded matrix operations. The absolute convergence established before localization applies at the literal complex parameter; localization and bar words are algebraic finite sums and require no specialization of a formal series. This completes the actual geometric parallel-Hom and binary comparison independently of any coherent-sheaf or theta comparison.