Elliptic Hamiltonian-copy completion and the full derived dg comparison

The cyclic input has the precise meaning in [F1]. In the supplier's cyclic-coderivation construction, differentiating the paired expression along $\exp (t\xi )$ gives zero by the cyclic coderivation identity; at any fixed bar length this is a finite polynomial calculation. Its arity/weight corrections therefore preserve the pairing, and its matrix trace and invariant-output descent retain the same cyclic identities. Thus the initial transversal comparison has this property before the completion construction below. The conclusion here extends that comparison as an ordinary full A-infinity functor; it does not require a new cyclic trace construction on every localized copy complex. 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 copy completion using the explicitly named smooth-end supplier.

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.

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. For ALL fixed-copy words, including mixed parents, the height/face-chain proof below now gives a direct quantitative bound. Every rigid lifted map is embedded, its positive finite-face multiplicities bound all edge winding, and each curve occurs once, so a chain cannot hide canceled winding. The number of actual maps is polynomial in energy; source unit-modulus/unipotent transports grow polynomially as well. Hence every operation sum is absolutely convergent for the literal complex parameter Im(tau)>0. No degeneration to the coincident minimal model, no uniform copy limit and no Novikov specialization are needed. The fixed irrational ell_0 ensures finite charts and their refinements use compatible height-monotone copies.

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.

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.

The elementary conformal-domain argument in step 5.1 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 5.1 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.

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 5.2 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.

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 6.2. 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 6.2. 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.

3. Uniqueness for a prescribed lifted boundary word Suppose two rigid polygons have the same oriented lifted edge paths and corners. step 6.3 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.

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 6.1. The graph-domain construction in step 5.1 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.

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 6.3 and 7.2 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.

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 6.2. 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.

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.

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 6.3. 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 5.1. 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 9.1 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.

Precise setup Let E be a semistable bundle on the elliptic curve (a line bundle is the rank-one case). Choose disjoint positive effective divisors D1,D2 with different degrees, in generic positions. Put K=E(−D1−D2), A=E(−D1), B=E(−D2), P=A plus B. Their evaluation maps give an exact sequence 0 -> K --a--> P --b--> E -> 0. (1) The factors are explicitly $a=(s_{D2},-s_{D1}),b=(s_{D1},s_{D2})$. Exactness follows at a point since the two sections have no common zero: if one section is a unit, eliminate that coordinate and solve the kernel equation; these local solutions agree. Tensoring the scalar sequence by the locally free E preserves exactness. Semistability implies Hom(P,K)=0 because each summand has strictly greater slope than K. For line bundles this is simply negative degree. The lead's isogeny/unipotent classification supplies this semistable bundle case and its exact comparison. Let c in Ext1(E,K) be the extension class of (1), with the sign fixed so its derived connecting triangle is the sequence (1). All objects and the finitely many probe bundles below are chosen pairwise transversal except the repeated endpoints in compositions. Write the same letters for their Fukaya correspondents and their theta-identified morphisms. The all-arity comparison on distinct-object words is the lead's actual constructed functor, including finite nilpotent/isogeny descent.

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 5.1 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 7.1.

The unital geometric category is constructed first. Let indices be finite ancestry charts. A chart records a finite set of original objects, all finite ancestor records, its fresh graph parameters, and the finite genericity conditions just described. It retains actual old curve labels; only fresh copies are chosen. Two incomparable charts have a common upper chart by taking the finite union of history and choosing fresh parameters against all of it. Incomparable histories need not themselves be mutually generic. Only objects on a comparable chain enter an operation. This is a filtered system of actual immutable finite data, not an integer enumeration of all objects. Form the directed A-infinity category $A_{ord}$ with objects (I,X). For $I<J$ use the actual geometric Floer complex of their fixed copies, with actual absolutely convergent polygon operations on every increasing word. At equality retain only the formal scalar identity; put backward Homs zero. The distinct-label polygon identities prove the nonunit equations. The strict-unit extension adds the two binary identity actions and makes every higher identity-containing operation zero; the unit-containing A-infinity equations reduce to the cancellation of the two adjacent binary unit contractions. No repeated geometric polygon is postulated. Mark the closed $c_{IJ}(X)$. Sections 2–4 prove directly, before any localization, that pre- and post-composition by every such c is a quasi-isomorphism on each actual forward Hom where defined. The proof is local to each finite diagram and uses no global angle or action bound. Pass to the explicit dg Yoneda envelope of $A_{ord}$, adjoin the finite mapping cones $C_c$, and form its algebraic dg quotient Q by adjoining ε_c of degree −1 with $d{\epsilon }_c=1$. This is the finite quotient-string construction spelled out in the subsequent localization argument, applied here using the independently proved forward continuation acyclicity. Its hypothesis on the coefficient $V_c=Hom(C_c,Y_l)$ is now proved directly: $V_c$ is the shifted cone of actual forward multiplication by c, which steps 7.1–11.1 make acyclic. It does not need an external full geometric category or either mirror target. Apply the existing finite epsilon-string contraction to any finite cone set and a later copy $Y_l$. It retracts the actual quotient Hom onto the forward Hom to $Y_l$. The original forward inclusion is a quasi-isomorphism: every quotient class, and every primitive witnessing its vanishing, has finite string support; postcompose by the invertible marked arrow $Y\to Y_l$, apply that finite contraction, and use the already proved forward continuation quasi-isomorphism to move the resulting forward class back. Injectivity uses the same finite primitive and the explicit inverse/homotopies of the marked arrow. This is the finite-support argument with the geometric-target hypothesis removed, not a new constant-nerve inference. Choose one representative of each original circle/local-system object in Q. The resulting full dg subcategory G is now an actual unital geometric A-infinity category (dg is a special case), with self, repeated and parallel words defined by its actual quotient-string composition. The Yoneda/bar map from every ordered finite polygon chart is an all-arity map that is a quasi-isomorphism on every strictly forward Hom, as proved independently in [F6]. Its formal scalar self-Homs are not asserted quasi-isomorphic to the localized two-degree self-Homs; those are computed only after replacing the target by a distinct later copy. It uses every original higher polygon operation. Therefore this construction agrees with the actual polygon source in the precise sense of these all-arity quasi-equivalences; it is not a transport of its binary cohomology algebra.

Minimal cochains and closed evaluation maps. Write $G_{\mathrm{d}\mathrm{g}}$ for the full geometric dg category just constructed, and form an actual minimal $A_{\mathrm{\infty }}$ model before using any Hom/Ext vanishing as a cochain equality. For each Hom complex $C$ over $\mathbb{C}$, full AC gives splittings $C=H\oplus B\oplus T$, with $B=\mathrm{im}d$, $H\subset \ker d$ representing cohomology, and $d:T^r\to B^{r+1}$ an isomorphism. Let $i,p$ be inclusion/projection and let $h$ invert this last map on $B$, with $h=0$ on $H\oplus T$. Then $dh+hd=1-ip$, $pi=1$, $h^2=hi=ph=0$. Choose identity representatives in $H$. Transfer the full dg composition by finite planar binary trees: put $i$ at leaves, composition at vertices, $h$ on internal edges and $p$ at the root; replacing the root $p$ by $h$ gives the higher components of the inclusion, with signs obtained by suspending degree by minus one. Equivalently solve the bar functor equation successively in word length, using $h$ on its residual term and $p$ for the corresponding transferred operation. Expanding $dh+hd=1-ip$ on each internal edge cancels adjacent contracted-edge terms; the surviving root terms are the transferred bar square and functor equations. This proves both identities at every length, since the original dg differential is a derivation and its composition is associative. Normalization at the chosen identities gives a strict unit: trees with a unit leaf cancel in pairs, except the two binary unit actions. There are finitely many trees for a fixed word, even though quotient Hom complexes themselves need not be finite-dimensional. This construction uses all geometric higher polygon data already present in the dg Yoneda envelope, not just its binary cohomology product. Denote this minimal geometric category hereafter by $G$, and retain its actual all-arity quasi-isomorphism $I:G\to G_{\mathrm{d}\mathrm{g}}$. Perform the same contraction on the coherent dg enhancement; its minimal Hom spaces are the actual Hom/Ext groups. Compose the known transversal functors with these minimal comparisons and their recursively constructed bar inverses. Thus the transversal comparison remains an all-arity comparison on every relevant generic word. Its first component is the same cohomological theta dictionary. In this minimal model form $R=(K[1]\oplus P,\delta =a)$, with $K$ in cochain degree $-1$ and $P$ in degree zero. Now ${\mu }_1=0$, and two consecutive delta factors have no composable matrix path, so its Maurer--Cartan equation holds. Define $f:R\to E$ by its $b$ component on $P$ and $g:E\to R$ by its $c$ component into $K[1]$. Their only possible twisted differential terms are respectively ${\mu }_2(b,a)$ and ${\mu }_2(a,c)$. These are literally zero in this minimal model because the corresponding cohomology products vanish: $ba=0$ in the exact sequence, the pushout class $a_{\ast }c=0$, and the pullback class $cb=0$. Their total degrees are zero, so $f,g$ are closed twisted morphisms. In $G_{\mathrm{d}\mathrm{g}}$ their images include the actual $I^2(b,a)$ and $I^2(a,c)$ homotopy components; no literal vanishing of boundary composites in the dg quotient is asserted. The finite matrix extension of $I$ preserves Hom quasi-isomorphisms by filtering source/target matrix indices; its associated graded is a finite direct sum of shifted $I^1$ maps. Consequently the evaluation equivalences proved below in the minimal model transfer to the original geometric dg category.

Degree-zero endomorphisms in the geometric quotient, before evaluation. The forward-Hom identification of step 12.1 applies also to two distinct Hamiltonian copies of the same parent circle with local-system fibre V and monodromy M. By step 6.1 their complex, after the exact action gauge, is $\mathrm{End}(V)\stackrel{T\mapsto MT{M}^{-1}-T}{\to }\mathrm{End}(V)$ in degrees zero and one. Hence every degree-zero class in the quotient self-Hom is represented uniquely by the minimum with coefficient $T$ satisfying $MT=TM$, after composing with the inverse of the marked identity-coefficient continuation. This is a calculation of the actual quotient Hom, not an identification assumed from a future mirror equivalence. To see faithfulness of negative probes directly, fix a straight rank-one probe Q of sufficiently negative slope and take the two copies sufficiently small relative to its fixed nonzero angle. All Q-to-copy intersections then have degree zero, so those forward complexes have zero differential. Repeat step 10.1 with coefficient T at the continuation corner. Its area ordering and its unique diagonal thin triangles are independent of coefficients. Boundary parallel transport replaces each diagonal identity block by a nonzero scalar times $U_{p}T{V}_p$, with $U_p,V_p$ invertible; all other blocks are strictly triangular in the same finite action ordering. Multiplication by the identity-coefficient continuation is invertible. Thus a self-Hom class acting as zero on $H^0{\mathrm{Hom}}_G(Q,E)$ has $U_{p}T{V}_p=0$ at every diagonal intersection and consequently $T=0$. There is at least one intersection since the slopes differ. Step 12.1 transports this calculation to the quotient; its minimal model has the same cohomology and binary action. For a finite semistable direct sum, group summands with the same underlying parent circle into one local system; between distinct parallel fluxes the Hom is acyclic by step 11.1. The same calculation detects every remaining intertwiner block. Finally this explicit binary dictionary agrees with the sheaf dictionary already proved by [F1]: its Statement identifies every graded Hom and binary product, its matrix-vertex formulas commute with coefficient intertwiners, and its finite isogeny descent retains this property. A flat intertwiner acts on each boundary coefficient by the same composition on either side; it commutes with the monodromy powers and all their finite jets. Therefore the same-parent classes just computed give the actual sheaf endomorphisms, and their binary products with transversal morphisms are the usual sheaf compositions. This verifies the previously missing bridge from the quotient to that binary dictionary without constructing any repeated-word higher comparison or assuming evaluation equivalences.

2. The first unit is detected on generic probes Let Q be any sufficiently negative line bundle in a generic position, transverse to K,A,B,E. Then H1(Q* K)=H1(QP)=H1(QE)=0. The twisted probe complex is literally H0(Q* K) --a--> H0(Q*P) in degrees −1,0. The map f_* to H0(QE), and the map g_ back, involve the binary terms and at most one delta insertion, hence only m2 and m3 on the DISTINCT quadruples Q,K,A/E and Q,K,B/E. The lead's full transversal A-infinity comparison identifies them with the coherent derived maps. In particular its twisted image of f has no F2(b,a) correction because the underlying output degree is −1; the possible F2(a,c) correction to g would lie in Hom(E,P), which is zero by the strict slope inequality. Thus the images are exactly the coherent b,c maps, not unidentified changes of their extension class. In the Cech model of (1) these maps are mutual quasi-isomorphisms, including the connecting correction, so f_* g_=identity on H0(QE). The A-infinity Yoneda identity in the actual full geometric category says that the action of the closed composition $h_E=m{2}^{Tw}(f,g)$ on probe cohomology is f_g_. This identity follows from its arity-three relation; it does not specify $h_E$ in advance. Therefore $h_E-1_E$ acts as zero on H0(Q*E). The independently proved geometric faithfulness in step 13.2 now gives $h_E-1_E=0$ in the actual quotient self-Hom. Its proof uses the same-parent two-cell complex, the forward-Hom localization calculation and coefficient-sensitive thin triangles; it does not treat an unidentified quotient-string class as a sheaf endomorphism or assume conservativity on arbitrary twisted complexes. Thus m2^Tw(f,g)=1_E in H0. (2) This is the required repeated-endpoint unit derived from probes, rather than assumed as a transversal m3 coefficient.

Binding the evaluation and resolution inputs. Use the finite twisted objects R=[K[1]→P] in the actual minimal geometric model G of step 13.1 and transport them through its all-arity inclusion into the original dg model. The finite twisted Hom filtration there preserves the constructed equivalences. The evaluation argument can use G's already constructed units and repeated compositions. Literal straight generic collections are retained as admissible labels in finite charts; later repeated occurrences use fresh perturbed labels. Thus the original transversal comparison on every fresh generic-resolution word is unchanged. Its finite twisted extension is the existing explicit matrix formula, and the actual geometric evaluation units are the existing probe/negative-degree argument. The directed resolution category constructed below consequently has an actual all-arity functor to G before its localization. Its forward Hom maps are quasi-isomorphisms by the existing finite matrix filtration, since every constituent polygon-chart map just proved is a quasi-isomorphism. The coherent functor is the repaired theta supplier's actual all-arity comparison. Evaluation maps are genuine equivalences in G, so every marked resolution arrow has invertible images in both targets. All hypotheses for the previously checked resolution localization 16.1–22.1 are now supplied without assuming the full geometric category in advance. The existing module/derived completion 23.1–28.1 is unchanged. The only analytic inputs used here are the actual smooth-end estimate, the existing local epsilon/regularity/gluing interfaces, the locally proved smooth-corner phase correction, and the explicit same-parent triangle calculation above. No clean/Morse limiting inverse, moving-boundary Floer theorem, enhancement uniqueness theorem, formal Novikov specialization, or binary-equivalence transport is inserted.

3. The second unit does not follow from probe cohomology alone Put $h_R=m{2}^{Tw}(g,f)$. Its degree-zero matrix block $P\to K[1]$ is precisely c b, hence zero in the minimal model by step 13.1: source P has no incoming delta and target K[1] has no outgoing delta, so no higher insertion contributes to that block. Its opposite block $K[1]\to P$ is zero by degree, since its underlying degree is −1 and the bundle Hom/Ext spaces have no negative degree. Thus $h_R$ has only diagonal blocks $u:K\to K$ and $v:P\to P$. Closedness gives v a=a u. (3) There are no extra delta terms in this equation, because only one delta direction exists and every remaining operation would have negative underlying degree. The independent binary dictionary of step 13.2 identifies these diagonal blocks and their products with a with actual sheaf maps. In particular $h_R$ is an actual diagonal chain map of the coherent two-term complex (1). Associativity in H0 and (2) give $f{h}_R=f$. The Hom complex from R to E has no negative-degree part: Hom(P,E) has no negative Ext, and $Hom(K[1],E{)}^{-1}=Ex{t}^{-2}(K,E)=0$. Consequently equality of these closed degree-zero maps in H0 is equality of their actual degree-zero components, not equality only modulo an unseen boundary. Higher delta insertions into their composition again have negative underlying degree. Hence b(v−1_P)=0. By exactness of (1), v−1_P factors as $P\to K\to P$. But Hom(P,K)=0, so $v=1_P$. Equation (3) and injectivity of a then give $u=1_K$. Thus m2^Tw(g,f)=1_R in H0. (4) This argument repairs the tempting but false shortcut that identity on the COHOMOLOGY of every two-term probe complex directly determines its diagonal blocks. What makes it true here is the concrete negative-slope vanishing Hom(P,K)=0 together with the absence of negative-degree boundaries. Equations (2),(4) prove the actual geometric quasi-isomorphism R≅E and the evaluation triangle. No appeal to a named surgery triangle, no replacement of normal translations by Hamiltonian isotopies, and no abstract assumption of faithful restricted Yoneda is used. The all-arity A-infinity identities of the actual full Hamiltonian-copy model are still indispensable; the proof is not a construction of those analytic operations from scratch.

4. Use in the nontransversal extension For each finite collection of objects choose fresh generic divisors so all components of their evaluation resolutions are mutually transversal with no triple concurrence. The choice excludes only finitely many slope/position conditions at each stage. Formula (1) and the just-proved quasi-isomorphisms allow repeated occurrences of E to be replaced by different genuine quasi-isomorphic twisted resolutions, not by non-Hamiltonian normal translates. Every word in the fresh resolutions is now evaluated using the known all-arity transversal operations, including all matrix/delta insertions. A directed-copy localization and the explicit bar extension therefore have an actual route to define all repeated/self words. The separate formal completion proof gives the exact twisted functor and perfect-module formulas; the companion Hamiltonian-copy construction gives the actual localization/cofinality maps and binds the frozen polygon provider.

4. Actual evaluation resolutions now identify objects In the full Hamiltonian-copy category take the generic evaluation resolution R_E=[E(−D1−D2) -> E(−D1) plus E(−D2)] for a semistable vector bundle E, with disjoint generic divisors. The evaluation-unit proof below constructs closed $f:R_E\to E$ and $g:E\to R_E$ and proves their actual inverse identities using generic probes, degree vanishing and Hom(P,K)=0. Its repeated-endpoint unit is a CONSEQUENCE of the existing A-infinity Yoneda identities in this geometric category. It is not inserted as a definition of those operations. Fresh divisors make every finite collection of resolution terms mutually transversal; nilpotent local systems and finite isogenies are covered by the lead's all-arity packet. Since the copied curves are Hamiltonian images of the same resolved straight terms, step 3.1 identifies their objects without changing their flux parameters. For a vertical indecomposable of length k, the lead's object-only rotation gives the corresponding cone identity without requiring a global Fourier-equivariant comparison. For K of class (1,a), P of class (1,a+k), T=(0,k), choose g=[[1−a,1],[-a,1]]. It sends these classes to (1,0),(k+1,k),(k,k), all nonvertical. Its coherent dg counterpart is tensor(L) after S after tensor(L^(1−a)), where S is the explicitly reconstructed Poincare-kernel transform. Tensoring first makes the two line bundles have degrees 1,k+1, so the transform of their torsion exact sequence is an actual exact bundle sequence. The transformed torsion is $L\otimes P_p\otimes U_k$, by the kernel's finite jets and the unipotent classification. The generic-probe unit proof applies to this bundle sequence, since Hom(P',K')=0 by slopes. Pulling back by the strict linear symplectic map $g^{-1}$ proves that the actual vertical circle with its length-k local system is $Cone(K\to P)$. The lead's binary theta-evaluation packet identifies the support p and nilpotent holonomy correctly. This proves an OBJECT identity only; no unsupported equality of two globally Fourier-conjugated higher functors is used.

A genuinely filtered directed resolution category. Fix for each original indecomposable circle/local-system object X the nonvertical base complex R0(X) from step 17.1; its coherent image has exactly the corresponding sheaf, up to the declared shift. Work with finite ancestry charts I: record a finite object set $S_I$, a finite downward-closed ancestry, and for each X in $S_I$ a FRESH generic evaluation resolution $R_I(X)$ of that fixed R0(X). Include all ancestor object sets and choose fresh terms avoiding all ancestor terms and every finite concurrence equation. This uses the concrete finite divisor resolution and finite upper-triangular transfer of the lead and steps 10.2, 12.1, 14.1, 15.2, 16.1 and 17.1. Two charts have a common upper chart by taking the finite union of ancestries and object sets, then fresh choices; charts containing a fixed finite diagram are cofinal. No injection of an uncountable object set into integer degrees is assumed. If $I<J$ in ancestry order, every fresh term of $R_J$ avoids all terms of $R_I$; the same holds along every finite increasing chain. Let D have objects (I,X), full actual twisted Hom complexes for $I<J$, only formal scalar identities at equality, and zero backward Hom. All operations on an increasing word are the already constructed transversal operations with finite delta insertions, because all its underlying terms are distinct generic labels. The suspended bar identities restrict, and the normalized strict-unit extension has higher unit-containing operations zero: the two adjacent binary unit contractions cancel. The comparison to coherent complexes is the actual transversal all-arity functor with those same finite insertions. Its first map on each forward Hom is a quasi-isomorphism: its ACTUAL mapping cone has a FINITE matrix filtration whose successive cones are the matched transversal Hom cones, so induction proves the full cone acyclic. The evaluation construction supplies actual geometric maps $f_I:R_I(X)\to R0(X),g_I$ in the opposite direction, inverse in H0. Mark $c_{IJ}=m2(g_J,f_I)$ for copies of the same X. The word $R_I,R0,R_J$ is generic termwise, so its coherent image is homotopic, by the actual F2 equation, to the corresponding coherent composition. Those coherent evaluation maps are quasi-isomorphisms: sufficiently negative generic Q identifies their probe maps with the exact coherent resolution; the cone is then zero because such twists detect a nonzero globally generated cohomology sheaf. Their composite on R0 is identity: R0 has just one coherent sheaf up to shift, so its degree-zero endomorphism is a sheaf endomorphism and negative globally generated probes are faithful on it. This uses faithful probes only for sheaf endomorphisms, not the false claim that they detect arbitrary ghost maps of complexes. Consequently both geometric and coherent functors send every marked $c_{IJ}$ to a genuine quasi-isomorphism. The next construction proves the localization fully faithful rather than inferring it merely from a filtered nerve.

Setup and actual inverse data on a marked arrow. Let D be the directed resolution category after its explicit dg Yoneda envelope. For a finite collection of its objects and arrows, there is a later copy $Y_l$ of any desired original target Y, later than every object used in that finite collection. Retain the full forward dg Hom complexes. Let $c:A\to B$ be a marked closed degree-zero arrow; its image under each target functor is a homotopy equivalence, and those functors are quasi-isomorphisms on forward Homs. Adjoin $C_c=Cone(c)$ and then the degree -1 endomorphism epsilon_c with $d{ϵ}_c=1$. This is the actual dg quotient Q, whose Homs are algebraic direct sums of finite strings through these cones. No completion is taken. Here are the inverse and homotopies, including the cone signs. Order $C_c=B$ direct-sum A[1], with differential matrix $(d_B,c;0,-d_A)$. Write the four entries of epsilon_c in that order. Its bottom-left entry is a closed underlying degree-zero map $v:B\to A$. Its diagonal entries give underlying degree -1 maps $h_B$ and $-h_A$. The diagonal parts of $d{ϵ}_c=1$ say $cv+d_{Hom}{h}_B=1_B,vc+d_{Hom}{h}_A=1_A$. (1) In the second equation the sign from the shifted A summand is precisely why $h_A$ is minus that diagonal entry. Thus each marked arrow is invertible on cohomology in Q, with an explicit representative v and explicit chain homotopies. This is not merely an assertion about its cone in a target.

The acyclic left coefficient is an actual forward complex. Fix finitely many marked cones $C_1,\mathrm{.}\mathrm{.}\mathrm{.},C_s$ and choose $Y_l$ later than ALL their endpoints, as well as the chosen source X. Then $V_j=Ho{m}_D(C_j,Y_l)$ is an acyclic complex. Indeed it is the shifted mapping cone of precomposition by $c_j$ between the two actual forward Hom complexes. Apply either target functor. Both Hom maps are quasi-isomorphisms and target precomposition by its actual invertible $c_j$ is a quasi-isomorphism. The commutative mapping-cone square therefore proves $V_j$ acyclic. This uses quasi-full-faithfulness only on already matched forward Homs, with all objects distinct and in order, not on localized or self Homs. Choose an actual square-zero contraction $h_j$ of $V_j$. Its precise algebraic construction over C is: for each degree split $V_j^n=Z_j^n$ direct-sum $T_j^n$; acyclicity makes $d:T_j^n\to Z_j^{n+1}$ a bijection. Define $h_j(z+t)$ to be the inverse image of z in $T_j^{n-1}$. Then $d{h}_j+h_{j}d=1$ and $h_j$ squared zero. These are splittings of the actual vector spaces and actual differentials; full AC in the item permits them. No inverse of a non-surjective differential on the entire space is claimed. Analytic boundedness is unnecessary: these are algebraic dg module Hom complexes after the polygon sums have already been proved convergent.

Explicit maps on every finite epsilon string. Let $Q_S(X,Y_l)$ be the subcomplex of $Q(X,Y_l)$ using only the finite cone set S. Every positive-length string is written with its leftmost factor first: $a_r{ϵ}_{C_r}{a}_{r-1}$ ... ${ϵ}_{C_1}{a}_0,a_r$ in $Ho{m}_D(C_r,Y_l)$. (2) Split its differential as $d=d_0+\delta$: $d_0$ is the tensor/internal differential with $d_0ϵ=0$; delta replaces one epsilon by its identity and composes its neighbours, with the graded Leibniz sign. Thus delta strictly lowers epsilon length, while $d_0$ preserves it. Define h on (2) by replacing $a_r$ with $h_{C_r}(a_r)$, and set $h=0$ on the length-zero summand $W_l=Ho{m}_D(X,Y_l)$. Tensor signs imply $d_0$ h+h $d_0=1-i$ p, $h^2=0,hi=ph=0$, p $i=1$, where i includes length zero and p projects onto it. The cross-terms in the tensor differential cancel because h has degree -1; it is not necessary that the other Hom factors be finite dimensional or acyclic. The FULL quotient contraction is $P_l=$ p ${\sum }_{k\ge 0}$ (-delta $h{)}^k,I_l=i,H_l=h{\sum }_{(}k\ge 0$) (-delta h)^k. (3) On a string of length r only $k\le r$ can contribute, and $H_l$'s last possible length-zero contribution is zero. Expanding $d_0$ h+h $d_0=1-i$ p and cancelling successive powers gives $P_l{I}_l=1,P_{l}d=d_W{P}_l,d{H}_l+H_{l}d=1-I_l{P}_l$. (4) Equivalently these follow by multiplying the finite geometric inverses (1+delta h)^-1 and (1+h delta)^-1; $\delta i=0$, so the induced differential on $W_l$ is exactly its original one. Formula (3) is the requested comparison on epsilon strings, and $I_l$ is its actual chain inverse up to the displayed homotopy. In particular $P_l$ of a positive-length word is an explicit finite sum of contractions/compositions; it is not its putative associated-graded image. All compositions of marked arrows occurring in delta are the actual dg compositions in D.

Actual normalized-bar retraction, including coherent components. For any finite ancestry diagram choose the above terminal index l. Let $J_l$ be that finite poset with the added terminal vertex l. Define $B_l=W_l\otimes N_{\ast }(N{J}_l)$, putting a simplicial p-chain in cochain degree -p. The differential on w tensor sigma is $d_{W}w\otimes \sigma +(-1{)}^{\mathrm{\mid }}w\mathrm{\mid }w\otimes boundary(\sigma )$. The augmentation $p_B$ sends every vertex coefficient to w and all positive simplicial degrees to zero. Its section $i_B$ is w -> w tensor [l]. For a normalized simplex $\sigma =[j_0,\mathrm{.}\mathrm{.}\mathrm{.},j_p]$, put $k(\sigma )=(-1{)}^{p+1}[j_0,\mathrm{.}\mathrm{.}\mathrm{.},j_p,l]$, with a repeated final vertex interpreted as zero. Then boundary k+k boundary is identity minus the section/augmentation. Consequently $H_B(w\otimes \sigma )=(-1{)}^{\mathrm{\mid }}w\mathrm{\mid }w\otimes k(\sigma )$ satisfies the corresponding total-complex identity. The ACTUAL maps and their homotopies are now $J_l=i_B{P}_l$ : $Q_S(X,Y_l)\to B_l,K_l=I_l{p}_B$ : $B_l\to Q_S(X,Y_l),1-K_l{J}_l=d{H}_l+H_{l}d,1-J_l{K}_l=d{H}_B+H_{B}d$. (5) This is an explicit epsilon-string/bar chain comparison and inverse. Its coefficient complex $W_l$ is the actual forward Hom reached by (3), rather than a constant coefficient substituted before the quotient was computed. It does not identify the original unstrictified forward coefficient diagram with a constant diagram without proof. The construction also gives the required finite compatibility, instead of requiring a globally functorial choice of AC complements. If two contractions have the same length-zero inclusion I and retractions P,P', then P-P'=d(P H')+(P H')d. (6) For a forward transition t of the full quotient complexes, write $t_0$ for its length-zero restriction, so $tI=I^{\mathrm{\prime }}{t}_0$. Then $P^{\mathrm{\prime }}t-t_{0}P=d(P^{\mathrm{\prime }}tH)+(P^{\mathrm{\prime }}tH)d$. (7) Both follow immediately from (4). The W transitions are STRICT: w(t)=P' t I, and $tI=I^{\mathrm{\prime }}w(t)$ implies $w(t_2)w(t_1)=P_2{t}_2{I}_1{P}_1{t}_1{I}_0=P_2{t}_2{t}_1{I}_0$. Consequently the putative transferred higher components $P_r{t}_r{H}_{r-1}$ ... $H_1{t}_1{I}_0$ are ZERO for $r\ge 2$, since $H_1{t}_1{I}_0=H_1{I}_{1}w(t_1)=0$. These are not the components of the coherent retraction. The actual retraction ends in $H_0$, as follows. The coherent retraction on actual marked transitions For a composable word of closed degree-zero forward transitions $t_1,\mathrm{.}\mathrm{.}\mathrm{.},t_r$, write $Q_j,W_j,I_j,P_j,H_j$ for its successive complexes and contractions. Define degree -r maps $R_j^0=P_j,R^r(t_r,\mathrm{.}\mathrm{.}\mathrm{.},t_1)=P_r{t}_r{H}_{r-1}{t}_{r-1}$ ... $H_1{t}_1{H}_0,r\ge 1$. (9) These are maps $Q_0\to W_r$. They retain the actual marked transition maps; no marked arrow is replaced by its cohomology class or by a formal poset identity. Set $d_{Hom}u=du-(-1{)}^{\mathrm{\mid }}u\mathrm{\mid }ud$. Directly differentiating (9) gives $d_{Hom}{R}^r(t_r,\mathrm{.}\mathrm{.}\mathrm{.},t_1)=-w(t_r)R^{r-1}(t_{r-1},\mathrm{.}\mathrm{.}\mathrm{.},t_1)+{\sum }_{j=1}^{r-1}(-1{)}^{r-1-j}{R}^{r-1}(t_r,\mathrm{.}\mathrm{.}\mathrm{.},t_{j+1}{t}_j,\mathrm{.}\mathrm{.}\mathrm{.},t_1)+(-1{)}^{r-1}{R}^{r-1}(t_r,\mathrm{.}\mathrm{.}\mathrm{.},t_2)t_1$. (10) For $r=1$ this reads $d{R}^1+R^{1}d=P_1{t}_1-w(t_1)P_0$, exactly (7). To verify every higher sign, the differential crossing the H factors to the left of $H_j$ contributes (-1)^(r-1-j). In $d{H}_j+H_{j}d=1-I_j{P}_j$, the identity term merges its neighbouring transitions (or gives the final right action when $j=0$). Every $I_j{P}_j$ term except $j=r-1$ vanishes because $H_{j+1}{t}_{j+1}{I}_j=H_{j+1}{I}_{j+1}w(t_{j+1})=0$. The surviving top projection term is $-w(t_r)R^{r-1}$. Thus (10) follows on the actual string complexes, not just after augmentation. For the homotopy from identity to I R define degree -(r+1) maps $S^r(t_r,\mathrm{.}\mathrm{.}\mathrm{.},t_1)=H_r{t}_r{H}_{r-1}$ ... $H_1{t}_1{H}_0,S_j^0=H_j$. (11) The same expansion gives $d_{Hom}{S}^0=1-IP$ and, for $r\ge 1,d_{Hom}{S}^r=t_r{S}^{r-1}(t_{r-1},\mathrm{.}\mathrm{.}\mathrm{.},t_1)-I_r{R}^r(t_r,\mathrm{.}\mathrm{.}\mathrm{.},t_1)+{\sum }_{j=1}^{r-1}(-1{)}^{r-j}{S}^{r-1}(t_r,\mathrm{.}\mathrm{.}\mathrm{.},t_{j+1}{t}_j,\mathrm{.}\mathrm{.}\mathrm{.},t_1)+(-1{)}^r{S}^{r-1}(t_r,\mathrm{.}\mathrm{.}\mathrm{.},t_2)t_1$. (12) Here the top $H_r$ supplies the first two terms, and each lower H supplies the displayed signed merge/right-action term; its projection term vanishes by the same side condition. These are all the coherent retraction and homotopy components, with their signs fixed by the actual differential. Insertion maps and homotopy on the normalized forward bar Use the normalized simplicial bar of the actual closed forward-transition category. For q in $Q_{i_0}$, denote a bar element by $[t_r\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_1]q,degree=\mathrm{\mid }q\mathrm{\mid }-r$. Its differential is $(-1{)}^r[t_r\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_1]dq$, plus the source face $[t_r\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_2]t_{1}q$, the internal faces $(-1{)}^j[t_r\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_{j+1}{t}_j\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_1]q$ for $1\le j<r$, and the terminal face $(-1{)}^r[t_{r-1}\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_1]q$. The terminal face drops the terminal vertex as well as its last arrow; each summand retains its actual source and target labels. Terms containing an identity arrow are zero in the normalized bar. Identity-containing components of (9) and (11) vanish: a terminal identity in R meets P H; every other identity meets H squared on the same complex. Thus these formulas descend to normalization. Let a_(r,k)=(-1)^(k(r-k)+k(k-1)/2) and b_(r,k)=(-1)^((k+1)(r-k)+k(k-1)/2). Define finite insertion sums $R_{bar}([t_r\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_1]q)={\sum }_{k=0}^r{a}_{r,k}[t_r\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_{k+1}]R^k(t_k,\mathrm{.}\mathrm{.}\mathrm{.},t_1)q,I_{bar}([t_r\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_1]w)=[t_r\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_1]I_{0}w,S_{bar}([t_r\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_1]q)={\sum }_{k=0}^r{b}_{r,k}[t_r\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{t}_{k+1}]S^k(t_k,\mathrm{.}\mathrm{.}\mathrm{.},t_1)q$. (13) For $k=0,R^0=P_0$ and $S^0=H_0$. Empty remaining words are permitted. The factor k(k-1)/2 accounts for the successive degree -1 homotopies; the other exponent moves the output of degree -k, respectively -(k+1), past the r-k remaining suspended transition slots. Expanding the specified bar differential and applying (10) gives $D_W{R}_{bar}=R_{bar}{D}_Q,D_Q{I}_{bar}=I_{bar}{D}_W$. (14) Indeed the two terms for an internal merged pair cancel between adjacent insertion lengths; the source-action terms are precisely the last terms of (10), and its first term cancels the W source face. Terminal faces cancel because a_(r,k)=(-1)^k a_(r-1,k). The coefficient differential has sign (-1)^(r-k) after k arrows are consumed, as accounted for by a_(r,k). The same expansion with (12) gives the actual chain homotopy $D_Q{S}_{bar}+S_{bar}{D}_Q=1-I_{bar}{R}_{bar},R_{bar}{I}_{bar}=1$. (15) The $r=0$ identity is exactly (4). For higher words the merged/source/terminal terms cancel in the same pairs; the surviving top term of (12) is minus I R and the identity part of $d{H}_0+H_{0}d$ survives only at insertion length zero. Finally $R^k{I}_0=0$ for $k\ge 1$ and $P_0{I}_0=1$ prove the second identity. This verifies the bar comparison and inverse/homotopy, rather than mistaking the vanishing higher T components for a coherent retraction. Each expression (9), (11) and (13) is finite on the actual input word; each $P_j,H_j$ in it is already finite on epsilon strings by (3). Finite upper-copy enlargements retain the actual marked arrows, and (10)–(15) supply their coherent comparison without requiring globally compatible choices of AC complements. Together with (5) this connects the actual quotient-string contraction to the normalized forward-resolution bar and the terminal augmented bar calculation. No arbitrary coefficient diagram has been replaced by a constant diagram before its chain contraction.

Passage to all quotient Homs, with no global-support shortcut. Let q be a closed element of Q(X,Y). It is a finite sum of finite strings. Choose l later than X,Y and every cone endpoint in that expression. The marked arrow $s:Y\to Y_l$ is a cohomological equivalence in Q by (1). The element s q lies in $Q_S(X,Y_l)$, where S is its finite cone set. Equations (3)–(4) give the actual relation $sq-I_l{P}_l(sq)=d{H}_l(sq)$. (8) Thus every localized cohomology class has a forward representative, after postcomposition by an explicitly invertible marked arrow. If its image in a target is zero, the forward class $P_l(sq)$ is zero because the forward Hom functor is a quasi-isomorphism. Choose a primitive a in that forward Hom; then $sq=d(I_{l}a+H_l(sq))$, and precompose with the closed inverse v of s from (1). The inverse homotopy in (1) shows q itself is a boundary. This proves injectivity, using an actual finite primitive. For surjectivity take a target cohomology class from X to Y. Postcompose it with the target image of s. The forward Hom quasi-isomorphism supplies a closed $a:X\to Y_l$ representing it. Then v a in Q(X,Y) represents the original class. This proves $Q\to target$ quasi-isomorphic on EVERY Hom complex. The argument applies to both the actual geometric and coherent targets using the same directed source and genuine all-arity functors. No associated- graded convergence, global poset contraction or abstract enhancement uniqueness is needed. The later inverse A-infinity functor is constructed on its actual mapping cones by bar-length recursion, as in the existing completion argument. This step still requires that source's genuine forward functors; it does not obtain them from a binary equivalence. All original object resolutions, vertical object cones and complex convergence remain the separately stated geometric inputs. Native HOLD disposition and consumer integration belong to root.

One actual full functor from the two localized comparisons. The preceding finite-string calculation proves that the two localized dg functors are quasi-isomorphisms on every Hom. The original object resolutions, including the separate vertical object cone, prove essential surjectivity onto the represented objects; finite cones, shifts and retracts give both perfect-module targets. To construct an A-infinity inverse of the geometric dg quasi-equivalence, choose representatives of inverse objects in H0 and actual closed equivalences. The cone of each first-component Hom quasi-isomorphism is an acyclic complex over C, contracted by the explicit cycle/complement construction in step 20.1. At bar length n, previously solved lower components yield a closed defect in this cone, because the length-n differential of the defect is the already satisfied lower functor equations. Apply its contraction to obtain the length-n component. The same normalized-bar recursion solves the unit and inverse-homotopy equations. Every fixed word has finitely many subdivisions, so each component is a finite expression. Compose this actual inverse with the coherent dg functor. The result extends the given transversal comparison up to the specified coherent inverse homotopies and supplies repeated/self/parallel and vertical operations through the genuine localized source; no binary-equivalence transport or global constant-nerve inference is used.

7. Finish at all higher/derived levels The preceding dg localization yields one full comparison whose restriction to each increasing generic-resolution word is the original higher comparison. Every original line, rational circle/local system and vertical object is represented by its actual quasi-isomorphic resolution or cone, so it is essentially surjective on the original object classes. The lead's coherent/isogeny/torsion classification and hereditary decomposition cover every bounded coherent complex. Finally apply the explicit Tw formulas, bar bimodule extension, and homotopy-idempotent splitting proved in the explicit bar-completion construction below. This constructs the full A-infinity comparison, its pretriangulated/split-closed extension, and its dg perfect-module enhancement. No binary-equivalence transport or unproved uniqueness of a dg enhancement is invoked.

1. Twisted complexes and all functor components A finite twisted complex is $E=({⨁}_{i=1}^r{X}_i[s_i],\delta )$, with a chosen finite ordering such that delta is strictly upper triangular, degree one in the unsuspended matrix convention, and satisfies the Maurer–Cartan equation ${\sum }_{n\ge 1}{m}_n(\delta ,\mathrm{.}\mathrm{.}\mathrm{.},\delta )=0$. Extend Hom spaces by finite matrices and shifts with the suspension convention. Because every delta factor strictly increases the matrix index, no nonzero summand uses r or more consecutive delta factors. Define delta_F=sum_{n>=1} F^n(delta,...,delta), and for composable matrix morphisms $a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_d$ define Tw(F)^d(a_d,...,a_1) =sum_{r_0,...,r_d>=0} F^{d+r_0+...+r_d}(delta_d^{r_d},a_d, delta_{d−1}^{r_{d−1}},...,a_1,delta_0^{r_0}). (1) The usual suspension is used inside this displayed word, so its signs are exactly those of the original bar-coalgebra equation. Every sum is finite: each block of inserted delta's follows a strictly increasing matrix path, bounded by its finite object length. Expand $b_{B}F=F{b}_A$ on a word decorated with all such delta strings. Terms in which b consumes only a delta string vanish by its Maurer–Cartan equation; all remaining terms are precisely the twisted-category functor identities for (1). On the empty external word the same equation proves the Maurer–Cartan equation for delta_F. Thus these formulas construct every operation and every higher functor component, rather than merely the induced triangulated functor. The cone of a closed degree-zero map $f:E\to G$ is the direct sum G plus E[1], with upper triangular differential having f as its off-diagonal block. Formula (1) identifies its image with the corresponding twisted cone through the same block formulas. Shifts and finite direct sums are also literal matrix operations. Hence the induced H0 functor is exact.

2. Full faithfulness survives every finite twisted extension The twisted Hom differential is the sum of all operations with one external morphism and arbitrary delta insertions on its two sides. Give its finite matrix space the filtration by the ordered source/target matrix indices, refined lexicographically so that every nonempty delta insertion strictly raises filtration. The same filtration on (1) has associated graded the direct sum of shifted maps $F^1$ on the original Hom complexes. They are quasi-isomorphisms by hypothesis. The cone of Tw(F)^1 has finite filtration with acyclic associated graded. Induction on the number of filtration pieces, using the long exact sequence of cohomology of a short exact sequence of complexes, proves this cone acyclic. Thus Tw(F)^1 is a quasi-isomorphism for ALL twisted Hom complexes. This argument includes arbitrary repeated entries $X_i$; the required repeated-word F-components were inputs to (1), not supplied by the filtration argument.

3. Actual dg module enhancement An A-infinity right module M consists of graded spaces M(X), with a degree-one coderivation on M tensor the reduced bar coalgebra whose square is zero. A module morphism is a comodule map of arbitrary degree; its differential is $b_N\varphi -(-1{)}^{\mathrm{\mid }\varphi \mathrm{\mid }}\varphi b_M$. Composition is the ordinary composition of comodule maps. These formulas define a genuine dg category Mod(A): $d^2=0$ by the two square-zero identities, and the graded Leibniz identity follows by expanding the same two compositions. The representable module $h_X$ has $h_X(Y)=Ho{m}_A(Y,X)$, with operations those of A. Yoneda is the all-arity map obtained by putting the external word into the remaining entries of $m_A$. To verify its quasi-full-faithfulness, evaluate a closed module natural transformation $h_X\to M$ at the strict unit 1_X. Its inverse sends x in M(X) to the comodule transformation whose components are the module operations with final input x. Their two composites differ from identity by the homotopy inserting 1_X at the terminal empty place of the augmented bar word. In the bar differential, contractions away from the inserted unit cancel in pairs; the contraction adjacent to the unit is the identity, and all higher operations containing a strict unit vanish. Thus the augmentation has an explicit contracting homotopy. The unit-insertion raises word length by one but is a single term on every word, so no infinite series or convergence assumption occurs. This proves the dg Yoneda enhancement and its agreement with the original Hom complexes. The perfect modules are the retracts, in H0(Mod(A)), of finite iterated cones and shifts of representables. A twisted complex maps to the corresponding finite cone of representable modules; the formulas of step 26.1 make this an all-arity Yoneda map, and step 27.1 proves it fully faithful. Conversely every finite cone of representables is the module of the corresponding upper triangular twisted complex, by the explicit cone construction. This gives a concrete dg enhancement of the finite pretriangulated envelope, followed by its retract closure.

4. Homotopy idempotents really split Here is the required construction; strict idempotents are not assumed. Let $p:M\to M$ be a closed degree-zero module map with $p^2-p=dh$. Form the dg mapping telescope Tel(p)=Cone(1−shift_p : direct-sum_{n>=0}M -> direct-sum_{n>=0}M), where shift_p sends the nth copy to the (n+1)st by p. All operations and cone matrices are well-defined on algebraic direct sums; every input has finite support. The inclusions $j_n$ of copies into this cone satisfy $j_n$ homotopic to $j_{n+1}p$, by the defining cone homotopy. Put $i=j_0$. Then i p is homotopic to i: use $j_0=j_1$ p and $p^2=p$ in the homotopy category. A map back r is obtained on each unshifted summand by p and on the shifted cone summands by h with the cone sign; its chain-map equation is exactly $p-p^2+dh=0$. Thus $ri=p$ in the homotopy category. The cohomology of the telescope is the direct limit under H(p), hence im H(p); this follows directly because 1−shift is injective on finite-support direct sums and its cokernel is the algebraic direct limit. On this space i r induces identity. Moreover (i r)²=i p $r=ir$ in the homotopy category. The map i r is a quasi-isomorphism and hence invertible in H0 of the A-infinity module category, so its idempotence forces $ir=identity$. Here is the latter module fact rather than a hidden cohomology-to-homotopy inference: a pointwise acyclic A-infinity module has a pointwise contracting homotopy over the field k; insert it into the module bar coderivation and correct by its positive-word-length part using the geometric perturbation series. Each correction lowers the remaining input word length, so every component is a finite sum. The identity d h+h $d=1$ follows by cancellation of successive terms. Apply this to the cone of a module quasi-isomorphism to get its homotopy inverse. Thus the claimed invertibility is actual in this dg category, not just at vector-space cohomology. This proves the homotopy image of p is a retract of M. It is perfect whenever M is perfect, by the stated retract definition. No strict matrix idempotent, extra coherent-idempotent assumption, or convergent analytic infinite sum is used.

5. The derived comparison is a genuine bimodule/bar construction The functor F defines an A–B bimodule $B_F$ with entries B_F(X,Y)=Hom_B(Y,F(X)). Its operations are obtained by inserting all components $F^j$ into the corresponding B operation, exactly as in the coalgebra functor equation. Define extension of modules by the two-sided bar complex F_!(M)=M tensor T(s Hom_A) tensor B_F, with direct sum over all finite composable words and differential the sum of: the module coderivation, all A multiplications, and the $B_F$ bimodule operations. Each summand of its square cancels either by a module/A/B identity or by $F{b}_A=b_{B}F$. On a fixed word there are finitely many placements. Thus F_! is an actual dg functor on module morphism complexes. It sends $h_X$ to a module quasi-isomorphic to $h_{F(X)}$; the contraction is the same augmented-bar unit insertion from step 3. On representables it induces the original $F^1$ quasi-isomorphism. By the finite-cone filtration of step 27.1 it is quasi-fully-faithful on the finite pretriangulated envelope. If P,Q are retracts of finite cones, their Hom complexes are the corresponding homotopy summands of the finite-cone Hom complexes; applying the telescope maps or their split cohomological projectors shows the induced map remains a quasi-isomorphism. Hence F_! is quasi-fully-faithful on perfect modules. If its representable images thickly generate Perf(B), every B-perfect module is an iterated cone, shift and retract of image objects; these same operations exist in Perf(A), and quasi-full-faithfulness lifts their closed maps and idempotents. Therefore F_! is essentially surjective in H0. This proves a dg quasi-equivalence of the chosen perfect-module enhancements and the exact equivalence of their homotopy categories.

6. Elliptic application and the input boundary The lead's coherent classification gives every indecomposable vector bundle and every torsion sheaf, and hereditary splitting gives every bounded coherent complex as a finite sum of such shifts. Once the ACTUAL full line/local-system functor (including parallel/self and vertical cases) is supplied, its image contains those source object classes; the module construction above therefore upgrades it to the full derived/split-closed dg comparison. All sums in the twisted upgrade are finite, and the bar tensor is an algebraic direct sum of finite words, so nothing here replaces the literal complex parameter q by a formal or Novikov parameter or tries to evaluate a divergent completion. This formal proof does not manufacture a full functor from a binary comparison. In the attached elliptic construction, genuine fixed Hamiltonian copies have convergent all-arity polygon operations; generic evaluation resolutions are actual quasi-isomorphisms by the separate probe/unit argument; and the directed-resolution dg localization extends the already proved transversal all-arity functor. Those completed constructions supply the full input required here. The local continuation maps use actual thin nonconstant triangles; constant tangent-line models occur only as separately regular local limits, not as illicit higher constant polygons in the no-three-concurrence analytic provider.