Signed opposite transport and homologically unital module Yoneda

At total arity n, expanding two suspended insertions gives $(b^2{)}_n=(-1{)}^{c_n}s[{\sum }_{r+k+t=n}(-1{)}^{r+kt}{\mu }^{r+1+t}(1^{\otimes r}\otimes {\mu }^k\otimes 1^{\otimes t})](s^{-1}{)}^{\otimes n}$, where $k\ge 1$ and $r,t\ge 0$. The tensor-input degree factors agree with those of the displayed composite; the remaining scalar equality is $c_k+c_{n-k+1}+(n-k)(k+1)\equiv c_n(\mathrm{m}\mathrm{o}\mathrm{d}2)$. The two common minus signs in the two b insertions cancel. Hence the exact unsuspended Stasheff identity implies $b^2=0$ in every arity. The right formula in [F1] follows by restricting the category bar differential to a word whose first suspended letter is a representable value: conjugate by suspension of that first coefficient and multiply the degree-one differential by $-1$. The two minus signs, this conjugation and the common minus in $b$, give $q_d^R(n;s{a}_1,\dots ,s{a}_d)=(-1{)}^{c_{d+1}+d(\mathrm{\mid }n\mathrm{\mid }-1)+{\sum }_j(d-j)(\mathrm{\mid }{a}_j\mathrm{\mid }-1)}{\mu }^{d+1}(n,a_1,\dots ,a_d)$. The suffix insertion has sign $(-1{)}^{\mathrm{\mid }n\mathrm{\mid }+{\sum }_{i\le r}\mathrm{\mid }{v}_i\mathrm{\mid }}$. For a last suspended coefficient the tensor map $1\otimes s$ itself has sign $(-1{)}^S$; the same negative conjugation gives $q_d^L=(-1{)}^{c_{d+1}+{\sum }_j(d-j)\mathrm{\mid }{v}_j\mathrm{\mid }}{\mu }^{d+1}(a_1,\dots ,a_d,n)$, with the left insertion formula of [F1]. Both squares are zero because the original restricted square is zero. Thus the representables have value differential ${\mu }^1$ and ordinary binary actions, rather than actions inferred from an unspecified bar convention.

The graded reversal of suspended words with factor $(-1{)}^d$ is an involutive coalgebra anti-isomorphism: its reversal-pair Koszul sign is $(-1{)}^P$, and for any cut the pair signs crossing that cut are exactly the tensor symmetry between its two blocks. Conjugating a degree-one coderivation by it gives another coderivation and preserves its square. The arity-$d$ component is $(-1{)}^{d+1}{b}_d$ applied in reverse suspended order with its Koszul sign. Undoing suspension gives exponent $d+1+(d-1){\sum }_i(\mathrm{\mid }{a}_i\mathrm{\mid }-1)+{\sum }_{i<j}(\mathrm{\mid }{a}_i\mathrm{\mid }-1)(\mathrm{\mid }{a}_j\mathrm{\mid }-1)$, which is $c_{d-1}+{\sum }_{i<j}\mathrm{\mid }{a}_i\mathrm{\mid }\mathrm{\mid }{a}_j\mathrm{\mid }$ modulo two. This proves the opposite formula, including ${\mu }_{\mathrm{o}\mathrm{p}}^1={\mu }^1$ and the usual graded opposite product for $d=2$.

Finite shifts with fixed signs. For a shift X[k] use a one-dimensional graded line $L_k$ of degree -k. Set $\mathrm{Hom}(X[k],Y[l])=\mathrm{Hom}(L_k,L_l)\otimes \mathrm{Hom}(X,Y)$, with its total grading. Identify s(f tensor a) with $(-1{)}^{\mathrm{\mid }f\mathrm{\mid }}f\otimes sa$; call this identification J. On homogeneous matrix words $f_i\otimes v_i$, put $p_i=\mathrm{\mid }{f}_i\mathrm{\mid },w_i=\mathrm{\mid }{v}_i\mathrm{\mid }$ and $C={\sum }_{i<j}{w}_i{p}_j$. Define the shifted suspended operation, before conjugating by J, as $(-1{)}^{C+\sum p_i}(f_1\mathrm{.}\mathrm{.}\mathrm{.}{f}_d)\otimes b_d(v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_d)$. Sum over matching finite matrix indices, and conjugate by J to obtain the shifted additive category. This is an explicit definition of all shift signs, rather than an instruction to choose them. Its square is zero in every arity: for an inner block I with prefix P and suffix S, its contraction changes the crossing exponent to $C-C_I+{\sum }_{j\in S}{p}_j$. Including the inner operation, the outer operation and the coderivation prefix sign yields total exponent $C+{\sum }_{i\in P}{w}_i$; the p-prefix and p-suffix terms each occur twice and cancel. Thus the entire two-insertion sum is the common matrix coefficient times the original b-square sum. This proves all Stasheff identities. At arity one the unsuspended differential is $(-1{)}^{\mathrm{\mid }f\mathrm{\mid }}f\otimes {\mu }^1(a)$, as required by the cohomological shifts.

Moving an unsuspended value $n$ through a word gives $(-1{)}^{\mathrm{\mid }n\mathrm{\mid }S}$. Combining this with the preceding reversal gives precisely ${\mathrm{\Theta }}_N=(-1{)}^{d+P+\mathrm{\mid }n\mathrm{\mid }S}$, which is involutive after reversing object labels. Conjugate the left suffix differential, including its prefix tensor signs, by this degree-zero map. The new core is $q_d^{R,\mathrm{o}\mathrm{p}}(n;v_1,\dots ,v_d)=(-1{)}^{d+P+\mathrm{\mid }n\mathrm{\mid }S}{q}_d^L(v_d,\dots ,v_1;n)$; its remaining terms are the right suffix insertions of $b^{\mathrm{o}\mathrm{p}}$. Substituting the two $q$ formulas cancels their $c_{d+1}$ factors and gives the unsuspended module exponent $c_d+{\sum }_{i<j}\mathrm{\mid }{a}_i\mathrm{\mid }\mathrm{\mid }{a}_j\mathrm{\mid }+\mathrm{\mid }n\mathrm{\mid }{\sum }_i\mathrm{\mid }{a}_i\mathrm{\mid }$ in [F1]. The identical core calculation gives the stated premorphism formula, with no extra $\mathrm{\mid }f\mathrm{\mid }$ factor; its suffix extension already has $(-1{)}^{\mathrm{\mid }f\mathrm{\mid }S}$. Conjugation therefore commutes with differential and composition. Its arity-zero component is identity, so it preserves objectwise quasi-isomorphisms, unit action and full Taylor products. The representable left module becomes exactly the right representable for the opposite category.

The curvature equation and the necessary public correction. In the shifted additive category of step 2.2 let q be a degree-one matrix strictly decreasing a fixed finite filtration, and set $t=-sq$, of degree zero. Put $\kappa (q)={\sum }_{d\ge 1}{b}_d(t,\mathrm{.}\mathrm{.}\mathrm{.},t)$. Koszul suspension on degree-one inputs has no input-degree factor, so $\kappa (q)=s{\sum }_{d\ge 1}(-1{)}^{c_{d-1}}{\mu }^d(q,\mathrm{.}\mathrm{.}\mathrm{.},q)$, since $d+1+c_d\equiv c_{d-1}$ modulo two. Consequently the curvature-zero equation in THIS convention is ${\sum }_{d\ge 1}(-1{)}^{c_{d-1}}{\mu }^d(q,\mathrm{.}\mathrm{.}\mathrm{.},q)=0$. It starts ${\mu }^1(q)+{\mu }^2(q,q)-{\mu }^3(q,q,q)-{\mu }^4(q,q,q,q)+$..., not the unsigned sum. Every sum is finite by the strict filtration. This is the necessary correction to the earlier unsigned equation, with no restriction on higher operations or matrix length. It agrees with dq+q²=0 for every DG category.

Here are the actual Yoneda maps. For a homogeneous $n\in N(X)$ of degree $h$ and $x$ a representable input, the right map is $I_R(n{)}_d(x;s{a}_1,\dots ,s{a}_d)=(-1{)}^{c_d+d(h+\mathrm{\mid }x\mathrm{\mid })+{\sum }_j(d-j)\mathrm{\mid }s{a}_j\mathrm{\mid }}{m}_N^{d+1\mathrm{\mid }1}(n;x,a_1,\dots ,a_d)$. The left map is $I_L(n{)}_d(s{a}_1,\dots ,s{a}_d;x)=(-1{)}^{c_d+{\sum }_j(d-j)\mathrm{\mid }s{a}_j\mathrm{\mid }+h(\mathrm{\mid }x\mathrm{\mid }+{\sum }_j\mathrm{\mid }s{a}_j\mathrm{\mid })}{\mathrm{\ell }}_N^{d+1\mathrm{\mid }1}(a_1,\dots ,a_d,x;n)$. Both have degree $h$. On the right the same formula is $(-1{)}^h{q}_{d+1}^R(n;sx,s{a}_1,\dots ,s{a}_d)$. To prove its chain equation, put $J_n(x;v)=n\otimes sx\otimes v$, a degree-$h-1$ map of the underlying word complexes, not a claimed comodule map. Direct expansion gives $D_N{J}_n-(-1{)}^{h-1}{J}_n{D}_{Y_X}=J_{dn}+F_n$, where $(F_n{)}_d=q_{d+1}^R(n;sx,v)$. The category terms agree because suspension of the representable coefficient conjugates its differential to $-b$. Applying the Hom differential once more gives $\mathrm{\partial }{F}_n=-F_{dn}$; multiplication by $(-1{)}^h$ proves $\mathrm{\partial }{I}_R(n)=I_R(dn)$. Conjugating by step 3.1 gives exactly the displayed left exponent and its chain equation, not a separately guessed action.

All insertion signs and all arities. For twisted objects indexed 0,...,d, insert any number $r_i$ of their $t_i$ into gap i of a suspended word, and denote the resulting finite map by $E_t$. Each gap has a bounded number of insertions since $t_i$ strictly decreases the relevant finite filtration. Deconcatenation splits each inserted string at every possible cut, so $E_t$ is a coalgebra isomorphism with inverse $E_{-t}$: for a string with r identical insertions the composite coefficient is ${\sum }_{j=0}^r\mathrm{binom}(r,j)(-1{)}^j$, equal to zero for $r>0$. Conjugate b by $E_t$, namely $b^t=E_{-t}b{E}_t$. Its curvature is κ, its square is zero, and its positive Taylor components are $b_d^t(v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_d)={\sum }_{r_0,\mathrm{.}\mathrm{.}\mathrm{.},r_d}{b}_{d+R}(t_0^{r_0},v_1,t_1^{r_1},\mathrm{.}\mathrm{.}\mathrm{.},v_d,t_d^{r_d}),R=\sum r_i$. Indeed projection to one output letter after $E_{-t}$ retains only the one-letter component of $b{E}_t$; inserted output letters only increase length. Thus κ=0 gives an uncurved category in all arities. For unsuspended homogeneous inputs $a_i$ of degrees ${\alpha }_i$, undoing the SAME suspension gives ${\mu }_{tw}^d(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_d)={\sum }_r(-1{)}^{\epsilon }{\mu }^{d+R}(q_0^{r_0},a_1,q_1^{r_1},\mathrm{.}\mathrm{.}\mathrm{.},a_d,q_d^{r_d})$, where $\epsilon =R+c_{d+R}+c_d+{\sum }_{i=1}^d(r_i+\mathrm{.}\mathrm{.}\mathrm{.}+r_d)({\alpha }_i-1)$ modulo two. This follows by comparing each original suspended $a_i$'s tensor exponent before and after the insertion; t contributes $(-1{)}^R$. In particular the morphism differential is the full $d=1$ formula. In a DG category it is $D(a)=da+q_{out}a-(-1{)}^{\mathrm{\mid }a\mathrm{\mid }}a{q}_{in}$, so its two-block cone is the usual cohomological cone. This verifies every insertion sign rather than testing finitely many arities.

Evaluation $E(f)=f_0(e_X)$ is a chain map because $e_X$ is closed. The displayed inclusions have $I_R(n{)}_0(x)=m_N^{1\mathrm{\mid }1}(n;x)$ and $I_L(n{)}_0(x)=(-1{)}^{h\mathrm{\mid }x\mathrm{\mid }}{\mathrm{\ell }}_N^{1\mathrm{\mid }1}(x;n)$. Hence $EI$ is the actual binary unit action, identity on cohomology for homologically unital N. In particular the left degree-$h$ DG specialization is $f(ax)=(-1{)}^{h\mathrm{\mid }a\mathrm{\mid }}af(x)$, while the right specialization is $f(xa)=f(x)a$. Taking the free graded algebra on odd generators $x,y$, $n=y$, gives $I_L(y{)}_0(x)=-xy$; this checks the essential sign which degree-zero examples miss.

An explicit reason the unsigned equation cannot be retained. Take four objects 0,1,2,3, degree-one arrows a:1→0, b:2→1, c:3→2 and r:3→0, and a degree-two arrow h:3→0. Let ${\mu }^1(r)=h$ and ${\mu }^3(a,b,c)=-h$; all other nonidentity operations vanish. Adjoin strict identities with the binary rules of [F1]. The Stasheff equations hold: ${\mu }^1$ squares to zero, the only nonzero ternary value is closed, any insertion of a nonzero unary value into that ternary operation is impossible by its source/target labels, and two ternary operations cannot be nested in the length-three directed graph. The identities with one formal unit cancel adjacent binary/unit insertions with their Stasheff signs; with two or more units they reduce to the ordinary associative unit identity or zero. For the finite upper-triangular matrix $q=a+b+c+r$ the unsigned sum is $h-h=0$, whereas the corrected sum is $h+h=2h$. The insertion formula of step 4.2 on the diagonal projector $p_0$ gives $D{p}_0=-a-r,Da=-{\mu }^3(a,b,c)=h$ and $Dr=h$, so $D^2{p}_0=-2h$. Over the stipulated field this is nonzero. Thus the earlier unsigned equation together with the fixed b convention can define a square-nonzero morphism differential. The correction is compulsory, rather than a silent weakening or an optional sign choice.

We prove that E is a quasi-isomorphism without assuming strictly unital higher operations or invoking a minimal-model comparison theorem. Split every value and category Hom complex into its cohomology representatives and contractible pairs, choosing $e_X$ itself as representative of its nonzero unit class. If its class is zero, all Hom spaces to and from X and N(X) have zero cohomology by the unit assumption; the same argument uses their zero cohomology and needs no representative for a zero vector. Write each contraction $pi=1$, $1-ip=dh+hd$, $h^2=hi=ph=0$. Tensor contractions on each finite input word use $h_{U\otimes W}=h_U\otimes 1+i_U{p}_U\otimes h_W$ with the tensor signs of [F2]. For the internal Hom complex from the finite input tensor U to a value N, set $p_H(f)=p_{N}f{i}_U$, $i_H(g)=i_{N}g{p}_U$, and $h_H(f)=h_{N}f+(-1{)}^{\mathrm{\mid }f\mathrm{\mid }}{i}_N{p}_{N}f{h}_U$. Expansion of its Hom differential gives $1-i_H{p}_H={\mathrm{\partial }}_0{h}_H+h_H{\mathrm{\partial }}_0$, with the side conditions. These operations act componentwise on the full Taylor product and preserve arity; unbounded or infinite-dimensional vector spaces cause no convergence problem.

Write the full Hom differential as ${\mathrm{\partial }}_0+\delta$, where ${\mathrm{\partial }}_0$ is unary/internal and every term of $\delta$ raises Taylor arity by at least one. The perturbation formulas $T=\delta (1+h_H\delta {)}^{-1}$, $i^{\mathrm{\prime }}=i_H-h_{H}T{i}_H$, $p^{\mathrm{\prime }}=p_H-p_{H}T{h}_H$, $h^{\mathrm{\prime }}=h_H-h_{H}T{h}_H$, and $d_C=p_{H}T{i}_H$ give a contraction onto the cohomology-input product with differential $d_C$. Every inverse is its geometric series and is finite at each prescribed output arity. Substitution of ${\mathrm{\partial }}_0^2=0$, ${\mathrm{\partial }}_0\delta +\delta {\mathrm{\partial }}_0=-{\delta }^2$ and the preceding contraction cancels adjacent factors to give $({\mathrm{\partial }}_0+\delta )i^{\mathrm{\prime }}=i^{\mathrm{\prime }}{d}_C$, $p^{\mathrm{\prime }}({\mathrm{\partial }}_0+\delta )=d_C{p}^{\mathrm{\prime }}$, $p^{\mathrm{\prime }}{i}^{\mathrm{\prime }}=1$ and $1-i^{\mathrm{\prime }}{p}^{\mathrm{\prime }}=({\mathrm{\partial }}_0+\delta )h^{\mathrm{\prime }}+h^{\mathrm{\prime }}({\mathrm{\partial }}_0+\delta )$. There is no infinite tail at a fixed arity. Its arity-one differential $d_1$ is precisely the binary graded-category/module differential on cohomology; $d_C-d_1$ raises arity by at least two, including all transferred higher unit-input terms.

On this cohomology-input product evaluation $E_0$ at $[e_X]$ has $E_0{p}^{\mathrm{\prime }}=p_{N}E$: terms of $p^{\mathrm{\prime }}$ beyond $p_H$ raise arity and cannot contribute to arity zero. Likewise $p_{N}E{i}^{\mathrm{\prime }}=E_0$. The binary actions are strictly unital on cohomology, even though higher operations containing the unit have not vanished. The binary Yoneda inclusion $I_0$ has only arity-zero term, given on the right by $nx$ and on the left by $(-1{)}^{\mathrm{\mid }n\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }}xn$. It satisfies $E_0{I}_0=1$. If $[e_X]=0$, the cohomology-input source representable and the target are both zero, so evaluation is already a quasi-isomorphism by step 7.1. We henceforth treat the nonzero class.

The direct-sum bar assertion has the opposite convergence direction. At K the augmented left word complex is $C_N^L(K)[-1]$: a positive word is written $(x;v_1,\dots ,v_d;n)$ with outer $x$ unsuspended, and identified with $(sx,v_1,\dots ,v_d;n)[-1]$; its degree-zero augmentation is $-q_{d+1}^L(sx,v_1,\dots ,v_d;n)$. Include the empty-word $N(K)[-1]$ summand. Its degree-one differential is exactly $-D_N^L$ under this identification; hence it squares to zero, includes the full higher augmentation and has the correct negative target differential on the shifted summand. This specifies every sign, including the binary augmentation $xn$. Transfer the unary part by the tensor contractions of step 6.1 on each finite word, including the empty word. Now nonunary terms lower word length, so the first perturbation series is finite on each direct-sum word. On cohomology its leading binary differential is $d_0$ and its higher terms lower length by at least two.

For the binary-only Hom differential, the right homotopy on a degree-$h$ premorphism has components $(k_0^{R}f{)}_d(x;v_1,\dots ,v_d)=(-1{)}^h{f}_{d+1}(e_X;sx,v_1,\dots ,v_d)$. It lowers Taylor arity by one. Its left conjugate is $(k_0^{L}f{)}_d(v_1,\dots ,v_d;x)=(-1{)}^{h+1+{\sum }_i\mathrm{\mid }{v}_i\mathrm{\mid }}{f}_{d+1}(v_1,\dots ,v_d,sx;e_X)$. To check the right contraction in all degrees, the arity-d component of $d_{1}f$ is the last-action term with exponent $h+\mathrm{\mid }x\mathrm{\mid }+{\sum }_{i<d}\mathrm{\mid }{v}_i\mathrm{\mid }$, the first-action term with exponent $h+\mathrm{\mid }x\mathrm{\mid }+1$, and each internal product term at i with exponent $h+\mathrm{\mid }x\mathrm{\mid }+{\sum }_{j\le i}\mathrm{\mid }{v}_j\mathrm{\mid }+1$. In $d_1{k}_0+k_0{d}_1$ these pair with opposite signs; the unpaired first face is $f_d(x;v)$ and the exceptional arity-zero face is $-f_0(e_X)x$. Thus $d_1{k}_0+k_0{d}_1=1-I_0{E}_0$. The left formula follows by inserting the exact reversal signs from step 3.1, giving the extra $1+{\sum }_i\mathrm{\mid }{v}_i\mathrm{\mid }$ rather than omitting it.

The binary extra degeneracy is $h_0(v;n)=(s{e}_K,v;n)$ in the shifted augmented coordinates; for the empty word it inserts just $s{e}_K$. Its degree is minus one. Since $q_1^L(s{e}_K;n)=-e_{K}n$ and $b_2(s{e}_K,sa)=-s(e_{K}a)$, the outer face of $D_N^L{h}_0+h_0{D}_N^L$ is minus identity, while all internal faces cancel across the odd $s{e}_K$. Therefore $d_0{h}_0+h_0{d}_0=1$ for the augmented differential $-D_N^L$. Replace $h_0$ by $h=h_0{d}_0{h}_0$ to have $h^2=0$ while still increasing length by one. If $\mathrm{\Delta }$ is the remaining transferred differential, it lowers length by at least two; thus $h_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}=h(1+\mathrm{\Delta }h{)}^{-1}$ is finite on each direct-sum word and contracts the full transferred augmented complex. Composing with its unary contraction contracts the original augmented complex. Reversal transports this result to the right convention. Empty object sets are vacuous; if the unit class at K is zero, the unary cohomology of every word ending at K and of N(K) is zero and the first contraction already suffices.

Normalize this contraction: with $P=I_0{E}_0$, put $k_1=(1-P)k_0(1-P)$ and $k=k_1{d}_1{k}_1$. On the complementary complex, $d_1{k}_1+k_1{d}_1=1$, whence $d_1{k}_1^2=k_1^2{d}_1$ and $k^2=0$; also $E_{0}k=k{I}_0=0$. It still lowers arity by one. Put $\eta =d_C-d_1$. Then $\eta k$ and $k\eta$ raise arity by at least one, so $K=k(1+\eta k{)}^{-1}$ and $I_{\ast }=(1+k\eta {)}^{-1}{I}_0$ are coefficientwise finite. Since $E_0\eta =0$, the projection stays $E_0$ and the transferred target differential is zero. The same perturbation identities give $E_0{I}_{\ast }=1$ and $d_{C}K+K{d}_C=1-I_{\ast }{E}_0$. This retains every higher unit-input operation; it uses no normalized-unit quotient.

Combine the two contractions: $i_C=i^{\mathrm{\prime }}{I}_{\ast }$, $p_C=E_0{p}^{\mathrm{\prime }}=p_{N}E$ and $H_C=h^{\mathrm{\prime }}+i^{\mathrm{\prime }}K{p}^{\mathrm{\prime }}$ give $p_C{i}_C=1$ and $1-i_C{p}_C=\mathrm{\partial }{H}_C+H_C\mathrm{\partial }$. Thus E induces a cohomology isomorphism. Explicitly a quasi-inverse as a map of complexes is $J=i_C{p}_N:N(X)\to {\mathrm{hom}}_{\mathrm{m}\mathrm{o}\mathrm{d}}(Y_X,N)$; $JE-1=-\mathrm{\partial }{H}_C-H_C\mathrm{\partial }$. The other composite has homotopy $h_{N}E{i}_C{p}_N-h_N$, since $p_{N}E{i}_C=1$ and $1-i_N{p}_N=d_N{h}_N+h_N{d}_N$. To compare the actual all-arity inclusion I of step 4.1 with J, let $u=EI-1$. It is a chain endomorphism inducing zero on cohomology; the internal-Hom contraction gives $H_u=h_{N}u+i_N{p}_{N}u{h}_N$ and $d_N{H}_u+H_u{d}_N=u$. Consequently $I-J=\mathrm{\partial }(H_{C}I+J{H}_u)+(H_{C}I+J{H}_u)d_N$. These are explicit product homotopies for the actual graded Yoneda inclusion, not a replacement of it by a unary action or by strict geometric units.

For an objectwise acyclic target module Q, choose contractions of its value complexes. On each arity of Hom(M,Q) the unary contraction is just postcomposition with that contraction; the perturbation raises arity, so the same coefficientwise formulas contract the full Hom product. Thus Hom(M,-) sends objectwise acyclic cones to acyclic complexes. An objectwise quasi-isomorphism f therefore induces a quasi-isomorphism on Hom(M,-) for every M. Surjectivity on $H^0\mathrm{Hom}(\mathrm{target}f,f)$ supplies a homotopy right inverse; injectivity on $H^0\mathrm{Hom}(\mathrm{source}f,f)$ then supplies the left inverse identity. Hence these quasi-isomorphisms are homotopy equivalences and the Hom model computes derived Hom. The proved Yoneda formula identifies derived Hom from a representable with evaluation; evaluation commutes with arbitrary coproducts. Sums of homologically unital modules remain such on value cohomology, and finite cones retain the unit equation. Representables and finite cones are consequently compact in the stated derived module category.

Finally fix a supplied parameter cell P of dimension r, a right opposite-A map $c_P$ of degree $-r$, and its supplied homotopies/comparison maps. Define $c_P^L={\mathrm{\Theta }}_N^{-1}{c}_P^{R,\mathrm{o}\mathrm{p}}{\mathrm{\Theta }}_M$, and identically define $R_P^L,H_P^L,K_{P,Q}^L$ and every higher family. In components, $(c_P^L{)}_d(v_1,\dots ,v_d;m)=(-1{)}^{d+{\sum }_{i<j}\mathrm{\mid }{v}_i\mathrm{\mid }\mathrm{\mid }{v}_j\mathrm{\mid }+\mathrm{\mid }m\mathrm{\mid }{\sum }_i\mathrm{\mid }{v}_i\mathrm{\mid }}(c_P^{R,\mathrm{o}\mathrm{p}}{)}_d(m;v_d,\dots ,v_1)$. This is a reindexing of the SAME tensor/count coefficient, not a new moduli count or a reflection of its domain. Apply the same reversal to its category arrows, opposite operations and module structure simultaneously. The degree-zero conjugation commutes with Hom differential, composition, scalar normalizations and parameter-boundary maps. Thus it preserves, with their original parameter signs, $\mathrm{\partial }{c}_P=c_{\mathrm{\partial }P}$, the composition $(-1{)}^{rs}{c}_Q{c}_P$, and every painted or separation equation. Parameter directions stay first: no extra r-dependent sign is introduced by moving them through category inputs. Arity zero is unchanged. For a continuation cochain $c:X^{+}\to X$ its actual left Yoneda map $Y_X^L\to Y_{X^{+}}^L$ is the full formula $I_L(c)$; in degree zero it uses ${\mu }^{d+2}(a_1,\dots ,a_d,x,c)$ with the sign in step 4.1. Every all-arity unit-flag comparison, if supplied as a counted family, is transported by this same formula; a unary comparison alone is not promoted to an all-arity one. This proves the precise count-component transport asserted, while leaving the existence and analytic validity of any such counted family to its actual geometric construction.

A full product-Hom comparison, without an invented geometric endpoint. Let $C={\mathrm{Hom}}_{mod}(Y_X,N),V=N(X),E(f)=f_0(e_X)$, and I the ACTUAL signed inclusion of step 4.1, using the same closed proper Morse/PSS representative $e_X$. Retain all higher operations containing this unit. With the choices of step 11.1 put $u=EI-1,H_u=h_{N}u+i_N{p}_{N}u{h}_N,L=H_{C}I+J{H}_u$ and $K=H_C-LE$. That step proves $1-JE=\mathrm{\partial }{H}_C+H_C\mathrm{\partial },I-J=\mathrm{\partial }L+L{d}_V$ and $d_V{H}_u+H_u{d}_V=u$. Since E is a chain map, subtraction gives ∂K+K∂=1-IE. These are explicit maps on the full Taylor PRODUCT: $H_C$ is the finite-at-each-arity perturbation formula from steps 7.1 and 10.1, L and E are their displayed composites, and each K coefficient is determined by those formulas. No division by arity or convergence of an infinite scalar sum occurs. The formula does not set $EI=1$ and does not replace I by its unary part.

Sums of homologically unital modules are homologically unital because cohomology of vector-space sums commutes with sums and the binary unit action is diagonal. For cones and finite-word module bars, filter by cone summand or word length. The binary unit action is a chain map preserving that filtration; on its associated graded it is the given unit action on a value or outer representable, tensored with the remaining factors. Hence it is a quasi-isomorphism on each finite filtration by induction on the filtration triangles. Filtered unions of vector spaces are exact, so it remains a quasi-isomorphism on the direct-sum bar. Its cohomology action is also idempotent: the module identity makes the action factor through the associative action of category cohomology, where $[e{]}^2=[e]$. An invertible idempotent is identity. This proves homological unitality of these cones and word-bar colimits without discarding a higher operation containing the unit.

Twisted modules, maps and all parameter faces. Extend a module to the finite shifted sums by the same graded-line tensor rule (leaving its coefficient unsuspended). On a right comodule insert t at every category gap, including the gap adjacent to its coefficient; on a left comodule use the corresponding suffix construction. Call these maps $E_{N,t}$. The coaction identity follows by the same deconcatenation cuts as in step 4.2. Set $D_N^t=E_{N,-t}{D}_N{E}_{N,t}$. Its square is zero and its base coderivation is $b^t$. For a degree-h premorphism F set $F^t=E_{P,-t}F{E}_{N,t}$. Direct multiplication gives ${\mathrm{\partial }}^t{F}^t=E_{P,-t}(\mathrm{\partial }F)E_{N,t},(GF{)}^t=G^t{F}^t$ and $(\mathrm{id}{)}^t=\mathrm{id}$. These identities prove all module, map and homotopy equations simultaneously. More explicitly, for d ordinary inputs of degrees ${\alpha }_i$ the right unsuspended $m^{d\mathrm{\mid }1}$ insertion exponent is $R\mathrm{\mid }n\mathrm{\mid }+c_{d+R+1}+c_{d+1}+{\sum }_i(r_i+\mathrm{.}\mathrm{.}\mathrm{.}+r_d)({\alpha }_i-1)$; the left ${\mathrm{\ell }}^{d\mathrm{\mid }1}$ exponent is $R+c_{d+R+1}+c_{d+1}+{\sum }_i(r_i+\mathrm{.}\mathrm{.}\mathrm{.}+r_d)({\alpha }_i-1)$. They follow from the two q-formulas of [F1]; for the right formula the difference of the d(|n|-1) factors and the $(-1{)}^R$ insertion factor is R|n|. Here in either formula $r_0$ is the gap before the first category input, $r_d$ is the gap after its last category input, and the module coefficient remains at its declared side. Twisting is degree-zero conjugation. For fixed twisting data it therefore commutes with the normalized parameter-face equations $\mathrm{\partial }{F}_P=F_{\mathrm{\partial }P}$, compositions and every higher prism, without an additional parameter-dimension sign. The opposite/reversal equations remain exact when applied simultaneously to the category, twisting matrix, module and every premorphism; one must transport t as well, rather than retaining q while changing only b.

Every parameter cell and every arity at once. Let $F_P\in C^{-r}$ be the ACTUAL counted full module map for an oriented r-cell, with its actual boundary equation $\mathrm{\partial }{F}_P=F_{\mathrm{\partial }P}$, including all ordinary/category/Morse/clock faces. Define $c_P=E{F}_P=F_{P,0}(e_X)$. It is an actual finite linear combination of the same rigid continuation solutions with the supplied proper PSS cochain as input, not an abstract cohomology replacement; linearity of counts in that finite input defines it. Then $d_V{c}_P=c_{\mathrm{\partial }P}$. Set $H_P=K{F}_P$. Step 12.3 gives $\mathrm{\partial }{H}_P+H_{\mathrm{\partial }P}=F_P-I(c_P)$ in the ENTIRE product C, hence on each arbitrary finite ordinary-input word. If a separately fixed actual continuation cochain family $s_P$ is used, and its ACTUAL unary flag comparison supplies $E{F}_P-U{s}_P=d_V{q}_P^e+q_{\mathrm{\partial }P}^e$ with $U=EI$, put $q_P=q_P^e+H_u{s}_P$. Because $d{s}_P=s_{\mathrm{\partial }P}$, this gives $E{F}_P-s_P=d_V{q}_P+q_{\mathrm{\partial }P}$; then replace $H_P$ by $K{F}_P+I{q}_P$. Its boundary is exactly $F_P-I(s_P)$. This explicitly states the precise unary geometric input needed for that separately chosen family. It does not assert its existence from a closed unit alone. For the canonical evaluation family $c_P$ the q-term is zero and no flag endpoint theorem is needed.

The finite hull and perfect-module interface retain their strength. The construction permits every finite shifted matrix with the corrected curvature equation, without strictifying a geometric category. Apply the actual all-arity Yoneda formulas to the finite matrices and then twist by step 13.2; they are compatible with the twisted category operations because every relevant finite string is the same original operation with all possible t insertions. Filter the comparison between morphism complexes of two twisted objects by target-row minus source-column index (refine by the two finite filtrations if needed). A target q strictly decreases its row and a source q strictly increases its column, so every term containing at least one q strictly lowers this filtration. On each associated quotient the comparison is a finite sum of shifted original Yoneda comparisons of step 11.1. Each is a quasi-isomorphism; finite induction on the filtration, using the long exact sequence of the short exact sequence of complexes, proves the full comparison is a quasi-isomorphism. The represented modules are finite iterated module cones and shifts: successively split off the last filtration piece; its off-diagonal terms are precisely the closed attaching module map supplied by D²=0, and its block differential is that cone. Conversely these block cones assemble such finite triangular matrices. Thus their cohomological retracts give the same perfect envelope; identities are the lifted identities of the module model, hence only cohomological units are needed. No strict DG enhancement of the original category or generation claim is inferred.

Orientation and physical/count scope of this bridge. For parameter-first counts normalized by ${\alpha }_r=(-1{)}^{r(r-1)\mathrm{/}2}$, the actual equation in step 13.3 already has $\mathrm{\partial }{F}_P=F_{\mathrm{\partial }P}$. Applying the fixed degree-minus-one operator K gives the plus $H_{\mathrm{\partial }P}$ above: ∂K $F_P=(1-IE)F_P-K\mathrm{\partial }{F}_P$. Conjugation by Θ has degree zero and leaves e and E at arity zero unchanged; conjugating K,I,E and every F therefore preserves this equation with no extra r-dependent sign. Likewise a cubical face is treated by exactly the same K as the parent cell, so all restrictions and corner relations are simultaneous. Products of maps retain their original Koszul parameter sign; conjugation and this linear Hom identity do not change a geometric determinant orientation or manufacture an endpoint count. In particular $H_P$ is an explicit ALGEBRAIC full-Hom homotopy between actual count maps, not a purported count of all-word unit-flag domains. The original assertion that such an all-word geometric family has only the claimed two ends is not a consequence of binary unit cells, and is not used here. All geometric $F_P$, their full count equations, ordinary unit counts and analytic hypotheses remain required in the history construction. The derived localization conclusion only needs the displayed full-Hom comparison and the value locality criterion; it does not require $H_P$ to be geometric. Thus this bridge preserves that entire Statement while correcting the unsupported assertion about the proof's comparison homotopy.

Actual identities and all unit actions. For this formal/PSS interface put $G=A$, retaining its FULL self-complexes and its fixed closed unit representative $e_X$; in the history application this is the actual proper Morse/PSS representative with every minimum, handle and coefficient-line term. Let $D$ be a finite directed copy category: off-diagonal operation strings are composable strictly decreasing copies from $G$, while its diagonal is the DIFFERENT formal scalar identity ${\nu }_X$. No self-complex is removed from $G$. The geometric representable is $Y_X$, and $I_X$ is the full signed Yoneda map already constructed. The full right comodule identity on $C_{Y_X}$ has prefix components $(I{d}_X{)}_0(x)=x$ and $(I{d}_X{)}_d=0$ for $d>0$. Its extension is identity on every word. In the left convention its suffix extension has exactly the same property, since its degree is zero. Hence $\mathrm{\partial }I{d}_X=DI{d}_X-I{d}_{X}D=0,FI{d}_X=F$ and $I{d}_{N}F=F$ for EVERY homogeneous full premorphism F, not merely in cohomology or at arity zero. Scalar multiples define a unital dg algebra map $k\to {\mathrm{End}}_{mod}(Y_X)$. A formal diagonal ${\nu }_X$ is realized by this map; no Floer input generator has yet been inserted. Its identity action on a formal representable is taken to the literal module identity action. This is the correct strict-unit realization, rather than assigning the formal unit the geometric cochain $e_X$.

Directed nonunit strings and the full realization. To make the preceding realization compatible with ordinary operations, here is its all-arity construction on the finite directed category D. It is convenient to use right representables; the exact Θ of the preceding construction gives the left/opposite version with no new sign. Put $V=sG$ and retain the frozen negative b. For k≥1, a composable string of suspended NONUNIT arrows $w_1,\mathrm{.}\mathrm{.}\mathrm{.},w_k$ has a bar-functor component $H_k(w_1,\mathrm{.}\mathrm{.}\mathrm{.},w_k)=s{F}_w$, where the full right-premorphism core is $(F_w{)}_d(x;v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_d)=(-1{)}^{(\mathrm{\mid }{w}_1\mathrm{\mid }+\mathrm{.}\mathrm{.}\mathrm{.}+\mathrm{\mid }{w}_k\mathrm{\mid })}{s}^{(-1)}{b}_{k+d+1}(w_1,\mathrm{.}\mathrm{.}\mathrm{.},w_k,sx,v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_d)$. Here the first $s^{(-1)}$ acts on the output, and all displayed suspended maps already carry the Koszul tensor signs. The degree of $F_w$ is $1+\sum \mathrm{\mid }{w}_i\mathrm{\mid }$, so $H_k$ has degree zero. For $k=1$ this is EXACTLY the actual frozen Yoneda $I_R(c)$: indeed $q_d(n;v)=-s^{(-1)}{b}_{d+1}(sn,v)$, and $I_R(c{)}_d=(-1{)}^{\mathrm{\mid }}c\mathrm{\mid }{q}_{d+1}(c;sx,v)$. For a formal unit set $H_1(s{\nu }_X)=sI{d}_X$, and set $H_k=0$ for $k>1$ whenever an input is formal. Extend H to the object-labelled coalgebra map by all finite consecutive partitions. No unary generator $e_X$ is substituted for ${\nu }_X$.

All-arity verification of the realization. Projection of b² to the output on the word $(w_1,\mathrm{.}\mathrm{.}\mathrm{.},w_k,sx,v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_d)$ divides its two-insertion terms into exactly three types: insertions wholly inside the w-block; insertions wholly inside the representable test word (sx,v); and insertions meeting the cut between those blocks. The first type is $H{b}_D$, including its prefix sign $(-1{)}^{({\sum }_{i<j}\mathrm{\mid }{w}_i\mathrm{\mid })}$ for an inner insertion starting after j-1 arrows. The second type is the source-module part of the differential of $F_w$; the output insertion is its target-module part. The remaining terms are the compositions $F_{w_1,\mathrm{.}\mathrm{.}\mathrm{.},w_j}{F}_{w_{j+1},\mathrm{.}\mathrm{.}\mathrm{.},w_k}$, with every prefix subdivision d of the test word, and are precisely the binary bar differential in the target dg category. To check scalar signs, let $A={\sum }_{i\le j}\mathrm{\mid }{w}_i\mathrm{\mid }$ and $B={\sum }_{i>j}\mathrm{\mid }{w}_i\mathrm{\mid }$. Its target binary sign is $(-1{)}^{(\mathrm{\mid }{F}_{w_1,\mathrm{.}\mathrm{.}\mathrm{.},w_j}\mathrm{\mid }-1)}=(-1{)}^A$, while the two F-core factors contribute $(-1{)}^{(A+B)}$; the common factor after extracting $(-1{)}^{(A+B)}$ is therefore the same prefix-crossing sign $(-1{)}^A$ as the original b insertion at the cut. For target unary insertion, $b_{\mathrm{End}},1(sF)=-s\mathrm{\partial }F$ supplies the same common negative as $q=-s^{(-1)}b$ on the representable coefficient. The suspended prefix factors inside the test word are its comodule Hom differential by definition. These three mutually disjoint groups exhaust EVERY two-insertion term; b²=0 proves $b_{\mathrm{End}}H=H{b}_D$ at arbitrary k,d, not a finite-arity test.

For strings containing formal identities the verification is the strict unital extension, which can be checked directly rather than assumed. A nonunit directed word has strictly decreasing copy labels, so no operation on it can output a formal diagonal. Terms applying an operation of arity other than two to a block containing a formal unit vanish by definition. If a unit is internal, its two possible adjacent binary contractions have opposite suspended prefix signs and give the same shortened word, so cancel. If it is at an outer end, the one surviving source contraction is exactly the target composition with Id of step 15.1. If two consecutive units occur, their binary contraction is the formal unit and its further action is the literal associative scalar identity; every higher H with such an input is zero. The unary differential of a formal unit is zero. This covers any number and location of units, and proves the full functor equations on all formal/nonformal strings. It does not discard any geometric higher-unit input in G: only the DIFFERENT formal-unit slots of D have the strict rules. Thus this construction is a unital realization D→dgMod(G), not a claim that the formal directed representable equals the geometric one or that D has G's self-cohomology.

The actual PSS evaluation. In the geometric Hom complex $C_X={\mathrm{Hom}}_{mod}(Y_X,Y_X),E_X$ is a chain map since $e_X$ is closed. The literal identity component gives $E_X(I{d}_X)=e_X$, proving the commuting scalar diagram in the Statement. Evaluation is linear in the ACTUAL PSS cycle: if $e_X=\sum {\lambda }_j{\xi }_j$ in the retained full generator/coefficient-line complex, then $(F_P{)}_0(e_X)=\sum {\lambda }_j(F_P{)}_0({\xi }_j)$. This is precisely the finite sum of actual count coefficients with that source input. It does not replace the chain complex by a single minimum, does not trivialize End(E) holonomy, and does not change a primitive or source-copy label. In particular ${\nu }_X$ and $e_X$ remain distinct elements of distinct spaces; the interface is ${\rho }_X$ followed by $E_X$. On formal scalar unit action the resulting geometric cochain is $e_X$ exactly, rather than a chosen cohomologous representative.

Full unit-action homotopy. Use the preceding module construction's EXPLICIT full product-Hom maps on $C_X$: $E_X,I_X,J_X,H_C$ and the value-complex contraction data $h_N,i_N,p_N$. With $u_X=E_X{I}_X-1$, let $H_u=h_N{u}_X+i_N{p}_N{u}_X{h}_N,L_X=H_C{I}_X+J_X{H}_u$, and $K_X=H_C-L_X{E}_X$. Step 12.3 gives $\mathrm{\partial }{K}_X+K_X\mathrm{\partial }=1-I_X{E}_X$. Thus $h_X=K_X(I{d}_X)$ has degree -1 and $\mathrm{\partial }{h}_X=I{d}_X-I_X(e_X)$, since $\mathrm{\partial }I{d}_X=0$ and $E_X(I{d}_X)=e_X$. Its EVERY Taylor component is supplied by the finite-at-output-arity perturbation formulas of the preceding construction; no implicit infinite scalar sum or strict geometric unit is used. $I_X(e_X)$ retains all higher operations having that PSS input. It is this chain homotopy, not an equality $I{d}_X=I_X(e_X)$, which connects the formal identity action to the geometric PSS action.

Arbitrary degrees and parameter faces. For an r-cell $F_P$ with the actual equation $\mathrm{\partial }{F}_P=F_{\mathrm{\partial }P}$, use the dg Leibniz rule $\mathrm{\partial }(F_P{h}_X)=(\mathrm{\partial }{F}_P)h_X+(-1{)}^r{F}_P\mathrm{\partial }{h}_X$. Consequently $Q_P=(-1{)}^r{F}_P{h}_X$ has $\mathrm{\partial }{Q}_P=(-1{)}^r{F}_{\mathrm{\partial }P}{h}_X+F_P(I{d}_X-I_X(e_X))$. Every oriented face has dimension r-1, so $Q_{\mathrm{\partial }P}=(-1{)}^{(r-1)}{F}_{\mathrm{\partial }P}{h}_X$, and these first terms cancel. This proves the exact parameter-first identity of this formal/PSS interface proposition, including $r=0$. For a degree-h F with no parameters, $\mathrm{\partial }((-1{)}^{h}F{h}_X)=F-F{I}_X(e_X)$ when F is closed. The corresponding left action on an actual target is $\mathrm{\partial }(h_{Y}F)=(I{d}_Y-I_Y(e_Y))F-h_Y\mathrm{\partial }F$. These two explicit formulas give full left and right unit-action homotopies with their degree signs. Applying E to the parameter identity gives a genuine VALUE-complex homotopy between the resulting evaluations, retaining every higher module component of F; it does not identify an independent graph/PSS cochain.

Other ordinary inputs and corner compatibility. Every product-Hom equation above is an equality on the ENTIRE comodule; projection to an arbitrary input word gives the full prefix/suffix sum, including the frozen Koszul sign of a degree-minus-one left suffix. Thus an ordinary geometric action is transported with all its Taylor components. Finite direct sums and graded-line shifts use the frozen shift conjugation, and the exact Θ transports Id,I,E,h,K,F simultaneously. Since Θ has degree zero it preserves the displayed r-sign and the parameter-first normalizations ${\alpha }_r$. Use the SAME $h_X$ on all faces and corners of a parameter family. The formulas then restrict identically at intersecting faces, without choosing a new contraction or a new cochain at each corner. Twisted finite objects use the frozen SAME $t=-sq$ insertion conjugation, preserving the full Hom identities. None of these algebraic homotopies is claimed to be a new geometric unit-flag count.

Exact join with the unary graph-count theorem. This realization tells the count author how a formal diagonal slot is represented: $I{d}_X$ on the geometric representable, whose proper-PSS evaluation is the literal $e_X$. If the unary theorem gives, for the SAME geometric fixture, $t_P$ with $d{t}_P+t_{\mathrm{\partial }P}=E_X{F}_P-s_P$, then the frozen all-arity homotopy is $K{F}_P+I_X{t}_P$ and has boundary $F_P-I_X{s}_P$. If that unary theorem instead gives $E{F}_P-U{s}_P$, use its explicit $H_u$ correction as stated in step 13.3. The identity/PSS interface supplies neither equality between independent fixtures nor a determinant comparison. In particular the formal directed identity is not itself a Floer generator. The unary endpoint theorem and its actual source-copy, primitive, proper PSS, coefficient and orientation identifications remain separately required.