Proof. Write the logarithm as \(\log_F(x)=\sum_{i\geq0}l_i x^{p^i}\), where \(l_0=1\). The Hazewinkel recursion is \[pl_n=v_n+\sum_{i=1}^{n-1}l_i v_{n-i}^{p^i}.\] Applying the logarithm to the formal sum for \([p]_F\) and comparing the coefficient of \(x^{p^n}\) shows that the coefficient of the indecomposable \(v_n\) in \(a_n\) is \[ c_n=1-p^{p^n-1}. \tag{4}\] Indeed the two terms involving \(l_n\) are \(pl_n\) and \(p^{p^n}l_n\), and the coefficient of \(v_n\) in \(l_n\) is \(1/p\). All other terms involve lower \(v_i\). The coefficients \(a_n\) are integral, so the grading and polynomial independence of the \(v_i\) give \(a_n-c_nv_n\in(v_1,\ldots,v_{n-1})R\). Each \(c_n\) is a \(p\)-local unit.

The invariance and primitivity follow inductively. The element \(p\) is invariant. Once the lower coefficients vanish, the first surviving term of the \(p\)-series is \(c_m v_m x^{p^m}\). A strict coordinate change has leading term \(x\), so comparison of this first surviving term on the two sides of the coordinate-change identity fixes \(c_m v_m\), and hence \(v_m\). This proves both invariance of the next ideal and the asserted primitivity.

After all base generators have been killed, the formal group is additive, and composition of additive \(p\)-polynomials gives Equation (3). Modulo \(I_m\), a correction to this coproduct must have a positive base coefficient, of weight at least \(u_m\). For \(i<m\) its weight is already too large. For \(i=m\) the only possibility is a constant multiple of \(v_m\) with no positive-weight cooperation factor. Applying either counit excludes that term. This proves the stated range. ◻

Proof. Modulo \(I_4\), the first two formal-sum coefficients are unit multiples of \(v_4,v_5\). There is no decomposable correction involving \(v_4\) to the second of these coefficients: \(u_5=pu_4+1\) is not a multiple of \(u_4\). The formal group law has no nonlinear term of total ordinary degree less than \(p^4\), because a coefficient of such a term would have positive weight less than \(u_4\). Thus, through exponent \(p^5\), its \(p\)-series is the ordinary sum of its \(x^{p^4}\) and \(x^{p^5}\) terms. The nonlinear terms in adding these two inputs have exponent greater than \(p^5\).

In the identity intertwining this \(p\)-series with a strict coordinate change, only \(x+t_1x^p\) from that change can affect these two exponents. On the side where the change is applied after the \(p\)-series, the coefficient of \(x^{p^5}\) acquires a term proportional to \(v_4^p t_1\). On the other side it acquires a term proportional to \(v_4t_1^{p^4}\). All nonlinear corrections to the strict formal-sum change enter too late after taking the \(p^4\)-th power. Consequently \[\eta_R(v_5)-v_5=c\,v_4t_1^{p^4}+d\,v_4^p t_1 \quad\text{in }\Gamma/I_4\Gamma,\] where \(c,d\in\mathbb F_p\) and \(c\ne0\). The nonzero coefficient follows also directly from the unit factors \(c_4,c_5\) in Equation (4); their reductions cannot annihilate the first displayed contribution. Divide this coboundary by \(c\) and put \(e=d/c\). Since \(t_1\) is primitive modulo \(I_4\), all its \(p\)-power iterates are primitive, and the resulting cohomology relation is Equation (5). ◻

Proof. Every reduced cooperation factor is a polynomial monomial containing at least one \(t_i\), and hence has weight at least one. The coefficient has nonnegative weight. Adding these weights proves the claim. ◻

Proof. In each of the three degrees, any positive coefficient monomial consumes at least \(u_m\) weight. It would leave at most \(w-u_m<s-1\) for at least \(s-1\) reduced bar factors, which is impossible. A coordinate \(t_i\) with \(i\geq m+1\) exceeds the entire weight budget. Thus these cochain groups have scalar coefficients and only \(t_1,\ldots,t_m\) in their bars. Their coproducts are additive by Lemma 2. This identifies the two groups adjacent to the desired cohomology group as well as the group itself, so both kernels and images agree. If \(w<u_m\), the same exclusion of coefficients and coordinates works without a cochain-length restriction. ◻

Proof. The coefficient and cooperation gradings give the assertion on the cobar page. A differential preserves the sparse support only if \(q\) divides \(r-1\). Subsequent pages are subquotients of earlier pages. Since \(r\geq2\) on or after the cohomology page, the first allowed length is \(q+1\). ◻

Proof. The first two inequalities are the rearrangement of \(w\geq s\). Here \[u_4=1{,}028{,}262{,}820,\qquad 2u_4=2{,}056{,}525{,}640<p^4=1{,}036{,}488{,}922{,}561,\] so the remaining inequalities in Equation (8) hold at the fixed prime. For a cochain length \(d\in\{s-1,s,s+1\}\), a coefficient containing \(v_4\) would leave at most \(k\) weight for \(d\geq qk>k\) reduced factors. It therefore cannot occur. A bar containing \(t_4\) would leave at most \(k\) weight for the other \(d-1\geq qk-1>k\) bars, and likewise cannot occur. Higher generators exceed the weight bound. The remaining coproducts are additive by Lemma 2; equivalently, apply Lemma 5. This identifies both adjacent differentials and proves the cohomology assertion. ◻

Proof. If \(w<s\), the degree-\((s,w)\) cochain groups for both \(Y\) and \(P\) are zero, as is the May page, and all the conclusions are immediate. We may therefore suppose \(w\geq s\). The normalized bound in Lemma 4 then gives \((q-1)k\leq u_4-1\). Consequently \[w\leq u_4+\frac{u_4-1}{q-1} <\frac q{q-1}u_4<4p^3<p^4 \quad (p=1009),\] where \(u_4<2p^3\) and \(q/(q-1)<2\). Thus no letter can reach position four. Also \(w<u_5\). A coefficient containing \(v_4\) leaves at most \(k\) weight for normalized bars, whereas the three lengths \(s-1,s,s+1\) are all greater than \(k\), because \(s-1=qk>k\). Higher coefficient generators are excluded by \(w<u_5\). The adjacent-chain comparison of Lemma 9 therefore identifies the required cohomology with that of \(P\).

There are ten possible exterior intervals on positions \(0,1,2,3\). Since \(s\) is odd, \(m\) is odd and hence \(m\leq9\). Every polynomial letter has weight at least \(p\), so \[u_4+k\geq pL\geq p((p-1)k-4), \qquad (p^2-p-1)k\leq u_4+4p.\] But \[(p^2-p-1)(p+3)-(u_4+4p)=p^2-9p-4>0 \quad (p=1009).\] It follows that \(k\leq p+2\).

For \(k\geq2\) we have \(L\geq2p-6\). If any letter reaches position three, that letter has weight at least \(p^3\), while all other polynomial letters have total weight at least \((L-1)p\). This is a valid lower bound whether the distinguished letter is exterior or polynomial. Thus \[w\geq p^3+2p^2-7p >p^3+p^2+2p+3=u_4+p+2\geq w,\] since the strict difference is \(p^2-9p-3>0\) at \(p=1009\). This contradiction excludes position three in that range. ◻

Proof. We prove exhaustion simultaneously for \(w=u_n+1\), with \(n=4,5\). Since \(u_n+1<p^n\), the available exterior intervals are \([a,b)\) with \(0\leq a<b\leq n\), and the polynomial intervals cover positions \(1,\ldots,n-1\). The target weight has base-\(p\) digits \((2,1,\ldots,1)\). At position zero, \(H_0\leq n<p\), so \(H_0=2\) and there is no carry into position one. As \(s=2p-1\) is odd, write \[m=2D-1,\qquad L=p-D.\] There are exactly two exterior intervals starting at zero; at most \(n(n-1)/2\) others are available. Hence \(m\leq11\), \(2\leq D\leq6\), and \[ p-6\leq L\leq p-2. \tag{24}\] The lower bound on \(D\) follows from \(m\) being odd and at least two.

Let \(c_r\) be the carry into position \(r\), with \(c_1=c_n=0\), and let \(B_r\) count polynomial intervals covering position \(r\). The largest possible exterior coverage at a position is nine. Inductively, the quantity \(H_r+B_r+c_r\) is at most \(9+(p-2)+1=p+8<2p\); therefore every carry is zero or one. The digit equations are \[ B_r=1-H_r-c_r+pc_{r+1},\qquad 1\leq r\leq n-1. \tag{25}\] If there were no carry, then \(B_r\leq1\) for every \(r\), and \(L\leq\sum B_r\leq4\), contrary to \(L\geq p-6=1003\).

Call a position high when \(c_{r+1}=1\). A high position has \(B_r\geq p-9\), and a nonhigh position has \(B_r\leq1\). There cannot be two blocks of high positions. Indeed, choose high positions \(a<b\) in different blocks and a nonhigh position \(c\) between them. An interval covering both \(a\) and \(b\) also covers \(c\), whence \[L\geq B_a+B_b-B_c\geq2(p-9)-1=2p-19>p-2.\] Here \(2p-19=1999\) and \(p-2=1007\). This contradicts (24). Thus the high positions form a single block \([l,r]\) with \(1\leq l\leq r\leq n-2\). At the next position (25) reads \(B_{r+1}=-H_{r+1}\), so \[ B_{r+1}=H_{r+1}=0. \tag{26}\] At the beginning of the block, \(B_l=p+1-H_l\leq L\), giving \(H_l\geq D+1\). If \(l<r\), its end gives \(B_r=p-H_r\leq L\), hence \(H_r\geq D\).

A singleton block is impossible. Every exterior interval covering its position \(l\) must end at \(l+1\) by (26). When \(l=1\) there are only two such intervals. When \(l>1\), at most one begins before \(l\), since \(H_{l-1}\leq1\), and at most one begins at \(l\). Thus in either case \(H_l\leq2\), contradicting \(H_l\geq D+1\geq3\). We henceforth have \(l<r\).

If \(l>1\), at most two exterior intervals cover both \(l\) and \(r\): they must end at \(r+1\), and at most one starts before \(l\), while at most one starts at \(l\). At least one of the two intervals starting at zero misses both positions, since \(H_{l-1}\leq1\). Consequently \[H_l+H_r =m+\#\{\text{intervals meeting both}\} -\#\{\text{intervals meeting neither}\} \leq m+1.\] This contradicts \(H_l+H_r\geq2D+1=m+2\). Therefore \(l=1\). There are now at most two exterior intervals covering both endpoints, namely \([0,r+1)\) and \([1,r+1)\). Equality in the same counting identity is forced: both are present, every exterior interval meets position \(1\) or position \(r\), and \[H_1=D+1,\qquad H_r=D.\] It follows that \(B_1=B_r=L\). Every polynomial interval therefore covers both positions, and by (26) every one is exactly \([1,r+1)\). No exterior or polynomial interval can occur after the block. The carry supplies the target digit at \(r+1\); all further digits would be zero. Since the required remaining digits are all one, necessarily \(r=n-2\). Thus \[ \text{polynomial factor }b_{n-2,0}^{p-D},\qquad H_0=2,\quad H_1=D+1,\quad H_2=\cdots=H_{n-2}=D, \quad H_{n-1}=0. \tag{27}\]

It remains to list a small, now forced set of exterior intervals. For \(n=4\), the intervals \([0,3)\) and \([1,3)\) are mandatory. The other start-zero interval must be \([0,2)\); \([0,1)\) would meet neither endpoint. The only optional intervals are \([1,2)\) and \([2,3)\). The coverage requirements in (27) require both or neither. This is exactly (23).

For \(n=5\), the mandatory intervals are \([0,4)\) and \([1,4)\). The remaining start-zero interval is \([0,t)\) with \(t=2\) or \(3\). The only optional intervals are \([1,2),[1,3),[2,4),[3,4)\); \([2,3)\) meets neither endpoint and is forbidden. Write their inclusion indicators as \(e_{12},e_{13},e_{24},e_{34}\in\{0,1\}\), and let their sum be \(o\). The three required coverage equations are \[ e_{12}+e_{13}=o/2,\qquad e_{24}+e_{34}=o/2,\qquad e_{13}+e_{24}+[t=3]=o/2, \tag{28}\] where \(o\in\{0,2,4\}\). For \(t=2\) the solutions are no optional intervals, the pair \(\{12,24\}\), the pair \(\{13,34\}\), and all four. For \(t=3\) the only solution is the pair \(\{12,34\}\). This proves the exhaustive Table 2, including its length-seven entry.

For completeness, the splitting calculation is also explicit. The length-three entry has, for example, a nonzero component on \(01,04,12,14\). For the three length-five entries in their displayed order, take the following four rows, ordering exterior intervals lexicographically: \[\begin{array}{c|rrr} 01,04,12,13,14,34&1&-1&0\\ 02,03,12,14,24,34&1&0&1\\ 02,04,12,13,24,34&0&1&1\\ 02,04,12,14,23,34&-1&-1&1 \end{array}\] The common polynomial factor is \(b_{3,0}^{p-3}\). The minor using the first, second, and fourth rows is \(3\), which is nonzero in \(\mathbb F_{1009}\). Thus these three images are linearly independent. Finally, (22) shows that \(acb\) has a nonzero splitting differential and that \(acybz\) is closed: every term in the latter differential repeats an exterior factor. This finishes the lemma. ◻

Proof. Use the filtration in Lemma 10 with the explicit choice \(M=3p\). If the source cohomology is nonzero, any nonzero class has leading detector a nonzero scalar multiple of \(acybz B^{p-3}\), by Lemma 16. Its filtration is \[F_0=3p^2-9p+13+M(2+p(p-3)).\] The cocycle \([t_1^{p^4}]\) represents \(h_{1,4}\), with degree, weight, and filtration shifts \((1,p^4,1)\). The proposed leading product therefore has filtration \[ F_*=3p^2-9p+14+M(2+p(p-3)). \tag{29}\]

Every possible incoming May differential to this product has source cohomological degree \(q+1\) and weight \(u_5+1\). Table 2 is its complete initial-page source list. The length-three and length-five source filtrations are respectively \[5p^2-10p+15+M(2+p(p-2)),\qquad 5p^2-15p+19+M(2+p(p-3));\] all three length-five sources have the latter filtration. The length-three and length-five entries disappear under \(d_1\), since their displayed splitting maps are injective. None can be a splitting boundary for the product itself, because its polynomial factor is a power of \(B=b_{2,0}\), whereas their polynomial factor is a power of \(b_{3,0}\). The length-seven source has filtration \[F_7=5p^2-20p+23+M(2+p(p-4)).\] For \(M=3p\) the difference is \[F_7-F_*=2p^2-11p+9-pM=-p^2-11p+9<0.\] As all May differentials lower filtration, this last source cannot hit the proposed product on any later page. There are no other incoming sources.

This argument does not require \(B\) to represent a permanent class by itself. Choose an actual cocycle for the given nonzero source cohomology class and multiply that entire cocycle by \([t_1^{p^4}]\). The result is an actual cocycle. Its leading associated-graded cup product is the nonzero monomial \(acybz B^{p-3}h_{1,4}\), since the new exterior letter is distinct from all the old ones. An actual cocycle has no outgoing May differential; its leading detector could disappear only by becoming a boundary, which the preceding incoming-source calculation excludes at every page. Its product cohomology class is therefore nonzero. ◻

Proof. Equation (21) and Proposition 17 give \(\sigma(d_{q+1}(v_4))=0\). Lemma 15 identifies this specialization with the original target group. Hence \(d_{q+1}(v_4)=0\). ◻

Proof. Lemma 15 allows only positions \(0,1,2\) and \(2\leq k\leq p+2\). Thus (12) applies. There are six exterior letters; \(m\) is odd, so write \[m=2f+1,\quad 0\leq f\leq2,\qquad L=(p-1)k-f.\] Reducing the weight modulo \(p\) gives \(H_0\equiv1+k\pmod p\), with \(0\leq H_0\leq3\). The only possibilities are \(k=2\) or \(k=p+d\) with \(-1\leq d\leq2\).

When \(k=2\), \(H_0=3\) forces all of \(x,a,c\). The other exterior letters are an even subset of \(\{y,b,z\}\). Put \(T=H_2+n_B+n_C\). Substitution in (12) gives \[ pT-n_C+H_1=p^2-p+3+f. \tag{30}\] For the four possible exterior subsets the value \(e=H_1-3-f\) is \(-2\) or \(-1\). Therefore \(n_C\equiv e\pmod p\). Since \(0\leq n_C\leq L\leq2p-3\), the only possible lift is \(n_C=p+e\): the lift \(e\) is negative, and \(2p+e\geq2p-2>L\). Equation (30) now gives \(T=p\). All the remaining counts follow as displayed: \[\begin{array}{c|c|c|c|c|c|c} \text{extra exterior letters}&f&H_1&H_2&n_C&n_B&n_A\\ \hline \varnothing&1&2&1&p-2&1&p-2\\ y,b&2&4&2&p-1&-1&p-2\\ y,z&2&3&2&p-2&0&p-2\\ b,z&2&3&3&p-2&-1&p-1 \end{array}\] The negative counts exclude the second and fourth rows. The first and third are exactly the \(k=2\) row of Table 3.

Now let \(k=p+d\), \(-1\leq d\leq2\). Then \(H_0=1+d\), and the weight equation becomes \[ pT-n_C+H_1=(2-d)p+2+d+f. \tag{31}\] Since \(n_C\leq T\) and \(H_1\geq0\), it follows that \[(p-1)T\leq(2-d)(p-1)+(4+f).\] Here \(4+f\leq6<p-1\), so \(T\leq2-d\leq3\) and \(n_C\leq3\). Rearranging (31) now gives \[p(T-(2-d))=2+d+f+n_C-H_1.\] The right side lies between \(-3\) and \(9\), strictly between \(-p\) and \(p\). Both sides therefore vanish, proving the exact equations \[ H_2+n_B+n_C=2-d,\qquad H_1=2+d+f+n_C. \tag{32}\]

For \(d\geq1\), these require \(3+f\leq H_1\leq\min(2f+1,4)\), impossible for each of \(f=0,1,2\). For \(d=0\), exactly one of \(x,a,c\) occurs, so \(H_1\leq3\). Equation (32) forces \(f=1\), \(n_C=0\), and \(H_1=3\). The only exterior triples are \(ayb\) and \(cyb\). Their \(H_2\) values are respectively \(1\) and \(2\), giving the two \(k=p\) monomials in the table.

For \(d=-1\), only \(y,b,z\) can occur. If \(m=1\), the choices \(y,b,z\) have \((H_1,H_2)\) equal to \((1,0),(1,1),(0,1)\). Equation (32) excludes \(z\) and gives respectively \((n_C,n_B)=(0,3),(0,2)\) for the other two. If \(m=3\), the exterior factor is \(ybz\), with \(f=1\), \(H_1=H_2=2\); the same equations give \((n_C,n_B)=(0,1)\). Since \(L=N_0-f\), these are exactly the three \(k=p-1\) monomials. Every nonnegative count and every exterior subset has now been considered. ◻

Proof. The subalgebra map includes a subset of the exterior and polynomial generators in (9); its initial-page map is therefore injective in every bidegree. Likewise, multiplication by the polynomial \(A^a\) is injective in every bidegree on that page. It is induced by a filtered cochain map: \(t_1\) is primitive, and \[ \sum_{i=1}^{p-1}\frac{\binom pi}{p} [t_1^i\mid t_1^{p-i}] \tag{33}\] is an actual cocycle, obtained by taking the integral divided reduced coproduct and then reducing modulo \(p\). It has leading class \(A\). For the second comparison shift the source cohomological degree, weight, and auxiliary filtration by \(2a\), \(pa\), and \(ap(1+M)\).

Take the filtered mapping cone of either cochain map. The long exact sequence of associated-graded cohomology, together with injectivity in all bidegrees, identifies its initial cohomology page with the cokernel of the specified page map; there is no extra kernel from the next degree. That cokernel is zero in the stated target bidegree. Finite filtration therefore gives zero cone cohomology there. The ordinary cone long exact sequence gives the required surjection. Transport through the stated commuting comparisons preserves that surjectivity. ◻

Proof. The inclusion of the Frobenius subring makes \(P_r\) free of rank \(p^r\), with basis \(t_1^{e_1}\cdots t_r^{e_r}\), \(0\leq e_i<p\). It is therefore faithfully flat. Its kernel has coordinate algebra \[O(K_r)=\mathbb F_p[e_1,\ldots,e_r]/(e_1^p,\ldots,e_r^p).\] All the mixed terms in its coproduct vanish, so each \(e_i\) is primitive. This proves the description of the kernel and the exactness of (35).

Here is the acyclicity calculation needed to derive invariants. Right multiplication by a kernel point is, by (34), simply \(t_i\BeginAccSupp{method=hex,unicode,ActualText=21A6}\OriginalMapsto\EndAccSupp{}t_i+e_i\): every mixed term contains a positive \(p\)-power of a kernel coordinate. Hence multiplication gives an isomorphism of \(K_r\)-comodules \[ O(Q_r)\otimes O(K_r)\xrightarrow{\ \cong\ }O(G_r), \qquad f\otimes e_1^{j_1}\cdots e_r^{j_r} \BeginAccSupp{method=hex,unicode,ActualText=27FC}\OriginalLongmapsto\EndAccSupp{}f t_1^{j_1}\cdots t_r^{j_r}, \quad 0\leq j_i<p. \tag{37}\] The first tensor factor has trivial \(K_r\)-coaction. The second is the regular comodule. In particular, \(O(G_r)^{K_r}=O(Q_r)\), with its regular quotient coaction.

A cofree comodule is injective: its cofree functor is right adjoint to the exact forgetful functor to vector spaces. Equation (37) shows that every cofree \(G_r\)-comodule is \(K_r\)-acyclic and that its \(K_r\)-invariants are cofree over \(Q_r\). Every injective is a summand of a cofree comodule, so these assertions also hold for injectives. Finally, \(M^{G_r}=(M^{K_r})^{Q_r}\). Resolving these two invariants functors successively, using the indicated cofree resolutions, gives (36). This argument derives \(K_r\)-invariants; it does not assert their exactness on arbitrary modules. The spectral sequence is first quadrant, so each fixed total cohomological degree has a finite diagonal. All coordinate weights are positive and each weight component is finite dimensional. These observations give the asserted convergence, also weight by weight. ◻

Proof. The dual algebra of a primitive truncated coordinate \(\mathbb F_p[e]/(e^p)\) is a truncated polynomial algebra after rescaling the dual basis by the invertible factorials \(j!\), \(j<p\). Its resolution alternates multiplication by its generator and by the \((p-1)\)st power of that generator. Applying the augmentation gives one class in each cohomological degree. Its algebra of extensions is an exterior degree-one generator and a polynomial degree-two generator. The shift by two of the periodic resolution represents the latter, and its iterates are nonzero in every even degree. The former has square \([e\mid e]=\tfrac12d[e^2]\), which is zero at this odd prime. Tensoring these resolutions gives (38), including the indicated weights.

For completeness, the naturality of the polynomial generators is Frobenius semilinear. A primitive coordinate \(T\) has degree-two cocycle \[\beta(T)=\sum_{j=1}^{p-1}\frac{1}{p}\binom pj [T^j\mid T^{p-j}],\] where the displayed coefficients are integers reduced modulo \(p\). For a linear coordinate \(T=\sum_i\lambda_i e_i\), the integral polynomial \[\frac{(\sum_i\lambda_i e_i)^p-\sum_i\lambda_i^p e_i^p}{p}\] has reduced coproduct \(\beta(\sum_i\lambda_i e_i)-\sum_i\lambda_i^p\beta(e_i)\). Thus a matrix \((\lambda_{ij})\) on degree-one coordinates induces \((\lambda_{ij}^p)\) on these degree-two cohomology classes. This is a polynomial identity in the \(\lambda_{ij}\), so it also applies when they are quotient coordinate functions.

If \(e\) is a kernel point, direct use of (34) gives \[(g^{-1}eg)_1=e_1,\qquad (g^{-1}eg)_2=e_2+e_1g_1^p,\qquad (g^{-1}eg)_3=e_3+e_1g_2^p+e_2g_1^{p^2}.\] These coefficients belong to \(O(Q_r)\), so the action descends to the quotient. The degree-one and degree-two naturality just proved gives (39) and (40). The exterior matrix is triangular with diagonal entries one, so its top exterior power \(xac\) is fixed. The formulas also prove separate preservation of the two degrees. ◻

Proof. In (39) and (40), every nonidentity term has lower coefficient weight, its difference being the positive weight of its quotient coordinate. The cobar differential therefore preserves the increasing coefficient filtration. In the associated graded complex only the trivial coefficient action remains, giving (41). At fixed total weight the coefficient weights lie between zero and \(w\), so the filtration is finite. The stated description of its first possible differential follows by taking the leading nonzero weight-lowering component of the cobar differential. Finally, expanding \[(B+A\xi^p)^jA^{n-j} =\sum_{l=0}^j\binom jl A^{n-l}B^l\xi^{p(j-l)}\] proves the stated action on (42). ◻

Proof. The next exterior weights are \(p^2\) and \(p^2+p\); the third-coordinate weight, if present, is \(p^2+p+1\). The next polynomial weight is \(p^2\). All are outside the asserted range. Lemma 10 therefore gives exactly (46). To verify the table, first reduce its weight equation modulo \(p\): its remainder is \(e+g\in\{0,1,2\}\). For weights divisible by \(p\) it forces \(e=g=0\), after which the degree equation fixes \(j\) and \(f\). For the listed weights congruent to two it forces \(e=g=1\); substitution into both equations gives the entries shown. The weight-two entries follow in the same way, since \(j=f=g=0\) at that weight. All weights in the table are within the asserted range, because \(3p+2<p(p-1)\) at \(p=1009\).

The cocycles \(z=[\xi^p]\) and \(C=\beta(\xi)\) come from the primitive coordinate \(\xi\). Their product \(zC^{p-2}\) is an actual cocycle with the indicated nonzero leading May monomial. The preceding cohomological degree at its weight has no May monomial in any filtration. It consequently cannot become a boundary on any May page. An actual cocycle has no outgoing May differential, proving its nonvanishing. The factor \(C^{p-2}\) is therefore nonzero as well.

For the other two classes, use the cobar convention \(d[u\mid v]=d[u]\mid v-u\mid d[v]\) with \(d[\eta]=[\xi\mid\xi^p]\). The corrected cocycle \[ U=[\xi\mid\eta]+\frac12[\xi^2\mid\xi^p] \tag{47}\] is closed: its two differentials are \(-[\xi\mid\xi\mid\xi^p]\) and \([\xi\mid\xi\mid\xi^p]\), respectively. In the ordinary May filtration its first summand has degree four and the correction has degree three. Its leading detector is thus \(yb\). The table excludes both an incoming and an outgoing May differential, so \(\upsilon=[U]\neq0\). Similarly \[U\mid\xi^p =[\xi\mid\eta\mid\xi^p] +\frac12[\xi^2\mid\xi^p\mid\xi^p]\] is a closed representative with leading detector \(ybz\). Its correction has smaller May filtration. The absence of a degree-two May monomial at weight \(2p+2\) excludes every possible incoming differential, and the explicit cocycle excludes outgoing differentials. Hence \(\upsilon z\neq0\). ◻

Proof. We prove vanishing for \(G_3\) by treating the first two rows of Table 4. Their kernel degrees are \(2p+1\) and \(2p-1\), so their coefficient differentials cannot mix.

For \(M=xac\operatorname{Sym}^{p-1}(A,B,E)\), we have \(\mathcal R=p(p-1)\) and quotient degree \(a=2p-4\). After the weight exclusion of \(E\), the finite module is \(xacA^{p-1}\langle1,D,\ldots,D^{p-1}\rangle\). The sole candidate from Table 3 is \(D C^{p-2}\), suppressing the invariant factor \(xacA^{p-1}\). Its quotient weight is \(p(p-2)\) by (44). The translation of \(D\) gives, on the first possible coefficient page, \[ D[C^{p-2}]\BeginAccSupp{method=hex,unicode,ActualText=27FC}\OriginalLongmapsto\EndAccSupp{}\pm[zC^{p-2}]. \tag{48}\] The target has quotient degree \(2p-3\) and weight \(p(p-1)\). Both source and target are nonzero actual quotient-cohomology classes by Lemma 24; they are present on this page because all earlier coefficient differentials vanish. Thus (48) removes the candidate.

We record the adjacent-degree checks. A differential on this page entering its source would have \(D\)-degree two, quotient degree \(2p-5\), and quotient weight \(p(p-3)\); this group is zero by Lemma 24. The target, having \(D\)-degree zero, has no outgoing coefficient differential. Its first-page incoming source can only have \(D\)-degree one, and (46) in degree \(2p-4\), weight \(p(p-2)\), gives only \(C^{p-2}\). Thus no other source can cancel the nonzero map in (48). In the original total degree, the same equation, or the exhaustive candidate table, leaves no other coefficient-leading term in this vertical summand. Hence a change of representative cannot replace the removed class.

Now take the full second summand \(M=xac\operatorname{Sym}^{p-2}(A,B,E)\), with \(\mathcal R=p^2-p+1\) and quotient degree \(2p-2\). Table 3 leaves only its \(D\)-degree-zero candidate, whose coefficient \(xacA^{p-2}\) is invariant. Its quotient detector \(yzC^{p-2}\) is already a splitting boundary, since \(db=yz\) and \(dC=0\) on the May page. This also has a direct cobar realization: \(yz=d[\eta]\) and \(C\) is an actual cocycle, so \(yzC^{p-2}\) is exact. It is the only May monomial in this quotient bidegree, as follows by putting \(a=2p-2\), \(w=p^2-p+1\) in (46). It therefore contributes no actual quotient cohomology in (41). These are all initial candidates of total degree \(4p-3\) and weight \(u_4+2\). The coefficient spectral sequences consequently vanish there, so all corresponding terms of (36) are zero. Its convergence proves the result. ◻

Proof. We treat the last two rows of Table 4 for \(G_2\), using the finite module (42) in each case. For \(n=N_0\), the residual weight is \(\mathcal R=3p+1\) and the quotient degree is one. We show directly that every one-cocycle vanishes. The leading candidates could survive only if lower powers of \(D\) supplied corrections. The crossed-cocycle equation will rule these out: it requires unequal mixed coefficients where every available correction has equal ones.

Write \(\xi(g)=r\), \(\eta(g)=u\), \(\xi(h)=s\), \(\eta(h)=v\), so that \[\xi(gh)=r+s,\qquad\eta(gh)=u+v+rs^p.\] A normalized one-cocycle is a polynomial \(f(D;g)\) satisfying \[ f(D;gh)=f(D;g)+f(D+r^p;h). \tag{49}\] Since \(D\) has weight \(p\) and \(4p>3p+1\), it must have the form \[f(D;g)=D^3f_3(g)+D^2f_2(g)+Df_1(g)+f_0(g).\] All these powers are in the finite module, since \(N_0\geq3\). For a polynomial function \(\varphi\) write \(\delta\varphi(g,h)=\varphi(gh)-\varphi(g)-\varphi(h)\). Comparing the four coefficients in (49) gives \[\begin{align*} \delta f_3&=0,\\ \delta f_2&=3r^pf_3(h),\\ \delta f_1&=2r^pf_2(h)+3r^{2p}f_3(h), \tag{50}\\ \delta f_0&=r^pf_1(h)+r^{2p}f_2(h)+r^{3p}f_3(h). \end{align*}\] Homogeneity forces \[\begin{align*} f_3&=a_3\xi,\\ f_2&=b_2\eta+c_2\xi^{p+1},\\ f_1&=d_1\eta\xi^p+e_1\xi^{2p+1},\\ f_0&=r_0\xi^{3p+1}+s_0\eta\xi^{2p} +t_0\eta^2\xi^{p-1}. \end{align*}\] Here the coefficients \(a_3,b_2,c_2,d_1,e_1,r_0,s_0,t_0\) lie in \(\mathbb F_p\). The second equation of (50), together with \[\delta\eta=rs^p,\qquad \delta(\xi^{p+1})=r^ps+rs^p,\] forces \(c_2=3a_3\) and \(b_2=-3a_3\). The next equation has coefficient \(2b_2\) on \(r^pv\) and zero on \(us^p\). On the other hand, \[ \delta(\eta\xi^p) =us^p+vr^p+r^{p+1}s^p+rs^{2p}, \tag{51}\] and \(\delta(\xi^{2p+1})\) contains neither \(u\) nor \(v\). Its left side must therefore have equal coefficients on \(r^pv\) and \(us^p\). Since \(2b_2=-6a_3\) and \(6\neq0\) in \(\mathbb F_{1009}\), we obtain \(a_3=0\), and then \(b_2=c_2=0\). No term in \(f_0\) can alter this coefficient-of-\(D\) equation.

Now \(f_1\) is primitive. Equation (51) first forces \(d_1=0\), and the coefficient of \(r^{2p}s\) in \(\delta(\xi^{2p+1})\) then forces \(e_1=0\). Finally \(f_0\) is primitive. In its reduced coproduct the monomial \(u^2s^{p-1}\) has coefficient \(t_0\), so \(t_0=0\). The monomial \(us^{2p}\) then has coefficient \(s_0\), so \(s_0=0\). The coefficient of \(r^{3p}s\) in \(\delta(\xi^{3p+1})\) is one, giving \(r_0=0\). Every normalized cocycle is thus zero, proving the required \(H^1\) vanishing, including all lower-\(D\) corrections.

For \(n=N_0-1\), the residual weight is \(\mathcal R=3p+2\) and the desired quotient degree is three. The first coefficient differential comes from the term \(2D\xi^p\) in \[(D+\xi^p)^2-D^2=2D\xi^p+\xi^{2p}.\] Using the actual class \(\upsilon\) of Lemma 24, it is \[ D^2\upsilon\BeginAccSupp{method=hex,unicode,ActualText=27FC}\OriginalLongmapsto\EndAccSupp{}\pm2D\upsilon z. \tag{52}\] The coefficient differential first prepends \(z\). Its product \(z\upsilon\) agrees with \(\upsilon z\): both are actual cocycles with the same leading May monomial \(ybz\), with coefficient \(+1\) after two exterior interchanges, and the list in Equation (46) leaves no other initial May monomial or lower-filtration cohomology in this bidegree. Thus the product order in Equation (52) is justified directly. Both powers of \(D\) lie in the finite module, since \(n\geq3\). Equation (44) gives quotient degree and weight \((2,p+2)\) for the source and \((3,2p+2)\) for the target. Both therefore have the required residual weight \(3p+2\). All earlier coefficient differentials vanish, so source and target are still their actual quotient-cohomology classes on this first possible page. They are both nonzero by Lemma 24.

A same-page incoming differential to the source would require \(D\)-degree three and quotient \(H^1\) of weight two, which is zero. The only same-page incoming source for the target is \(D^2H^2(Q_2)_{p+2}\), the one-dimensional span displayed in (52). A same-page outgoing differential from the target is proportional to \(\upsilon z^2=0\). Indeed \([\xi^p\mid\xi^p]=\tfrac12d[\xi^{2p}]\), so this last vanishing also follows directly in the cobar complex. The term \(\xi^{2p}\) in the expansion of \(D^2\) lowers two \(D\)-levels and cannot cancel its first differential. Since \(2\neq0\) in \(\mathbb F_p\), (52) removes the target.

There is no other degree-three coefficient-leading term. In fact, the possibilities for \(D\)-degrees \(0,1,2,3\) require quotient degree three at respective weights \(3p+2,2p+2,p+2,2\); by Lemma 24, only the second has a candidate, namely \(ybz\). Larger \(D\)-degrees exceed the residual weight. Thus there is no replacement in a lower coefficient filtration. The initial candidate table excludes every other kernel degree or exterior summand at this total bidegree. Both possible terms of (36) consequently vanish, and the proposition follows. ◻

Proof. Apply Propositions 25 and 26, and the exhaustive list in Table 3. ◻

Proof. By \(q\)-sparsity, a positive differential out of \((5,W)\) has target \[(s,w)=(6+qj,W+j),\qquad j\geq 1.\] All possible Adams–Novikov terms in the stem in (57) have \[(s,w)=(6+qj,W+1+j),\qquad j\geq 0.\] We treat both by writing \[ (s,w)=(6+qj,W+\delta+j),\qquad \begin{cases} \delta=0,&j\geq1,\\ \delta=1,&j\geq0. \end{cases} \tag{58}\]

First we give explicit bounds permitting the additive calculation. The normalized-cobar inequality of Lemma 4 implies \[(q-1)j\leq W+\delta-6.\] At the specified prime, \[q-1=2015,\quad W=2{,}040{,}199,\quad 2015\cdot1013=2{,}041{,}195>W+1-6.\] Consequently \(j\leq1012\) and \[ w\leq2{,}041{,}212 <p^3=1{,}027{,}243{,}729 <u_4=1{,}028{,}262{,}820. \tag{59}\] Thus no nonconstant coefficient of \(R/I_4\) occurs, in any cobar degree at these weights. The only possible digit positions are \(0,1,2\), and the cooperation formulas are additive. Lemma 5 therefore identifies the relevant cobar calculation with the indicated range of \(P\).

Use the six exterior letters \(x,a,c,y,b,z\) and the three polynomial letters \(A,B,C\) of Equation (12). If \(m\) is the number of exterior letters and \(L=(s-m)/2\) the number of polynomial letters, then \(m\leq6\) and \[L\geq (p-1)j,\qquad w\geq pL.\] It follows that \[\bigl(p(p-1)-1\bigr)j\leq W+\delta.\] Since \(p(p-1)-1=1{,}017{,}071\) and \(3\cdot1{,}017{,}071>W+1\), this improves the bound to \(j\leq2\).

For completeness, we enumerate this remaining finite calculation. Let \(H_i\) count the exterior letters covering position \(i\), and let \(n_A,n_B,n_C\) be the polynomial exponents. The weight equation is \[ w=H_0+p(H_1+L-n_C)+p^2(H_2+n_B+n_C). \tag{60}\] In (58), \(m\) is even and reduction modulo \(p\) gives \(H_0=1+\delta+j\), whenever a solution exists. Put \(\kappa_2=H_2+n_B+n_C\). All position contributions are nonnegative; (59) also gives \(w<3p^2\), so \(\kappa_2\leq2\) and \(n_C\leq2\). Substituting \(L=3+(p-1)j-m/2\) into (60) yields \[p(\kappa_2+j-2)=1+j+m/2+n_C-H_1.\] For \(0\leq j\leq2\), its right side lies between \(-3\) and \(8\). Its only multiple of \(p=1009\) is zero. Hence \[ H_0=1+\delta+j,\qquad \kappa_2=2-j,\qquad H_1=1+j+m/2+n_C. \tag{61}\]

For \(j=2\), either \(H_0=4\), which is impossible, or \(H_0=3\) forces \(c\) to occur whereas \(\kappa_2=0\). For \(j=1,\delta=1\), all of \(x,a,c\) occur and \(\kappa_2=1\) excludes \(b,z\). Thus \(H_1\leq3\), but the last equation of (61), with even \(m\geq4\), requires \(H_1\geq4\). For \(j=1,\delta=0\), one has \(H_0=2\). If \(m=2\), the required inequality \(H_1\geq3\) is impossible. If \(m=4\), one must have \(H_1=4\) and \(n_C=0\); the four letters covering position one are exactly \(a,c,y,b\), which have \(H_2=2\), contrary to \(\kappa_2=1\). Finally \(m=6\) would give \(H_0=3\). All positive-\(j\) groups in (58) therefore have empty initial May pages.

It remains to examine \(j=0,\delta=1\), where \(H_0=2\) and \(m\) is \(2\) or \(4\). For \(m=2\) both letters start at zero, and (61) permits only \(ac\), followed by \(n_B=1,n_C=0,n_A=1\). For \(m=4\) choose two of \(x,a,c\) and two of \(y,b,z\). The equations become \[n_C=H_1-3,\qquad n_B=2-H_2-n_C,\qquad n_A=1-n_B-n_C.\] The following table lists all nine choices; a dash means that at least one exponent is negative.

Thus the complete initial-page list at \((6,W+1)\) is \[ acAB,\qquad xaybB,\qquad xcybA,\qquad acyzA. \tag{62}\] With exterior order \(x,a,c,y,b,z\) and \(da=xy\), \(db=yz\), \(dc=xb+az\), their splitting differentials are \[\begin{aligned} d(acAB)&=(xab-xcy)AB, &d(xaybB)&=0,\\ d(xcybA)&=-xaybzA, &d(acyzA)&=-xaybzA. \end{aligned}\] The first image has polynomial factor \(AB\) and cannot cancel either of the other images. The two remaining cycle directions are boundaries, since \[d(acyB)=-xaybB,\qquad d(acbA)=-xcybA+acyzA.\] The splitting cohomology of (62) is zero. The May spectral sequence, finite in each of these weights, proves all the claimed \(E_2\) vanishings.

For orientation, the same equations also enumerate the source \((5,W)\). Here \(H_0=1\), \(m\) is odd, and \(L=(5-m)/2\). They force \(H_2+n_B+n_C=2\) and \(H_1+L-n_C=4\). If \(m=1\), then \(H_1+L\leq1+2<4\); if \(m=5\), then at least two of \(x,a,c\) occur, contrary to \(H_0=1\). Thus \(m=3\), which forces \(n_C=0\) and \(H_1=3\). The only monomials are \(aybB\) and \(cybA\). We need no assertion that either one survives the May spectral sequence.

No topological Adams–Novikov differential can hit filtration five: its length would have to be at least \(q+1>5\). The outgoing target groups have just been proved zero. Thus every actual class of \(E_2^{5,W}(Y)\) survives and is represented by a class \(\gamma\in\pi_{qW-5}Y\). The vanishings for \(\delta=1\) exhaust all filtrations in the stem \(q(W+1)-6\); strong convergence proves (57). ◻

Proof. The cooperation \(t_1\) is primitive. In weight one the normalized sphere cobar is \[\mathbb Z_{(p)}\{v_1\}\xrightarrow{\ p\ } \mathbb Z_{(p)}\{t_1\},\] because \(\eta_R(v_1)-v_1=pt_1\). Hence its degree-one cohomology is \(\mathbb Z/p\) on \([t_1]\).

The integral polynomial \[ b(x,y)=\frac{(x+y)^p-x^p-y^p}{p} =\sum_{i=1}^{p-1}\frac{\binom pi}{p}x^iy^{p-i} \tag{63}\] defines the two-cobar cocycle \(b_{1,0}\) by substituting the two primitive copies of \(t_1\). Its differential is zero: the two-cocycle identity follows by expanding \((x+y+z)^p\), or by dividing the identity \(d^2(t_1^p)=0\) in the torsion-free integral cobar. Its image in the additive specialization is nonzero in cohomology. At weight \(p\) the only coordinate available is \(t_1\); the only one-cochain of that weight is \(t_1^p\), whose differential is zero in characteristic \(p\), whereas (63) is a nonzero two-cochain. Thus it is not a boundary.

The first positive topological differential length permitted by sparsity is \(q+1\). From \((2,p)\) an outgoing differential has target \((qj+3,p+j)\) for \(j\geq1\), violating \(w\geq s\); no incoming differential can reach filtration two. The same argument applies to \((1,1)\). Therefore these cocycles survive to the stated sphere classes.

For any positive stem \(i\) with a nonzero normalized-cobar term, either \(s=0\) and \(i=qw\geq q\), or \(s\geq1\) and \[i=qw-s\geq(q-1)s\geq q-1.\] This proves the lower sphere vanishing. In stem \(q-1\) the same inequality permits only \((s,w)=(1,1)\), so there is no extension ambiguity and \(\pi_{q-1}S^0_{(p)}=\mathbb Z/p\{\alpha\}\). Finally, the sphere action and its cobar pairing in Proposition 7 identify the detector of multiplication by \(\beta\) with \(b_{1,0}=A\). ◻

Proof. Let \(C\) be the cofiber of \(b_0:S^0\to U_0\). The assumed integral homology groups and successive applications of connective Hurewicz show that \(C\) is \((q-2)\)-connected and \[\pi_{q-1}C=H_{q-1}(C;\mathbb Z_{(p)})=\mathbb Z/p.\] Choose a generator \(S^{q-1}\to C\). Its obstruction to lifting to \(U_0\) lies in \(\pi_{q-2}S^0\), which is zero by Lemma 29. A lift, together with \(b_0\), gives \(S^0\vee S^{q-1}\to U_0\). Its cofiber \(D\) is \((q-1)\)-connected. The integral homology exact sequence gives \[H_q(D;\mathbb Z_{(p)}) =\ker\bigl(\mathbb Z_{(p)}\longrightarrow\mathbb Z/p\bigr) =p\mathbb Z_{(p)}.\] In particular this is a free rank-one module. Use Hurewicz to choose a generator of \(\pi_qD\) and take its boundary in \(\pi_{q-1}(S^0\vee S^{q-1})\). Attaching the corresponding cell gives a map \(Q\to U_0\) extending the two lower cells. Its component on \(S^{q-1}\) is multiplication by a unit times \(p\), since its homology image is exactly \(p\mathbb Z_{(p)}\). Its component on the bottom belongs to \(\pi_{q-1}S^0=\mathbb Z/p\{\alpha\}\), so it is \(a\alpha\) for some \(a\in\mathbb F_p\).

The map \(Q\to U_0\) induces isomorphisms on integral homology in degrees zero and \(q-1\). The universal coefficient sequence then gives isomorphisms on mod-\(p\) cohomology in degrees zero and \(q\); the latter is the class arising from the torsion in degree \(q-1\). The assumed nonzero reduced power therefore pulls back to a nonzero reduced power of the bottom class of \(Q\). If \(a=0\), the actual attaching map would give a wedge \[Q\simeq S^0\vee\operatorname{cofib} \bigl(up:S^{q-1}\to S^{q-1}\bigr).\] The degree-zero class in this wedge is pulled back from \(S^0\), so its positive reduced powers vanish by naturality. This is a contradiction. Hence \(a\ne0\). The construction used an integral homology gap and relative Hurewicz in a finite range; no assertion that \(U_0\) itself is finite is involved. ◻

Proof. Write \(d=pq-2=2m\), where \[m=p(p-1)-1\equiv-1\pmod p.\] Let \(\rho_{\mathbb C}\) be the complex regular representation of \(C_p\). The cyclic extended power of the even sphere \(S^{2m}\) is the Thom spectrum of \(m\rho_{\mathbb C}\) over \(BC_p\): \[(EC_p)_+\wedge_{C_p}(S^{2m})^{\wedge p} \simeq\operatorname{Th}(m\rho_{\mathbb C}).\] Applying this functor to a sphere map representing \(\beta\), then collapsing \(BC_p\) in the extended power of \(S^0\), gives a map \[ \Phi:\operatorname{Th}(m\rho_{\mathbb C})\longrightarrow S^0 \tag{64}\] whose restriction to a fiber sphere \(S^{pd}\) is \(\beta^p\). Only the symmetric multiplication of the sphere spectrum is used in constructing this map.

The group \(\mathbb F_p^\times\) acts by permutations of the cyclically labelled smash coordinates and by the corresponding automorphisms of \(C_p\). This normalizer action is compatible with (64), on whose target it is trivial. Normalize the bottom degree by setting \[U=\Sigma^{-pd}\operatorname{Th}(m\rho_{\mathbb C}),\qquad e_0=\frac1{p-1}\sum_{a\in\mathbb F_p^\times}a:U\longrightarrow U.\] The idempotent \(e_0\) splits in spectra. Denote its image by \(U_0\), with maps \(i_0:U_0\to U\), \(r_0:U\to U_0\) satisfying \(i_0r_0=e_0\) and \(r_0i_0=1\). The projected Thom fiber \(b_0:S^0\to U_0\) induces an isomorphism on \(H_0(-;\mathbb Z_{(p)})\): the complex orientation makes every normalizer permutation act as one on the bottom class. Moreover, normalizer invariance of \(\Phi\) gives a map \[\Phi_0:\Sigma^{pd}U_0\longrightarrow S^0, \qquad \Phi_0\Sigma^{pd}b_0=\beta^p.\]

We describe the homology and the reduced power of this summand. The complex bundle is integrally oriented, so the integral Thom isomorphism identifies \(H_*(U;\mathbb Z_{(p)})\) with \(H_*(BC_p;\mathbb Z_{(p)})\). The periodic cyclic-group resolution, with maps \(g-1\) and \(1+g+\cdots+g^{p-1}\), gives \[H_0(BC_p;\mathbb Z_{(p)})=\mathbb Z_{(p)},\quad H_{2j-1}(BC_p;\mathbb Z_{(p)})=\mathbb Z/p\ (j\geq1),\quad H_{2j}(BC_p;\mathbb Z_{(p)})=0\ (j\geq1).\] These groups are finitely generated degreewise, as are their direct summands for \(U_0\). The same resolution, or the circle bundle of Chern class \(p\) over infinite complex projective space, gives \[H^*(BC_p;\mathbb F_p)=\Lambda(s)\otimes\mathbb F_p[t], \qquad |s|=1,\quad |t|=2,\quad t=\operatorname{Bockstein}(s).\] An automorphism \(a\in\mathbb F_p^\times\) multiplies both \(s\) and \(t\) by \(a\). Equivalently it acts by \(a^j\) on integral homology in degree \(2j-1\). Averaging consequently retains, through degree \(q\), exactly the mod-\(p\) classes in degrees \(0,q-1,q\) and the integral groups \[ H_0(U_0;\mathbb Z_{(p)})=\mathbb Z_{(p)},\quad H_{q-1}(U_0;\mathbb Z_{(p)})=\mathbb Z/p, \quad H_j(U_0;\mathbb Z_{(p)})=0\ (0<j\leq q,\ j\ne q-1). \tag{65}\]

Let \(u\) be the normalized Thom class. We claim \[ \mathcal P^1u=-m\,t^{p-1}u\ne0. \tag{66}\] For a universal complex line bundle with first Chern class \(l\) and Thom class \(u_l\), write \(\mathcal P^1u_l=c\,l^{p-1}u_l\) by the Thom isomorphism. Pullback along the zero section and the identity \(\mathcal P^1(l)=l^p\) imply \(c=1\) in the polynomial cohomology of infinite complex projective space. Naturality gives the formula for every line bundle. The Cartan formula for a direct sum of lines therefore gives the sum of their \((p-1)\)st Chern powers as the multiplier of \(\mathcal P^1\) on its Thom class. The regular representation splits into the characters with first Chern classes \(it\), \(i\in\mathbb F_p\). Thus the multiplier here is \[m\sum_{i\in\mathbb F_p}(it)^{p-1} =-m\,t^{p-1}.\] Both this class and \(u\) are normalizer-invariant. The reduced power commutes with the idempotent, so the same nonzero formula holds in \(U_0\).

Apply Lemma 30 to (65) and (66). It supplies a three-cell spectrum \[Q=(S^0\vee S^{q-1})\cup_{(a\alpha,\,up)}e^q\longrightarrow U_0\] with \(a\) nonzero modulo \(p\) and \(u\) a \(p\)-local unit.

Restrict \(\Phi_0\) to \(\Sigma^{pd}Q\). Its bottom value is \(\beta^p\); let its value on the other lower cell be \(\zeta\). The top cell gives the relation \[a\alpha\beta^p+up\zeta=0 \quad\text{in }\pi_{pd+q-1}S^0.\] The action of \(p\) on \(Y\) is null by Corollary 14. Apply the unit to this relation and use \(a\ne0\) modulo \(p\) to obtain \(\alpha\beta^p=0\) in \(\pi_*Y\). ◻

Proof. The two-cell coaction. We first compute the coefficient comodule needed for duality. The dual \(DJ\) is the cofiber of \(\alpha':S^{q-1}\to S^0\), where \(\alpha'=\pm\alpha\) after fixing cell orientations, and its bottom inclusion is \(D\varepsilon\). Put \(\mathcal F=\operatorname{fib}(S^0\xrightarrow{\eta_{BP}}BP)\). A chosen \(BP\)-nullhomotopy of \(\alpha'\) gives a lift \(\widetilde\alpha:S^{q-1}\to\mathcal F\) and a map \(H:DJ\to BP\) extending the unit, in the following map of cofiber sequences: \[\begin{array}{ccccccc} S^{q-1}&\xrightarrow{\alpha'}&S^0&\longrightarrow&DJ& \longrightarrow&S^q\\[2pt] \big\downarrow\mathrlap{\,\scriptstyle\widetilde\alpha}&& \big\Vert&&\big\downarrow\mathrlap{\,\scriptstyle H}&& \big\downarrow\mathrlap{\,\scriptstyle\Sigma\widetilde\alpha}\\[2pt] \mathcal F&\longrightarrow&S^0&\xrightarrow{\eta_{BP}}&BP& \longrightarrow&\Sigma\mathcal F . \end{array}\] The \(BP\)-module \(BP\wedge DJ\) splits because \(BP_{q-1}=0\). Let \(e_0\) be its bottom generator and choose an upper generator \(z\in BP_qDJ\) whose top quotient is \(1\) and for which \(\rho_*z=0\), where \(\rho=\mu_{BP}(BP\wedge H):BP\wedge DJ\to BP\) retracts the bottom inclusion. With \(\epsilon:\Gamma\to R\) the counit, the element \(g=(BP\wedge H)_*z\) satisfies \[\epsilon(g)=0,\qquad \Gamma_q=\eta_L(R_q)\oplus\mathbb Z_{(p)}\{t_1\}, \qquad g=ct_1.\] Here \(\eta_L\) is induced by the bottom map \(BP\wedge\eta_{BP}:BP\to BP\wedge BP\). After smashing the cofiber diagram with \(BP\), its connecting map \(\partial:\Gamma_q\to\pi_{q-1}(BP\wedge\mathcal F)\) has kernel \(\eta_L(R_q)\) and gives \[\partial g=(BP\wedge\widetilde\alpha)_*(1).\] Thus \(g\) represents the first Adams–Novikov detector of \(\alpha'\). It is also the actual off-diagonal coaction: writing \(\psi(z)=1\otimes z+\omega\otimes e_0\), naturality under \(H\) gives \(\Delta g=1\otimes g+\omega\otimes1\). Apply \(\operatorname{id}\otimes\epsilon\) and use \(\epsilon(g)=0\) to obtain \(\omega=g\). By Lemma 29 this detector is \(\pm[t_1]\), so \(c\not\equiv0\pmod p\).

For completeness, fix the left-comodule convention \(\psi(am)=\eta_L(a)\psi(m)\) with tensor product using \(\eta_R\), so \(\eta_R(v_1)-\eta_L(v_1)=pt_1\). Replacing \(z\) by \(z+av_1e_0\) changes its off-diagonal term to \(g+a(\eta_L-\eta_R)(v_1)=g-pa\,t_1\). Changing \(H\) by the composite \(DJ\to S^q\xrightarrow{av_1}BP\) instead changes \(g\) by \(\eta_R(av_1)\); renormalizing the upper generator to \(z-av_1e_0\) subtracts \(\eta_L(av_1)\), giving \(g+pa\,t_1\). Opposite conventions reverse these signs, and dual cell orientations change the detector by a sign; none changes its nonzero residue modulo \(p\). The underlying \(BP\)-module splitting therefore need not split the comodule. Replacing \(z\) by \(c^{-1}z\) now makes the off-diagonal term exactly \(t_1\). Its top quotient is then \(c^{-1}\) rather than \(1\); we no longer need the earlier top normalization. This change fixes \(e_0\) and the bottom inclusion \(D\varepsilon\), which are the data used in the following symmetric-power calculation.

The symmetric coefficient module. Duality changes each permutation into its inverse, and hence preserves the average. Consequently \[DT\simeq\operatorname{Sym}^{n}(DJ).\] Under this identification \(D\varepsilon_T\) is the pure bottom tensor \(e_0^{\otimes n}\). All the cell degrees of \(DJ\) are even. If \(z_S\) is the tensor with upper generator in positions \(S\) and bottom generator elsewhere, define its normalized symmetric basis by \[ r_k=\binom nk^{-1}\sum_{|S|=k}z_S, \qquad 0\leq k\leq n. \tag{68}\] Every denominator is a \(p\)-local unit. Expanding the translation of each tensor gives the coefficient \[\frac{\binom{n-j}{k-j}\binom nj}{\binom nk}=\binom kj\] on \(r_jt_1^{k-j}\) in the coaction of \(r_k\). Thus we can denote this basis by \(1,r,\ldots,r^{p-1}\), with coaction (67). Each \(r^k\) has weight \(k\) because it is represented by tensors with \(k\) upper generators of degree \(q\).

The module \(BP_*DT\) is free. Smashing its free \(BP\)-module splitting with \(Y\) gives \[BP_*(DT\wedge Y)=BP_*DT\otimes_R(R/I_4),\] with the same translated basis and with \(D\varepsilon_T\) identified with constants. This argument uses a free factor, so it requires no flatness of \(R/I_4\). ◻

Proof. Finite duality identifies \[[\Sigma^dT,Y]\cong[S^d,DT\wedge Y]\] and carries \(f\) to \((D\varepsilon_T\wedge1_Y)(\eta\beta)\). By Lemma 32, its induced \(E_2\) class is represented by the constant-valued beta cocycle. We first show that this cocycle is a boundary in the translated coefficient module. Consider the integral polynomial \[ F(r,x)=\frac{(r+x)^p-r^p-x^p}{p} =\sum_{i=1}^{p-1}\frac{\binom pi}{p}r^{p-i}x^i. \tag{70}\] The displayed sum defines it before reduction modulo \(p\); it uses only the allowed coefficient powers \(r,\ldots,r^{p-1}\). The exact integral identity \[ F(r,x)+F(r+x,y)-F(r,x+y) =\frac{(x+y)^p-x^p-y^p}{p} \tag{71}\] follows by cancellation of the four \(p\)th powers. After reduction, the coefficient coaction (67) and primitivity of \(t_1\) make the left side the cobar differential of \(F(r,t_1)\), with the common overall sign determined by the cobar convention. The right side is the constant cocycle (63). Thus the bottom image of \(\beta\) has zero image on \(E_2\).

There is no possible higher-filtration homotopy value hidden by this boundary. The spectrum \(DT\wedge Y\) is connective and its coefficient weights are nonnegative. In stem \(pq-2\), its \(q\)-sparse terms have \[(s,w)=(2+qj,p+j).\] For \(j\geq1\) the inequality \(w\geq s\) would give \(p-2\geq(q-1)j\), which is impossible. Naturality places the image of \(\beta\) in filtration at least two, and the boundary just found raises it to filtration at least three. All later associated-graded groups are zero, so convergence with finite filtration makes the actual image zero.

The bottom beta image is therefore zero, and finite duality makes \(f\) itself nullhomotopic. ◻

Proof. Lemma 28 gives \(\alpha\gamma=0\) in precisely the stem \(q(W+1)-6\). Proposition 31 gives \(\alpha\beta^p=0\) after applying the unit to \(Y\). By (56), choose lifts \[\widetilde\gamma:S^{qW-5}\longrightarrow J\wedge Y, \qquad \widetilde b:S^{pd}\longrightarrow J\wedge Y\] of \(\gamma\) and \(\eta\beta^p\), respectively. Fix any parenthesization of the \(n\)-fold iteration of \(\mu\). Smash one copy of \(\widetilde\gamma\) with \(p-2\) copies of \(\widetilde b\), collect the \(J\) factors, and apply that chosen iteration to the \(Y\) factors. This gives \[L:S^{qW-5+(p-2)pd}\longrightarrow J^{\wedge n}\wedge Y.\] Its top projection is \(\beta^{p(p-2)}\gamma\). To verify this without an associativity assumption, for any sphere scalar \(a\) use the two unit homotopies to identify \[\mu(\eta a,x)=a x,\qquad \mu(x,\eta a)=(-1)^{|a||x|}a x.\] The scalar \(\beta^p\) is even. In the chosen binary tree, every subtree without \(\gamma\) is a scalar times the unit, and the unique subtree containing \(\gamma\) is the product of its scalars times \(\gamma\). This proves the asserted top value by induction on the tree.

Now project to the symmetric retract after those \(Y\) factors have already been multiplied: \[L_T=(\operatorname{pr}\wedge1_Y)L: S^{qW-5+(p-2)pd}\longrightarrow T\wedge Y.\] For every permutation \(\sigma\) of the \(J\) factors, naturality of the symmetry and the zero-dimensional top sphere give an equality of stable maps \[\varepsilon^{\wedge n}\sigma=\varepsilon^{\wedge n}.\] Consequently \[ (\varepsilon_T\wedge1_Y)L_T = (\varepsilon^{\wedge n}e\wedge1_Y)L = (\varepsilon^{\wedge n}\wedge1_Y)L =\beta^{p(p-2)}\gamma. \tag{73}\] No permutation of the multiplied \(Y\) factors occurs in this identity.

The actual null map \(f\) of Proposition 33 now gives the following homotopy-commutative diagram: \[ \begin{array}{ccc} \Sigma^d(T\wedge Y)&\xrightarrow{\ f\wedge1_Y\ }&Y\wedge Y\\[2pt] \mathllap{\scriptstyle\Sigma^d(\varepsilon_T\wedge1_Y)\,}\big\downarrow &&\big\downarrow\mathrlap{\,\scriptstyle\mu}\\[2pt] \Sigma^dY&\xrightarrow{\ \beta\wedge1_Y\ }&Y. \end{array} \tag{74}\] Commutativity is the left unit identity for \(\mu\) and the definition of \(f\). Its top arrow is null, so composing the diagram with \(\Sigma^dL_T\) and using (73) proves \(\beta^{p(p-2)+1}\gamma=0\). ◻

Proof. Let \(\theta\in E_2^{5,W}(Y)\). By Lemma 28, it is a permanent cycle represented by a homotopy class \(\gamma\). The sphere class \(\beta\) is also permanent. The pagewise product \(A^K\theta\) therefore supports no nonzero outgoing differential; if either factor has zero image on a page, the product is already zero. By Proposition 34 its filtered homotopy product is zero. Product compatibility and finite convergence then imply that \(A^K\theta\) is zero at \(E_\infty\).

If it is nonzero at \(E_2\), it must consequently be hit by an incoming differential. Its cohomological filtration is \[S=5+2K=2p^2-4p+7.\] An incoming differential of length \(r\) has source filtration \(S-r\geq0\), so \(r\leq S\). This bounds the page of disappearance, rather than merely its eventual abutment. The exact comparisons are \[ qp+1-S=2p-6>0,\qquad N-K=p-3>0. \tag{75}\] Numerically, \[K=1{,}016{,}064,\quad N=1{,}017{,}070,\quad S=2{,}032{,}133<qp+1=2{,}034{,}145.\] Thus \(A^K\theta\) is zero before the page of the prospective \(d_{qp+1}(v_4)\). Multiplication by the further power \(A^{N-K}\) is defined on every page by the sphere permanent cycle \(\beta\); it shows that \(A^N\theta\) is zero there as well. The surjection (54) accounts for every \(E_2\) class of the target, proving the assertion. ◻

Proof of Theorem 1. The only possible positive differentials from \(v_4\) have target \((s,w)=(qk+1,u_4+k)\). Proposition 18 rules out \(k=1\). Lemma 15 and Table 3 leave, for \(k\geq2\), only \(k=2,p-1,p\). Corollary 27 eliminates the first two target groups already at \(E_2\), and Proposition 35 makes the last target zero before its possible differential. No differential can hit filtration zero. Hence \(v_4\in E_2^{0,u_4}(Y)\) is permanent.

The edge homomorphism of the convergent Adams–Novikov spectral sequence supplies a map \(a:S^{qu_4}\to Y\) whose \(BP\)-Hurewicz image is exactly \(v_4\). Extend it by the existing binary multiplication to \[\widetilde v_4:\Sigma^{qu_4}Y \xrightarrow{\ a\wedge1_Y\ }Y\wedge Y \xrightarrow{\ \mu\ }Y,\] and take its cofiber \(X\). The right unit identity shows that the bottom generator is sent to \(v_4\). Since \(BP_*Y=R/I_4\) is cyclic as an \(R\)-module, this determines the entire induced map as multiplication by \(v_4\). That element is a nonzerodivisor in \(R/I_4=\mathbb F_p[v_4,v_5,\ldots]\). The homology exact sequence of the cofiber therefore gives the short exact sequence \[0\longrightarrow\Sigma^{qu_4}R/I_4 \xrightarrow{\ v_4\ }R/I_4 \longrightarrow BP_*X\longrightarrow0.\] The composite bottom map \(S^0\to Y\to X\) induces the quotient map \(R\to R/(I_4,v_4)\). It is a comodule map and is surjective, so it identifies the induced coaction with the canonical quotient coaction, and identifies its generator in degree zero.

Finally, \(X\) is the cofiber of a map between finite \(p\)-local spectra. It is finite \(p\)-local in the sense defined in the introduction; equivalently, the finite cell construction can be realized by a finite spectrum before localization by clearing prime-to-\(p\) denominators in its attaching maps, as in Lemma 12. More explicitly, the mapping cone adds to the cells of \(Y\) their translates by \(qu_4+1\). Together with Equation (18), this gives a finite \(p\)-local CW model for \(X\) with one cell in each dimension \[\sum_{i\in S}(qu_i+1)=\sum_{i\in S}(2p^i-1), \qquad S\subseteq\{0,1,2,3,4\}.\] In particular, this model has \(32\) cells. ◻

Proof of Proposition 6. We first identify the ordinary totalization tower exactly. Let \(\mathcal P_0([n])\) be the poset of nonempty subsets of \(\{0,\ldots,n\}\). The functor \[u_n:\mathcal P_0([n])\longrightarrow\Delta_{\leq n}, \qquad S\BeginAccSupp{method=hex,unicode,ActualText=27FC}\OriginalLongmapsto\EndAccSupp{}[|S|-1],\] sends an inclusion to the corresponding order-preserving injection. It is homotopy-initial. To prove this, for \(m\leq n\) identify \((u_n\downarrow[m])\) with the nonempty face poset of the simplicial complex \(L_{n,m}\) whose vertices are \((i,j)\), \(0\leq i\leq n\), \(0\leq j\leq m\), and whose simplices have strictly increasing first coordinates and weakly increasing second coordinates. Its nerve is the barycentric subdivision of \(L_{n,m}\).

The complex \(L_{n,m}\) is contractible for \(n\geq m\), by induction on \(n\). The case \(n=m=0\) is a point. For the inductive step start with the cone with vertex \((n,m)\) on the entire subcomplex \(L_{n-1,m}\). Adjoin the cones with vertices \((n,j)\), \(0\leq j<m\), on their bases \(L_{n-1,j}\). Two distinct vertices with first coordinate \(n\) cannot belong to a simplex. Each new cone therefore intersects the preceding union in exactly its base. That base is contractible by induction, since \(n-1\geq j\). Attachment along this simplicial subcomplex is a homotopy pushout and preserves contractibility. This proves the claim.

Homotopy-initiality, applied to the cosimplicial unit resolution, now gives, naturally in \(n\) and \(Z\), \[ K_n(Z)\simeq \operatorname*{holim}_{\varnothing\ne S\subseteq\{0,\ldots,n\}} BP^{\wedge|S|}\wedge Z. \tag{76}\] This is the standard cubical description of the partial Amitsur totalizations; compare Mathew et al. (2017, Propositions 2.11 and 2.14). Indeed derived partial totalization is the homotopy limit over \(\Delta_{\leq n}\): the restricted standard simplex is a Reedy-cofibrant resolution of the constant point diagram. Adjoining the empty vertex \(Z\) on the right of Equation (76) gives the \((n+1)\)-fold cube formed by smashing copies of the unit \(S_{(p)}\to BP\). If \(\mathcal F\) is its fiber, iterated fibers and exactness of smash identify its total fiber with \(\mathcal F^{\wedge(n+1)}\wedge Z\). Thus \[ \operatorname{fib}(Z\longrightarrow K_n(Z)) \simeq\mathcal F^{\wedge(n+1)}\wedge Z. \tag{77}\] The comparisons commute with restriction from \(n+1\) to \(n\): on the fibers this is the map induced by the last factor \(\mathcal F\to S_{(p)}\). There is no suspension in Equation (77).

The unit induces an isomorphism on \(\pi_0\), and \(\pi_1BP=0\), so \(\mathcal F\) has no homotopy in degrees at most zero. If \(Z\) is \(c\)-connective, \(\mathcal F^{\wedge s}\wedge Z\) is at least \((c+s)\)-connective. Equation (77) therefore identifies the coaugmentation \(Z\to K(Z)\) as an equivalence, and identifies the relative totalization tower as \[ G^s(Z):=\operatorname{fib}(K(Z)\longrightarrow K_{s-1}(Z)) \simeq\mathcal F^{\wedge s}\wedge Z \quad(s\geq0). \tag{78}\] Its homotopy groups vanish in each fixed stem for sufficiently large \(s\). The exact couple consequently has the asserted abutment, no inverse-limit ambiguity, and finite filtration in each stem.

For completeness, the successive layer is \[ \operatorname{cofib}(G^{s+1}(Z)\longrightarrow G^s(Z)) \simeq BP\wedge\mathcal F^{\wedge s}\wedge Z \simeq\Sigma^{-s}BP\wedge\overline{BP}^{\wedge s}\wedge Z, \tag{79}\] where \(\overline{BP}=\operatorname{cofib}(S_{(p)}\to BP)\). This is also \(\operatorname{fib}(K_s(Z)\to K_{s-1}(Z))\), with the usual normalization and the displayed \(s\) desuspensions.

The iterated extended terms compute the cobar complex. Indeed \(BP\wedge BP\), as a \(BP\)-module, is a wedge of suspended free \(BP\)-modules with basis the polynomial monomials in the \(t_i\). Iteration identifies their homotopy with the iterated tensors of \(\Gamma\) over \(R\). There are no derived tensor corrections, whether or not \(BP_*Z\) is flat over \(R\). Cofaces give the cobar differential; the codegeneracy splittings give its normalized form. The exact-couple grading gives the displayed differential shift. Finally the first map is \(Z\to BP\wedge Z\). Its image is exactly the permanent filtration-zero part by the finite-filtration convergence just proved. ◻

Proof of Proposition 7. We construct maps \[G^s(Z_1)\wedge G^t(Z_2)\longrightarrow G^{s+t}(Z_3)\] which induce the stated cobar pairing and recover the actual map \(f\).

The box pairing. We use Batanin’s cosimplicial box product in the form of McClure and Smith (2004, Definition 2.1, Proposition 2.3, and Remark 2.4); they attribute its Kan-extension description to Cordier and Porter. We construct the pairing before any fibrant replacement and then transport the entire pairing. For cosimplicial spectra define \(A\mathbin\square B\) as the left Kan extension of \((a,b)\BeginAccSupp{method=hex,unicode,ActualText=21A6}\OriginalMapsto\EndAccSupp{}A^a\wedge B^b\) along ordinal concatenation \[\Delta\times\Delta\longrightarrow\Delta, \qquad([a],[b])\BeginAccSupp{method=hex,unicode,ActualText=27FC}\OriginalLongmapsto\EndAccSupp{}[a+b+1].\] Equivalently, in degree \(n\) it is the zigzag pushout of the terms \(A^a\wedge B^b\) with \(a+b=n\), with neighboring terms identified by the maps \[A^a\wedge B^b\xrightarrow{d^{a+1}\wedge1} A^{a+1}\wedge B^b, \qquad A^a\wedge B^b\xrightarrow{1\wedge d^0} A^a\wedge B^{b+1}, \qquad a+b=n-1.\] The same definition applies to cosimplicial based spaces. For the original unit resolutions, concatenating the two \(BP\) blocks and applying \(f\) once gives maps \[T^a(Z_1)\wedge T^b(Z_2)\longrightarrow T^{a+b+1}(Z_3).\] They are natural in both ordinals: cofaces and codegeneracies within a block insert a unit or multiply adjacent factors within that block. Kan-extension adjunction therefore gives an actual cosimplicial map \[ m:T(Z_1)\mathbin\square T(Z_2)\longrightarrow T(Z_3). \tag{80}\] On the degree-\(n\) summand \(a+b=n\) in the pushout description, this map multiplies the last factor of the first \(BP\) block with the first factor of the second block. This is the single overlapping slot. The two descriptions agree by applying the codegeneracy that identifies the two junction vertices of \([a+b+1]\).

Deriving the pairing. We record the model-category check that permits us to derive this construction. The box product is a Quillen bifunctor for the Reedy model structures. Let \(h_a([n])=\Delta([a],[n])_+\), and let \(\partial h_a\) be the subdiagram of noninjective maps. These are the representables and boundaries defining the generating Reedy cells. Indeed, the boundary of the covariant representable consists of maps factoring through a nonidentity inverse map out of \([a]\), hence of noninjective maps by the unique surjection–injection factorization in \(\Delta\). This representable boundary is distinct from the latching subobject in target degree \(n\), which consists of maps factoring through a proper injection into \([n]\). Kan extension gives \(h_a\square h_b=h_{a+b+1}\). In the relative box boundary \[\frac{h_{a+b+1}} {\operatorname{im}(\partial h_a\square h_b) \cup\operatorname{im}(h_a\square\partial h_b)},\] the remaining maps are exactly the monotone maps injective on each of the two blocks. Such a map is either globally injective or identifies only the two junction vertices. A map that is not surjective onto its target lies in that target degree’s latching subobject. The only remaining surjections are the junction identification onto \([a+b]\) and the identity of \([a+b+1]\). Thus the junction-identifying and globally injective non-latching cells occur in degrees \(a+b\) and \(a+b+1\), respectively. Attach the junction cell first; every further collapse of it is already in the boundary, since it collapses two vertices within one block. Attach the globally injective cell second: its junction codegeneracy lands in the first cell, and its other codegeneracies land in the boundary. This gives a two-cell Reedy filtration of the relative box boundary. For two generating Reedy cofibrations, each of these attachments is tensored with the ordinary spectral pushout-product of their underlying generating cofibrations. The spectral pushout-product axiom therefore makes both attachments cofibrations, and makes them trivial cofibrations if either input is trivial. This proves the Reedy pushout-product axiom, including the acyclic case, for generating cofibrations. Closure under pushouts, composition, and retracts proves the assertion for all cofibrations.

Choose Reedy-cofibrant replacements \(A_i\to T(Z_i)\), \(i=1,2\), that are trivial fibrations, and trivial cofibrations \(A_i\to R_i\) with \(R_i\) Reedy fibrant. For the target choose a Reedy fibrant replacement \(T(Z_3)\to R_3\) directly. Equation (80) gives \(A_1\square A_2\to R_3\). By the preceding Quillen check, \(A_1\square A_2\to R_1\square R_2\) is a trivial Reedy cofibration. The lifting axiom extends the map to \[ m_R:R_1\square R_2\longrightarrow R_3. \tag{81}\] The space of such extensions is contractible up to the usual relative homotopy. This transports the original pairing; it places no condition on the literal \(BP\) slots of the fibrant replacements. All resulting tower and spectral-sequence pairings are transported along the equivalences with the original diagrams. In particular, when some \(Z_i\) coincide, their source and target replacements need not be identical.

The filtered diagonal. We use the simplex subdivision map given in McClure and Smith (2004, Lemma 3.6), credited to Grayson in their Remark 3.7. We record its formula to verify the relative filtration. For \(t=(t_0,\ldots,t_n)\in\Delta^n\), put \[I_i=\left[\sum_{j<i}t_j,\sum_{j\leq i}t_j\right],\qquad x_i=2\,\operatorname{length}(I_i\cap[0,1/2]),\qquad y_i=2t_i-x_i.\] For a cut index \(k\), \(x\) is supported in \(0,\ldots,k\) and \(y\) in \(k,\ldots,n\). The cut region is a copy of \(\Delta^k\times\Delta^{n-k}\), with inverse \(t=(x+y)/2\). Its boundary seams identify the last face of the first factor with the first face of the second factor, precisely as in the box pushout. Thus the cut defines \[\delta:\Delta^\bullet_+\longrightarrow \Delta^\bullet_+\square\Delta^\bullet_+.\] It is natural under all monotone maps: they aggregate consecutive intervals, and aggregation commutes with the cut. In particular this is a map of ordinary cosimplicial objects, including codegeneracies.

Define the relative simplex object \[P_s^\bullet=\Delta^\bullet_+/\operatorname{sk}_{s-1} \Delta^\bullet_+\qquad(s\geq0),\] where the skeleton for \(s=0\) is the basepoint. The simplex, its skeleta, and these quotients are Reedy cofibrant: their latching maps are the corresponding boundary inclusions or identities. Consequently \[ G^s(Z_i)=\operatorname{Map}_\Delta(P_s,R_i) \tag{82}\] is a derived relative end and the actual fiber of the fibration \(\operatorname{Tot}R_i\to\operatorname{Tot}_{s-1}R_i\). The degree-\(n\) box summands have dimensions \(k\) and \(n-k\). If \(n<s+t\), at least one is below its respective skeleton cutoff. Naturality applies the same observation to every face of a larger simplex, including faces with missing ambient vertices. Hence the cut descends to maps \[ \delta_{s,t}:P_{s+t}\longrightarrow P_s\square P_t. \tag{83}\] Enriched evaluation, followed by \(m_R\) and precomposition with \(\delta_{s,t}\), gives the actual relative pairing \[\begin{align*} \operatorname{Map}_\Delta(P_s,R_1)\wedge \operatorname{Map}_\Delta(P_t,R_2) &\longrightarrow \operatorname{Map}_\Delta(P_s\square P_t,R_1\square R_2)\\ &\longrightarrow \operatorname{Map}_\Delta(P_{s+t},R_3). \end{align*}\] It commutes with the tower inclusions in both inputs. By Equation (78), these are the exact relative fibers of the unit-resolution filtration, with filtration indices \(s,t,s+t\).

The cobar and homotopy pairings. Put \(L_s=\operatorname{cofib}(G^{s+1}\to G^s)\). The relative pairing induces \(L_s(Z_1)\wedge L_t(Z_2)\to L_{s+t}(Z_3)\): raising either input filtration by one raises the output filtration by one. The boundary formula for the product of relative cells is \(\partial(a\cdot b)=\partial a\cdot b+(-1)^{|a|}a\cdot\partial b\). It gives a pairing of the exact couples and the Leibniz rule on every derived page. On the layer of dimension \(s+t\), the only contributing cut cell has front and back dimensions \(s,t\). The inverse \(t=(x+y)/2\) gives its usual oriented front-face/back-face cup map, with coefficient one. The normalized cobar identification therefore gives exactly the asserted \(E_2\) cup pairing.

We finally verify the homotopy target without assuming that a replacement preserves units. For cofibrant \(Z_i\), the diagram \(\Delta^\bullet_+\wedge Z_i\) is Reedy cofibrant and maps levelwise equivalently to the constant diagram \(Z_i\). Its original all-unit map to \(T(Z_i)\) lifts to a map \(\alpha_i:\Delta^\bullet_+\wedge Z_i\to A_i\) along the trivial fibration \(A_i\to T(Z_i)\). The composite of \(\alpha_i\) with \(A_i\to R_i\) represents the coaugmentation. The box product of the \(\alpha_i\), precomposed with \(\delta\), lands in \(A_1\square A_2\). Since \(m_R\) extends the original map on that object, the composite is exactly the target coaugmentation composed with \(f\), transported through \(T(Z_3)\to R_3\). On totalizations the pairing therefore satisfies \[m_{\mathrm{Tot}}\circ(e_1\wedge e_2)=e_3\circ f.\] The coaugmentations \(e_i:Z_i\to K(Z_i)\) are the equivalences proved above. Thus the limit pairing is the actual homotopy map \(f\), and the \(E_\infty\) pairing is its associated-graded pairing for the filtration by the images of \(\pi_*G^s\to\pi_*K\).

For the final assertion, the coefficient pairing is \(R\)-bilinear and sends the two units to the unit. Every element of \(R/I\) is a scalar multiple of that unit. The pairing is consequently \((r\bmod I,r'\bmod I)\BeginAccSupp{method=hex,unicode,ActualText=21A6}\OriginalMapsto\EndAccSupp{}rr'\bmod I\). This uses neither flatness of the quotient nor associativity of the target map. ◻