From Proposition 7.9, at least one of statements \((3)\) or \((5)\) in Proposition 7.8 is false. Consequently, statement \((2)\) is also false, which confirms that statement \((1)\) in Proposition 7.8 is true.

We only need to show that statement \((4)\) implies statement \((4')\).

According to [ ,

and is generated by \(\theta _5\) and classes of \(\mathrm{AF}=6,8,10\). Therefore, the indeterminacy of the classical \(\theta _5\), or the difference of any two choices of classical \(\theta _5\), lies in \(\mathrm{AF}\ge 6\).

As explained in Remark 7.5, \(\pi _{62,62+2}S^{0,0}\) contains no \(\lambda \)-torsion, and therefore, the indeterminacy of the synthetic \(\theta _5\) also belongs to \(\mathrm{AF}\ge 6\). Since every \(\theta _5\) has order 2, the indeterminacy of the synthetic \(\theta _5^2\) lies in \(\mathrm{AF}\ge 12\).

Therefore, if for some \(\theta _5\), \(\theta _5^2\) is detected by this specific element \(\lambda ^6 h_0^2x_{124,8}\), which is nonzero in \(\mathrm{AF}=10\), then for any \(\theta _5\), \(\theta _5^2\) is nonzero and detected by \(\lambda ^6 h_0^2x_{124,8}\).

We only need to show that statement \((5)\) implies statement \((5')\), which is sufficient to show that the indeterminacy of \([h_0^2x_{124,8}]\), or the difference between any two homotopy classes \([h_0^2x_{124,8}]\), when multiplied by \(\lambda ^3 \eta \), belongs to \(\mathrm{AF}\ge 15\), given that \([h_1h_4x_{109,12}]\) has \(\mathrm{AF}=14\).

Since \([h_0^2x_{124,8}]\) has \(\mathrm{AF}=10\), the indeterminacy is generated by cycles in \(\mathrm{AF}\ge 11\) of stem 124. We observe that classical cycles in \(\mathrm{AF}=11, 12\) of stem 124 are all killed by Adams \(d_2\) or \(d_3\)-differentials, and cycles in \(\mathrm{AF}=13\) are all annihilated by \(h_1\) in Ext. Therefore, we conclude that the indeterminacy, when multiplied by \(\lambda ^3 \eta \), belongs to \(\mathrm{AF}\ge 15\).

From the proof of Axiom 7.3 [ we know that

where \(\delta _1\) is the map \(S^{0,0}/\lambda \rightarrow S^{1,-1}\). Suppose that \(\eta \theta _5^2\) is detected by \(\lambda ^{n-5} T_n\) in \(\mathrm{AF}= n\) of the synthetic Adams spectral sequence for some \(T_n \in \mathrm{Ext}_A^{n, 125+n}\). This implies a differential in the \(\delta _1\)-ESS:

By Axioms 0.119, 0.120 and Corollary 4.7, this is equivalent to a synthetic Adams differential

and a classical Adams differential

This differential is nonzero if and only if \(T_n\) is nonzero on the classical \(E_{n-2}\)-page, i.e., not the image of a differential \(d_{\le n-3}\). In particular, when \(\lambda \eta \theta _5^2=0\), we conclude that \(h_6^2\) is a permanent cycle.

We first show that statements \((3)\), \((4')\) and \((5')\) together imply statement \((2)\).

By statements \((4')\) and \((5')\), we have that for any \(\theta _5\),

Recall from Axiom 7.6(2) that \(h_1h_4x_{109,12}\) is a permanent cycle, and can only be killed classically by

Therefore, statement \((3)\) implies that the expression \(\lambda ^{10} [h_1h_4x_{109,12}]\) is nonzero in synthetic homotopy, leading to a nonzero classical \(d_{12}\)-differential:

We will complete the proof of Proposition 7.8 by showing that if one the statements \((3)\), \((4)\) or \((5)\) is false, then statement \((2)\) is also false, and in this case,

making statement \((1)\) true. This conclusion is reached by estimating the Adams filtration of \(\theta _5^2\) and subsequently that of the expression \(\eta \theta _5^2\).

We begin by estimating the Adams filtration of \(\theta _5^2\) in \(\pi _{124,124+4} S^{0,0}\). First, we observe that this group \(\pi _{124,124+4} S^{0,0}\) does not contain any \(\lambda \)-torsion classes. This is because, in the 125-stem, the group (from Table 7 in the Appendix)

and thus, by the rigidity Axioms 0.119 and 0.120 for the synthetic Adams spectral sequence for \(S^{0,0}\), in the 124-stem, \(\lambda \)-torsion classes can only appear in \(\pi _{124,124+j} S^{0,0}\) for \(j \geq 7\).

Additionally, by analyzing classical Adams differentials, the Adams filtration of \(\theta _5^2\) is at least 10. In the case where it is of Adams filtration 10, it is detected by the element \(h_0^2 x_{124,8}\), which, according to Axiom 7.6(3), is a permanent cycle and cannot be killed.

Upon further inspection of the differentials associated with elements in stem 124 and filtration between 10 and 13, we are left with three possibilities:

1. \(\theta _5^2 = \lambda ^6 \cdot [h_0^2 x_{124,8}]\), which is statement (4), or

2. \(\theta _5^2 = \lambda ^9 \cdot [e_0 \Delta h_6 g]\), where \(e_0 \Delta h_6 g\) is permanent cycle in \(\mathrm{AF}=13\), or

3. \(\theta _5^2\) is a \(\lambda ^{10}\)-multiple.

Since in Ext, we have \(h_1 \cdot e_0 \Delta h_6 g = 0\), we deduce that in either possibilities (b) or (c), \(\eta \theta _5^2\) is a \(\lambda ^{10}\)-multiple. In other words, \(\eta \theta _5^2\) has \(\mathrm{AF}\ge 15\). In the group \(\pi _{125,125+5} S^{0,0}\), the only class that is a \(\lambda ^{10}\)-multiple is actually \(\lambda \)-free: By Axiom 7.6(4), it is \(\lambda ^{20} g^4 \Delta h_1 g\) in \(\mathrm{AF}=25\) and is detected by tmf (see [ for the Hurewicz image of tmf). Since \(\theta _5\) maps to zero in tmf, we have that \(\eta \theta _5^2\) maps to zero in tmf and therefore must be zero in this case.

This shows that if statement (4) is false, then \(\eta \theta _5^2 = 0\), and therefore \(h_6^2\) is a permanent cycle.

Hence, we focus on the remaining possibility (a), assuming statement (4) holds:

By the proof of Lemma 7.11, the only nonzero possibility for \(\lambda ^3 \eta [h_0^2x_{124,8}]\) is \(\lambda ^6 [h_1h_4x_{109,12}]\). Therefore, if statement (5) is false, then using the same tmf detection argument, we conclude that \(\eta \theta _5^2 = 0\), and consequently, \(h_6^2\) is a permanent cycle.

Finally, if statement (3) is false, an inspection reveals the only alternative classical differential:

This, combined with the tmf detection argument, would again imply \(\eta \theta _5^2 = 0\), and consequently, \(h_6^2\) is a permanent cycle. This completes the proof.

From Axiom 7.13(1), the element \(x_{123,9}+h_0x_{123,8}\) survives to the Adams \(E_{12}\)-page, and is not killed by any classical differential.

Let \(\alpha _1 = [x_{123,9}+h_0x_{123,8}] \in \pi _{123,123+9} S^{0,0}/\lambda ^{11}\) denote any homotopy class detected by \(x_{123,9}+h_0x_{123,8}\). For simplicity, we also use \(\alpha _1\) to denote its images in \(\pi _{123,123+9} S^{0,0}/\lambda ^r\) for \(1 \leq r \leq 10\), under the following sequence of maps:

From the nonzero \(d_2\)-differential in Axiom 7.13, we have

on the classical \(E_3\)-page. This implies synthetically, for \(2 \leq r \leq 11\),

lies in \(\mathrm{AF}\ge 11\) for any \([h_0^2 x_{124,8}]\).

By inspection, as in the proof of Lemma 7.11,

has \(\mathrm{AF}\ge 13\). The only possibility for it to have \(\mathrm{AF}=13\) is that it is detected by the element \(\lambda ^6 e_0 \Delta h_6g\). (Note that the class \([\lambda ^6 h_4 x_{109,12}]\) is irrelevant due to the nonzero differential \(d_3(\lambda ^6 h_4 x_{109,12}) = \lambda ^8 h_1x_{122,15,2}\).) Since in Ext we have

we may choose \(\alpha _2 \in \pi _{124,124+13} S^{0,0}/\lambda ^9\) to be any class detected by \(\lambda ^6 e_0 \Delta h_6g\), with the property that \(\eta \alpha _2\) is \(\lambda \)-divisible. Thus, there exists an \(\alpha _3\) such that \(\eta \alpha _2 = \lambda \alpha _3\).

This proves the required property \((1)\).

For the relation in property \((2)\), we will first prove it in \(\pi _{123,123+7} S^{0,0}/\lambda ^{11}\) and then map it to \(\pi _{123,123+7} S^{0,0}/\lambda ^9\).

By Proposition 0.122 for the synthetic Adams spectral sequence for \(S^{0,0}/\lambda ^{11}\), the expression

has \(\mathrm{AF}\ge 17\). In particular, the values

can be ruled out due to the nonzero Adams differentials

in the synthetic Adams spectral sequence for \(S^{0,0}/\lambda ^{11}\), which are zero in the spectral sequence for \(S^{0,0}/\lambda ^{9}\).

Therefore, the expression \(\lambda ^3 \cdot \alpha _1 \cdot [h_0]\) is \(\lambda ^{10}\)-divisible in \(\pi _{123,123+7} S^{0,0}/\lambda ^{11}\). Mapping it further to \(\pi _{123,123+7} S^{0,0}/\lambda ^9\), we conclude that it is zero.

We assume both statements \((3)\) and \((5')\) in Proposition 7.8 are true. From statement \((5')\), we have

Mapping this relation to \(S^{0,0}/\lambda ^9\), and applying the following relations from Lemma 7.14

we have

which, from statement \((3)\) and Remark 7.17, is nonzero and detected by \(\lambda ^6 h_1h_4x_{109,12}\) in \(\mathrm{AF}=14\).

On the other hand, since \(\eta ^2 = \langle [h_0], \eta , [h_0] \rangle \), we have

Therefore, the synthetic Toda bracket \(\langle \lambda ^3 \alpha _1, [h_0], \eta \rangle \) does not contain zero, and its \([h_0]\)-multiple is detected by \(\lambda ^6 h_1h_4x_{109,12}\) in \(\mathrm{AF}=14\). Since \(h_1h_4x_{109,12}\) is not \(h_0\)-divisible in Ext, the synthetic Toda bracket is detected by an element in \(\mathrm{AF}\le 12\).

This synthetic Toda bracket \(\langle \lambda ^3 \alpha _1, [h_0], \eta \rangle \) lies in \(\pi _{125,125+7} S^{0,0}/\lambda ^9\), whose \(\mathrm{AF}\le 12\) part is generated by

From Table 9 in the Appendix, we have

In both scenarios, \(\lambda ^2 [h_6(\Delta e_1 +C_0+h_0^6h_5^2)]\) remains and is in \(\mathrm{AF}=9\).

For the rest of the proof, we only need to rule out the cases \([h_0^2 x_{125,5}]\) in \(\mathrm{AF}=7\) and \(\lambda ^2 [h_6(\Delta e_1 +C_0+h_0^6h_5^2)]\) in \(\mathrm{AF}=9\).

For \([h_0^2 x_{125,5}]\) in \(\mathrm{AF}=7\), due to the nonzero \(d_3\)-differential

it can be chosen to be annihilated by \(\lambda ^2\).

However, from statement \((3)\) and Remark 7.17, \(\lambda ^8 [h_1h_4x_{109,12}]\) remains nonzero in the homotopy of \(S^{0,0}/\lambda ^9\). Therefore, the synthetic Toda bracket is not annihilated by \(\lambda ^2\). As discussed earlier, its \([h_0]\)-multiple is detected by \(\lambda ^6 h_1h_4x_{109,12}\), and thus, this case of \([h_0^2 x_{125,5}]\) can be ruled out.

For \(\lambda ^2 [h_6(\Delta e_1 +C_0+h_0^6h_5^2)]\) in \(\mathrm{AF}=9\), we first consider a classical Toda bracket in stem 125:

From [ , \(\pi _{62} \cong \mathbb {Z}/2 \oplus \mathbb {Z}/2 \oplus \mathbb {Z}/2 \oplus \mathbb {Z}/2\), so in particular both \(\theta _5\) and \([\Delta e_1 +C_0+h_0^6h_5^2]\) have order 2, and this classical Toda bracket is well defined.

From classical \(d_2\)-differentials:

we obtain the following Massey product on the \(E_3\)-page

and we check that it has zero indeterminacy. Further analysis shows that there are no crossing differentials, as per the criteria of Moss’s theorem [ (noting that crossing differentials in Moss’s theorem have a different meaning from our definition). Therefore, we have a classical Toda bracket

Synthetically, by inspection, we also have \(2\theta _5=0\) and \(2 [\Delta e_1 +C_0+h_0^6h_5^2] = 0\). It follows that there is a corresponding synthetic Toda bracket

Multiplying by \(\lambda ^2 [h_0]\), we get:

Note that all expressions in the above equation have zero indeterminacy.

If the synthetic Toda bracket \(\langle \lambda ^3 \alpha _1, [h_0], \eta \rangle \) were \(\lambda ^2 [h_6(\Delta e_1 +C_0+h_0^6h_5^2)]\), mapping the above equation to \(S^{0,0}/\lambda ^9\), we would then have a nonzero equation in \(\pi _{125,125+8} S^{0,0}/\lambda ^9\).

For this equation to be nonzero, we must have a nonzero equation

However, in Ext, we have

so this equation is not possible.

We have ruled out the possibility that the synthetic Toda bracket \(\langle \lambda ^3 \alpha _1, [h_0], \eta \rangle \) is detected by \(\lambda ^2 [h_6(\Delta e_1 +C_0+h_0^6h_5^2)]\) or \([h_0^2 x_{125,5}]\). Therefore, we conclude that it must be detected by \([\lambda ^4 h_0^2 x_{125,9,2}]\).

From the proof of Lemma 7.16, there exists a homotopy class \([\lambda ^4 h_0^2x_{125,9,2}]\) contained in the synthetic Toda bracket \( \langle \lambda ^3 \alpha _1, [h_0], \eta \rangle \), and we have

Since \(\lambda ^7 \alpha _3\) is in a strictly higher filtration than \(\lambda ^6 [h_1h_4x_{109,12}]\), we may choose another class \([h_1h_4x_{109,12}]\) such that the right-hand side of the above equation is simply \(\lambda ^6 [h_1h_4x_{109,12}]\). From the discussion of this synthetic Toda bracket in the proof of Lemma 7.16, we know that the difference of any two classes detected by \(\lambda ^4 h_0^2x_{125,9,2}\), when multiplied by \([h_0]\), is not detected by \(\lambda ^6 [h_1h_4x_{109,12}]\). Therefore, the corollary holds for any choice of \([\lambda ^4 h_0^2x_{125,9,2}]\).

From Axiom 7.19, we know that the element \(h_1x_{121,7}\) may only support a nonzero \(d_r\)-differential for \(r \ge 6\). By Proposition 4.5, \(\lambda ^4 h_1 x_{121,7}\) detects nonzero homotopy classes in \(\pi _{122,122+4}S^{0,0}/\lambda ^9\), and hence the required existence of such a homotopy class.

For the desired relation, we apply the Generalized Mahowald Trick Theorem 6.9. Consider the distinguished triangle

The short exact sequence on \(\mathrm{H}{\mathbb F}_2\)-homology

induces a long exact sequence on Ext-groups

Also recall for notations, if an element \(x\) in \({\mathrm{Ext}}_A^{*,*}(S^0/\nu )\) satisfies

we denote \(x\) by \(a[4]\); otherwise, due to exactness,

and in this case, we denote \(x\) by \(b[0]\).

We consider

For conditions in Theorem 6.9, we have:

1. \(d_0^{q, E_2}(h_1 x_{121,7} [4] + x_{126,8}[0] + x_{126,8,2}[0]) = h_1 x_{121,7}\).

2. \(d_3 (h_1 x_{121,7} [4] + x_{126,8}[0] + x_{126,8,2}[0]) = h_0^2 x_{125,9,2} [0]\). This is a classical Adams \(d_3\)-differential for \(S^0/\nu \), and is obtained from our computations.

3. 1. The differential in (1) has no crossing, as it is a \(d_0\)-differential, and

   2. Upon inspection, the differential in (2) has no crossing.

4. \(d_0^{i, E_2} (h_0^2 x_{125,9,2}) = h_0^2 x_{125,9,2}[0]\), as \(h_0^2 x_{125,9,2}\) is not divisible by \(h_2\) in Ext.

Since all conditions of Theorem 6.9 are satisfied, we conclude that there is an \(([h_2], E_4)\)-extension:

In other words, we have the following relation:

Using the map \(\rho :S^{0,0}/\lambda ^5 \rightarrow S^{0,0}/\lambda ^3\) from Notation 4.1, we lift the above relation and obtain:

In fact, since \(h_1 x_{121,7} \cdot h_2 = 0\) in \(\mathrm{AF}=9\) of Ext, we might obtain a relation of the form:

for some element \(x\) in \(\mathrm{AF}=10\). However, upon inspection, for all \(x \in \mathrm{Ext}_A^{10,125+10}\), the homotopy class \([\lambda x]\) either does not exist or can be chosen to be zero.

Using the map \(\lambda ^4: \Sigma ^{0,-4}S^{0,0}/\lambda ^5 \rightarrow S^{0,0}/\lambda ^9\) from Notation 4.1, we further push the above relation and obtain the following relation:

This completes the proof.

We assume that statement \((3)\) in Proposition 7.8 is true. For the sake of a contradiction, we also assume statement \((5')\) is true.

From Lemma 7.20 and Corollary 7.18, there exists a homotopy class
\([\lambda ^4 h_1 x_{121,7}] \in \pi _{122,122+4}S^{0,0}/\lambda ^9\), such that

for some \([h_1h_4x_{109,12}]\). Note that statement \((3)\) in Proposition 7.8, Axiom 7.6(2), and Proposition 4.5 imply that the expression \(\lambda ^8 [h_1h_4x_{109,12}]\) is nonzero in the homotopy of \(S^{0,0}/\lambda ^9\).

Since the element detecting \(h_1h_4x_{109,12}\) in not an \(h_2\)-multiple in Ext, we must have the expression

in \(\pi _{122,122+5} S^{0,0}/\lambda ^9\) be nonzero, and have \(\mathrm{AF}\le 12\). Upon inspection, the only possibilities are:

From Axiom 7.21 both \(h_6 M d_0\) and \(h_5x_{91,11}\) are nonzero permanent cycles in the classical Adams spectral sequence, so either possibility would lift to a relation in the homotopy groups of \(S^{0,0}\).

For the case of \(\lambda ^6 [h_6 M d_0]\), consider the expression

Since \(h_6 M d_0 \cdot h_2\) in \(\mathrm{AF}=12\) is killed by a classical \(d_2\)-differential, we know that

is in \(\mathrm{AF}\ge 13\). Upon inspection, the permanent cycles in \(\mathrm{Ext}_A^{13, 125+13}\) are all killed by \(d_2\) or \(d_4\)-differentials, and thus,

is in \(\mathrm{AF}\ge 14\). Therefore, for the above relation in the homotopy of \(S^{0,0}/\lambda ^9\) to hold, we must have

For the case of \(\lambda ^7 [h_5x_{91,11}]\), consider the expression

Since \(h_5x_{91,11} \cdot h_2\) in \(\mathrm{AF}=13\) is killed by a classical \(d_2\)-differential, we know that

is in \(\mathrm{AF}\ge 14\). Therefore, for the above relation in the homotopy of \(S^{0,0}/\lambda ^9\) to hold, we must have

In both cases, \(\lambda ^4 [h_1h_4x_{109,12}]\) is a \(\lambda [h_2]\)-multiple in the homotopy of \(S^{0,0}\). Since the classical \(\nu \in \pi _3\) has \(\mathrm{AF}=1\), we have

By the rigidity Axiom 0.119 of the synthetic Adams spectral sequence for \(S^{0,0}/(\lambda [h_2])\), we know that the element \(\lambda ^4 h_1h_4x_{109,12}[0]\) must be killed by a synthetic Adams differential, which corresponds to to a statement that in the classical Adams spectral sequence of \(S^0/\nu \), the element \(h_1h_4x_{109,12}[0]\) must be killed by a \(d_r\)-differential for \(r \leq 5\).

However, from Table 1 (obtained from [ [ , and can be visualized from [ ), in the classical Adams spectral sequence of \(S^0/\nu \), the element \(h_1h_4x_{109,12}[0]\) is not killed by any \(d_r\) for \(r \leq 5\) (from the range of \(9 \le s \le 14\) in stem 126). Therefore, we arrive at a contradiction.