Sorry I didn't get to write a reply sooner, I have been busy and haven't been frequenting the forum very much lately.
First of all, Ambrose was cited in response to the claim in the OP that the "Latin Fathers of the Church has NEVER described Jesus carrying a patibulum"; Ambrose was certainly a Latin Father regardless of whether he was early or not. I also don't understand your reference to Christian art as pertinent to Ambrose's use of the term. Those equally late (or later) pictorial depictions portray as you say "Jesus carrying a complete cross, not the 'patibulum' [alone]", whereas Ambrose uses the (partitive) genitive patibulum crucis. So Ambrose (and others who used this expression, such as Apponius, Rufinus, and a Latin edition of Josephus) understood the patibulum as an object belonging to the crux (in the case of a possessive genitive) or as a portion of the crux itself (in the case of the partitive). This is not representative of an understanding of patibulum as a synonym for crux, nor is it reflective of the depiction in Christian art of Jesus carrying a composite cross to Golgotha.
Although Tertullian used the term antemna to refer to the crossbeam, he similarly understood that Jesus' crux was a composite (tota crux), with the upright beam being only a part of the crux (pars crucis), made complete (integrae crucis) only with the crossbeam (cum antemna). This pictures the antemna as a part of the crux as well, and one which was secondarily added to it, which accords well with the way Ambrose, Rufinus, and others referred to the patibulum as belonging to the crux or a part of the crux. In fact, Tertullian explicitly designates the antemna as a part of the crux, with crux realized as a partitive genitive (antemna crucis pars est in Adversus Marcionem 3.18), a parallel to the expression patibulum crucis in Ambrose (cf. also antemna crucis in Jerome, Epistula 14.18, Paulinus of Nola, Carmina 17.107, etc.).
There was also a usage of the word patibulum as a synonym for crux (and in Late Antiquity as a term for a "gallows" used in hanging by suffocation as in Isidore of Seville, Etymologiae 5.27.33-34, a usage that still survives in Romance languages like French patibule or Spanish patíbulo), but this fact does not imply that this was the only sense; the use of the term in Ambrose and elsewhere shows that it was not always synonymous with crux in the Latin Fathers. Rather this would be an instance of metonymy (specifically, synecdoche), with the word for a salient portion standing in for the object as a whole. This use may be an indication of how ubiquitous the patibulum was in crucifixion, if in patibulo suspendere could easily stand in for in crucem suspendere. There are several potential motives for a metonymic use of the word patibulum. One is euphemism. Cicero referred to crucifixion as "the most cruel and disgusting penalty" (In Verrem 2.5.165) and Josephus wrote that it was "the most wretched of deaths" (Bellum Judaicum 7.203) and the apostle Paul referred to Jesus' crucifixion as a "humiliation" (Philippians 2:8), a "curse" (Galatians 3:13), and perceived by the world as a folly and scandal (1 Corinthians 1:23, Galatians 6:14). It is well known that ancient writers wrote euphemistically about crucifixion (such as using the term "tree" as in the expression arbor infelix), and tended to avoid describing its gruesomeness (with a few exceptions). The term crux highlights the torturous nature of this form of execution (as it is derived from cruciare "to torture"), whereas patibulum was originally a word for rather ordinary domestic objects. Another likely reason for Christians was the desire to reserve the term crux to the unique cross of Jesus Christ and avoid using it with profane or derogatory reference. A rather clear instance of this can be found in Paulinus of Nola's account of the discovery of the "True Cross" (Epistulae 31.5). After excavating the site of the Crucifixion, "three crosses (tres cruces) were found together, as they had once stood together"; these were identified as the crosses of Jesus Christ and the two thieves executed with him. But "the thanksgiving for their discovery began to be compounded with troubled doubts. The devoted faithful were rightly afraid that they might perhaps choose the patibulum of a thief (patibulum latronis) in mistake for the Lord's cross (cruce domini)". Although the word crux was used of the thief's cross when referred collectively with the "True Cross", when the crosses were specified individually the two inferior crosses were designated patibula in contrast to the crux of the Lord. The story also does not construe the three crosses as obviously distinct in appearance, as it took a resurrection miracle to prove which of the three crosses was the Lord's (cf. also Zeno of Verona, Tractatus 1.2 for the use of patibulum to refer to the thieves' crosses).
That this usage is most likely a case of synecdoche can be seen not only in the fact that the patibulum was sometimes related to the crux as either a part or as something belonging to it (patibulum crucis), but from its usage in the earliest sources onward, which strongly emphasizes lateral and/or horizonal extension, as would be expected with a crossbeam, as well as from references to it being brought to the crux (as something separate from it). The term derives from the verb patere "spread out, open up" (cognate with Greek petannumi "spread out, spread wide"), cf. "they [the Helvitii] extended (patebant) in length 240 and in breadth 180 miles" (Julius Caesar, De Bello Gallico, 1.2), "as far as the earth extends (qua terra patet), the fierce Erinys reigns supreme" (Ovid, Metamorphoses 1.240-241). It originally referred to a kind of wooden bar (sera) that was used to shut folding doors (valvae) in Roman houses. These dual doors (affixed to door posts on both sides of the entrance) opened inward and they were secured by a wooden bar on the inside that fit into holes in both door posts (such holes are found in entrances in houses excavated at Pompeii), such that the bar extended beyond the length of the door through the door posts. It is unclear if the term patibulum (lit. "extender, opener") was used because it extended across and beyond the door frame or because its removal opened the door (cf. the use of patere in reference to the opening of doors, e.g. Ut populo reditus pateant ad bella profecto, total patet demta janua nostra sera, pace fores obdo in Ovid, Fasti 1.279-280). In order to open the door from the outside, a hole was carved into the door to allow the person to remove the bar by either sticking the hand through the hole (cf. Apuleius, Asinus Aureus 4.10, Petronius, Satyricon 94) or with a key. The patibulum was thus a horizontal crossbeam that had to be removed to open such doors: "The patibulum is a bar that closes the doors, by means of it doors open up out of the way (patibulum sera qua ostia obcluduntur, quod hac remota valvae pateant)" (Nonius Marcellus, De Compendiosa, 366). Compare also Petronius: "While we were speaking, the bar slipped and fell of its own accord and the door suddenly swung open and let in our visitor (sera sua sponte delapsa cecidit reclusaeque subito fores adminserunt intrantem)" (Satyricon 16). This bar upon its removal could be used as a weapon by the householder if the person at the door is a threat. Thus in Titinius (second century BC) a character standing at the backdoor (posticum) tries to drive off an intruder by saying he would "smash his head with this patibulum (patibulo hoc ei caput diffringam)" (Fullones 30). There was likely a direct connection between this household item and the punitive patibulum: Lying by the door, it would have been a handy object to punish slaves with. The punitive patibulum mentioned in Plautus, Miles Gloriosus 359, in fact, may have been just such an object. It was mentioned by Palaestrio to the slave Sceledrus just as he was standing in the doorway with his arms outstretched blocking access through the door. Plautus also may have alluded to the door bar in Rudens, 429. In that passage, Sceparnio opens his door to woman named Ampelisca holding a water jar seeking access to his well, and he makes sexual advances to her. She resists and shows him her jar to show her intent, and then he holds up some unstated object and says: "To a shrewd woman, this equipment too of mine (meus hie ornatus), would give indication of what I want". The equipment in question was possibly the door bar, held up as a phallic object. The term patibulum was also used to refer to a certain prop used in arboreal viticulture (Cato, De Agricultura 26, 68, Pliny, Naturalis Historia 17.212); it was not described but Pliny shows that it probably had a staff that supported it (impositis) and it was used to train vines (patibulis palmites circumvolvit) that project to the side (lateribus excurrant) from tree to tree. The patibula used here were thus likely "extenders" that supported vines growing laterally.
When we look at references to the punitive patibulum, it was frequently mentioned with references to a stretching out of the hands or arms. In your reply to my post, you say in reference to Seneca that it is not explicit whether this extension of the arms is horizontal or rather vertically on a pole. But the vocabulary used shows quite clearly that the extension is lateral, not vertical. (1) First we have the above-mentioned passage from Plautus, Miles Gloriosus, 359. The slave Sceledrus stands in the doorway trying to block access to the old man's room. Naturally, his arms would have been extended laterally to the two door posts. Palaestrio says to him: "I believe you'll have to walk out of the city gate in that pose very soon, hands spread out as you carry the patibulum (dispessis manibus patibulum quom habebis)". The situation itself makes clear that it is a side-to-side extension of arms, and the verb used here is dispandere "to spread out, extend, expand". (2) Aulus Gellius (Noctes Atticae 15.15.4) quotes this passage from Plautus and comments: "We say passis manibus of hands and velis passis of sails when they are stretched out and spread (diductis atque distentis)". This comment explains the expression used by Plautus with the verbs distendere "stretch in different directions, stretch apart" and diducere "draw apart, part, split, separate"; the image is one of hands being stretched apart in different directions. This again supports the understanding of patibulum as a crossbeam with the arms stretched in opposite directions, than a vertical raising of the arms together on a pole. (3) Next we have the two references in Seneca. First we read in Consolatio ad Marciam 20.3 that some who fashion "crosses" (cruces) "stretch out the arms on a patibulum" (alii brachia patibulo explicuerunt). This uses explicare "to spread out, unroll", which suggests that the arms (brachia) were spread out and not brought up together along a beam, while (4) Seneca elsewhere (in Fragmenta 124.6) makes reference to someone made "to stretch out his hands with a patibulum" (extendendae per patibulum manus), here using extendere "to stretch out, widen". (5) Lactantius (Epitome Divinarum Institutionum, 51) in reference to the Crucifixion mentions Jesus' "hands stretched out on a patibulum (extendit in patibulo manus), which reached out their wings west to east (in orientem occidentum que porrexit)". The first clause uses extendere with respect to the stretching of the hands on a patibulum, just as Seneca did, and the second clause makes clear that this is a lateral extension from side to side, both in the metaphor of birds' wings (which stretch out to the side when a bird is in flight) and the directionals of west and east (the verb in this clause is porrigere "reach out"). (6) Arnobius (Commentarii in Psalmos 137.33) also connects the patibulum to a stretching out of hands: "He stretched forth his hands upon the patibulum of the cross (extendit manus suas crucis patibulum)", here using the same verb extendere that Lactantius described as a lateral extension in the same situation, and here the patibulum is not the cross as a whole but a part thereof. (7) As a final example, Augustine of Hippo (Enarrationes in Psalmos, 87.9) wrote: "If in fact that he says [by the prophet], 'I stretched out my hands (extendi manus meas),' we understand the patibulum of the cross (crucis patibulum)". This again presents the patibulum as the part of the cross where the hands are outstretched, with the same verb used by Seneca, Lactantius, and Arnobius. These passages show that the word patibulum was closely associated with a stance of outstretched hands, which was rather clearly depicted as lateral in Plautus and Lactantius, and the use of distendere and diducere by Aulus shows it was a stance that had the hands stretched apart in different directions, as it would be if stretched laterally from side to side. Extendere also suggests the expansion of distance between the hands, and patibulum crucis in the references by Arnobius and Augustine indicate that the patibulum is not synonymous with the cross but the portion on which the hands are stretched out. To these examples may be added many others using the word crux instead of patibulum, which is what would be expected if the patibulum was a part of the crux or something added to it. These references also suggest a lateral extension of the arms: (8) Seneca (De Ira 1.2.2) elsewhere refers to "another to have his limbs stretched upon the crux (alium in cruce membra distendere)"; here he uses the same word distendere that Aulis Gellius used to illustrate the use of dispandere manibus in Plautus. The limbs are not brought together or near each other but are stretched apart in different directions. Since this passage describes a similar situation as that in the other two Seneca passages in (3) and (4), it further suggests the use of extendere there should be understood as referring to extension along a lateral beam than up together along a vertical beam. (9) Tertullian (Ad Nationes 1.12) stated: "If you position a man with his arms outstretched (manibus expansis), you have created an image of the cross (imaginem crucis). This text uses the verb expandere "to spread across, widen" and the context shows clearly that this pose is one with the arms outstretched laterally since this image of the cross includes a crossbeam or antemna. (10) Minucius Felix (Octavius 29.6) made a very similar comparison: "A crossbeam (iugum) set up forms the sign of the cross (crucis signum), and so too does a man with outstretched hands (homo porrectis manibus)". This passage uses porrigere "to reach out", the same verb used by Lactantius, and although it can be used more easily with upward extension than many of these other words discussed here (since it involves an act of reaching), it clearly refers to a reaching to the side since the pose is directly compared with the crossbeam of the cross. (11) Rufinus (Commentarius in Symbolum Apostolorum, 14): "These words, height and breadth and depth are a description of the cross (descriptio crucis).... by breadth the right hand and the left that extend outwards (latitudinem quoque illam quae distenta in dextram laevamque manus)". This image here is one of the two hands extended laterally, one on the right side and one on the left. The verb distendere, again, is the same one used by Seneca and Aulis Gellius, and it emphasizes that the hands are stretched apart in different directions. (12) Finally, we may quote Augustine of Hippo (De Doctrina Christiana 2.41.62): "Of the cross of the Lord (crucem domini), its breadth (latitudo) is signified by the transverse beam on which the hands are stretched out (in transverso ligno quo extenduntur manus)". This is another clear reference to the hands being stretched out laterally from one side to another, for it explicitly mentions the hands being stretched out upon a transverse beam (in transverso ligno). Taken together, the verbs extendere, distendere, dispandere, diducere, explicare, expandere, and porrigere collectively fit an expansion along a crossbeam much better than an upward extension on the upright. Porrigere, for example, may be used of upward extension, but the two instances here are clearly described as lateral extension. A further passage in Seneca (De Vita Beata 19.3) is noteworthy, tho it doesn't mention the hands or arms unambiguously: "They are stretched upon as many crosses as their own desires (quot cupiditatibus tot crucibus distrahuntur)"; the verb here is distrahere "draw in different directions, separate forcibly, divide apart", a meaning very similar to that of distendere. Since "stretching" in crucifixion was otherwise almost unanimously in reference to hands/arms/limbs, distrahere could possibly be added to the list as another word that more felicitously pertains to the stretching of the arms on a crossbeam than on the upright post. What is conspiciously missing in this group are words that more clearly suggest upward extension of the hands, such as erigere "reach up, make erect" (like porrigere a form of rigere "to reach"), surrigere "to raise, lift up", or elevare, allevare, levare "to lift up, raise". This may be illustrated in the use of such words in passages that do discuss the upright. So Rufinus (Commentarius in Symbolum Apostolorum, 14) wrote: "By height he meant the part which stretches above the earth and towers upwards (altitudunum vero illam quae super terram porrecta sublimis erigitur)". In addition to porrigere, Rufinus uses erigere to refer to the extension upwards of the upright pole. So in short, the word patibulum pertained both to a horizontal beam used with doors and a beam used in the lateral extension of the hands from side to side in the punishment of slaves and criminals, especially in the case of crucifixion.
Another matter questioned in your response is the connection between the carrying of the patibulum and crucifixion, whether these are two separate punishments such that the patibulum was carried by the prisoner on a separate occasion, or was not itself used in the execution. This is supported by the observation that the punishment of patibulum-bearing did not always result in capital punishment. This last point is something of a non sequitur since no one is claiming that patibulum-bearing always concluded with execution (nor is it claimed that crucifixion always involved the use of a crossbeam). In fact, my previous post made mention of the fact that the humiliation of slaves by compelling them to publically carry a beam of wood or a furca (probably originally a fork used with wagons, cf. also the separate terms used to refer to the bearer: furcifer vs. patibulatum) existed in pre-Republican times and antedated the Roman adoption of crucifixion as a form of capital punishment. As Dionysius of Halicarnassus described this practice: "A Roman citizen of no obscure station, having ordered one of his slaves to be put to death, ... directed them to drag him through the Forum and every other conspicuous part of the city as they whipped him ... having stretched out both his hands and fastened them to a piece of wood (tas kheiras apoteinantes amphoteras kai xulò prosdèsantes) which extended across his chest and shoulders as far as his wrists (para ta sterna te kai tous òmous kai mekhri tòn karpòn), they followed him tearing his naked body with whips" (Antiquitates Romanae, 7.69.1-2). Although the practice was later combined with crucifixion, it did not necessarily accompany it. But there is sufficient evidence to show that there was a close connection between patibulum-bearing and crucifixion. First of all, we see above that ancient writers mentioned the hands-outstretched-to-the-sides pose both in references to the carrying of the patibulum (as is the case in Plautus) and in references to the victim mounted or nailed to the cross (as is the case in Seneca, Lucian, Lactantius, Augustine, etc.); this is a continuity that is naturally explained by the addition of the patibulum to the cross with the victim attached to it. This is not merely hypothetical since we have already seen that Ambrose refers to Jesus carrying the "patibulum of the cross (patibulum crucis)" (De Abraham 1.8.72) as well as being nailed to the patibulum (patibulo adfixus) and suspended from it (in patibulo pendere) (Adversus Nationes 1.40, 62). That Ambrose pictured Jesus as carrying the crossbeam is also suggested by his statement that "because his neck was tender and not stiff (tener non dura cervice) but he knew the yoke of the law (iugum legis) he did not refuse the patibulum of the cross (crucis patibulum)" (De Abraham 1.5.40). The metaphor here pictures the patibulum as resting on Jesus' neck which is what would be case if it was supported laterally across his shoulders rather than over a single shoulder as depicted in late Christian art. Moreover, the expression iugum legis, while drawing on a biblical expression, evokes the use of iugum as a term for the crossbeam of the cross (see the citation above from Minucius Felix, Octavius 29.6). Ambrose uses the same metaphor in De Patriarchis, 2.9 when he makes reference to the necessity to "carry the yoke of Christ (iugum Christi portare) without a stiff neck (dura nesciens cervice)".
Furthermore, if the patibulum did come to stand in for the term crux as I have argued above, then it would have to have been used in the course of crucifixion. This would explain why the patibulum was so frequently mentioned in connection with crucifixion. In the case of Plautus, we have three explicit references to patibulum-bearing. So in the passage from Miles Gloriosus, 359 quoted above, the slave Sceledrus is threatened while blocking access to a room with outstretched arms: "I believe you'll be walking (eundum) out of the city gate (ex portam) in that pose very soon, hands spread out as you carry (habebis) the patibulum". Here the outstretched pose is one Sceledrus would have while walking and carrying the patibulum on the way to his execution. The phrase ex portam especially raises the spectre of exeuction because the ancient Roman custom was to conduct executions outside the Esquiline Gate (Tacitus, Annales 2.32), and it was where bodies would be left to be consumed by vultures and wolves (Horace, Epodes 5.97-102). Elsewhere Plautus used the phrase ex portam with similar effect: "It's true, he'll see you dead, burning outside the gate (videre ardentem te extra portam mortuam)" (Casina 2.6). Moreover, Sceledrus connects this threat to crucifixion when he responds after a second threat: "Stop threatening me, I know that the cross will be my tomb (scio crucem futuram mihi sepulcrum). It where all my ancestors met their ends: my father, my grandfather, my great-grandfather, my great-great-grandfather. My eyes cannot be torn out by any more threats of yours" (Miles Gloriosus, 372-375). Thus within the story, the character threatened with patibulum-bearing understood it as pertaining to crucifixion. And just before this scene, immediately before the entrance of the character who would threaten him with patibulum punishment, Sceledrus stated his fear that his master would "put me up on the cross (sustollat in crucem)" (line 309). So it is clear here that patibulum-bearing is connected with crucifixion. The second reference to patibulum-bearing in Plautus can be found in Mostellaria, where the jealous slave Grumio threatens his rival Tranio: "Oh sieve of the executioners (carnuficium cribrum), I believe they will pierce you with goads through the streets (per vias) with you attached to a patibulum (patibulatum), as soon as the old man returns" (Mostellaria, 55-57). The reference to executioners indicates that capital punishment is in view here. Then in line 352, Tranio announces the return of his master Theopropides (the old man referred to by Grumio) and he is sure that he is doomed to execution, just as predicted by Grumio (erus advenit peregre, periit Tranio). Then he offers money to anyone at the party willing to take his place: "I’ll give a talent to that man who shall be the first to run to the cross (in crucem excucurrerit) for me but on the condition that his arms and legs are double-nailed (offigantur bis pedes bis brachia)" (lines 359-360). The context thus relates Tranio’s expected carrying of a patibulum through the streets with Tranio’s expected hastening forwards (excurrere) to the crux where his arms and legs would be nailed. And then at that end of the play, Theopropides himself declares that Tranio would be crucified: "I’ll have you carried to the cross (ego ferare faxo in crucem), as you deserve" (line 1133). All of this shows that Grumio’s reference to patibulum-bearing pertains to crucifixion. The third reference is in the play Carbonaria (Fragmenta, 2): "Let him carry his patibulum through the city (patibulum ferat per urbem), and then be fastened to the cross (deinde adfigatur cruci)". This makes explicit what was implicit in the other two passages; patibulum-bearing for the punished slave ends with crucifixion. The historian Licinius (first century BC) also made a similar comment: "Bound to patibula they are led around (deligati ad patibulos circumferuntur) and they are fastened to the cross (et cruci defigntur)" (Historiae Romanae, 21). If these people are fastened to the cross while still bound to patibula, then this implies the addition of the patibula to the cross; there is no mention here of their removal. Since Plautus describes patibulum-bearing as involving a pose of hands spread out to the side, the addition of the patibulum to the cross would produce the same pose on the cross itself, which is precisely the kind of pose described in crucifixion on a stauros or crux by Seneca, Lucian, Tertullian, and others.
The Lex Puteoli Inscription (first century BC) is somewhat ambiguous because it is unclear whether it describes patibulum-bearing or workers bringing patibula to the execution site: "Whoever will want to exact punishment on a male slave or female slave at private expense, as he who wants the punishment to be inflected, he exacts the punishment in this manner: If he wants to bring the patibulum to the cross (in crucem patibul[um] agere), the contractor will have to provide wooden beams (asseres), chains, and cords for the floggers and the floggers themselves. And anyone who will want to exact punishment will have to give four sesterces for each of the workers who bring the patibulum (patibul[um] ferunt) and for the floggers and also for the executioner. Whenever a magistrate exacts punishment at public expense, so shall he decree; and whenever it will have been ordered to be ready to carry out the punishment, the contractor will have gratis to set up crosses (cruces statuere), and will have gratis to provide nails, pitch, wax, candles, and those things which are essential for such matters" (II.8-12). As you point out, this may simply be a matter of workers bringing the patibulum along with other materials to set up the execution apparatus, in which case it wouldn't refer to patibulum-bearing. Even if this is the case, this is still a matter of the patibulum being brought to the crux, which is itself set up (statuere) at the execution site, so it is clear here that patibulum is not synonymous with crux. But John Cook (NT, 2008) makes a pretty convincing case that the inscription refers to patibulum-bearing by the victim. The verb agere, which is loosely translated "bring" by Cook, has more of a sense of "impel, push", which is intelligible in the case of forced patibulum-bearing involving floggers (indeed, it is the usual word for referring to the driving of animals under a harness or yoke). The floggers may thus have been the workers who move or impel the patibulum to the cross by flogging the slave carrying it. Since patibul[um] is incomplete in the text, it is also possible that the word was patibulatum and the sense is "If he wants to impel the person attached to the patibulum to the cross". The phrase in crucem agere, in fact, occurs elsewhere, where it pertains to the person condemned to the cross: "You dared to lead someone off to the cross (in crucem tu agere ausus es)" (Cicero, In Verram 2.5.163), "He was led off to the cross (in crucem ageretur)" (Cicero, In Verram 2.5.165), "The student is led off to the cross (agitur paedagogus in crucem)" (Calpurnius Flaccus, Declamationum 23), "A prostitute leads off to the cross her slave who is in love with her (meretrix servum suum amantem se in crucem agit)" (Calpurnius Flaccus, Declamationum 33), etc. As for Seneca, he describes (in Consolatio ad Marciam 20.3) the spreading out of the arms on a patibulum as a pose that can be beheld in crosses (cruces), and as mentioned earlier, he also mentions this same pose when referring to the crux itself (De Ira 1.2.2). In your reply to my post, you make reference to Seneca's De Vita Beata, 19.3 as using the word crux interchangeably with stipes. If true this does not militate against viewing the patibulum as an object brought to the crux, for we have already seen that Plautus, Licinius, and the Lex Puteoli speak of the patibulum being brought to the cross, whether bound to a prisoner or not, and since crucifixion did not necessarily involve crossbeams either, the simple stipes was just as legitimately a crux as a cross with a crossbeam. But as argued above, I do not believe that Seneca conceives of a simple cross in this passage nor is necessarily using the word stipes interchangeably with crux. The stipes was mentioned in a reference to the compelling of prisoners to the crucifixion site: "When brought to punishment (ad supplicium acti) they suspend each individual on a stipes (stipitibus singulis pendent)". This uses a form of agere, the same verb used in Cicero, Calpurnias Flaccus, and in the Lex Puteoli. If Seneca was thinking of the compelling of a patibulatum (a person bound to a patibulum), as suggested by the use of patibulum later in the same passage, then indeed the patibulatum would have probably been taken to a stake (stipes) set up at the execution site (cf. Cicero, In Verram 2.5.66, 169 on the rather permanent installation of crosses at execution sites outside the city). At any rate the stipes is what the person was suspended on, whether bound to a patibulum or not. And since Seneca elsewhere used the term patibulum in connection with hand-stretching and since the given passage uses the word distrahere "to draw in different directions, divide apart" to describe the stretching out of the victim on the crux (reminiscent of the use of distendere in reference to the stretching apart of limbs on the crux in De Ira), I do think indeed that Seneca is envisioning a crux that has a crossbeam in place. Finally, it is not clear that Seneca had a crux simplex in mind in Epistula 101. If he did, it would not have been a reference to suspension on a stake by nailing the hands and feet but rather a literal impaling of the body internally on a pointed stake (which Seneca did call a crux in in Consolatio ad Marciam 20.3), for Maecenas' prayer and Seneca's comment on it concern the victim sitting (sedere) on the piercing cross (acutam crucem). But Justus Lipsius' interpretation of this passage as pertaining to internal impalement on a sharpened stake is not conclusive. It is equally feasible to interpret the passage as relating to the thorn-like sedile ("seat") on which the victim rested his or her weight. Justin Martyr (Dialogue, 91) described the crucified (hoi stauroumenoi) as riding atop a horn (keras) in the middle (en tò mesò), Irenaeus (Adversus Haereses 2.24.4) similarly referred to the victim as reposed (requiescat) on one of the five points of the cross (crucis summitates habet quinque), and Tertullian (Adverses Nationes 1.12) described the cross as having both a crossbeam (antemna) and a "projecting seat" (sediles excessu). A cross with such a resting block installed would appropriately be a "piercing cross". Maecenas also uses the verb suffigere in his reference to crucifixion (suffigas licet et acutam sessuro crucem subas, "You may nail me up and set my seat upon the piercing cross"), which could feasibly refer to impalement but which normally (along with adfigere and figere) has reference to nailing in crucfixion contexts. In his comment on Maecenas, Seneca describes this kind of execution as a "lingering death" (diu mori), where one would "waste away in pain (tabescere), dying limb by limb (perire membratim), letting out his life drop by drop, rather than expiring once for all" (Epistula 101.13-14). This too seems like a more appropriate description of crucifixion (which most agree involved a rather long, gradual death) instead of internal impalement (which probably brought death quickly). The reference to "dying limb by limb" is also indicative of crucifixion, as it involves the nailing of limbs, something not involved in internal impalement.
Another pertinent text is Firmicus Maternus' treatise on astrology Matheseos Libri Octo written in the early fourth century (when he was still a pagan). This work makes a number of references to crucifixion as an unlucky circumstance predicted by certain astrological conjunctions. He regularly uses the expression in crucem tollere "to put someone on a cross" as a term for crucifixion (6.31.59, 6.31.73, 8.6.11, 8.17.2, 8.25.6, etc., cf. Plautus, Miles Gloriosus 309, Livy, Ab Urbe Condita, 38.48.13, Cicero, In Verrem 5.168-170, Epistlae Ad Atticum 7.11.2, Quintilian, Institutio Oratoria 8.4.4, Calpurnias Flaccus, Declamationum Excerpta 50, etc.). One passage however also mentions the patibulum: "If however Saturn is found in conjunction with them, it means a deadly fate. For someone guilty of these sins is punished with a severe sentence; he is put on the cross attached to a patibulum (in patibulo suffixus crucem tollitur)" (6.31.58). Here the patibulum is something brought to the crux and the victim is put on the cross attached (or "nailed," as suffigere usually means with respect to crucifixion, cf. Lucius Apuleius, Asinus Aureus 6.31-32) to it.
Another fact mentioned by earlier scholars is that classical writers never used expressions like crucem portare, crucem ferre, or crucem habere, like they did with the term patibulum. This is unusual, if in fact crucifixion victims usually carried the whole cross. Instead, as we have seen, the crux was usually mentioned as the destination of the person compelled (agere), i.e. led off to the cross. These expressions instead occur only in Latin versions of the NT and Christian writers influenced by them. It is in Greek where we have a corresponding synecdoche that utilizes the general term for cross (stauros) to refer to what the prisoner carries: "They were duly brought out, chained together at foot and neck, each carrying his own cross (ton stauron ephere)....Chaereas said nothing as he was led off with the others, but Polycharmus yelled out upon taking up his cross (ton stauron bastazòn)" (Chariton, Chaereas and Callirhoe 4.2.6-7), "Every criminal who goes to execution must carry his own cross (ekpherei ton hautou stauron) on his back" (Plutarch, De Sera Numinus Vindicta 554A), "He who does not take his cross (lambanei ton stauron) and follow me is not worthy of me" (Matthew 10:38), "Jesus was led away and carrying the cross by himself (bastazòn hautò ton stauron) went out to what is called the Place of the Skull" (John 19:17), "For the cross (stauros) is like death and the man who is to be nailed carries it beforehand (proteron baztazei)" (Artemidorus Daldianus, Oneirocritica 2.56). This is a usage in Greek we do not find in the Latin of the time, whereas later Christian writers who used expressions like crucem ferre were influenced by biblical Greek. The synecdoche however is understandable since Greek did not borrow the Latin term patibulum as a loanword, so the word stauros was used to refer to the cross whether in whole or in part. Moreover we find references in Greek sources to the same outstretched stance mentioned in Latin re the patibulum, as well as to the composite cross (or descriptions that make better sense if they pertain to such a device). (1) So Epictetus (first century AD) writes: "When you have undressed in a bath and stretched yourself out like men who have been crucified (ekteinas seauton hòs hoi estauròmenoi), you may be massaged on this side and the other (enthen kai enthen), and the attendant may stand by and say, 'Take him away and wipe the table' " (Dissertationes 3.26.22). This statement makes a general comparison of the posture of athletes being massaged with the typical posture of those crucified. The description is suggestive of crucifixion with a crossbeam, with the arms stretched out hither and thither, i.e. out to one side and also out to the other. (2) Josephus in his account of the fall of Jerusalem wrote that "the soldiers out of rage and hatred amused themselves by nailing their prisoners in different postures (allon allòi skhèmati, or "from one form to another"), and so great was their number that space could not be found for the crosses nor crosses for the bodies (tois staurois kai stauroi tois sòmasin)" (De Bellum Judaicum 5.451-452). Here the victims are described as positioned on the stauros in a variety of styles or forms (skhèma) to suit the sadistic amusement of the soldiers. Since the body cannot be arranged in a wide variety of ways on a linear stake, this statement suggests a cross that allows the limbs to be nailed in different postures at different angles. (3) Lucian (Prometheus, 1.12, 19, 2.3-8) also describes Prometheus' arms nailed individually (first the right hand, then the left) to different rocks on both sides of a precipice, "Let him be crucified (anestauròsthò) over this precipice with his arms stretched across from crag to crag (ekpetastheis to kheire pros ton enantion)", a posture that "will make a cross (ho stauros genoito)". This is the same kind of lateral extension of the hands mentioned as early as Plautus with respect to the patibulum and this is clearly the configuration of the victim after he has been nailed in place. Elsewhere Lucian compared the shape of the stauros with the Greek letter Tau: "Men weep and bewail their lot, and curse Cadmus with many curses for putting Tau into their alphabet, for they say that their tyrants, taking his body as a model (somati phasi akolouthésantas) and imitating his shape (mimèsamenous autou to plasma) have fashioned similar looking timbers (skhèmati toioutòi xula) to crucify (anaskolopizein) men upon them, and the vile device is even named after him...The only just thing to do would be to punish Tau on what has been made in his own shape (tò skhèmati tò hautou)" (Lis Consonantium, 12). Here stauros (punningly linked to Tau by phonetic resemblence) is described as shaped like a T; not only does stauros have reference to crosses with crossbeams but the author takes this form as the device's most common, typical shape. (4) The author of Barnabas makes the same comparison, finding in the LXX of Genesis 14:14 a "representation of the cross" (tupon staurou) in the letter Tau, revealing "the cross in the Tau (ho stauros en tò T)" (9:7). Then later the author finds another "representation of the cross" (tupon stauros) in the LXX of Exodus 17:8-12: "Moses piled one shield upon another in the midst of the battle and standing high above them all he stretched out his hands (exeteinen tas kheiras), and so Israel was again victorious" (Barnabas 12:1-2). This clearly a lateral extension of the arms to the side, as "Aaron and Hur held up his hands, one on one side and one on the other (enteuthen eis kai enteuthen)" (Exodus 17:12). Furthermore the author quoted Isaiah 65:2 as another prophecy of the cross: "I stretched out my hands (exepetasa tas kheiras) the whole day to a disobedient people" (Barnabas 12:4-5). (5) Justin Martyr made these same exegetical comparisons as the author of Barnabas, but added several other comparisons that reinforce the idea that the typical shape of the cross was one with a crossbeam. First he wrote that "the lamb, when it is cooked, is roasted and fashioned into the form of the cross (skhèmatizomenon homoiòs tò skhèmati tou staurou). For one upright spit is transfixed (heis orthios obeliskos diaperonatai) right through from the lower parts up to the head, and one across the back (heis palin kata to metaphrenon), to which areattached the limbs (prosartòntai hai kheires) of the lamb" (Dialogue 40.3.3-7). The shape (skhèma, the same word used by Josephus and Lucian) of the stauros is here compared with the intersection of two wooden spits, one which goes upward from the lower parts to the head and the other which goes across the back where the "hands" (kheires) are attached. This clearly alludes to the crossbeam on which the hands are attached. Later he goes on: "No one can assert or prove that the horns of a rhinoceros represent any other matter or form(skhèmatos) than that of the cross (tupou hos ton stauron). For one of its beams stands upright (orthion to hen esti xulon), the highest extremity of which is raised like a horn when the other beam is fitted onto it (hotan to allo xulon prosarmosthè), and the ends appear on both sides (hekateròthen ta akra phainètai) like horns joined to the first horn (hòs kerata tò heni kerati parezeugmena)" (Dialogue 91.2). This is another depiction of the "form" (skhèma) of the stauros as consisting of two beams (the second being the allo xulon), one standing upright and the other fitted at the top. Also Justin Martyr (Apology 1.55.1-4) said that "this figure" (tou skhèmatos toutou) can be found in the trophy "which is called a sail" (ho kaleitai histion), "tools of this form" (mèdia tòn to skhèma touto ekhontòn) such as ploughshares, and the "human figure" (anthròpeion skhèma), when a man "stands erect" (orthon einai) and "stretches out his hands" (ektasin kheiròn). And finally (6) Artemidorus wrote that "the cross (ho stauros) like a ship is made from wood and nails and the ship's mast resembles a cross (hè katartios autou homoia esti staurò)", "if he is a criminal, he will be crucified in his height and in the extension of his hands (kheiròn ektasis)" (Oneirocritica 2.53, 76). This is another general statement about the form of the stauros, indicating that it typically resembled (homoios) the shape of the mast of a ship, which consisted of an upright pole and a transverse yardarm (antemna in Latin), and with the extension of the hands being distinguished from the upward extension of the cross. This again suggests that the use of the crossbeam was very common in Roman crucifixions, such that the average seafarer dreaming about being crucified would picture himself on a cross shaped like this.
And I think the depictions in post-Nicene Christian art hardly constitute historical evidence as strong as literary references to crucifixion practice. Such art is typically stylized and representation in compressed narrative (notice how the 5th century Maskell ivory illustrates Pilate washing his hands and the cock crowing in the same panel) is also influenced by concerns of economy. In order to visually tell the story of salvation, the cross of victory needs to be visually recognizable; I doubt an artist motivated to express the meaning of the cross would sacrifice narrative clarity for verisimilitude, and that's assuming that he or she was even aware of historical crucifixion practice (as crucifixion was abolished by Constantine and later generations were more often unfamiliar with it outside of tradition). The depiction in fact parallels depictions of the resurrected, victorious Christ holding aloft the cross of victory, whether in a generic scene or in heaven (cf. the idea that the cross was taken up to heaven). One example from Ravenna has Christ hold the whole cross over one of his shoulders; this is not the humbled Christ on the way to his execution but the trimphant Christ wearing military garb and crushing the Serpent under his feet. Christian iconography is generally very stylized and intended to convey symbolic meaning. The details in these depictions are thus historically quite inaccurate; for instance, it is unlikely that a crucifixion victim compelled to carry an entire composite cross would hold it over his shoulder with only one hand and have the other hand free, as can be seen in the Domitilla sarcophagus, the Maskell ivory panel, the Ravenna mosaics, etc.
I am sure that many of these points may be debated and contested individually but the cumulative weight of the evidence imo is much stronger than "fragile". Of course sources generally are laconic and most of the evidence is circumstantial and must be interpreted, and so one must exercise caution and restraint, but in terms of which conclusion is best supported by the available evidence, I do not think it is the one that claims that the patibulum was the entire composite cross, or that is was (exclusively) the upright, or that there is no evidence that the prisoner carried only the crossbeam, or that crucifixion involving crossbeams did not exist before the second or third century AD, or that stauros did not refer to crosses with crossbeams until the time of Constantine (as sometimes stated in the literature of the Watchtower Society).
