The other three-layer at polity scale is not family / charter / executive. It is Montesquieu's.
De l'esprit des lois (1748), Book XI, chapter 6, "Of the Constitution of England." Nugent's translation: "In every government there are three sorts of power: the legislative; the executive in respect to things dependent on the law of nations; and the executive, in regard to things that depend on the civil law." He names the third the judiciary. The first enacts, amends, or abrogates laws. The second makes war and peace. The third punishes crimes and decides disputes.
His failure is fusion, not a missing continuity job. "There would be an end of every thing, were the same man, or the same body whether of the nobles or of the people, to exercise those three powers." The topology is from XI.4: "by the very disposition of things power should be a check to power." The people are the source of the legislative, not a layer that has to keep producing members. That three has no slot for the nuclear-envelope job.
Those two threes disagree on Congress, and they disagree on what a failure looks like.
Montesquieu's legislature is the body that makes and unmakes law. The essay's governance section puts "deliberative bodies" in the strategic layer and puts the judiciary with the charter. That is not a translation. It moves law-making out of the rule layer.
The discriminator is already in the US text. Article I lets a majority of both houses, with presentment, write the statutes other branches consult. Article V writes the Constitution: two-thirds of both houses, then three-fourths of the states (or the convention route). An ordinary statute rewrites a large part of "the rules" without the Article V channel. If the essay is right, that is fine only if those statutes are a cache, not the core. If Montesquieu is right, the legislature is doing its one job, and "core versus cache" is a different architecture sitting on top.
They also predict different repairs. Montesquieu: do not let the same office make, execute, and judge. The essay: protect the write path to the core. An independent commission that writes rules, enforces them, and hears its own cases fails Montesquieu even if Article V still holds. A parliament that can repeal its own constitutional statutes by ordinary majority fails the essay even if the offices are split.
He already draws the polity map that way: family and civil society as the reactive substrate, constitution and courts as the rule layer, executive and deliberative bodies as strategy. He already cites Ackerman's We the People on dualist higher lawmaking, and he already splits core / cache / overlay for messy write channels.
What's still open is the comparison with the three everyone already has. That map is not Montesquieu with different labels. It adds a continuity branch the classical three never had, and it files the legislature under strategy. Ackerman's dualism is about ordinary politics versus constitutional moments, which is closer to Article V than to three co-equal powers. The leftover is whether those branches are the same object or a competing partition. The essay treats extra offices as implementations of three jobs. Montesquieu's three are the jobs.