I recently finished Amir Alexander's Proof!: How the World Became Geometrical, which I expected to be a social history of geometry, and instead I got a lot of only rather tangentially related French military history and exact descriptions of garden layouts. His central argument is that the hierarchical, timeless, and universal structures of Euclidean geometry provided the perfect spatial metaphor for European monarchs and governments seeking to set up unyielding and unchallengeable hierarchical political structures, and so they employed geometrical designs in art and architecture to express that sentiment.
Ironically for a book with that title, however, he seems rather light on primary sources. His chapters on the development of perspective in Italian Renaissance art and the symbolism incorporated into Italian architecture of the period (as well as the later chapter on Washington, DC) are well-supported by references to contemporary sources, but as soon as the discussion moves to fifteenth-century and later France, he does not appear to cite primary sources explicitly connecting the ideas of Euclidean geometry and hierarchies of political power until Louis XIV's reign. Rather, he seems to project motives onto the intervening French kings based on his own extensive aesthetic interpretation of the gardens they had built at their palaces. I'm not seeing anything contemporary that explicitly connects the geometrical aesthetic of gardens of the period to ideas about political organization in the minds of decision-makers.
Rather, the sources he does quote from link the geometrical aesthetic explicitly to imitations of nature. I see nothing to distinguish Alexander's interpretation of the meaning behind the spread of that style from the more quotidian interpretation that Charles IX saw a style of garden he liked and associated with the privileged culture of the Italians, copied that style at home for the cultural caché, and subsequent kings carried on with that style of garden and palace layout because it was (arbitrarily) popular; its subsequent spread would then be attributable to imitation of the French monarch by other crowned heads and nobles in Europe, no metaphors for political or social organization necessary. Are there more primary sources that back up Alexander's argument, or is he out on a limb?
There are a couple of architectural treatises dating from the middle of the sixteenth century that Alexander fails to identify and analyze in his book and that might fill the gap you identify.
The first is Sebastiano Serlio’s Tutte l'opere d'architettura et prospetiva, a series of eight books created between 1537 and 1575 that explore the creation of ideal forms, perspective, and architectural symmetry. This work features, among many other things, a plan of Poggio Reale and designs for garden labyrinths. It served as a textbook and pattern book for architects throughout Europe well into the seventeenth century.
The second is Le Premier tome de l’architecture by Philibert de L’Orme, published in 1567. De L’Orme was architect to Henri II, Charles IX, and Catherine de Medici. His design for the Tuileries Palace is discussed briefly in Alexander’s book.
Both de L’Orme and Serlio were closely involved in the development of the château and grounds at Fontainebleau, which, as Alexander notes, is an important link between the pleasure gardens of the Italian Renaissance and the axial geometry of Louis XIV’s Versailles.