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The Saecular technology described in the novel spans a range from somewhat more advanced versions of recognizable 21st century Earth technology (smartphones, GPS devices, computers) to technologies which indicate significant advancement (page trees, fuel trees, vrone oak, library grapes, smartmatter devices such as tires which display photographs, photomnemonic tablets, etc). The novel implies that these are settled technology and that innovation has largely been replaced by simple imitation by Saecular. Isaac Asimov also uses this trope in Second Foundation where his Traders encounter populations on isolated worlds who can use existing technology but who lack the knowledge to repair or reconstruct it.
The state of travel technology appears to have stablized at a level akin to current Earth norms. Planes, trains, automobiles, missiles and intrasystem rockets exist and are, as noted, largely settled technology, though as noted above, innovation requires evocation ("Voco") of avout as Saeculars lack adequate theoretical knowledge to advance the state of the art. .
The most prominent example of the necessity for avout in the Saecular world is the Reticulum (a network which we are given to understand as akin to a 2010-level Internet). Newmatter (matter with unusual and useful physical properties, derived from avout practicum on cosmological constants) and products of genetic engineering also fall into this category.
A with much other technology computers have stabilized at about 2010 levels, with no mention made of machine sentience or highly interactive systems.
The constant bleeding-off of the brightest minds into the maths slows the pace of technological change. Successor Saecular societies reimplement technologies of earlier Saeculums, and due to abundant energy resourcing provided by avout, technology has stabilized at a late-20th-century Earth first-world level. This is not without its drawbacks, as we repeatedly hear narrators referring to "after a nuclear war" or "when the ozone level is more robust" as relatively common matters, not as significant aberrations.
Technology available to the avout in concents is limited. Each avout has three newmatter items (bolt, a bolt of cloth for clothing, chord, a newmatter rope used for adornment, and sphere, a newmatter sphere with variable physical properties). Newmatter is matter composed of atoms which conform to different physical constants (speed of light, etc) than those that naturally obtain in the Arbran cosmos.
Chalk, ink, rock, and wood make up much avout technology. Presumably there are smithies, physicians, and other necessary functions within the walls of a concent, but they use only pre-industrial technologies.
The avout are the beneficiaries of earlier avout efforts, and so have access to page trees which provide media for writing, the vrone oak and library grape for wine, and the tangle - a mix of plants which grow food without diminishing soil quality - to feed themselves.
Given Mathic proscriptions against computers and other Saecular devices, the avout are limited to cosmology - specifically, astronomy. The telescopes at the top of St. Edhar's mynster (tower) use newmatter lenses and guidestar lasers to focus and to ameliorate the effects of viewing the stars through an atmosphere, and the telescopes record their images not on film but on Photomnemonic Tablets, devices which are essentially slow motion video cameras.
There is a reputation-based internet equivalent called the Reticulum to which the avout are denied access.
Reticulum technology exists in a liminal space between avout and saecular, and consequently is maintained by an artisanal techologically focused order called the Ita. In general, the level of technical understanding among the avout and Ita far exceeds that of the Saecular world, although terms of the various Mathic reforms disallow avout instantiation of forbidden technologies.
There are technologies which appear late in the narrative and which are never explained. Some of the more advanced avout have developed the ability to alter events without tools by mentally accessing alternate reality paths and fixing a chosen path as the actual path of reality. These avout are the source of saecular legends of Incanters and Rhetors.