# Echoes of the Future

### Lillian Patch - đťŚ´đťŤŹ

Take a deck of playing cards and spread them out before you; search for the kings and cast them out. Cast out the jacks and the tens, and lay the jokers aside, bearing in mind the red joker is katak. Decide if you want to keep your queens or kill them, if you'd prefer 36 cards or 40. Now shuffle the deck and place it within arms reach. The time is ripe for a game of Decadence.

Draw five cards from the deck face up; number them from 0 to 4. The zeroth card is placed at the base of the cross, what is called the Root Pylon or the Greater Depths. Moving up the cross, place the first card above the Root: this Pylon is the Centre, also called the Lesser Depths. A little higher up, on the left and right respectively, place down the fourth and second cards. These Left and Right Pylons are the Drifts: the Rising Drift on the right hand, the Falling Drift on the left. Finally, above these and between them, set down the third card in the Pylon of the Crown, in the utmost Twin Heavens. All together, this cross is the Atlantean Cross, the Pentazygon. The ritual is prepared.

Draw five cards again, but to begin with, keep this hand face-down. Reveal the cards in the hand one by one, dealing with each before passing to the next. You will have to seek a pair for these within the Pentazygon which is yet unmatched and adequate in sum. Did you keep your queens? They count for nothing, so treat them as if they were zeroes; their effect is to change the overall method of pairing. If you killed your queens, you are playing Decadence of the ordinary type, and so the pair you must seek for the card you've revealed is that with which it sums to ten. Fives pair together, and sixes pair with fours, sevens with threes, eights with twos, nines with ones, regardless of which hand either card is from. But if you kept your queens, the game you are playing is Subdecadence, and your pairs must sum to nine. Thus, your nines are paired with queens, your eights with ones, sevens with twos, sixes with threes, and fives with fours. Either way, if a Pylon forms a valid pair, place the newfound card upon it; it cannot be paired again. If there is no valid pair, remove the card from consideration. It is of no lasting import.

Having exhausted the face-down hand, consider the cross you've made. Sum together the positive differences between the created pairsâ€”for example, add nothing for five and five and add nine for nine and zero. Subtract from this the values of all the unpaired Pylons, from queens which make no difference to nines which reduce by nine. At a minimum, your score will be -44. If you killed your queens, it maxes out at +38; if you kept them, it may reach as high as +43. Is your score greater than zero? If so, then mark it down as contributing to the Aeon's sum. Your Aeon, and its task, is not yet completed, so reshuffle the deck and play again until a game ends in zero or a negative score. Once this happens, the Aeon is complete, and this instance of Decadence is over. If the Aeon had accrued a positive score before the final game, that score is an Angelic Index, and invokes an angel; either way, your final value is a Mesh-Number, and invokes a lemur.

đź™˘

Of Neo-Lemurian rituals, Decadence was the first uncovered and is the most well-known. Supposedly, it arose in its ordinary form with an organization called the AOE, or the Architectonic Order of the Eschaton. The AOE is alleged to be opposed to Lemuria, an "opposition" which mostly consists of trying to ignore it. However, Decadence of this this type has been widely appropriated and continues to be regularly practiced by Neo-Lemurians. Subdecadence is an additional, specifically Lemurian form of the Decadence rite; whether it was derived from regular Decadence or is actually its long-forgotten source is a matter of debate. However, whichever version of the game one playsâ€”"playing" being meant in a loose sense of the word, seeing as this is a game which can neither be influenced nor wonâ€”the Neo-Lemurian attitude toward it is distinct from that of the AOE.

According to Architectonic teachings, not only is a game of Decadence unwinnable, but it almost always ends in a loss. Any negative value is considered a lost Aeon, meaning that only Aeons which end with a game whose score is zero are considered not to be failuresâ€”and this is not very common. In light of this, AOE players hope that although most Aeons end after only one game, the given Aeon being played will stretch on indefinitely, accruing an unlimited score. This length can be seen as the production of an angel through continued modifications of the Aeon's Angelic Index, an angel which ideally is never quite complete and so is never invoked. The AOE aims for a project which is never quite done and so dislikes the lemurs for bringing finality. Only the zeroth lemur can really be tolerated, and this only because it can be seen as making no difference, as an end which might also be imagined as contributing to the Indexâ€”contributing nothing.

Being infinite in number, the qualities of any given angel are necessarily elusive. Neo-Lemurians are not uninterested in these qualities, but they tend to focus more on the attributes of the lemurs. Unlike the unfathomable hosts of angels who may manifest over the course of an Aeon, a lemur always emerges unexpectedly and without precedent, always emerging from the very last moment. As such, every lemur can be known and understood in relation to each other: while there is always infinite other angels to reach, the Mesh of Decadence has 45 lemurs which can be listed and contemplated. Consequently, of these are known a rich manifold of qualities, from which even the study of the angels can begin; every angel, after all, quickens through a call to a lemur.

Spending time with the lemurs is the task of all Neo-Lemurians, and that task can begin with Decadence. Although there is a great deal which is left to be uncovered, a surprising amount is implicit here. Just from the Mesh-tags derived from games of Decadence, a lemur's most important qualityâ€”its net-spanâ€”can be intuited. A lemurâ€™s net-span refers to a conjunction of two non-identical numerals, called zones. If you have ever played Subdecadence, these zones are already familiar to you: the inclusion of queens among the ranks of the Subdecadence deck means that all the zones, from 0 to 9, are accounted for and at play. 45, the number of the lemurs derived from 0 to -44, is a triangular number, indexing all possible pairs of ten elementsâ€”and thus, all possible net-spans. These progress in the following manner.

Beginning with the lowest net-span (1::0), one increments the lower zone of the two until incrementing it again would make it equal to the upper zone. At this point, one instead increments the upper zone and returns the lower zone to zero. The implication of this process is that, for every net-span ending in 0, the number of lemurs preceding it is equal to the (u-1)th triangular number, where u is the net-spanâ€™s upper zone. Triangular numbers refer to the sum of every integer from 0 to some limit n, with that sum being the nth triangular number; because one lemur starts with 1, two start with 2, three start with 3, and so on, their number before a specific upper-increment is always triangular. The mesh-tag of a lemur from 0 to 44 refers to how many increments have taken place to derive it. As such, it is possible to derive a mesh-tag from a net-span by adding the triangular number of the higher zone minus one to the value of the lower zone. Likewise, the process can be performed in reverse. It is not necessary to fully understand how to do so at this stage; the point is only to understand that the core quality of a lemur proceeds from the end of an Aeon.

Defined simply, a Neo-Lemurian is a person who seeks out the lemur of an Aeon, someone willing to play Decadence and lose. It is for this reason that the originators of Decadence can only be called â€śallegedlyâ€ť anti-Lemurian. Regardless of oneâ€™s reasons and motivations in playing the game, there is no mechanism by which to oppose a lemurâ€™s invocation. Lemuria can be procrastinated through the construction of its angels, but once they emerge, these angels are always the angels of Lemuria. Decadence invariably establishes the conditions of an encounter with Lemuria, revealing it in the lemurs and the angels, the pylons and the hands. In this sense, Decadence is the model for all numogrammaticism to come, which must always consist of encountering the Continentity through its immanent modes. The rest is all a question of how.

đź™˘

Decadence and Subdecadence are often misunderstood as two alternate systems, the subjects of a disagreement over the proper set of rules. In fact, they are two sides of a combined system, the shape of which is a spiral. We refer to this spiral by two principal names: named for its uncoverer, we call it the Barker Spiral, and named for its motion, the Diplozygotic Spiral. To draw it, one begins with the number line, writing the numerals from 0 to 9 at equal distances from each other. From there, a curve is drawn along the outermost Subdecadence (i.e. nine-sum) pair, passing from 0 to 9 in a semicircle; the spiral must begin or end with 0 because it lacks a valid pair in Decadence. Having reached 9, another semicircle must be drawn from 9 to its pair in Decadence, 1, ending up one numeral higher than where it began. The spiral proceeds onward in this manner, alternating between nine-sum and ten-sum pairs and so forming coils equidistant from each other. Finally, the spiral terminates once 4 passes to 5 by the rules of Subdecadence. As 5 pairs with itself in Decadence, there is consequently no further movement, a zero-width coil; to then carry on making nine-sum pairs would only reprise the already completed spiral from within rather than from without.

The term â€śdiplozygoticâ€ť derives from the double (â€śdiploâ€ť) system by which the numerals are paired (â€śzygoâ€ť). Were the pairs of Decadence and Subdecadence to be drawn on their own, they would each constitute a set of concentric circles of varying widths: 1, 3, 5, 7, and 9 units wide for Subdecadence; 0, 2, 4, 6, and 8 units wide for Decadence. Connected, with each set of circles appearing on one side of the spiral, they form a continuous line. The relation between Decadence and Subdecadence, which previously appeared as an exclusive disjunction, becomes evidently complementary in the Barker Spiral. By combining the system of even circles with that of odd circles, each of the ten numerals acquires a corresponding circle, a zone. Further, Decadence and Subdecadence each provide what the other lacks: there is no card for 0 in Decadence, and there is no difference of 0 in Subdecadence. For each numeral to refer to both a point in the spiral and to one of its diameters, both Decadence and Subdecadence must be present. What begins with Decadence must end with Subdecadence, and vice versa.

In the terms of more developed numogrammatics, the two sides of the Barker spiral are understood as two different kinds of operations on â€śpitch.â€ť Pitch is a concept referring to modular arithmeticâ€”that is to say, situations in which numbers, once they reach a given limit (or â€śradixâ€ť), wrap around to the lowest value again. For example, a standard 12-hour clockface has a radix, or base, of 12: if one attempts to reach the 13th hour, one is returned to the 1st hour again. The radix with which the Barker Spiral has affinity is 9, in which one can count from 0 to 8 as normal but, upon attempting to reach 9, one is returned to 0 once again. As a result, 9 and 0 are of equal value in this system; adding 9 necessarily passes through the whole cycle and returns to whatever value 9 was added to. Furthermore, the numerals of every other nine-sum pair are equal in their absolute value, i.e. their value if values are considered without their sign. Within modular arithmetic, a specific sum, when added, is equivalent to another sum when subtracted. For example, adding 8 to a number with a radix of 9 will always pass through almost the whole cycle of 9 integers to arrive at the exact place which subtracting 1 would reach (e.g. 4 + 8 = 12; 12 - 9 = 3). Necessarily, adding any given integer is equivalent to subtracting its radix-sum pair, such that +8 = -1, +7 = -2, +6 = -3, +5 = -4, and so on. This is the reason why 9 = 0 with a radix of 9: the nine-sum pair of 9 is 0, so 9 must be equal to -0, and -0 is equal to +0.

In numogrammatic terms, a zone has a â€śpitchâ€ť corresponding to the shortest route within the cycle, whether positive or negative, by which one can pass from any radix-9 number to its sum with that zone. For the zones from 0â€“4, this pitch is equal to its face value; for the zones from 5â€“9, it is equal to the negative of its nine-sum pair. Where values would normally be called positive or negative, pitches are instead called â€śAnaâ€ť or â€śCth,â€ť e.g. â€śAna-4â€ť or â€śCth-3â€ť; this is to emphasize that there is neither beginning nor end in this cycle, only greater and lesser escalation. Pitches which amount to 0 are called â€śPitch Null,â€ť being neither Ana nor Cth. The operations which the Barker Spiral performs on pitch are best understood by drawing the spiral from within, starting at 5, whose pitch is Cth-4. From 5, one must move on to its ten-sum pair: to 4, Ana-4. 4 then moves on to 6, Cth-3, and then to its nine-sum pair, 3, Ana-3. The pattern here is already fairly obvious. Either form of pairing switches the previous pitch between Ana and Cth, but while nine-sum pairs keep the pitch otherwise unchanged, ten-sum pairs decrement (or increment, depending on perspective) the pitch by 1. Because ten-sum pairs always follow nine-sum pairs regardless of where one starts, ten-sum pairing can be seen as removing (or adding) a coil of the spiral. The total number of completed coils encasing a zoneâ€™s point in the spiral corresponds to its pitch.

By means of nine-sum pairing, the Barker spiral can be seen as effecting a fold in the number line with which it is drawn. While the Decadence side of the spiral does not connect each of the ten numerals, missing 0 and arguably 5, the Subdecadence side does, and so it is possible to transpose each numeral onto these coils. If the zones were labelled in the middle of the semicircles rather than at their ends, one would see two parallel, opposite progressions: one Cth, counting up from 5 to 9; one Ana, counting down from 4 to 0, each on top of the other. It is as though the number line were rotated 90 degrees, then one side was bent back, snapping over itself. The result is a numeracy which, rather than progressing forward, radiates outward, losing pitch like heat and light dissolving into the void. Nullity is a horizon.