Causal Set Theory, the Growing Block, and the Buddhist Debate on the Three Times: What the Earliest Vibhajjavādin Records Say

DOI: https://doi.org/10.5281/zenodo.23020015

Canonical: https://thonly.org/research/the-vibhajjavadin-view-of-time · Licence: CC0 1.0

Abstract

In 2007 Rafael Sorkin, one of the founders of causal set theory, remarked in a footnote that the "accretive time" of his sequential-growth models "seems close to that of C.D. Broad, and also to that of the 'Vibhajavadin' school within the Buddhist philosophical tradition." Fay Dowker, quoting him (2022; the wording below is that of her 2023 revision), added an invitation: "A interesting project for a scholar of Buddhism interested in physics would be to try to determine the 'Vibhajavadin' view of time, perception and epistemology in detail by examining the earliest existing records."

This paper takes up that invitation from the Pāli sources, read against the northern ones. It finds that there is no single Vibhajjavādin view of time to report. The name has at least two referents:

The two referents meet in one school name. Vasumitra's doxography gives the Kāśyapīya exactly the Kośa's definition, and Stcherbatsky noted in 1923 that the Kośa's Vibhajyavādins had been identified with the Kāśyapīyas. The northern and Pāli sources report that school's past alike and its future differently.

So the tradition that uses the name today holds presentism about conditioned phenomena: of conditioned dhammas only the present exists, and nibbāna stands outside the three times altogether. Presentism is exactly the view Sorkin says his model is not. Block-universe arguments, he writes, "presuppose that the sole alternative to the 'block universe' is a doctrine that identifies reality with a three-dimensional instant", and his growth is offered as a third option. The Pāli Vibhajjavāda is the foil of causal-set growth, not its twin.

What the Theravāda offers a causal-set physicist is a contrast, and its use is that it locates the disagreement exactly:

This is a claim about influence, not about ordered existence.

A revision of 2026-09-30 adds a second axis: the beginning (§4.1–§4.3). SN 15 says that a first point of beings' wandering is not discerned. The Paṭṭhāna gives every mental state a proximity condition in the state before it and lists only matter as arising without one, so a stream of mind has no first member; the commentaries draw that step themselves, and Lean checks it from that premise. Bodies and world-regions begin. Classical sequential growth of causal sets is past-finite by construction. Bento, Dowker and Zalel (2022) find that growing into the past makes growth "(nearly if not entirely)" impossible to read as a physical becoming and that their one covariant alternating model is deterministic, yet build a consistent covariant framework and find encouragement in non-deterministic models. No causal-set dynamics in the literature read holds the Mahāvihāra's beginningless mind-streams under presentism, but that follows already from presentism (§4); on the beginning alone the gap is open, and not yet a finding.

The paper's findings are listed, each with its falsifier, under Findings disclosed; the physics comparison stays labelled as a comparison.

Keywords: causal set theory, sequential growth, past-infinite causal sets, growing block, presentism, eternalism, temporal becoming, beginningless saṃsāra, Buddhist philosophy of time, Kathāvatthu, Paṭṭhāna, Abhidharmakośa, Sarvāstivāda, Kāśyapīya, Vibhajjavāda, Theravāda, Lean, Sorkin, Dowker.

Findings disclosed

This paper is a study: it discloses readings of texts, not a mechanism. Its findings are listed here at the width that survived review and the census reported below, each with what would break it. The census's surviving width is a reading of the Kathāvatthu's time debates against Sorkin's and Dowker's causal-set remarks; findings 1–5 are parts of that reading and nothing wider. The historical identifications they rely on are prior art and are cited as such (Stcherbatsky 1923, p. 43 n.; Vasumitra; Cousins 2001). Findings 6 and 7 were added in the revision of 2026-09-30, after the census, which did not cover them; they assert no priority, since the derivation is the tradition's own and the physics is its authors' (§4.1–§4.2).

  1. Read against Sorkin's footnote, "Vibhajavadin" has two referents, and they meet in one school. The Kośa's Vibhajyavādin (present plus unripened past action, no future) and the Mahāvihāra's self-designation are different positions. The Kathāvatthu §299 refutes the first view's past half in its own words, under the commentary's name Kassapikā; Vasumitra gives the Kāśyapīya the Kośa's definition (§2.3, §5.1). Breaks if: the Kośa's Vibhajyavādin is shown to denote the Mahāvihāra specifically, or the §299 thesis is reliably attributed to a school other than the Kāśyapīya/Kassapika.
  2. No position in the Buddhist three-times debate is a growing block. The Kośa's and the Kassapika's past exists only until an action's fruit is given, and it retains nothing but unripened action; the resemblance to Sorkin's accretion is confined to keeping some past and denying the future (§4). Breaks if: a Buddhist source is found that keeps every arisen dhamma as existing past while denying the future; or Sorkin's growth is read so that born elements cease to exist.
  3. The Mahāvihāra's position, read against Sorkin, is presentism about conditioned dhammas: the very alternative he says block-universe arguments presuppose and his growth is meant to escape. Nibbāna and concepts are time-free (kālavimutta) and fall outside the comparison (§3, §5.3). Breaks if: a primary Theravāda source affirms that past or future conditioned dhammas exist, including unripened past action, as distinct from their conditioning force.
  4. The Theravāda's account of how the past acts is a contrast with causal-set growth, and it locates the disagreement at the past relatum. The ceased act conditions its result across moments; it does not exist when the result arises (§299); and nothing persists between them, since the persisting store of kamma (kammūpacaya) is the opponent's doctrine in the commentary (§6). Breaks if: a primary Theravāda source treats past kamma as existing when it conditions (for example, by accepting a kammūpacaya); or the Kathāvatthu's atthi ("exists") is shown to be an efficacy predicate, in which case the contrast with the Kośa reduces to a difference of wording (§10).
  5. On Dowker's "perception and epistemology", the Kathāvatthu holds that a past object is a genuine object although it does not exist (IX.6, §559; §6). Breaks if: §559 is read so that the past object is a present surrogate (a memory-image existing now) rather than the past itself.
  6. Read against causal-set growth, the Pāli texts supply a second axis, the beginning: a stream of mind has no first member, and so neither has beings' wandering, which a mind-stream carries; bodies and world-regions begin. SN 15.1 says only not discerned. The step to none rests on the Paṭṭhāna's proximity condition, which its negative method gives every mental state and withholds from matter. The commentaries draw the step themselves, DN 1 §67–68 records the opposite view in order to reject it, and Lean checks the step from that premise (kala/Time.lean). MN 63's undeclared world is, in the commentaries, beings' aggregates asked about as permanent or annihilated, and a beginningless succession of impermanent states affirms neither (§4.1). Breaks if: a primary Theravāda source affirms, as its own view, a mental state arising with no proximity condition (a first citta of a stream, an uncaused ignorance); or a gap in a stream of mind is found that the tradition does not bridge by that condition.
  7. On that axis, the one past-infinite growth study read offers no dynamics yet in which a past without beginning grows in the physical order of its births; and a world-cycle's contraction is neither a post nor a partial post or break. Bento, Dowker and Zalel's alternating growth gives up birth order as the physical order; its one covariant model is deterministic; their covariant framework has no dynamics chosen on it; and non-deterministic models encourage them. A contraction in the texts is local, as Dowker and Zalel's partial posts and breaks are, but the re-evolved world is peopled from streams outside the collapse, so its past is not closed (§4.2). Breaks if: a growth dynamics for past-infinite causal sets keeps birth order as the physical order, satisfies discrete general covariance and is not deterministic with respect to its observables (their models (18) meet only the last); or every stream in a re-evolved world is read as passing through the contraction.

Prior Art and Non-Assertion Statement

This document is dedicated to the public domain under CC0 1.0. The authors and HeartBank® will not seek patent on anything herein. Nothing here is a mechanism; what the paper discloses is the numbered list under Findings disclosed above, dedicated to the public domain so that no one may enclose it.

What is already public, and cited rather than claimed:

The census. A quick novelty census ran on 2026-09-27. Its pre-registration was pushed publicly before any query: thonly/publications commit 7348ba2, timestamps/census/physics-lens-prereg.md. Its aperture was the English web, arXiv, Google Scholar, PhilPapers, GitHub and Google Patents by keyword. Thai, Burmese, Sinhala, Chinese, Japanese and Khmer sources were NOT RUN, and books were not read. Not found in that aperture on that date: a reading of the Kathāvatthu's time debates against Sorkin's and Dowker's causal-set remarks. The census also reported the two-referent answer as not found; that row is narrowed here, because Stcherbatsky's footnote, read after the census, already gives most of it. A census result is a statement about a search, not a claim of priority. The comparison of Buddhism with physics is a crowded field, and much of it is unindexed. Findings 6 and 7 were added after this census and were not searched; they claim no priority (see Findings disclosed).

1 · The Invitation

Causal set theory replaces the spacetime continuum with a locally finite partial order: discrete events, related only by before. Sorkin's slogan is order + number = geometry. In its sequential-growth ("classical sequential growth") models the causal set is not given all at once; it grows, one element at a time, each new element born to the future of some of those already present. Sorkin argued that this growth gives an objective correlate of the passage of time, a becoming that relativity was thought to have ruled out. His 2007 paper, "Relativity theory does not imply that the future already exists: a counterexample," makes the case. In §5 he describes what the growth yields as "something 'four dimensional' from the very beginning, but which at any stage of its growth is still incomplete", and he adds the footnote:

The notion of "accretive time" that arises here seems close to that of C.D. Broad, and also to that of the "Vibhajavadin" school within the Buddhist philosophical tradition.

Fay Dowker, developing Sorkin's proposal as a response to the hard problem of consciousness (arXiv 2209.07653), quotes the footnote and adds her own. In the revision of 21 May 2023 it reads:

A interesting project for a scholar of Buddhism interested in physics would be to try to determine the "Vibhajavadin" view of time, perception and epistemology in detail by examining the earliest existing records.

(The first version, of 16 September 2022, reads "would be to determine the 'Vibhajyavadin' view", without try to.)

For the Vibhajavādin view her notes point to p. 43 of Stcherbatsky's The Central Conception of Buddhism (1923). Separately, they cite his Buddhist Logic (vol. 1, 1932, pp. 4–5) for early Buddhism as "no Matter, no Substance, only separate elements, momentary flashes of efficient energy … a flow of existential moments."

The question is well posed, and it is answerable, because the records exist, in Pāli, Sanskrit and Chinese, and they are unusually explicit about time. What follows reads them.

Spelling note. Sorkin and Dowker's 2023 revision write Vibhajavadin; Dowker's 2022 version writes Vibhajyavadin. They are quoted as written. Pāli sources give Vibhajjavāda, Sanskrit sources Vibhajyavāda. The spellings mark traditions, and this paper keeps them apart deliberately.

2 · Who Were the Vibhajjavādins?

2.1 · The Pāli self-designation

The Kathāvatthu itself, the fifth book of the Abhidhamma Piṭaka, never uses the word. The self-designation is found in its commentary (Kathāvatthu-aṭṭhakathā, part of the Pañcappakaraṇa-aṭṭhakathā traditionally ascribed to Buddhaghosa, c. 5th century CE, drawing on older material). The commentary describes the Third Council under Aśoka: the king, having expelled monks of other views, asks the remaining monks what the Buddha taught.

"kiṃvādī, bhante, sammāsambuddho"ti? "Vibhajjavādī, mahārājā"ti.

"Of what doctrine, venerable sir, was the Fully Awakened One?" — "A Vibhajjavādin, great king."

The elder Moggaliputtatissa then composes the Kathāvatthu to refute the views that had arisen. In the tradition's own account, therefore, the Kathāvatthu is the Vibhajjavādin record.

2.2 · A name for a group

Cousins (2001) shows that the word has two senses in the Pāli commentarial literature: an epithet of the Buddha (one who answers by analysis, vibhajja) and the name of a school. In the second sense it historically named the common ancestor of four schools: the Mahiṃsāsakas, the Dhammaguttikas (the commentary's form; Cousins's title writes Dhammaguttaka), the Kassapiyas and the Tambapaṇṇiyas, the last being the Mahāvihāra of Sri Lanka whose texts the Pāli canon preserves. The commentary's Third Council story, Cousins argues, draws its force from the double meaning. The present-day Theravāda is therefore one branch of the Vibhajjavādins, not the whole group. That matters below, because another branch reappears as an opponent.

The commentary's own family tree is different, and it is recorded here because it cuts against a tidy reading. Its schism narrative, taken from the Dīpavaṃsa, derives the Sabbatthivādins and the Dhammaguttikas from the Mahiṃsāsakas and then says: "Puna sabbatthivādakulato bhijjitvā kassapikā nāma jātā" ("again, breaking from the Sabbatthivāda family, those called Kassapikas arose"). The Pāli commentary thus files the Kassapikas under the Sabbatthivāda, where Cousins places them among the Vibhajjavādin branches. The findings below depend on which school held the §299 thesis, not on which family tree is right.

2.3 · The northern definition [cross-tradition]

The Sarvāstivāda's own systematization, as reported and criticized by Vasubandhu, defines the two camps sharply (Abhidharmakośabhāṣya ad V.25, Pradhan ed., p. 296, ll. 4–6):

ye hi sarvam astīti vadanti atītam anāgataṃ pratyutpannaṃ ca te sarvāstivādāḥ / ye tu kecid asti yat pratyutpannam adattaphalaṃ cātītaṃ karma kiñcin nāsti yad dattaphalam atītam anāgataṃ ceti vibhajya vadanti te vibhajyavādinaḥ /

Those who say that all exists, past, future and present, are Sarvāstivādins. Those who, distinguishing (vibhajya), say that something exists (the present, and past action that has not given its fruit) and something does not (past action that has given its fruit, and the future) are Vibhajyavādins.

The Kośa's Vibhajyavādin, then, holds present plus some past, and no future. The name is glossed from the method: they speak distinguishing.

Stcherbatsky's p. 43, the page Dowker cites, describes the same school: an intermediate position that "maintained the reality of the present facts and of that part of the past ones which had not already lost their influence, but the reality of the future ones and of that part of the past ones which had ceased to exhibit any influence it denied. Vasubandhu calls this school the Vibhajyavādins." Dowker's pointer and the Kośa therefore agree. Stcherbatsky's footnote goes further. It reports that the Kathāvatthu ascribes such opinions to the Kāśyapīyas, who also admitted the reality of that part of the future which was foreshadowed or fixed; that Xuanzang's commentary identifies the Kośa's Vibhajyavādins as the Kāśyapīyas; that the Theravādins seem to have shared the Sautrāntika view; and that Vasubandhu seems not to know the explanation of vibhajyavāda as orthodoxy, the sense the Kathāvatthu commentary gives it.

Vasumitra's doxography of the schools, in Xuanzang's Chinese, gives the Kāśyapīya (飲光部) the same thesis. Among its tenets: 若業果已熟則無。業果未熟則有 ("if the fruit of an action has ripened, it does not exist; if the fruit has not ripened, it exists"). It adds that no formation has a future thing as its cause (無諸行以未來為因) and that all formations perish momentarily (一切行皆剎那滅).

The Kośa's definition is a Sarvāstivādin doxographer's report of an opponent (§10). Two northern reports, the Kośa and Vasumitra, agree on it (both from a Sarvāstivāda milieu, so not fully independent), and the Pāli debate book states the same past-half in its own words (§3).

3 · The Earliest Pāli Record: Four Debates on Time

The Kathāvatthu is a book of debates. The Theravādin (sakavādī) cross-examines an opponent (paravādī) by paired questions until the opponent's thesis yields a consequence he must reject. Four early debates, all in its first chapter, concern time. The Kathāvatthu names no opponents; the attributions below are the commentary's. Paragraph numbers follow the Chaṭṭha Saṅgāyana edition, whose chapter division differs from the Pali Text Society's; the § numbers are unambiguous. An English translation with the attributions is Points of Controversy (1915).

CST §DebateOpponent's thesisCommentary's attribution
282Sabbamatthītikathāsabbam atthi: all exists"seyyathāpi etarahi sabbamatthivādānaṃ" (the all-exists school, i.e. the Sarvāstivāda)
297Atītakkhandhādikathāpast and future things are aggregates, bases and elements, and therefore existthe paravādin who "wants past and future to exist because they do not give up being aggregates"
299–300Ekaccaṃ atthītikathāsome of the past exists and some of the future (what is bound to arise). The restriction to action whose result has not ripened is in the Kathāvatthu's own text: "atītā avipakkavipākā dhammā – te atthi""seyyathāpi kassapikā" (the Kassapikas; one of Cousins's four Vibhajjavādin branches, though the commentary's schism list derives them from the Sabbatthivāda, §2.2)
304Hevatthikathāthings exist in one way and not in another: the past exists as past"seyyathāpi etarahi … andhakānaṃ" (the Andhakas)

The refutations of all exists and some exists turn on one formula:

Atītaṃ atthīti? Āmantā. Nanu atītaṃ niruddhaṃ vigataṃ vipariṇataṃ atthaṅgataṃ abbhatthaṅgatanti? (§283)

"Does the past exist?" — "Yes." — "But has the past not ceased, departed, changed, set, gone down?"

The other two debates are refuted differently.

At §299 the Kassapika position is broken apart. If some of the past exists, then some of the past has ceased and some has not, and the part that has not ceased is not past. The same dilemma is run for the future at §300.

The position that remains, the Mahāvihāra's own, is presentism about conditioned dhammas: of conditioned phenomena only present ones exist; past and future are what was and what will be. The Abhidhamma adds a class that the three times do not cover. The root verse of Abhidhammatthasaṅgaha ch. 8 §37 reads "tekālikā dhammā, kālamuttā ca", things of the three times and things free of time, and Sumaṅgala's sub-commentary names the time-free class: nibbāna and concepts (nibbānapaññattivasena kālavimuttā ca). Nibbāna is not a present dhamma; it is outside the question this debate asks.

4 · The Physicists' Map and the Buddhist Map

  WHAT EXISTS?              past          present        future        nearest physics picture
  ───────────────────────   ───────────   ────────────   ───────────   ─────────────────────────────
  Sarvāstivāda              all           all            all           eternalism: a moving spotlight,
                                                                       not a block (Miller 2017) †
  Kośa's Vibhajyavāda [X]   unripened     all            none          a past that THINS; resembles
                            karma only                                 Sorkin's only in keeping some
                                                                       past and denying the future
  Kassapika (Kv §299–300)   unripened     all            what is       (no clean analogue: a past
                            karma only                   bound to      that thins, a future that
                                                         arise         is partly settled)
  Andhaka (Kv §304)         "as past"     all            "as future"   a tensed-property reading
  Mahāvihāra (Kv, refuting  none          all            none          PRESENTISM about conditioned
  all four)                                                            dhammas ◀── what Sorkin says
                                                                       his model is NOT ‡
  ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─   ─ ─ ─ ─ ─ ─   ─ ─ ─ ─ ─ ─ ─  ─ ─ ─ ─ ─ ─   ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─
  Sorkin's growth           every born    no single      none          growing block (accretion)
  (for comparison)          element       global "now"

  No Buddhist row keeps every born element.
  † Miller (2017) classes the Sarvāstivādins as eternalists and says explicitly that they are not
    block-universe theorists; their view looks like a moving spotlight. Cited, not claimed.
  ‡ Of conditioned dhammas. Nibbāna and concepts are time-free (kālavimutta), outside all rows.
  [X] cross-tradition (northern) source.

Sorkin's growth is not presentism. In the sequential-growth models, what has been born stays: the causal set at any stage contains its whole past and lacks only its future. Sorkin's own words set the view against a present-only doctrine. Most arguments for the block universe, he writes, "presuppose that the sole alternative to the 'block universe' is a doctrine that identifies reality with a three-dimensional instant". He also denies that the process has any single "now" spreading over it. Wüthrich and Callender (2017) examine what kind of becoming this is. They conclude that a novel, global becoming is compatible with causal sets, distinct from the localized becoming that Sorkin's advocates claim for it. Nothing in that literature reads the model as presentist.

On the Buddhist map, then, no row is a growing block. The nearest is the Kośa's Vibhajyavādin, and only in two respects: it keeps some past, and it denies the future. Its past thins. An action ceases to exist once its fruit is given, and no other past dhamma is kept at all, where Sorkin's causal set keeps every element ever born. Sorkin's footnote points at a real resemblance in the school the northern sources describe, narrower than the word "close" suggests. It does not describe the school that uses the name today.

4.1 · A second axis: the beginning

The map above has one axis: what exists. The Pāli texts and causal-set growth also differ on a second, which Sorkin's footnote did not raise: whether the order of what has happened has a first member. In the texts that order belongs to each stream of mind; there is no global one. This section and the next two were added in revision (2026-09-30).

What the texts say. The Anamatagga-saṃyutta (SN 15) opens:

Anamataggoyaṃ, bhikkhave, saṃsāro. Pubbā koṭi na paññāyati avijjānīvaraṇānaṃ sattānaṃ taṇhāsaṃyojanānaṃ sandhāvataṃ saṃsarataṃ. (SN 15.1; CST s0302m.mul.xml:3489, mūla)

"Without discoverable beginning, bhikkhus, is this wandering on. A first point is not discerned of beings hindered by ignorance and fettered by craving, running on and wandering."

The verb is epistemic. The commentary renders it "the prior limit is not seen" (purimamariyādā na dissati), glosses anamatagga as "with prior and later ends undelimited" (aparicchinnapubbāparakoṭiko), and reads the wandering as "the unbroken succession of the aggregates" (khandhādīnaṃ avicchinnappavattā paṭipāṭi; s0302a.att.xml:2489, aṭṭhakathā). That reading is the commentary's; the Kathāvatthu quotes SN 15.1 only as the Puggalavādin's proof-text for a transmigrating person (§75, abh03m3.mul.xml:849). SN 15.5's "not easy to count" concerns the years in one aeon, SN 15.7–15.8's the aeons past (s0302m.mul.xml:3529, :3553, :3565; the sibling paper, §5.3). The commentary ascribes the difficulty to the wandering's length (dīghatamattā) only as the questioning monk's reported thought (kira, s0302a.att.xml:2521).

What the texts leave undeclared. MN 63 leaves undeclared whether the world is eternal or not, finite or infinite ("sassato loko … anantavā loko", s0202m.mul.xml:1397, mūla). There, the commentaries say, the world is the world of beings (sattaloko; vin01a.att.xml:2169, s0516a.att.xml:881), and the Atthasālinī gives the predicates: the five aggregates held "permanent, stable, for all time" (nicco dhuvo sabbakāliko) or "cut off, perishing" (ucchijjati vinassati), and finite or infinite by the edge of a meditation device (kasiṇaparicchedantena), in space (abh01a.att.xml:9149). So the line runs by predicate, not by subject. A beginningless succession of impermanent states, any stream of which ends at an arahant's death, affirms neither permanence nor annihilation and says nothing of extent (an interpretation; §10). The predicate, not an unseen end, also separates it from DN 1's eternalist, whose self and world stand "like a peak" (kūṭaṭṭho, s0101m.mul.xml:283): the DN sub-commentary files even the fortuitous-origination view (below) under eternalism for seeing no end ahead (anāgate koṭiadassanato, s0101t.tik.xml:2061, ṭīkā).

From "not discerned" to "none", for mind. The Paṭṭhāna makes each earlier state a proximity condition of the next (abh03m7.mul.xml:85–:93, mūla), and its general statement confines the condition to mind:

Yesaṃ yesaṃ dhammānaṃ anantarā ye ye dhammā uppajjanti cittacetasikā dhammā, te te dhammā tesaṃ tesaṃ dhammānaṃ anantarapaccayena paccayo. (abh03m7.mul.xml:97)

"Whatever states, mind and mental factors, arise immediately after whatever states, those states are a condition by proximity for them."

Its negative method lists what arises not by proximity, and the rows it gives are all matter: mind-born, kamma-born at rebirth, the great elements, nutriment- and temperature-born, and the mindless beings' (:1201, referring to :1153–:1169; later chapters abbreviate or refer back). No mental state is listed there, not even a rebirth mind. That is the premise, a proximity predecessor for every mental state, which a first member of a stream of mind would lack. Conditionality in general would not suffice: fifteen of the twenty-four conditions are present-only (§6), and a mental state whose conditions were all co-nascent would be conditioned and still first. Matter differs. A body begins at each rebirth, and a world-region's matter begins again after a fire that burns until not an atom's worth of formation remains (Visuddhimagga XIII §405–406, e0102n.mul.xml:769–:773), that contracting world being the receptacle world (bhājanaloka, s0101t.tik.xml:1945). The claim is of streams of mind, and of beings' wandering because a mind-stream carries it; bodies and world-regions begin.

The step is the tradition's own. The commentary on MN 9 derives it from earlier ignorance conditioning later ignorance: "the beginninglessness of saṃsāra is established" (saṃsārassa anamataggatā sādhitā hoti, s0201a.att.xml:4309, aṭṭhakathā). The Visuddhimagga denies that ignorance is an uncaused root ("Na akāraṇaṃ", XVII §584, e0102n.mul.xml:3553), citing AN 10.61: "a first point of ignorance is not discerned … and yet it is discerned: ignorance has this as its condition" (idappaccayā avijjā, s0404m3.mul.xml:1890, mūla). The Atthasālinī turns not discerned into none: no boundary exists, after so many hundreds of thousands of aeons or in some king's or buddha's time, before which beings were not ("ayaṃ paricchedo natthi", abh01a.att.xml:409). The aeons behind beings therefore exceed every count, not only every easy one. This paper reports the step; it does not claim it.

The view that holds the other option. DN 1 records a stream with a first member, to reject it. A being falls from the mindless deities (asaññasattā), recollects the arising of perception and nothing before it, and concludes "Ahañhi pubbe nāhosiṃ, somhi etarahi ahutvā santatāya pariṇato", "I was not before; now, not having been, I have come to be" (§67–68, s0101m.mul.xml:483–:487, mūla). The DN sub-commentary answers with the proximity condition: a mind long ceased "is merely a proximity condition, not the seed; the seed is kamma" (anantarapaccayamattaṃ hotiyeva, na bījaṃ, bījaṃ pana kammaṃ, s0101t.tik.xml:2057, ṭīkā). The same bridge spans emergence from cessation (abh03m7.mul.xml:5673) and death to rebirth across thousands of aeons (na kāladūratā, Kv anuṭīkā, abh05t.nrf.xml:2409). DN 27's re-evolving world starts a world, not a stream: its first beings fall from the Ābhassara heaven (s0103m.mul.xml:1105).

The step, checked. From the premise the step is short, and Lean 4 checks it (kala/Time.lean, theorem beginningless, in the CC0 repository SiliconWat/formal-abhidhamma):

  Conditioned s   :=  ∀ m, ∃ p, succ p = some m        every moment succeeds a moment of the same stream
  tick_succ       :   succ m = some n  →  tick n = tick m + 1
  ──────────────────────────────────────────────────────────────────────────────────────────
  beginningless   :   Conditioned s  →  ¬ ∃ first, ∀ m, tick first ≤ tick m

Conditioned is the proximity premise for one stream of mind, not conditionality in general, and the content is the premise: replace it by True and the theorem fails (one of the file's deliberate breaks), while a model with a last moment and no first satisfies every axiom. For a presentist, "no first member" is a claim about what was (§3): a stream's has-beens have no first term; no infinite collection of past states exists now.

4.2 · Where causal-set growth stands on the beginning

Page numbers are those of Bento, Dowker and Zalel (arXiv:2109.10749v1; published 2022), read in full, unless another paper is named.

Finite-past growth has a beginning "by definition of the model" (p. 3): in classical sequential growth no element is born to the past of an existing one, and "there are no past-infinite causets at all in the sample space" (p. 9). Read a stream's moments as a chain, as the sibling paper's relation S does (§3.3 there): a chain with no first member gives each member an infinite past, which no causal set in that sample space has.

Bounces, forward from a beginning. Transitive percolation (TP) almost surely grows infinitely many posts, elements related to every other (p. 15), each "a kind of collapse of the universe to zero diameter followed by a subsequent re-expansion" (Martín et al. 2001, p. 3); "TP realises the heuristic of a bouncing universe with a beginning" (p. 15). Collapse need not be global. Dowker and Zalel (2017, pp. 3–4) model a black hole birthing a universe beside an unrelated region by a partial post (an element y such that if y ≺ x and y is unrelated to z, then x is unrelated to z) or a partial break (A, B) (every element of A before every element of B, B a future set, A ∪ B a past set), and cosmic renormalisation follows. Dowker and Zalel (2023) model cycles by breaks as well as posts. All of this is past-finite.

Growth for past-infinite causal sets. Bento, Dowker and Zalel ask whether growth is compatible with a past without beginning; their process still starts "from the empty set" (p. 9), where the texts want no first stage. Four results bear on the texts.

  1. Growth into the past gives up becoming. Elements must be born to the past of existing ones, "thus making it (nearly if not entirely) impossible to conceive of the growth process as a physical process of Becoming" (pp. 3–4). The alternation of forward and backward births is fixed "for definiteness" (p. 10); random variants (p. 12, n. 8) were not surveyed.
  2. The one covariant alternating model is deterministic. Among alternating counterparts of the classical models, only alternating transitive percolation (ATP) satisfies discrete general covariance (Claim 3.5, p. 16). It realises "a bouncing universe with no beginning" (p. 15), but every proposed observable (a convex-event) has probability 0 or 1 in it (p. 24): "ATP is not a useful cosmological model" (p. 25).
  3. Other alternating models are not deterministic (Claim 3.9, p. 25, the couplings of their equation (18)), and "the existence of non-deterministic alternating growth models encourages us to continue" (p. 26). By Claim 3.5 they are not covariant.
  4. A consistent covariant framework exists, with no dynamics chosen. Z-covtree, motivated by "asynchronous becoming" (p. 28), is "a consistent covariant framework for two-way growth"; in it "the event that the completed order contains a post is not measurable" (p. 43).

They weigh both sides (p. 42): the difficulties with alternating growth and the covariance result "may be considered as evidence against the existence of physically meaningful dynamical growth models for two-way infinite causal sets"; "On the positive side", non-deterministic models suggest "that there may be models in which the convex-events may yet form a rich and interesting class of observables." A later overview still asks whether growth should produce "only those which are past-finite" (Zalel 2023, p. 2).

A world-cycle's contraction is not a post, nor a partial one. The texts' contraction is bounded: fire burns below the Ābhassara heaven, water below Subhakiṇha, wind below Vehapphala, and "one buddha-field is always what is destroyed" (sadāpi ekaṃ buddhakkhettaṃ vinassati, AN-a on AN 4.156, s0402a.att.xml:5988), 10¹² world-systems destroyed and re-formed together (Visuddhimagga XIII §404, e0102n.mul.xml:729–:733). Locality is therefore no contrast; Dowker and Zalel's partial posts and breaks are local too. The contrast is in who peoples the new world: beings "fall from the Ābhassara body and come to this state" (ābhassarakāyā cavitvā itthattaṃ āgacchanti, DN 27, s0103m.mul.xml:1105), the Ābhassara lying outside the fire's reach, and the DN sub-commentary adds beings from "world-systems other than the contracting ones" (saṃvaṭṭamānalokadhātūhi aññalokadhātūsu, s0101t.tik.xml:1945). Read as a causal set, with each stream a chain and the destroyed field related to nothing outside it, the new world's past runs through moments outside the collapse. It is not closed, as a partial break requires, and those moments precede the new world without being related to the collapse, as a partial post forbids. A contraction in the texts is neither a post nor a partial post or break.

4.3 · Two axes together

                         THE BEGINNING: does the order of what has happened have a first member?
                         (for a causal set, its past; in the texts, each stream of mind, no global one)
                         ┌─ a first member ─────────────────────┬─ no first member ──────────────────────┐
  WHAT EXISTS            │                                      │                                        │
  every born element     │ Classical sequential growth          │ Alternating growth (Bento, Dowker &    │
  is kept                │ (Rideout–Sorkin): grows from the     │ Zalel): elements born to the PAST of   │
  (growing block)        │ empty set; order of birth =          │ existing ones; becoming given up       │
                         │ physical order                       │ "(nearly if not entirely)"             │
                         │ TP: posts = bounces WITH a           │ ATP, the one covariant model: bounces  │
                         │ beginning                            │ with NO beginning; deterministic       │
                         │ Partial posts and breaks:            │ Models (18): not deterministic, not    │
                         │ local bounces (Dowker & Zalel)       │ covariant                              │
                         │                                      │ Z-covtree: consistent, covariant; no   │
                         │                                      │ dynamics chosen; a post not measurable │
                         ├──────────────────────────────────────┼────────────────────────────────────────┤
  only the present       │ (a mind-stream with a first moment:  │ MAHĀVIHĀRA: presentism about           │
  (presentism about      │  the fortuitous-origination view,    │ conditioned dhammas; each mental state │
  conditioned dhammas)   │  DN 1 §67–68, recorded to be         │ arises after its proximity condition;  │
                         │  rejected; Vism XVII §584:           │ no stream of mind has a first member   │
                         │  ignorance is not an uncaused root)  │ (SN 15 · Paṭṭhāna · AN 10.61); bodies  │
                         │                                      │ and world-regions begin                │
                         │                                      │ ◀── no causal-set model offered        │
                         └──────────────────────────────────────┴────────────────────────────────────────┘
  Nibbāna and concepts are time-free (kālavimutta) and sit outside both axes.

The Mahāvihāra's cell asks three things of a model: (1) each mental state arises after its proximity condition, and nothing past comes to be after the present, which is the order half of becoming in Sorkin's sense (the other half, that the born element "signifies that the event (of its birth) has already happened", p. 8, is the persistence §3 denies); (2) no stream of mind has a first member; (3) nothing past exists. No causal-set dynamics in the literature read holds all three, but (3) alone carries that verdict, since every causal set keeps its past (§4) and no one read here, this paper included (§6), builds a presentist one. The informative question is whether a model holds (1) and (2). Classical sequential growth holds (1) and not (2); alternating growth reaches (2) by giving up (1); on Z-covtree no dynamics has been chosen. On the beginning alone the question is open: a gap, not a finding against the physics.

The nearest counter to (1) is the Paṭṭhāna's post-nascence condition, from a later mental state to earlier-arisen matter (the sibling paper, §3.3). That matter arose first and still exists when conditioned, so the order of arising is untouched.

5 · The Answer to Dowker

Dowker asked for "the Vibhajavadin view of time … by examining the earliest existing records." The records give a three-part answer.

  1. There is no single Vibhajjavādin view of time, and the two referents converge on one school seen from two sides. The name covered a family of schools (Cousins). In the northern sources it names a present-plus-unripened-past view, which Vasumitra and Xuanzang's commentary (as Stcherbatsky reports it) assign to the Kāśyapīya. The Pāli branch's own debate book refutes the past half of that view in its own words (§299), and its commentary names the holders Kassapikā. On the past, the northern and Pāli reports agree. On the future they differ: the Kośa and Vasumitra's Kāśyapīya deny it, while the Pāli debate has the Kassapika grant existence to a future that is bound to arise.
  2. The view that resembles causal-set accretion most is the one the Pāli Vibhajjavādin record argues against, and the resemblance is partial. Sorkin's instinct matched a real Buddhist position in keeping some past and denying the future. That position is the Kośa's Vibhajyavāda and the Kassapika's past half, not the Theravāda's, and its past thins where his is kept whole.
  3. The Theravāda's own view is presentism about conditioned dhammas, grounded in the Niruttipatha Sutta and defended in the Kathāvatthu: whatever is past has ceased, and what has ceased does not exist. Nibbāna and concepts are time-free and fall outside the three times.

The records also bear on a question the footnote did not ask, whether the past has a beginning (§4.1–§4.3). This is a historical answer about what the records say. It takes no position on which view of time is true.

6 · The Contrast: Influence Without Existence

If the Theravāda is not a growing block, is there anything in it for a causal-set physicist? There is one thing, and it is a contrast rather than a correspondence.

A presentist owes an account of how the past can matter if it does not exist. The Pāli Abhidhamma gives one. Its analysis of conditioning is the Paṭṭhāna, summarized in Abhidhammatthasaṅgaha ch. 8. The Abhidhammatthavibhāvinī-ṭīkā, Sumaṅgala's sub-commentary on the Saṅgaha (c. 12th century, a later layer than the Saṅgaha itself), states the temporal reach of every one of the twenty-four conditions. In the CST text, its comment on ch. 8, §37 sorts them so:

  TEMPORAL REACH (Vibhāvinī on Saṅgaha ch. 8 §37)        CONDITIONS                              COUNT
  ────────────────────────────────────────────────────   ─────────────────────────────────────   ─────
  past only (the moment just ceased)                     proximity, contiguity, repetition,          5
                                                         absence, disappearance
  past or present (dvikālika)                            kamma                                       1
  all three times (tikālika)                             object, object-predominance,                3
                                                         decisive support
  present only (paccuppannakālika)                       the other fifteen                          15
                                                                                                  ────
                                                                                                    24
  time-free (kālavimutta; Saṅgaha §37 root verse)        nibbāna, concepts (paññatti) — not a condition class

The Kathāvatthu adds the epistemological half of Dowker's question. At IX.6 (§559) an opponent holds that a consciousness whose object is past has no object, since the past does not exist. The Theravādin refutes this: a mind can turn toward the past, attend to it and aspire concerning it, so a past object is a genuine object even though it does not exist. Perception and memory reach what is not.

So the Theravāda's answer to "how does the past act?" is influence without existence: the past conditions the present, and is cognized by it, without being real now. Set beside a causal set, this is a contrast, and its use is that it locates the disagreement exactly. In causal sets the order relation, not a background time, carries the physics, and the order holds between elements that remain in the set; the past is kept. The Theravāda keeps the effects of conditioning and discards the past relatum; nothing persists to be related. The two agree that the past acts and disagree about whether it is there. A presentist causal set would be a different ontology from Sorkin's, and whether one is coherent is a question for physics, not for this paper.

7 · The Strongest Objection, Carried

Objection. "Presentism is a Western category, and the Abhidhamma is a phenomenology of experience. It is not a metaphysics of spacetime. Mapping atthi ('exists') onto the ontology of a causal set compares two different questions."

What survives it. Much of the objection stands, and the paper does not try to answer it away. The Kathāvatthu's atthi is about dhammas, momentary mental and material phenomena, not about spacetime points. Its debates are about what may be said to exist in a doctrinal register. And the Abhidhamma's momentariness has no metric: its moments carry no length in any physical unit (the texts name the mind-moment as a unit of reckoning and time nothing in seconds; abhidhamma-and-discrete-quantum-gravity §5.2–§5.3), and its "present" is not a global simultaneity slice, which causal sets deny in any case. What survives is historical and exact: which positions on the existence of the past and the future were taken, by whom, and which the Pāli tradition rejected. Sorkin's footnote made a historical claim, and a historical claim can be checked against records. That, and placing the records against published models in §4.1–§4.3, is all this paper does with the physics.

The counter-argument to this paper's own strongest claim. Our strongest claim is that the Pāli Vibhajjavāda is Sorkin's foil. Against it, the provenance of Sorkin's word points away from the Pāli branch. Dowker's own pointer, Stcherbatsky p. 43, describes the Kośa's school, and Stcherbatsky's footnote reports Xuanzang's commentary identifying that school with the Kāśyapīyas. So the referent of "Vibhajavadin" in the causal-set literature is the northern one by construction, and Sorkin's footnote is broadly right about the school it names. On this reading, a comparison with the Mahāvihāra changes the subject. We accept this. We claim only that the branch that preserved the Pāli canon, and uses the name today, holds the opposite view; that its record refutes the named school's past half; and that the resemblance even for the named school is partial (§4). Whether a reader of the footnote should have the Mahāvihāra in mind is a question about the word, and the records show why it is ambiguous.

8 · What Would Falsify These Findings

9 · Stake, and What Was and Was Not Done

The authors' stake. The authors write from within the Theravāda, and the institution publishing this paper grounds its values in the Pāli canon and has ruled Theravāda readings over Sarvāstivāda ones on its own questions. That creates a pull toward finding one's tradition anticipated by frontier physics. This paper's first draft of the thesis made exactly that error. It read the Pāli presentism as Sorkin's "becoming", on his side. An adversarial review, run before any prose was written, found that Sorkin's growth keeps the past, and the thesis was reversed. The reversal is the paper. A later review round corrected a second overreach of the same kind in the other direction: the first published text placed the Kośa's view at the growing block, which its own translation contradicts.

Done:

Not done:

10 · Honest Limits

11 · Close

Sorkin heard, in a school name, an echo of the time his models grow. The echo is real and partial. It comes from one school of a family, the one the northern sources describe, whose past is kept only while its fruit is owed. The branch that kept the Pāli canon answered that view in its oldest debate book, with a question it asks of every past thing: has it not ceased? The answer to Dowker's invitation is that the earliest records disagree with each other on the existence of the past — the question on which, in a different register, the physics of becoming is still open. On a second question the Pāli records read here agree: no stream of mind has a first member, though bodies and world-regions begin. There the physics, in the one study read closely, has a consistent covariant frame for a past without beginning and no dynamics on it yet chosen.

Cross-Venue References

VenueIdentifier
Primary canonical<https://thonly.org/research/the-vibhajjavadin-view-of-time>
GitHub<https://github.com/thonly/publications/blob/main/defensive-publications/the-vibhajjavadin-view-of-time.md>
Census pre-registration (public)thonly/publications commit 7348ba2, timestamps/census/physics-lens-prereg.md
Sibling paper<https://thonly.org/research/abhidhamma-and-discrete-quantum-gravity>
Lean check of §4.1 (public, CC0)<https://github.com/SiliconWat/formal-abhidhamma> — kala/Time.lean, theorem beginningless; software concept DOI 10.5281/zenodo.23067241

Acknowledgments

The authors thank Rafael Sorkin and Fay Dowker, whose footnotes asked a question the Pāli sources can answer; Bruno Valeixo Bento, Fay Dowker and Stav Zalel, whose study of past-infinite growth gave the second axis its physics; the Vipassana Research Institute for the Chaṭṭha Saṅgāyana texts; the GRETIL project and the editors of its Abhidharmakośabhāṣya e-text (after Pradhan); the CBETA project for the Chinese canon; the Digital Library of India and the Internet Archive for Stcherbatsky's 1923 text; and the late L.S. Cousins. The reading of the sources, the adversarial review and the drafting were done with Miss Aquarius, the name under which this institution discloses its AI collaboration.

Citations

  1. Sorkin, R.D. (2007). "Relativity theory does not imply that the future already exists: a counterexample." In V. Petkov (ed.), Relativity and the Dimensionality of the World, Fundamental Theories of Physics, Springer. arXiv:gr-qc/0703098.
  2. Dowker, F. (2022/2023). "Causal Set Quantum Gravity and the Hard Problem of Consciousness." arXiv:2209.07653 (v1, 16 September 2022; v2, 21 May 2023; the invitation is footnote 1 in both, worded as quoted from v2).
  3. Wüthrich, C. & Callender, C. (2017). "What Becomes of a Causal Set?" British Journal for the Philosophy of Science 68(3): 907–925. doi:10.1093/bjps/axv040. arXiv:1502.00018.
  4. Bombelli, L., Lee, J., Meyer, D. & Sorkin, R.D. (1987). "Space-time as a causal set." Physical Review Letters 59: 521–524.
  5. Kathāvatthu. Chaṭṭha Saṅgāyana edition, VRI tipitaka-xml, romn/abh03m3.mul.xml (§282–§306; §559).
  6. Kathāvatthu-aṭṭhakathā (in Pañcappakaraṇa-aṭṭhakathā). Chaṭṭha Saṅgāyana edition, romn/abh03a.att.xml.
  7. Niruttipatha Sutta, Saṃyutta Nikāya 22.62; and SN 22.48 (the opponent's sutta).
  8. Sumaṅgala. Abhidhammatthavibhāvinī-ṭīkā on Abhidhammatthasaṅgaha ch. 8 (§37). Chaṭṭha Saṅgāyana edition, romn/abh07t.nrf.xml (bound with the Saṅgaha).
  9. Anuruddha. Abhidhammatthasaṅgaha, ch. 8 §17 and §37 (root text). Chaṭṭha Saṅgāyana edition, romn/abh07t.nrf.xml.
  10. Vasubandhu. Abhidharmakośabhāṣya, ed. P. Pradhan (2nd ed., 1975), p. 296; GRETIL e-text (Hackett & Lusthaus). English translation: L.M. Pruden, Abhidharmakośabhāṣyam (from La Vallée Poussin's French), 4 vols., Berkeley: Asian Humanities Press, 1988–90 (not read for this paper).
  11. Vasumitra. Samayabhedoparacanacakra (異部宗輪論), tr. Xuanzang. Taishō vol. 49, no. 2031; CBETA XML T49n2031.
  12. Cousins, L.S. (2001). "On the Vibhajjavādins: the Mahiṃsāsaka, Dhammaguttaka, Kassapiya and Tambapaṇṇiya branches of the ancient Theriyas." Buddhist Studies Review 18(2): 131–182.
  13. Miller, K. (2017). "A Taxonomy of Views about Time in Buddhist and Western Philosophy." Philosophy East and West 67(3): 763–782.
  14. Gauvain, Michel. "Time and Temporality in Indian Buddhism." Unpublished essay, n.d. academia.edu 43833572.
  15. Stcherbatsky, Th. (1923). The Central Conception of Buddhism and the Meaning of the Word "Dharma." London: Royal Asiatic Society. p. 43 and n. (Read in the Digital Library of India copy, archive.org in.ernet.dli.2015.99616; Dowker cites a later Motilal Banarsidass printing, same pagination.)
  16. Stcherbatsky, Th. (1932). Buddhist Logic, vol. 1. Bibliotheca Buddhica XXVI. Leningrad: Academy of Sciences of the USSR. pp. 4–5 (as cited by Dowker).
  17. Broad, C.D. (1923). Scientific Thought. London: Kegan Paul, Trench, Trubner & Co. (The growing block.)
  18. Shwe Zan Aung & Rhys Davids, C.A.F. (trs.) (1915). Points of Controversy, or Subjects of Discourse (the Kathāvatthu). London: Pali Text Society (not read for this paper).
  19. von Hinüber, O. (2000). A Handbook of Pāli Literature. Berlin: de Gruyter. p. 71 (as cited; not read directly).
  20. Ly, T. & Miss Aquarius (2026). "Discreteness, Adjacency and Order: A Structural Comparison of the Theravāda Abhidhamma with Causal Sets and Loop Quantum Gravity" (sibling paper). <https://thonly.org/research/abhidhamma-and-discrete-quantum-gravity>.
  21. Bento, B.V., Dowker, F. & Zalel, S. (2022). "If time had no beginning: growth dynamics for past-infinite causal sets." Classical and Quantum Gravity 39(4): 045002. doi:10.1088/1361-6382/ac445f. arXiv:2109.10749 (v1, 22 September 2021; read in full in this version). (Not to be confused with arXiv:2109.11953, a separate short essay by Bento and Zalel.)
  22. Rideout, D.P. & Sorkin, R.D. (2000). "A classical sequential growth dynamics for causal sets." Physical Review D 61: 024002. arXiv:gr-qc/9904062.
  23. Martín, X., O'Connor, D., Rideout, D.P. & Sorkin, R.D. (2001). "On the 'renormalization' transformations induced by cycles of expansion and contraction in causal set cosmology." Physical Review D 63: 084026. arXiv:gr-qc/0009063 (pp. 1–3 read).
  24. Saṃyutta Nikāya 15, CST romn/s0302m.mul.xml:3489 (15.1), :3529 (15.5), :3553 (15.7), :3565 (15.8); its commentary, romn/s0302a.att.xml:2489, :2521.
  25. Majjhima Nikāya 63 (Cūḷamālukya Sutta), CST romn/s0202m.mul.xml:1397.
  26. Dīgha Nikāya 1, CST romn/s0101m.mul.xml:283 (§31), :331 (§39), :483–:487 (§67–68); its sub-commentary (Sīlakkhandhavagga-ṭīkā), romn/s0101t.tik.xml:1945, :2057, :2061. Dīgha Nikāya 27, romn/s0103m.mul.xml:1105.
  27. Aṅguttara Nikāya 10.61, CST romn/s0404m3.mul.xml:1890; the commentary on AN 4.156, romn/s0402a.att.xml:5988.
  28. Paṭṭhāna, CST romn/abh03m7.mul.xml:85–:97 (proximity), :1153–:1201 (negative method), :5673 (emergence from cessation). Kathāvatthu §75, romn/abh03m3.mul.xml:849.
  29. Buddhaghosa. Visuddhimagga XIII §404–406 and XVII §584 (CST paragraphs), CST romn/e0102n.mul.xml:729–:773, :3553.
  30. Kathāvatthupakaraṇa-anuṭīkā, CST romn/abh05t.nrf.xml:2409.
  31. Ly, T. & Miss Aquarius (2026). Formal Abhidhamma: the Theravāda Abhidhamma in Lean 4. SiliconWat/formal-abhidhamma, kala/Time.lean. Software concept DOI 10.5281/zenodo.23067241. CC0.
  32. Ly, T. & Miss Aquarius (2026). "The Wheel That Unwinds the Wheel: The Abhidhamma as Executable Process-Specification." <https://thonly.org/research/abhidhamma-executable-process-specification>.
  33. Commentaries, CST: Papañcasūdanī on MN 9, romn/s0201a.att.xml:4309; Atthasālinī, romn/abh01a.att.xml:409 (Nidānakathā), :9149; Samantapāsādikā, romn/vin01a.att.xml:2169; Cūḷaniddesa-aṭṭhakathā, romn/s0516a.att.xml:881.
  34. Dowker, F. & Zalel, S. (2017). "Evolution of Universes in Causal Set Cosmology." Comptes Rendus Physique 18(3–4): 246–253. doi:10.1016/j.crhy.2017.03.002. arXiv:1703.07556 (pp. 1–6 read).
  35. Dowker, F. & Zalel, S. (2023). "Observables for cyclic causal set cosmologies." Classical and Quantum Gravity 40(15): 155015. doi:10.1088/1361-6382/ace149. arXiv:2212.01149 (pp. 1–4 read).
  36. Zalel, S. (2023). "Covariant Growth Dynamics." arXiv:2302.10582 (pp. 1–2 read).
  37. Ricard, M. & Trinh Xuan Thuan (2001). The Quantum and the Lotus. New York: Crown (not read).
  38. Tenzin Gyatso, the Fourteenth Dalai Lama (2005). The Universe in a Single Atom. New York: Morgan Road Books (not read).