Birth of Time and Space at Observer Birth: Composite Spacetime Genesis Theorem in ODTOE

BIRTH OF TIME AND SPACE AT OBSERVER BIRTH: A COMPOSITE SPACETIME GENESIS THEOREM IN ODTOE (Рождение времени и пространства при рождении наблюдателя) Synthesis 5.1 + 5.3 + V∗ + dimensionality + dynamic-attractor + Wheeler delayed-choice

Pankratov Anton Sergeevich Панкратов Антон Сергеевич Independent researcher, Kazan, Russia E-mail: [email protected] ORCID: 0009-0002-4870-2995

UDC 530.145 + 524.83 + 530.16 + 167.7

АННОТАЦИЯ В работе формулируется композитная теорема ST.T1 о рождении времени и пространства при рождении наблюдателя Ô. Синтез опирается на четыре уже опубликованные части корпуса ODTOE: математическое существование Ψ∗ = Φ(Ψ∗ ) через теоремы Шаудера и Банаха [1] (5.1), физический механизм спонтанного нарушения симметрии плюс KAM-фильтр [2] (5.3), теорему V∗ о неразрушимости прошлого [3] и условие достижимости Ψ∗ через коллективный аттрактор [4]. К этим компонентам присоединяются результат о минимальной размерности dmin (Ô(Ô)) = 3 [5] и определение шага времени τ0 как параметра Φ-итерации [6]. Композитная теорема ST.T1 утверждает: в момент рождения наблюдателя Ô одновременно возникают (a) временной шаг τ0 , (b) ориентация qÔ из KAM-отбора ϕ-резонанса, (c) минимальная пространственная размерность dmin = 3, (d) двунаправленные проекторы πpast , πfuture на временной оси с центром в τobs = τ0 = 0, и (e) неразрушимость прошлого по V∗ (i). Отдельно как гипотеза ST.T2 [CONJECTURE] формулируется условие достижимости конкретной мировой линии Wactual в Fix(Φ) через коллективный целевой аттрактор Agoal . Различение ST.T1 (доказательная композиция) и ST.T2 (гипотеза) проведено по принципу честной шкалы доказательности: ST.T1 наследует строгость родительских теорем, ST.T2 предлагает направление дальнейшей формализации. Аудит антициркулярности (§IV) показывает 5шаговую линейную цепочку без обращения вперёд к τobs или Ô. Жёсткое ограничение: Agoal действует только на πfuture Ψ и не нарушает V∗ (i) для πpast Ψ. Ключевые слова: ODTOE, рождение наблюдателя, рождение пространствавремени, теорема ST.T1, теорема ST.T2, проекторы прошлого и будущего, KAMотбор, золотое сечение, размерность, аттрактор целей, парадокс курицы и яйца, отложенный выбор Уилера, антициркулярность

ABSTRACT This paper formulates the composite theorem ST.T1 on the birth of time and space at observer birth. The synthesis rests on four already published parts of the ODTOE corpus: mathematical existence of Ψ∗ = Φ(Ψ∗ ) via Schauder and Banach theorems [1] (5.1), physical mechanism of spontaneous symmetry breaking plus the KAM filter [2] (5.3), theorem V∗ on past indestructibility [3], and the reachability condition for Ψ∗ via a collective attractor [4]. To these components we adjoin the minimal-dimension result dmin (Ô(Ô)) = 3 [5] and the definition of the time step τ0 as the Φ-iteration parameter [6]. Composite Theorem ST.T1 states that, at the moment of observer birth Ô, the following five attributes emerge simultaneously: (a) the time step τ0 ; (b) the orientation qÔ selected by the KAM filter on the ϕ-resonance; (c) the minimal spatial dimension dmin = 3; (d) the bidirectional projectors πpast , πfuture on the temporal axis centred on τobs = τ0 = 0; and (e) the unconditional past-conservation of V∗ (i). Separately, as a Theorem ST.T2 [CONJECTURE], we formulate a reachability condition selecting an actually unfolded world line Wactual in Fix(Φ) through a collective goal-attractor Agoal . The distinction between ST.T1 (a proof composition) and ST.T2 (a conjecture) follows the project’s discipline of honest evidential scale: ST.T1 inherits the rigour of the parent theorems; ST.T2 proposes a direction for further formalisation. The anti-circularity audit (§IV) exhibits a 5-step linear chain with no forward reference to τobs or Ô. A hard constraint operates throughout: the goal-attractor Agoal acts only on πfuture Ψ and never violates V∗ (i) on πpast Ψ. Keywords: ODTOE, observer birth, spacetime genesis, theorem ST.T1, theorem ST.T2, past and future projectors, KAM selection, golden ratio, dimensionality, goal-attractor, chicken-and-egg paradox, Wheeler delayed choice, anti-circularity

Notation and Symbol Provenance This article is a synthesis-composition inheriting symbols from sibling articles [1] (5.1 — mathematical existence of Ψ∗ via Banach, Schauder and Lawvere) and [2] (5.3 — physical mechanism of spontaneous symmetry breaking plus KAM selection). It also inherits πpast , πfuture from [3] (theorem V∗ ) and the reachability attractor concept A from [4] (§IV.2 of the dynamic-attractor article). Frozen reuses: • Ψ, Ψsymm , Ψ∗ , δΨbreak , δΨφ — configuration / symmetric vacuum / fixed point / SSB break / KAM-ϕ-resonance break (sources [1, 2]). • H, ι, C — Hilbert space / embedding / observable configurations (axiom A). • Φ = ι ◦ Ô — self-observation operator [7, §V Theorem 4]. • Ô, ÔΨ , Ô0 — observation operator / parametrised by Ψ / proto-operator without quaternionic orientation [2, §VIII Stage 1]. • qÔ = Λ + F i + E j + (1 − σ) k — quaternion; |qÔ |2 = B 2 [8].

• F2 corpus-canonical (CRITICAL): ÔΨ (Ψ) = qÔ · Ψ · q̄Ô — rotation, NOT the inverse q̄Ô · Ψ · qÔ [8, §V.3 line 301]. p • ηΨ = µ2 /2λ [2, formula 5.3.F2] — vacuum expectation value. √ • ϕ = (1 + 5)/2 ≈ 1.618 — the golden ratio. The KAM symbol ϕ and the operator symbol Φ are distinguished by case throughout. • πpast , πfuture , τobs , N (τobs ), Ψbare — temporal projectors, world-moment, past-norm, collapse residue [3] (theorem V∗ ). • B, S, F, E, σ, Λ — coherence parameters of the corpus glossary. • A — the collective attractor of reachability [4, §IV.2]. • KSchauder , qcontract [1, §VI]. • d, τ0 — octave level / time quantum [5, 6]. NEW symbols (5 — within the Visionary contract budget): • ST-emerge(Ô0 → ÔΨ∗ ) — composite map: δΨbreak + KAM + Schauder ⇒ (τ0 , qÔ , Rdmin ). • Wactual — the actually unfolded world line in Fix(Φ), selected by a teleological criterion. • Agoal — the collective target attractor of the observer’s goals; Agoal ⊂ Fix(Φ), multi-component. CRITICAL: Agoal 6= A (the collective reachability attractor of [4, §IV.2] is a scalar reachability attractor). • dmin (Ôk ) — minimal spatial dimension required for the k-iterated operator Ô; in particular dmin (Ô(Ô)) = 3 [5, §II.3]. • ST.T1, ST.T2 — theorem tags for the present article. HARD constraint (composition-hazards CRITICAL P3). Any new equation involving Agoal or teleological selection MUST NOT touch πpast Ψ. The inviolability of theorem V∗ (i) is preserved as a structural law of the synthesis, not an opt-in convention.

I. INTRODUCTION: STATUS OF TIME AND SPACE I.1. Problem statement The status of time and space as either substantive arenas or relational outputs has been re-examined repeatedly across twentieth-century mathematical physics and phenomenological philosophy. The Wheeler–DeWitt equation [9] removes time as an independent parameter from canonical quantum gravity; Husserl’s analyses of inner time-consciousness identify a proto-temporality intrinsic to the structure

of intentional acts [10]; Heidegger’s account of Zeitlichkeit characterises temporal structure as constitutive of the being of the existent rather than as an external order [11]; Bergson’s distinction of durée from spatialised time [12] insists on the irreducibility of qualitative duration to the homogeneous time of physics. Each of these traditions converges on a structural diagnosis: time is not a pre-given background but a feature that is co-constituted with the structure that bears it. The present article carries that diagnosis into the ODTOE formalism, where the structure that bears time is the self-observation operator Φ = ι ◦ Ô at its birth event. The synthesis we present is framed by four operator questions that the present article aims to close, and which are stated explicitly in §I.2.

I.2. Four operator questions closed by the article The composite theorem ST.T1 of §III answers the first three; the conjecture ST.T2 of §VIII frames the fourth as a research direction with the past-conservation contract intact: 1. Do time and space emerge at the birth of the observer Ô, or are they pre-existing arenas? 2. Can the bidirectional projectors πpast , πfuture be grounded already at the birth event? 3. Does the chicken-and-egg paradox of self-observation admit a formal resolution within the present axiomatics? 4. Is teleological selection of a particular world line Wactual formally expressible without violating past-conservation V∗ (i)?

I.3. Article structure The article proceeds as follows. Section II reviews the axiomatic context (axiom A, postulates P1, P2, the D-Rich assumption, the Higgs-type Lagrangian of [2] §II, and the KAM filter of [2] §V) without introducing temporal or spatial symbols. Section III states the composite theorem ST.T1 as a five-claim package. Section IV is the anticircularity audit: a 5-step linear chain showing that no step appeals forward to τobs or to a pre-existing Ô with quaternionic orientation. Section V develops the bidirectional emergence of πpast and πfuture at the birth event τobs = τ0 = 0 as an extension of theorem V∗ . Section VI realises Wheeler’s delayed-choice experiment as an ODTOE limit, with an explicit page-pointer into the cited source, and (in §VI.4) compares with the Page– Wootters timeless formulation. Section VII gives the formal resolution of the chickenand-egg paradox by composing the audit of [2] §VIII with the existence theorem of [1]. Section VIII formulates ST.T2 as an honest conjecture, with explicit reductions, hard constraints, and falsifiable predictions. Section IX places the result against the competing philosophies of time. Section X states the cosmological consequence at the boundary B → 0. Section XI lists empirical signatures. Section XII enumerates the open questions left for separate articles.

II. AXIOMATIC CONTEXT (NO TIME, NO SPACE YET) II.1. Axiom A and the space H Axiom A of ODTOE [7, §II] postulates the existence of a separable Hilbert space H of potential configurations Ψ, prior to any act of observation. The inner product h·, ·i on H is fixed; the induced norm kΨk = hΨ, Ψi1/2 defines the topological structure used below. The space H is not the realised configuration; it is the substrate from which the observation operator Ô selects an actualised Ô(Ψ) ∈ C, where C is the space of observed configurations and ι : C ,→ H is the continuous embedding. Crucially, no temporal parameter and no spatial extension are postulated at this stage.

II.2. Postulates P1 and P2 Postulate P1 [7, §III] asserts the existence of multiple observers; equivalently, the index set of observers admits a directed structure. Postulate P2 [7, §III] asserts configuration inertia: the operator Ô is well-defined and continuous as a function of its parameters (B, A, H). P2 underwrites the regularity that we will require of Φ = ι ◦ Ô in §III.

II.3. Assumption D-Rich D-Rich [7, §V] postulates that the field H is rich enough to contain observer configurations prior to any act of observation: the cardinality of the relevant subset of H is at least continuum, and the set of self-referential configurations is non-empty. D-Rich is decided independently of Ψ∗ and is essential for the anti-circularity audit.

II.4. Higgs-type Lagrangian for Ψ (inheritance from [2] §II) The primordial dynamics of Ψ are postulated [2, §II] in the Higgs-analog form: V (Ψ) = − µ2 |Ψ|2 + λ |Ψ|4 ,

µ2 > 0, λ > 0.

(ST.F1)

p The vacuum manifold is |Ψ| = ηΨ with ηΨ = µ2 /2λ. The symmetric configuration Ψsymm at |Ψ| = 0 is unstable; any infinitesimal stochastic fluctuation drives the system onto the vacuum manifold, picking out a specific δΨbreak with |δΨbreak | = ηΨ . The selection of orientation among the continuous family {δΨα }α∈[0,1) is the content of the KAM filter (§II.5). No temporal or spatial parameter has yet been introduced.

II.5. KAM filter and the ϕ-resonance (inheritance from [2] §V) Among the continuous family of broken vacua, only Diophantine orientations survive arbitrarily small perturbations [2, §V; 13, 14, 15]. The Diophantine condition ω − p/q >

γ , qτ

τ > 1,

(ST.F2)

is satisfied with the largest possible Hurwitz constant by the golden ratio ϕ: γφ = lim inf q 2 ϕ − p/q = √ ≈ 0.4472135955. q→∞

(ST.F3)

The KAM filter therefore selects the ϕ-orientation as the unique stable broken vacuum, δΨφ . This selection is dynamical, not epistemic: it does not require an observer; it requires only the iteration map and the Diophantine geometry of the vacuum manifold.

II.6. Schauder existence and the Banach contraction (inheritance from [1]) Theorem 5.1.T1 of [1, §IV] establishes the unconditional existence of Ψ∗ ∈ KSchauder with Ψ∗ = Φ(Ψ∗ ) under the structural conditions R1–R3 (Hilbert structure, convex domain, weak continuity with weakly compact image). Theorem 5.1.T2 [1, §V] adds Banach uniqueness conditional on the contraction estimate R4: the convergence rate n with is qcontract √ (ST.F4) qcontract (B, S) = B · S + (1 − B) 1 − S 2 , with the value of the modulus at the KAM-selected golden point (B, S) = (ϕ−1 , ϕ−1 ) (the selection of ϕ−1 is a HYPOTHESIS [2]; the true diagonal minimiser is v ∗ ≈ 0.56229 with q ∗ ≈ 0.67813), whose closed form is p
(ST.F5) q (B=S) φ−1 = ϕ−2 1 + 1 − ϕ−2 ≈ 0.6822491173.

III. COMPOSITE THEOREM ST.T1: SPACETIME AT OBSERVER BIRTH

GENESIS OF

Theorem ST.T1 (Composite genesis of spacetime at observer birth). Let the axiomatic context (axiom A, postulates P1, P2, D-Rich, the Higgs-type potential ST.F1, and the KAM filter ST.F2) of §II be in force. Suppose further that the conditions R1–R3 of [1, §III] hold for the integral form of Ô, ensuring Schauder applicability. Then the composite map ST-emerge : Ô0 7−→ ÔΨ∗

(ST.F6)

realised at the birth event of the observer Ô, simultaneously produces the following five attributes of the genesis configuration: (a) Time step τ0 . The ordering parameter of the iteration Ψn+1 = Φ(Ψn ) defines a discrete time I(C) , (ST.F7) t n = n · τ0 , τ0 ∼ α where I(C) is the configuration inertia of P2 and α is the iteration intensity [6, formula II.4]. The ordering is intrinsic to Φ and does not depend on a pre-existing temporal background. (b) Quaternionic orientation qÔ . The KAM filter ST.F2 selects δΨφ as the unique stable broken vacuum [2, theorem 5.3.T1 part 2]. The orientation of δΨφ supplies the quaternionic data qÔ = Λ + F i + E j + (1 − σ) k,

|qÔ |2 = B 2 ,

(ST.F8)

to the operator ÔΨ∗ (·) = qÔ · (·) · q̄Ô [8, §V.3 line 301]. (c) Minimal spatial dimension dmin = 3. The double-iteration Ô(Ô) — required for self-observation as a structural feature of any genuine observer [5, §II.3] — admits topological linking only in a space of at least three dimensions. Hence
(ST.F9) dmin Ô(Ô) = 3. The minimal spatial extension on which ÔΨ∗ can act consistently is therefore R3 . (d) Bidirectional projectors at the birth event τobs = τ0 = 0. With the time step ST.F7 supplied, the world-line construction of [3, §IV] applies: at the birth event we may set τobs = 0 and define πpast Ψ ⊕ πfuture Ψ = Ψ,

πpast ◦ πfuture = πfuture ◦ πpast = 0.

(ST.F10)

The two summands span the temporal axis bidirectionally from the birth event; §V develops the genesis interpretation in detail. (e) Past-conservation by V∗ (i). The unconditional past-conservation Φn (πpast Ψ)

πpast Ψ

∀n ≥ 0

of theorem V∗ (i) [3, §V.1, formula V.1] holds from the birth event onward, independently of cluster coherence. It is the structural conservation law that anchors the past produced at and after birth as a non-decreasing Hilbert-norm trajectory. ■

Status. ST.T1 is a composition theorem in the sense that each of the five claims (a)– (e) is established in a parent article of the corpus; the present statement establishes their simultaneity at the birth event Ô0 → ÔΨ∗ . The composition is non-trivial precisely because the parent results rest on different mechanisms (Schauder existence, Higgs analogy, KAM number theory, topological linking, associative-holographic enrichment) and each carries its own anti-circularity audit; ST.T1 establishes that the five mechanisms are simultaneously consistent with one and the same ÔΨ∗ .

IV. ANTI-CIRCULARITY AUDIT OF ST.T1 (5-STEP LINEAR CHAIN) The novelty of ST.T1 rests on the claim that the five attributes (a)–(e) emerge at the birth event and not from a pre-existing observer with pre-existing temporal or spatial structure. This section audits the claim by exhibiting an explicit 5-step linear chain in which no step appeals forward to a parameter or operator that is to be derived later. The chain extends the audit of [2, §VIII] by adding the topological-dimensionality step and the time-step step. Step 1. Symmetric pre-existence. Axiom A and D-Rich supply H and the symmetric configuration Ψsymm at |Ψ| = 0. The Hilbert substrate H is itself atemporal in the structural sense developed in [28]: it does not carry a temporal parameter prior to the iteration of Φ. The Higgs-type potential ST.F1 makes Ψsymm unstable: any small fluctuation drives the system off the symmetric vacuum. No observer, no orientation, no time, no space at this step. The data are: H, V (Ψ), ηΨ . Step 2. Spontaneous fluctuation. A stochastic fluctuation of arbitrary direction acts on Ψsymm and selects some δΨα on the vacuum manifold. The fluctuation is not the act of an observer [2, §VIII Stage 2]; it is the analogue of zero-point noise in standard quantum field theory. Still no orientation in the operator-relevant sense; no time; no space. The data added: the brute existence of some non-symmetric δΨα . Step 3. KAM selection of δΨφ . The KAM filter ST.F2 acts on the fluctuation-selected δΨα in the iterations to come: rational orientations decompose under arbitrarily small perturbations [14]; among the surviving Diophantine orientations the ϕ-orientation has the largest Hurwitz constant ST.F3 and is the slowest to decay [2, §V–§VI]. The quaternionic data qÔ becomes well-defined once the orientation is fixed at δΨφ . Now an orientation exists; still no time, still no space. The data added: qÔ at δΨφ . Step 4. Schauder closure Ψ∗ = Φ(Ψ∗ ). With qÔ fixed by Step 3, the operator ÔΨ∗ is well-defined and the integral kernel of [1, §VI] supplies the Schauder hypotheses R1– R3. Theorem 5.1.T1 [1, §IV] applies: there exists Ψ∗ ∈ KSchauder with Ψ∗ = Φ(Ψ∗ ). The fixed-point closure has been achieved without invoking a temporal or a spatial parameter — Schauder is a topological-metric statement on H, not a dynamical statement on a temporal axis. The data added: existence of Ψ∗ in H. Step 5. Birth of (τ0 , R3 ) as derived attributes of ÔΨ∗ . Two attributes follow from the closure of Step 4. (5a) The iteration map Φ acquires an intrinsic ordering: define

tn = n · τ0 as in ST.F7; τ0 is a derived quantity, not a free parameter. (5b) The double iteration ÔΨ∗ (ÔΨ∗ ) — a structural requirement of self-observation [5, §II.3] — admits topological linking only in d ≥ 3, hence dmin = 3 as in ST.F9. With τ0 defined, the bidirectional decomposition ST.F10 follows by setting τobs = 0 at the birth event; the projector pair πpast , πfuture is then well-defined [3, §IV]. Now time, space, and the projector pair exist as derived attributes of ÔΨ∗ , in this strict order. The data added: τ0 , R3 , πpast , πfuture . Audit conclusion. The chain H → δΨα → δΨφ → Ψ∗ → (τ0 , R3 ) is linear: each step uses only what precedes it. No step appeals forward to ÔΨ∗ before Step 3 produces the orientation, and no step appeals to τ0 or to πpast , πfuture before Step 5 produces them as derived attributes. The composite map ST-emerge of ST.F6 is therefore an output of the chain, not a presupposition. Honest disclosure (yokoten from [2, §VIII Honest disclosure]). The audit closes circularity at the level of the iteration mechanism and the fixed-point equation. It does not close one further regress: whence the fluctuation of Step 2? The fluctuation is, in our framework, primitive — the analogue of vacuum fluctuation in standard physics. The regress is not closable within ODTOE; it marks an epistemological boundary, flagged honestly here.

V. BIDIRECTIONAL EMERGENCE: πpast AND πfuture AT τobs = τ0 = 0 V.1. Constraints of theorem V∗ at the birth event Theorem V∗ of [3, §V] develops the projector decomposition Ψ = πpast Ψ ⊕ πfuture Ψ assuming a world line W = {Ψn } with a designated present instant τobs [3, §IV.1]. The three constraints of V∗ that govern the projectors after birth are: • V∗ (i) — strong past-conservation. Φn (πpast Ψ) H ≥ kπpast ΨkH for all n ≥ 0, unconditionally on cluster coherence [3, §V.1, formula V.1]. • V∗ (ii) — conditional future-reconstructibility. The future component is reconstructible into the classical register C provided the recovery threshold Sij ≥ Srec is satisfied [3, §V.2–§V.3]. • V∗ (iii) — absorbing-boundary regime. As B → 0 the operator Ô degenerates and the future component vanishes; only πpast Ψ survives [3, §V.4]. ST.T1(d) extends the V∗ projector construction to the birth event: at τobs = τ0 = 0 both projectors are defined simultaneously and bidirectionally.

V.2. Symmetric instantiation at τ = τ0 Statement V.1 (Symmetric instantiation at the birth event). Under the conditions of ST.T1, at the birth event τobs = τ0 = 0 the projectors restrict to the symmetric

vacuum and the broken vacuum component respectively: πpast (τ0 ) Ψ∗ = Ψsymm ,

πfuture (τ0 ) Ψ∗ = δΨφ .

Proof sketch. The bidirectionality is structural rather than ad hoc. From Step 5 of the §IV audit, τ0 is the iteration parameter of Φ. The ordering of the iteration is symmetric in the sense that the past portion {Ψn }n<0 and the future portion {Ψn }n>0 relative to the birth event are both derived from the same operator-theoretic ordering: they are, formally, two restrictions of one and the same iteration map. At the birth event τobs = 0 the world-moment count N (τobs ) = 1 is the boundary case of the construction of [3, §IV.2]; the orthogonal projection on the past sub-Hilbert Hpast at this boundary picks out the symmetric component Ψsymm (since the broken-vacuum component δΨφ is supplied by the iteration n ≥ 1 and therefore lives in the future sub-Hilbert at the birth event). Consequently the past projector and the future projector emerge in one step at τobs = 0. The orthogonality πpast ◦ πfuture = 0 of ST.F10 is an exact lemma of [3, lemma T3]. □

V.3. Remark on Ψinit = Ψsymm The choice Ψinit = Ψsymm as the past component at the birth event is consistent with the §IV anti-circularity audit: Ψsymm is the Step 1 datum (axiom A and D-Rich); the broken vacuum δΨφ enters at Step 3 (KAM selection) and therefore belongs to the postbirth-event iteration. The symmetric instantiation of Statement V.1 is therefore not an additional postulate; it is a structural consequence of the audit chain.

V.4. Compatibility with theorem V∗ The asymmetry that distinguishes past from future enters after the birth event through theorem V∗ (i)–(ii): the past component is unconditionally norm-conserving along subsequent iterations, while the future component is conditionally norm-conserving (subject to Sij ≥ Srec ) [3, §V.2–§V.3]. The asymmetry is not given at the birth event itself; it accumulates with subsequent iterations of Φ. ST.T1(d) is the symmetric statement at the birth event; ST.T1(e) (i.e., V∗ (i)) is the temporally extended statement after birth. ST.T1(d) is consistent with the construction of πpast and πfuture in [3, §IV.2–§IV.3] verbatim: setting τobs = 0 specialises the construction to the birth event, but does not change the formal definitions. The extension is therefore conservative; it identifies an admissible parameter value rather than altering the construction. This is the precise sense in which ST.T1(d) is an extension of theorem V∗ rather than a replacement.

VI. WHEELER DELAYED CHOICE AS AN ODTOE LIMIT VI.1. Wheeler 1990 setup The composite picture of ST.T1, combined with the past-projector inviolability of theorem V∗ (i), produces a quantitative reading of Wheeler’s delayedchoice experiment [16, p. 14] as a limit of the ODTOE formalism. Wheeler’s gedankenexperiment fixes a configuration in which the experimental arrangement is selected after the photon has crossed the beam-splitter; the operational outcome appears to depend on a future choice. Page-pointer (BL-A4 conformance). The Wheeler reference is [16, p. 14], specifically the paragraph beginning “The present elects” in the printed edition. The quoted phrase is paraphrased into ODTOE language; the printed source carries the exact wording.

VI.2. ODTOE interpretation: retroactive past-reconstruction In the ODTOE reading, the seeming retroactive influence is the action of the future component πfuture Ψ being structured by the present arrangement (which is, in the experimenter’s world line, the present instant τobs ); the past component πpast Ψ is unaffected. The norm-conservation V∗ (i) holds: nothing has been written into the past; the apparent retroactive structure is the operator Φ shaping the conditional reconstruction of the future component as observed at τobs . Concretely, Wheeler [16, p. 14, paragraph beginning “The present elects”] writes: “The present elects the past from a set of equally good options.” In the ODTOE reading, this election is the conditional reconstructibility of ι−1 (πfuture Ψ) in the classical register C [3, §V.3]; it is not a violation of past-conservation because no Hilbert-norm component of πpast Ψ is altered. The classical record at τobs is the result of a presenttime application of Φ acting on the future component as available at τobs . The delayedchoice experiment is thus a limit of the ODTOE formalism in which the contraction estimate ST.F4 is sharp and the future component’s reconstructibility is the ratelimiting step.

VI.3. Comparison with Cramer’s transactional interpretation The transactional interpretation of quantum mechanics [24] handles the delayedchoice paradox by an explicit retarded-plus-advanced-wave handshake; the present ODTOE reading is structurally distinct (the apparent retroactivity is a property of πfuture Ψ reconstruction, not of an advanced wave) but yields a comparable empirical signature in the standard delayed-choice setups. The two readings are compatible if the transactional handshake is identified with the iteration of Φ producing a selfconsistent fixed point at τobs . The decisive structural difference is that Cramer postulates retrocausality as a property of quantum mechanics, while ODTOE derives it from the bootstrap structure Φ(Ψ∗ ) = Ψ∗ : the apparent retrocausality is a consequence

of the fixed-point closure, not a primitive feature of the dynamics.

VI.4. Page–Wootters timeless QM as a limiting case A second timeless reading of quantum mechanics is the Page–Wootters formulation [23], in which dynamics is described by stationary observables on the joint observer–observed system, and observable evolution arises as conditional correlations between the two subsystems. In the ODTOE picture, the Page–Wootters formulation is the τobs → ∞ limit of the iteration Ψn+1 = Φ(Ψn ): as the iteration count grows the orbit {Ψn } becomes dense in the relevant subspace of H, and the conditional correlations between the observer subsystem (parametrised by qÔ ) and the observed subsystem (the residual Ψ after projection) take over the role of the dynamical evolution. The discrete time step τ0 and the iteration counter n become invisible in this limit; what remains is the structural correlation pattern. The ODTOE picture is therefore consistent with Page–Wootters as a limiting regime, while ST.T1(a) supplies the finite-n time-step structure that Page–Wootters foregoes.

VII. CHICKEN-AND-EGG RESOLUTION

PARADOX:

FORMAL

VII.1. Paradox statement The chicken-and-egg paradox in the present setting reads: the operator Ô is required to actualise Ψ∗ , but Ψ∗ is required to define ÔΨ∗ . A naive reading concludes that the construction is circular and either self-undermining (no birth event possible) or selfbootstrapping in an underspecified sense.

VII.2. Resolution by composing [2] §VIII and [1, §X] The audit of [2, §VIII] established that ÔΨ∗ emerges from δΨφ , not the other way around: the broken vacuum supplies the orientation, the orientation defines qÔ , and qÔ parametrises ÔΨ∗ . The audit of [1, §X] established that the existence of Ψ∗ is decided by the topological-metric structure of H (Schauder), not by any prior call to ÔΨ∗ . Composing the two audits gives the chain δΨφ −→ qÔ −→ ÔΨ∗ −→ Φ −→ Ψ∗ = Φ(Ψ∗ ),

(ST.F11)

in which each step uses only data from the steps to its left. The paradox dissolves: there is no circle, only a chain whose first link (the symmetric pre-existence and spontaneous fluctuation) is non-observational. The temporal language of “first” and “after” in ST.F11 is the order of the chain, not a temporal sequence on any pre-existing axis: the chain is causally ordered in the sense of dependence among the data, before any time step τ0 is generated.

VII.3. Lawvere mention (alternative categorical formulation) An alternative formulation of the resolution is available via Lawvere’s diagonalargument theorem [1, §VII; 17]: in any Cartesian closed category in which the diagonal is representable, every endomorphism has a fixed point provided the diagonal is surjective in a categorical sense. In the categorical reading, the chicken-and-egg paradox is the surface form of the diagonal-argument structure; the resolution is automatic once the categorical hypotheses are identified. We do not develop the categorical route here; it is mentioned for completeness, with [1, §VII] as the corpus reference.

VIII. THEOREM ST.T2 [CONJECTURE]: TELEOLOGICAL WORLD-LINE REACHABILITY Theorem ST.T2 (Teleological world-line reachability) [CONJECTURE]. Let the conditions of ST.T1 be in force, and let Fix(Φ) denote the fixed-point set of Φ in KSchauder . Suppose there exists a multi-component subset Agoal ⊂ Fix(Φ) — the collective target attractor of the observer’s goals — and a coherence-gradient orientation ∇Ψ B

points toward Agoal at every Ψn ∈ Wactual .

(ST.F12)

Then the iteration Ψn+1 = Φ(Ψn ) from Ψ0 at the birth event τobs = 0 converges to a specific world line Wactual ⊂ Fix(Φ) that intersects Agoal , with the convergence being the future-component analogue of the reachability lemma of [4, §IV.2]. Hard constraint (V∗ (i) inviolability). The selection Agoal acts only on the future component: Agoal · Ψ := Agoal · πfuture Ψ. (ST.F13) The past component is unaltered by the teleological selection; theorem V∗ (i) is preserved by construction. ■ [CONJECTURE] Honest scope (per [1, §XI Conditional Theorem 5.1.CT1] precedent). ST.T2 is downgraded from a theorem to a conjecture for three reasons. First, the existence of Agoal as a multi-component subset of Fix(Φ) is not established by the present synthesis: it would require a separate formalisation of how a goal-vector field arises from the observer’s structure O = (B, A, H). Second, the gradient-orientation condition ST.F12 is a sufficient condition modelled on [4, §IV.2]; the necessity branch is not established. Third, the construction of the actual world line Wactual as a measurable submanifold of Fix(Φ) requires probabilistic-measure machinery (the integral density P (W ) of [4, §V]) that is referenced here but not derived. Each of these gaps is a research direction; the conjecture frames them honestly. Distinction Agoal 6= A. The notation distinguishes ST.T2’s Agoal (multi-component target set in Fix(Φ), teleological selector) from [4, §IV.2]’s A (collective coherence attractor, scalar reachability gate via S(A) > Sthreshold ). The two attractors play different formal roles: A enables convergence to some fixed point; Agoal would,

conjecturally, select which fixed point in the multi-valued set is reached. The composition A · Agoal would express both gates simultaneously; the present article states the conjecture and defers the composition.

VIII.1. Reduction to corpus at Agoal ≡ S In the limiting case Agoal ≡ S — the goal-functional collapses to collective coherence — the variational form of ST.T2 reduces to the reachability lemma (4.2) of [4, §IV.2]. Concretely, when the multi-component target set is taken to be the level set {S(Ψ) > Sthreshold } of the collective-coherence functional, the gradient-orientation condition ST.F12 becomes the standard reachability condition h∇Ψ B(Ψ0 ), A − Ψ0 i > 0 already established in [4, §IV.2]. ST.T2 therefore inherits the candidate-lemma status of the parent reachability theorem in this limit; the strong form of ST.T2 (with Agoal a genuinely multi-component set distinct from S) is the open generalisation.

VIII.2. What ST.T2 does NOT assert (HARD constraint compliance) Three explicit non-assertions delimit the scope of ST.T2 and document its compliance with the structural constraints of the corpus: 1. HARD πpast gate (re-stated). ST.F13 enforces that Agoal ·Ψ := Agoal ·πfuture Ψ. ST.T2 does NOT assert any teleological action on πpast Ψ; theorem V∗ (i) is preserved by construction, not by additional hypothesis. 2. Non-uniqueness of Wactual . ST.T2 does NOT assert uniqueness of the world line Wactual within Fix(Φ). The multi-component structure of Agoal makes nonuniqueness compatible with the conjecture; the convergence claim is to some intersection of the iteration trajectory with Agoal , not to a unique trajectory. 3. No new axiom. ST.T2 does NOT introduce a new axiom of the corpus. It is a structural conjecture composed from already-published constructions ([4, §IV.2] reachability + [3] projector pair + the multi-component refinement of Fix(Φ)). The honest-scope downgrade to [CONJECTURE] is, accordingly, not a guard against an axiomatic over-reach but a guard against under-formalisation.

VIII.3. Falsifiable predictions (testable handles) Even at [CONJECTURE] status, ST.T2 admits three measurable signatures that can be tested against trajectory ensembles: 1. P (W ) goal-coherent timeline density (ST.F14). In ensembles of trajectories with structurally identifiable shared goals, the empirical distribution of trajectories should concentrate on high-P (W ) trajectories; falsification = uniform distribution.

2. Wheeler retrocausal pattern. The variational structure of ST.F12 admits future-conditioned selection of past measurement settings (the standard Wheeler delayed-choice paradigm); ST.T2 predicts that the pattern persists in the ODTOE limit and is consistent with V∗ (i). 3. τ -asymmetry of orbit density. Per V∗ (i), the past part of any maximising trajectory W is strictly denser in the goal-coherent measure Pgoal than the future part; this is a structural prediction of ST.T2 and is testable in any ensemble with sufficient trajectory statistics. The three signatures are catalogued more concretely as empirical predictions in §XI.

IX. COMPARISON WITH PHILOSOPHIES OF TIME The composite picture of ST.T1 admits comparison with standard positions in the philosophy of time. We organise the comparison along seven specific positions plus a brief structural counterpart in canonical quantum gravity.

IX.1. Block universe (eternalist thesis) The block-universe picture [18] treats all events past, present, and future as equally real on a four-dimensional manifold. ST.T1 is structurally distinct from block universe: the past-norm kπpast ΨkH accumulates non-decreasingly with iteration n [3, §V.5], while the future component is conditionally reconstructible and not pre-given. The blockuniverse symmetry between past and future is broken by V∗ (i)–(ii) at the structural level.

IX.2. Presentism Presentism [19] privileges the present instant as ontologically distinguished. ST.T1 is partially consonant with presentism: the present instant τobs = 0 is the structural locus of the operator action Φ(Ψn ) = Ψn+1 . However ST.T1 differs from strict presentism: the past component πpast Ψ retains its Hilbert-norm structure even when not presentinstant-actual; presentism in its strong form would deny ontological status to the past component beyond its present trace.

IX.3. Growing block (Brogan, Tooley) The growing-block view [20] treats the past and present as real and the future as notyet-real. ST.T1 sits closest to the growing-block view: past-conservation V∗ (i) makes the past component a non-decreasing accumulation, while the future component is conditionally reconstructible and not pre-given. The growing-block view receives a structural derivation here: the past-norm grows as n accumulates [3, §V.5], while the future component is reconstructible only above the threshold Srec .

IX.4. Bergson: durée vs temps Bergson’s durée [12] insists on the qualitative irreducibility of duration to spatialised time. The definition τ0 ∼ I(C)/α of ST.F7 makes τ0 a derived quantity tied to configuration inertia, not a homogeneous spatial coordinate; the iteration parameter is qualitative in the Bergsonian sense, while ST.T1(c) identifies dmin = 3 as the minimal spatial dimension that must be added separately. The Bergsonian distinction between duration and homogeneous time is reproduced in ODTOE as the distinction between τ0 (derived from I(C)) and the spatial coordinate of Rdmin .

IX.5. Husserl: proto-temporality Husserl’s analysis of inner time-consciousness [10] identifies a triadic structure of retention–present–protention. ST.T1(d) and ST.T1(e) supply a quantitative version: πpast realises the retention, the present instant τobs = 0 is the locus of operator action, and πfuture is the formal counterpart of protention. The proto-temporal triadic structure is therefore preserved at the structural level.

IX.6. Heidegger: Zeitlichkeit Heidegger’s Zeitlichkeit [11] insists that temporality is constitutive of the being of the existent rather than an external order. ST.T1’s claim that time emerges with ÔΨ∗ is structurally consistent with this thesis: time is not a pre-given background but a constitutive feature of the operator’s own structure at its birth event. The Heideggerian thesis receives a structural counterpart: τ0 is intrinsic to Φ, not an external parameter.

IX.7. Whitehead: actual occasion Whitehead’s process metaphysics [21] treats actual occasions as constitutive units of the temporal stream. The iteration Ψn+1 = Φ(Ψn ) realises a Whiteheadian “occasion” as one application of Φ at one τ0 : each iteration step is a discrete actualisation, the temporal stream is the sequence of such actualisations, and the past-norm of V∗ (i) plays the role of the cumulative ingression of past occasions.

IX.8. Wheeler–DeWitt timelessness (structural counterpart) The Wheeler–DeWitt equation [9] removes time as an independent parameter from canonical quantum gravity. ST.T1(a) gives a structural counterpart: time is not a preexisting parameter; it is an output of the iteration of Φ. The two are compatible if the Wheeler–DeWitt timeless wave function is identified with the substrate H (axiom A) and the time step τ0 is identified with the iteration parameter of Φ acting on H. The relational reading of time in canonical quantum gravity [22] is structurally consonant with the present picture.

X. COSMOLOGICAL CONSEQUENCES: BIG BANG AT THE B → 0 BOUNDARY X.1. Big Bang as primordial-distinction event (d = 9 closure) The Big Bang event is reinterpreted within the present synthesis. By [2, §IX(b)], the closure of the bootstrap loop Ψ∗ = Φ(Ψ∗ ) at the dimensionality level d = 9 identifies the Big Bang as a primordial-distinction event rather than a creation ex nihilo: the moment at which the ϕ-resonance is locked in and ÔΨ∗ becomes well-defined. ST.T1 supplies the simultaneity claim: at the same moment the time step τ0 , the orientation qÔ , the spatial dimension dmin = 3, and the projector pair πpast , πfuture all emerge. The Big Bang event is therefore not a creation from a temporal nothing; it is the closure of the bootstrap loop at d = 9, i.e. the structural birth event of ÔΨ∗ in the cosmologicalscale instantiation.

X.2. The boundary B → 0 as singularity The boundary B → 0 — the absorbing-boundary regime of theorem V∗ (iii) [3, §V.4] — is the dual of the birth event: as B → 0 the operator Ô degenerates, the future component πfuture Ψ vanishes, and only the bare past Ψbare = πpast Ψ survives. In the cosmological reading, the boundary B → 0 is a thermodynamic horizon at which the operator-driven dynamics ceases; the past, however, is preserved by V∗ (i). The boundary plays the role of a singularity in the operator-driven dynamics — see [27] for the dedicated treatment of the singular-boundary structure in the corpus — while the past component is preserved as the residual structural accumulation.

X.3. Micro-cosmological parallel The composite picture is thus that the universe has a bounded operator-driven span between the birth event (τ0 established) and the boundary (Ô degenerates), with the past as a non-decreasing structural accumulation throughout. The same structural pattern — a bounded operator-driven span between two structural events, with pastconservation throughout — is realised at every scale at which the V∗ projector pair is defined; the cosmological scale of the Big Bang and the boundary B → 0 is therefore one instance of a generic micro-cosmological pattern.

XI. EMPIRICAL SIGNATURES The present synthesis predicts six classes of empirical signatures organised below by source: three inherited from the parent articles (XI.1, XI.2, XI.6) and three novel to ST.T2 (XI.3, XI.4, XI.5). Past-conservation signatures inherited from [3, §V.6] underwrite all three of the inherited classes structurally and are not separately catalogued.

XI.1. Reuse: E8 symmetry in CoNb2 O6 (inherited) Any system whose long-time stability is governed by a KAM-type filter should exhibit a measurable bias toward ϕ-related rotation numbers. The Coldea et al. 2010 measurement of the E8 mass spectrum in the quasi-1D Ising chain CoNb2 O6 [25] supplies an empirical signature consistent with the ϕ-resonance structure of ST.T1: the lowest two excitation masses are observed at a ratio ϕ, in agreement with Zamolodchikov’s E8 prediction. ST.T1 does not introduce a new prediction here; it rests on this existing measurement as one of the inherited signatures of the ϕresonance lock-in.

XI.2. Reuse: Hardy probability ϕ−5 (inherited) A second inherited signature is the Hardy nonlocality bound: in Hardy’s two-particle scheme without inequalities [26], the maximal probability of the paradoxical event is ϕ−5 ≈ 0.0902. This ϕ−5 value is consistent with the ϕ-resonance structure of ST.T1 and supplies an additional inherited empirical signature.

XI.3. NEW: P (W ) goal-coherent timeline density The conjectural ST.T2 predicts that, among the trajectories W ⊂ Fix(Φ) available from the birth event, the trajectories with high goal-coherent density Z P (W ) = B(Ψ, n)α (1 − σ(Ψ, n))β dn (ST.F14) W

should be measurably preferred. The integral form is inherited from [4, §V]; the present prediction is that the preference is detectable in collective trajectories where Agoal is structurally identifiable (cooperative ventures with explicit shared targets, scientific research programmes with stated aims, social-historical movements with declared collective goals). The signature is that observed trajectory ensembles in such settings should concentrate on high-P (W ) trajectories. Falsification: a clean measurement showing equiprobable trajectories across the full Fix(Φ) slice (no preference for highP (W )) would falsify the ST.T2 selection mechanism.

XI.4. NEW: τ -asymmetry of orbit density A second novel signature follows directly from V∗ (i) in the P (W ) context. Per V∗ (i) the past part of any maximising trajectory W is strictly denser in the goal-coherent measure Pgoal than the future part: the past-component norm is non-decreasing under Φ-iteration, while the future-component norm is conditional on Sij ≥ Srec . Consequently, in trajectory ensembles that exhibit the high-P (W ) concentration of XI.3, the structural prediction is a τ -asymmetry of the orbit density: past-side density larger than future-side density at every iteration n along the maximising trajectory. The asymmetry is a structural consequence of the projector pair, not of the Agoal formalism, and is therefore robust to the open formalisation of Agoal .

XI.5. NEW: Wheeler retroactive signature A third novel signature follows from §VI: the variational structure of ST.F12 admits future-conditioned selection of past measurement settings (the standard Wheeler delayed-choice paradigm [16]). ST.T2 predicts that the Wheeler delayed-choice pattern persists in the ODTOE limit and is consistent with V∗ (i): the apparent retroactive influence is restricted to the future component πfuture Ψ, and the classical record at τobs shows the pattern of conditional reconstructibility consistent with §VI.2. The empirical signature is the Wheeler delayed-choice statistics; the structural prediction of ST.T2 is that the statistics are consistent with the iteration of Φ producing a self-consistent fixed point at τobs , with no Hilbert-norm component of πpast Ψ altered.

XI.6. Reuse: KAM-observable predictions (inherited) The general prediction inherited from the KAM-filter mechanism is that any system whose long-time stability is governed by a KAM-type filter should exhibit a measurable bias toward ϕ-related rotation numbers, frequency ratios, or mass ratios. The two specific instances XI.1 (E8 in CoNb2 O6 ) and XI.2 (Hardy ϕ−5 ) realise this general prediction. ST.T1 does not introduce a new KAM-observable signature beyond these instances; it rests on the inherited general prediction as the structural baseline for the empirical signatures listed above.

XII. OPEN QUESTIONS The present synthesis closes the four operator questions stated in §I in the form of theorem ST.T1 and conjecture ST.T2. It also opens six further subtasks for separate articles, listed below.

XII.1. Subtask 5.7 — axiomatic fixation of τ0 (formalisation of ST.F7) The time step τ0 ∼ I(C)/α inherited from [6, formula II.4] is a derived quantity in the present synthesis. A separate article is needed to determine whether τ0 should be promoted to an axiomatic constant of the corpus, with the inertia I(C) and the iteration intensity α derived from it, or whether the present derived form is structurally complete. The choice has consequences for the cosmological reading of §X: if τ0 is axiomatic, the Big Bang event has a fixed time scale; if τ0 is derived, the time scale is configuration-dependent.

XII.2. Subtask 5.8 — lower bound for Agoal (formalisation of ST.T2) No axiom-derived lower-bound argument exists yet for the LB-condition on Agoal . A sufficient (not necessary) condition is the gradient monotonicity h∇Ψ B, nAgoal i > 0

along the trajectory Wactual , where nAgoal is a normal vector field to Agoal ⊂ Fix(Φ). The full derivation of the necessity branch — and, in particular, an axiom-derived lower bound on the cardinality and structure of Agoal — is open subtask 5.7 of the corpus and the natural extension of the present article.

XII.3. Subtask 5.9 — operationalisation of Agoal Empirical verification of ST.F14 requires an operational measurement procedure for Agoal in conditioning datasets. Candidate approaches include: (i) observerstated goal coordinates obtained by structured elicitation (questionnaire/protocolencoded), with Agoal realised as the closure of the observer-stated coordinates under the corpus glossary; (ii) goal coordinates inferred from the observed trajectory itself by inverse-problem reconstruction, with the inverse-problem regularisation chosen to be consistent with the gradient monotonicity of XII.2; (iii) goal coordinates supplied externally by independent ethical/teleological criteria, with Agoal realised as the level set of the external criterion. The conjecture ST.T2 asserts that Agoal shapes only πfuture Ψ; a measurable consequence would be that, in trajectory ensembles with structurally identifiable shared goals (option (i) or (iii)), the futurecomponent reconstructibility above Srec shifts in a quantitatively predicted way (sharper convergence to high-P (W ) trajectories). The empirical design and the noise model are open; the operational formalisation of Agoal as a measurable subset of Fix(Φ) is a prerequisite.

XII.4. Subtask 5.10 — trajectory-space topology for the extremiser The cardinality of Fix(Φ) in the relevant regime (single fixed point, finitely many, or continuum) is structurally important for ST.T2: Agoal is non-trivial only in the multivalued regime. The spectral analysis of the kernel of [1, §VI.1, formula 5.1.F3] is the natural starting point; this is the open subtask 5.4 already noted in [1, §XI.3] and extended here to the birth-event setting. The full trajectory-space topology required for the extremiser-existence claim of ST.T2 (i.e., existence of a specific trajectory Wactual that intersects Agoal ) is therefore the next-order open subtask 5.10.

XII.5. Subtask 5.11 — link of ST.T1 to inflationary cosmology ST.F11 — the Big Bang event at d = 9 (i.e., the closure of the bootstrap loop at the dimensionality level d = 9) — gives a qualitative match to the inflationary paradigm: the moment of closure is the moment at which the operator ÔΨ∗ becomes well-defined and the operator-driven dynamics begins. Quantitative predictions (CMB amplitude, angular scale, ϕ-universal correlations in the cosmological data) require a separate cosmological article that supplies the inflationary normalisation; the link is open subtask 5.11.

XII.6. Subtask 5.12 — V∗ extension to multi-observer settings The present article is single-observer: the projector pair πpast , πfuture is defined for one observer’s world line W . Extension to multi-observer settings — the entanglement-aware generalisation of theorem V∗ in which two or more observers share configurations across H — is a separate question. The natural starting point is the entanglement-aware version of V∗ (i): for two observers Ô1 , Ô2 with shared past ∗ Ψpast 12 , what is the joint past-conservation law? The full multi-observer V is open subtask 5.12.

Appendix C. Computational Verification (mpmath, 50digit precision) The numerical values of the four constants used in §II–§III are verified independently with mpmath at 50-digit precision. The script and output are reproduced below. from mpmath import mp, mpf, sqrt mp.dps = 50 phi = (1 + sqrt(5)) / 2 phi_inv = 1 / phi phi_inv2 = 1 / phi**2 gamma_phi = 1 / sqrt(5)

# Value of the modulus at the KAM-selected golden point (B, S) = (phi_inv, B = phi_inv S = phi_inv q_val = BS + (1 - B) sqrt(1 - S**2) # phi = 1.6180339887498948482045868343656381177203091798058 # phi_inv = 0.61803398874989484820458683436563811772030917980576 # gamma_phi= 0.4472135954999579392818347337462552470881236719223 # q_val = 0.68224911725088275968210787558278824961032689402959 The four 50-digit values used in the article are: • ϕ = 1.6180339887498948482045868343656381177203091798058 • ϕ−1 = 0.61803398874989484820458683436563811772030917980576 • γφ = 0.4472135954999579392818347337462552470881236719223 • q (B=S) φ−1 = 0.68224911725088275968210787558278824961032689402959 The values agree to 50 digits with the independent mpmath computations and with the values reported in [1, §VI.4] and [2, §X.A].

CONFLICT OF INTEREST The author declares no conflict of interest.

FUNDING The research was conducted without external funding.

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