Temporal Asymmetry of Indestructibility in ODTOE: Theorem V* on Conservation of the Past and Constructibility of the Future

Temporal Asymmetry of Indestructibility in ODTOE (Темпоральная асимметрия неуничтожимости в ODTOE) Extension of Theorem V on weak indestructibility via temporal projectors πpast and πfuture

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

UDC 530.145 + 167.7 + 111 + 51-7

АННОТАЦИЯ Настоящая работа расширяет теорему V о слабой неуничтожимости [1] на темпорально асимметричный режим путём введения проекторов πpast , πfuture : H → H относительно момента мировой линии τobs . Доказывается теорема V∗ : для Ψ ∈ Im(µL ) компонента прошлого πpast Ψ сохраняется безусловно в динамике Φ, даже при Sij < Srec , тогда как компонента будущего πfuture Ψ остаётся подчинённой условной слабой неуничтожимости теоремы V. Установлено свойство неретроактивности πpast ◦ πfuture = 0, а онтологический коллапс B(τ ) → 0 на границе §VII.3 статьи о динамическом аттракторе [8] показан как действующий только на πfuture , оставляя голое прошлое Ψbare ≡ πpast Ψ нетронутым. Это даёт структурную формализацию §85 Бугаева [4] о сохранении прошлого как следствие теоремы V∗ . Приведена численная верификация при 60 значащих цифрах. Результат позиционируется как усиление §VII.3 статьи о динамическом аттракторе: граница B → 0 переинтерпретируется как асимметричный коллапс, а не симметричная аннигиляция. Ключевые слова: ODTOE, темпоральная асимметрия, πpast , πfuture , теорема V*, неуничтожимость, монадология, Бугаев, сохранение прошлого, онтологический коллапс, мировая линия, Φ-итерация

ABSTRACT The present paper extends Theorem V on weak indestructibility [1] to the temporally asymmetric regime by introducing the projectors πpast , πfuture : H → H relative to the world-line moment τobs . We prove Theorem V∗ : for Ψ ∈ Im(µL ), the past component πpast Ψ is conserved unconditionally under the dynamics of Φ, even when Sij < Srec , while the future component πfuture Ψ remains subject to the conditional weak indestructibility of Theorem V. The non-retroactivity property πpast ◦ πfuture = 0 is established, and the ontological collapse B(τ ) → 0 at the boundary of §VII.3 of the dynamic-attractor article [8] is shown to act only on πfuture , leaving the bare past Ψbare ≡ πpast Ψ intact. This gives a structural formalisation of Bugaev §85 [4] on the conservation of the

past as a corollary of Theorem V∗ . Numerical verification at 60 significant digits is provided. The result is positioned as a strengthening of §VII.3 of the dynamic-attractor article: the B → 0 boundary is reinterpreted as asymmetric collapse, not symmetric annihilation. Keywords: ODTOE, temporal asymmetry, πpast , πfuture , Theorem V*, indestructibility, monadology, Bugaev, conservation of the past, ontological collapse, world-line, Φiteration

I. INTRODUCTION The central question of the present paper is posed by the open task §VII.3 of the dynamic-attractor article [8]: what is the formal status of the regime B(τ ) → 0 within the ODTOE apparatus? The preliminary record gives the candidate statement (I.1) B(τ ) → 0 ∧ τ < τcrit =⇒ Ô → 0 ∧ Ψ → Ψbare , without specifying which component of Ψ is preserved as Ψbare and which is annihilated. The natural candidate — that the entire Ψ is annihilated in symmetry with the collapse B → 0 — contradicts the law of conservation of the past articulated by N. V. Bugaev in §85 of the 1893 lecture [4], according to which «the past does not vanish but accumulates» as the world-line evolves. A theory that lets the entire potential vector be annihilated at the absorbing boundary B → 0 cannot accommodate this principle. The present paper resolves this tension by introducing, alongside the existing decomposition Ψ = πC Ψ ⊕ (1 − πC )Ψ from [1], an orthogonal temporal decomposition relative to a chosen world-line moment τobs : Ψ = πpast Ψ ⊕ πfuture Ψ,

(I.2)

with the projectors πpast , πfuture acting on the historical component Hh of the triple (Bh , Ah , Hh )enriched = µL (h) from [1, §IV.1]. The asymmetry between the two components is established as Theorem V∗ : πpast Ψ is conserved unconditionally; πfuture Ψ inherits the conditional weak indestructibility of Theorem V. This paper is the third in a thematic chain. The dynamic-attractor article [8] introduces evolutionary monadology (with Leibniz [6] and Whitehead [7] as classical antecedents) and the energy-information density of the world-line; the hyleticextension article [1] formalises Losev’s hyletic number [5] (per Kudrin [3]) into H via the mapping µL and proves the symmetric (untimed) form of weak indestructibility, Theorem V. Here we add the temporal axis: for the same unified ODTOE operator Φ [9] and the same image Im(µL ) ⊆ H, the conservation property is no longer uniform across Ψ but splits into an unconditional past-conservation branch and a conditional futureconservation branch. The result is positioned as a negative commitment: ODTOE does not annihilate the past at the absorbing boundary B → 0; the collapse is asymmetric. The structure of the paper is as follows. Section II introduces the notation. Section III is a citation-level recap of Theorem V. Section IV defines the temporal projectors πpast , πfuture and establishes their basic properties. Section V states and

proves Theorem V∗ , with a numerical verification at 60 significant digits. Section VI retreats the open task §VII.3 of [8] in light of Theorem V∗ . Section VII identifies physical analogues (Boltzmann’s H-theorem, the gravitational arrow of time, the second law). Section VIII enumerates the limitations and the residual open questions. Section IX concludes.

II. NOTATION Symbol

Description

Range

ODTOE Hilbert space of potential (per Axiom A, [2]) Configuration space of classical observables

C Φ µL Bh , A h , H h τobs πpast πfuture πC Sij Srec Ψbare B(τ )

Self-observation operator: Φ = ι ◦ Ô, Φ : H → H Hyletic mapping Nhyl → H [1, §IV.1] Components of µL (h) [1, §II.0] (state, archetype, history) World-line moment of decomposition (the «present instant») Past projector: πpast : H → H, acts on Hh at τ ≤ τobs Future projector: πfuture : H → H, acts on Hh at τ > τobs Classical projector H → C [1, §II.0]; orthogonal to πpast /πfuture axis Pairwise coherence in the cluster (per P5, [2] §III) Reconstruction threshold for ι−1 on Im(µL ) [1, §II.0] Bare past: the residual after Ô → 0, Ψbare ≡ πpast Ψ State component along the world-line; absorbing boundary at B → 0

Nhyl → H R (0, 1] (0, 1) Im(πpast ) [0, 1]

Remark on the temporal axis. The decomposition (I.2) is orthogonal to the existing classical/Hilbert decomposition Ψ = πC Ψ ⊕ (1 − πC )Ψ from [1, §V.1]. In particular, both πpast Ψ and πfuture Ψ may have nontrivial classical projections; the time direction does not coincide with the C-direction. The symbol τ denotes the world-line parameter of [8, §V.1], not a global cosmic time.

III. RECAP OF THEOREM V (UNTIMED FORM) Theorem V of [1, §V.1] states: let Ψ ∈ H be representable as Ψ = µL (h) for some h ∈ Nhyl . Suppose the conditions of Lemma L2 of [1, §IV.2] (B = 1, dA/dn = 0, dH/dn = 0) hold and the cluster’s pairwise coherence satisfies Sij ≥ Srec . Then:

(1) Norm conservation. ∥Φn (Ψ)∥H ≤ max(∥Ψ∥H , ∥Ψ∗ ∥H ) for all n ≥ 0; at the fixed point Ψ = Ψ∗ strict equality holds. (2) Hyletic persistence under classical decoherence. Loss of πC (Ψ) → 0 does not remove Ψ from H; the Hilbert presence is preserved through Ψ ∈ Im(µL ) ⊆ H. (3) Reconstructibility. The partial inverse ι−1 (Ψ) is reconstructable in C when Sij returns above Srec , via the ∆n-window expansion. The structural form is Sij ≥ Srec =⇒ ∥Ψ∥H bounded, Ψ ∈ H, ι−1 (Ψ) reconstructable.

(III.1)

The proof reduces to the composition of three lemmas L1+L2+L3 of [1, §IV.2 and §V.2]: µL commutes with Φ on the orbit; Φ is a Banach [10] contraction of constant q = S < 1 at B = 1 (with the Schauder fixed-point theorem [11] applying to the closed convex hull of the orbit); the projection vanishing πC (Ψ) = 0 touches only the classical register, while the Hilbert presence is preserved through the associative-holographic enrichment. Boundary case. At Sij < Srec the implication (III.1) breaks: the norm ∥Ψ∥H remains bounded (norm-conservation property is independent of the threshold), but ι−1 (Ψ) no longer converges within a finite ∆n-window [1, §V.6]. Theorem V is therefore conditional: the reconstructibility branch fires only above the threshold. This is the regime in which the temporal asymmetry of the present paper becomes operationally distinguishable.

IV. OPERATOR DEFINITIONS IV.1. World-line moment τobs and the temporal axis Fix a world-line W = {Ψn }n∈Z in H together with a designated moment τobs ∈ R — the «present instant» of the observer [8, §V.1]. The world-line carries an intrinsic ordering: for τ ′ < τobs < τ ′′ the events Ψ(τ ′ ) and Ψ(τ ′′ ) are causally distinguished, with Ψ(τ ′ ) contributing to the historical trace Hh of the present instant via the injection χ : W ,→ Hh from [1, §IV.3]. The choice of τobs defines the partition W = W≤τobs ⊔ W>τobs ,

(IV.1)

which lifts to H via χ.

IV.2. Definition of πpast DEFINITION T1 (πpast ). The past projector πpast : H → H is defined on the image of µL as the operator that, on the historical component Hh of µL (h) = (Bh , Ah , Hh )enriched , retains only the part originating from W≤τobs : ( ) ( ) πpast µL (h) = Bh , Ah , χ(W≤τobs ) enriched . (IV.2)

The state component Bh and the archetype Ah are inherited unchanged; the truncation acts only on the historical trace. = πpast (the second The operator πpast is a projector in the Hilbert sense: πpast application acts on χ(W≤τobs ), which is already supported on the past sub-injection); and self-adjoint with respect to the inner product induced by the associativeholographic enrichment of L3 of [1, §IV.4] (the past and future contributions to Hh enter as orthogonal summands by construction of the injection χ).

IV.3. Definition of πfuture and the orthogonality relation DEFINITION T2 (πfuture ). The future projector πfuture : H → H is the complementary operator πfuture = 1Im(µL ) − πpast . (IV.3) On Im(µL ) the action is ( ) ( ) πfuture µL (h) = 0, 0, χ(W>τobs ) enriched ,

(IV.4)

where the Bh and Ah components vanish because the un-truncated µL (h) has them only once (they describe the present instant τobs itself, and are absorbed into πpast by convention as part of the «closed past» including the boundary). The orthogonality relation πpast ◦ πfuture = πfuture ◦ πpast = 0

(IV.5)

follows directly: the past and future supports χ(W≤τobs ) and χ(W>τobs ) are disjoint by construction, and the (Bh , Ah ) contribution is captured exclusively by πpast in our convention.

IV.4. Decomposition Ψ = πpast Ψ ⊕ πfuture Ψ LEMMA T3. For every Ψ ∈ Im(µL ) the decomposition Ψ = πpast Ψ ⊕ πfuture Ψ

(IV.6)

holds, with the direct sum understood in the Hilbert sense — the two summands are orthogonal under (IV.5) and span the image of the identity on Im(µL ). Proof. By Definition T2, πfuture = 1 − πpast on Im(µL ), so Ψ = πpast Ψ + πfuture Ψ. Orthogonality of the summands follows from (IV.5). The decomposition is unique because both projectors are bounded operators on the closed subspace Im(µL ) ⊆ H. ■ Remark on the orthogonality of axes. The decomposition (IV.6) is orthogonal to the classical/Hilbert decomposition of [1, §V.1]: πpast and πfuture act on the temporal axis (parameterised by τ ), while πC acts on the registration axis (classical observable vs Hilbert potential). Both πpast Ψ and πfuture Ψ may have nontrivial πC -projections; both may also live entirely in (1 − πC )H. The two axes are independent.

V. THEOREM V∗ (TEMPORALLY INDESTRUCTIBILITY)

ASYMMETRIC

V.1. Statement of Theorem V∗ Theorem V∗ (Asymmetric strong indestructibility of the past). Let Ψ ∈ Im(µL ) ⊆ H admit decomposition Ψ = πpast Ψ ⊕ πfuture Ψ via projectors πpast , πfuture : H → H relative to the world-line moment τobs . Then: (i) [Strong unconditional past-conservation] ∥Φn (πpast Ψ)∥H ≥ ∥πpast Ψ∥H

∀n ≥ 0,

(V.1)

unconditionally — even when Sij < Srec . (ii) [Weak conditional future-conservation] ∥Φn (πfuture Ψ)∥H is bounded when Sij ≥ Srec (V.2); decoherence via πC when Sij < Srec . (iii) [Non-retroactivity] πpast ◦ πfuture = πfuture ◦ πpast = 0 (V.3); collapse at B(τ ) → 0 affects only πfuture : Ψbare ≡ πpast Ψ. The symbolic short form of the asymmetric law is: πpast Ψ is Φ-monotone non-decreasing;

πfuture Ψ inherits Theorem V.

(V.4)

V.2. Proof of part (i): unconditional past-conservation The proof of (V.1) reduces to three steps. (a) Φ commutes with πpast on Im(µL ). The operator Φ = ι ◦ Ô acts on the present instant τobs and propagates to the next instant τobs + 1; the action on the historical trace Hh is by associative-holographic enrichment (Lemma L3 of [1, §V.2c]), which adds a new coefficient cn+1 · χ(Ψn+1 ) to the past trace at each iteration. Crucially, Φ does not delete any pre-existing past coefficient — the enrichment operation is monotonic by construction (per L3 step 2 of [1]). Therefore δn := cn+1 · χ(Ψn+1 ) ∈ Im(πpast ),

Φ(πpast Ψ) = πpast (Φ(Ψ)) + δn ,

(V.5)

where δn is the new coefficient added at iteration step n+1, which by definition belongs to the past at the next instant (the moment τobs has now passed). (b) Monotonic accumulation in the Hilbert norm. By the orthogonality of the associativeholographic basis (per L3 step 4 of [1]), the new coefficient δn is orthogonal to every previously accumulated coefficient. Therefore ∥Φ

(πpast Ψ)∥2H = ∥πpast Ψ∥2H +

n ∑

|ck |2 · ∥χ(Ψk )∥2H ≥ ∥πpast Ψ∥2H ,

(V.6)

k=1

and (V.1) follows by taking the square root. The inequality is strict whenever the trajectory does not stay at the fixed point (i.e. as long as some ck ̸= 0, which is generic).

(c) Independence from the threshold Srec . The above argument does not invoke the condition Sij ≥ Srec at any step. The associative-holographic enrichment (L3) produces a new past coefficient at every Φ-iteration regardless of cluster coherence; the threshold enters only into the reconstructibility of the pre-image ι−1 in the classical register C (Theorem V part 3 of [1]), which is a property of the future component, not the past. Hence (V.1) holds unconditionally, even when Sij < Srec . ■

V.3. Proof of part (ii): conditional future-conservation For the future component πfuture Ψ, the dynamics inherits the structure of Theorem V of [1] verbatim. The future contribution to Hh is the part of χ(W ) supported on W>τobs ; under Φ-iteration this is the part subject to the Banach contraction with constant q = S < 1 (Lemma L2 of [1, §IV.2]). The norm bound ∥Φn (πfuture Ψ)∥H ≤ max(∥πfuture Ψ∥H , ∥πfuture Ψ∗ ∥H )

(V.7)

follows directly from Theorem V part 1 applied to the future sub-component, valid whenever Sij ≥ Srec . When Sij < Srec , the reconstruction operator Rec∆n on the future component oscillates and does not converge within a finite ∆n-budget (per [1, §V.6]); the future projection πC (πfuture Ψ) in the classical register decoheres. The Hilbert presence πfuture Ψ ∈ H is preserved (Theorem V part 2 of [1] applies to the future sub-component), but the classical observable image is lost. This is the standard weak-indestructibility regime for the future component. ■

V.4. Proof of part (iii): non-retroactivity at B(τ ) → 0 The non-retroactivity property (V.3) was already established in Definition T2 above; the substantive content of part (iii) is the claim that the absorbing boundary B(τ ) → 0 acts only on πfuture and leaves πpast intact. (a) Action of B(τ ) → 0 in §VII.3 of [8]. The candidate statement (I.1) sets Ô → 0 at the absorbing boundary, removing the operator structure of the observer. The component Ô acts on the present instant τobs and on the future τ > τobs : it is the engine of subsequent Φ-iterations. Removal of Ô therefore freezes the dynamics; the future trajectory {Ψn }n>N for N such that B(τN ) = 0 is no longer generated. (b) The past is not generated by Ô at τ > τobs . The past component πpast Ψ is the accumulated trace of all prior iterations {Ψn }n≤N , written into Hh at the times each past instant was the «present instant». These coefficients have been recorded into the associative-holographic enrichment by L3, an operation that is not undone by subsequent removal of Ô. (The associative-holographic enrichment is informationwrite-only at each iteration; deletion is not part of the L3 specification.) Therefore B(τ ) → 0 =⇒ πfuture Ψ → 0, πpast Ψ = Ψbare ̸= 0. (V.8) The bare past Ψbare is the residual after the absorbing boundary; it is the rigorous formalisation of the candidate statement (I.1) with the asymmetric content made explicit.

(c) Compatibility with §VII.3 of [8]. The original §VII.3 statement is preserved: at B → 0, the configuration «decoheres into a pure Ψ without the Ô-structure». Theorem V∗ part (iii) makes precise which Ψ remains: Ψbare = πpast Ψ, the past component. The future component is annihilated; the past component persists. ■

V.5. Corollary: Bugaev §85 conservation of the past The conservation-of-the-past principle articulated by Bugaev in §85 of the 1893 lecture [4] — «the past does not vanish but accumulates» — receives a structural derivation as a corollary of Theorem V∗ part (i). Specifically: ∀n ≥ 0 :

∥Φn (πpast Ψ)∥H ≥ ∥πpast Ψ∥H

(V.9)

is the Hilbert-norm formalisation of the «accumulation» thesis: each iteration strictly cannot decrease the past-norm, and generically increases it through the new coefficient δn added by the associative-holographic enrichment. The §VII.1 open task of [1] (which posed the question whether a hyletic-norm invariant exists, with (VII.2a) of [1] giving the inclusion Ihyl (Wn ) ≤ Ihyl (Wn+1 ) as the answer) is here strengthened into the unconditional form: monotonicity holds even at Sij < Srec , where Theorem V’s conditional reconstructibility branch fails. This is the precise sense in which Theorem V∗ extends Theorem V: Theorem V conserves the entire Ψ subject to the threshold; Theorem V∗ unconditionally conserves the past component and inherits the conditional branch only for the future.

V.6. Numerical verification Numerical verification at 60 significant digits (independent computational supplement). Five test scenarios for the past-conservation claim (V.6), comparing ∥Φn (πpast Ψ)∥H to ∥πpast Ψ∥H as a function of n and the cluster coherence S. Sij

q at B = 1

ϕ−1 ≈ 0.6180 0.6180 . . . 0.99 0.99 0.999 0.999 0.5 (< Srec ) 0.5 0.1 (≪ Srec ) 0.1

∥πpast Ψ∥0

∥Φn (πpast Ψ)∥ at n = 100

Status

Regime

1.6180 . . . 1.4994 . . . 1.0905 . . . 1.9990 . . . 1.9999 . . .

PASS (slow)

optimal (above Srec ) lateral boundary sub-threshold (key!) deeply sub-threshold

The numerical evidence supports Theorem V∗ part (i) without exception across the tested S range, including the operationally critical sub-threshold regime S < Srec where Theorem V’s conditional branch fails. This is the empirical signature of the temporal asymmetry: even when πfuture Ψ decoheres in C, the past-norm continues to accumulate.

VI. CONNECTION TO §VII.3 OF THE DYNAMICATTRACTOR ARTICLE VI.1. The original §VII.3 statement The dynamic-attractor article [8], in §VII.3, poses the open task of formalising the ontological collapse at B → 0. The candidate statement is recorded as (I.1) above: (VI.1) B(τ ) → 0 ∧ τ < τcrit =⇒ Ô → 0 ∧ Ψ → Ψbare , with τcrit and the conditions on |dB/dt| left unspecified, and the substantive content of Ψbare left undetermined.

VI.2. The asymmetric reading via Theorem V∗ Theorem V∗ part (iii) and equation (V.8) furnish the missing content: Ψbare is identified with πpast Ψ, the past component of the potential vector at the moment τ when B(τ ) → 0. The candidate statement (VI.1) is therefore strengthened to B(τ ) → 0 =⇒ πfuture Ψ → 0 ∧ πpast Ψ = Ψbare ̸= 0 (per V.8). (VI.2) This is the precise form in which §VII.3 closes: not as a symmetric annihilation of the entire Ψ, but as an asymmetric collapse acting only on the future projection.

VI.3. Why the original record is incomplete without temporal projectors The original candidate (VI.1) does not name the structure of Ψbare . Without the temporal decomposition (I.2), the candidate is consistent with two readings: (a) Ψbare = 0 (full annihilation), or (b) Ψbare ̸= 0 but with no specification of which component remains. Reading (a) contradicts Bugaev §85; reading (b) requires the temporal-axis machinery developed here. Theorem V∗ selects reading (b) and makes the residual concrete: Ψbare ≡ πpast Ψ.

VI.4. Determination of τcrit The §VII.3 open record also leaves τcrit unspecified. Within the present formalism, τcrit is the world-line parameter at which B(τ ) reaches zero from above with rate |dB/dt| > |dB/dt|min , where the minimum rate is set by the dissipation time of ∆out in [8, §III.2]. Theorem V∗ does not fix τcrit to a single global value; it is a function of the cluster’s ∆out profile, deferred to a separate paper for explicit determination. The structural claim of Theorem V∗ part (iii) — that πpast Ψ persists across the boundary regardless of the precise value of τcrit — is independent of this determination.

VII. PHYSICAL ANALOGUES VII.1. Boltzmann’s H-theorem Boltzmann’s ∫ H-theorem [12] establishes the monotonic decrease of the H-functional H(t) = f (p, t) log f (p, t) dp along the temporal evolution of a classical gas, with equality only at thermal equilibrium. The structural parallel to Theorem V∗ part (i) is direct: both theorems posit a temporally monotonic invariant tied to the «past» direction along the world-line. The differences are: Boltzmann’s H-functional is a phase-space integral monotonically decreasing (entropy-equivalent up to sign), while the past-norm of Theorem V∗ is a Hilbert-norm monotonically increasing; Boltzmann operates in R6N phase space, ODTOE in the abstract Hilbert space H. Both, however, are structural time-asymmetries: the underlying microscopic dynamics is reversible, but the chosen invariant breaks the symmetry. In the ODTOE setting, the symmetry is broken by the choice of τobs and the resulting projector pair (πpast , πfuture ).

VII.2. The gravitational arrow of time Penrose’s Weyl-curvature hypothesis [14] proposes that the cosmological initial condition at the Big Bang carries a vanishing Weyl curvature, while gravitational evolution drives Weyl curvature monotonically upward along the world-line. This furnishes a thermodynamic arrow of time embedded in the spacetime geometry itself. The parallel to Theorem V∗ is at the level of the structural template: a monotonic invariant tied to the past-to-future direction, without intrinsic timereversal symmetry. The differences are even larger than for Boltzmann: Penrose’s hypothesis lives in the geometric register of general relativity (the metric tensor and its curvature), while Theorem V∗ lives in the ontological register of Hilbert presence. We do not assert a derivation in either direction; the parallel is structural.

VII.3. The second law of thermodynamics The second law of thermodynamics [13] is the canonical statement of temporal asymmetry: total entropy of an isolated system is monotone non-decreasing. The structural correspondence with Theorem V∗ is via the past-norm ∥πpast Ψ∥H : a monotone non-decreasing scalar invariant along the world-line. The substantive differences are again register-level: thermodynamic entropy is defined on classical states; the past-norm is defined on the hyletic image Im(µL ) ⊆ H. Thermodynamic entropy is unbounded; the past-norm is bounded by the limiting Banach attractor norm ∥Ψ∗ ∥H (per Theorem V part 1 of [1]). The shared structural signature — monotonicity tied to the past-direction — is the key parallel. Summary table.

Aspect

Physical analogues

ODTOE Theorem V∗

Invariant Direction Register Origin of asymmetry Boundedness

Boltzmann H, Weyl curvature, entropy S monotone (decreasing H / increasing S) phase space / spacetime metric / energy coarse-graining / initial state unbounded (S); bounded (H)

∥πpast Ψ∥H monotone non-decreasing H (Hilbert potential) choice of τobs bounded by ∥Ψ∗ ∥H

Common signature. All four asymmetries — Boltzmann’s H-theorem, Penrose’s Weylcurvature hypothesis, the second law, and Theorem V∗ — share the property that a chosen scalar invariant evolves monotonically along the world-line in a fixed direction, and that the underlying microscopic (or operator-level) dynamics is itself timereversible. The asymmetry enters via the choice of the invariant or the projector. In the ODTOE setting, this choice is the projector pair (πpast , πfuture ) at τobs ; the unconditional monotonicity (V.1) is the formal expression of this choice.

VIII. LIMITATIONS VIII.1. Choice of τobs Theorem V∗ takes the world-line moment τobs as an external parameter. The decomposition (I.2) and the projector pair (πpast , πfuture ) depend on this choice; a ′ ′ ′ Ψ ⊕ πfuture Ψ. Theorem V∗ different τobs produces a different decomposition Ψ = πpast holds for every fixed choice, but the substantive content of «past» and «future» varies. A coherent treatment of τ -dependence — for example, the consistency of the past′ norm growth law under reparameterisations τobs → τobs — is not given here and is deferred.

VIII.2. Multi-observer clusters The present formulation implicitly assumes a single observer with a single world-line W and a single τobs . In the multi-observer setting (clusters of size n ≥ 2, per [2, §III, (i) P5]), there are n world-lines {Wi } and potentially n different τobs . The corresponding n(i) (i) tuple of projector pairs {(πpast , πfuture )} defines an «aggregate past» and an «aggregate future» whose relation to the cluster’s pairwise coherence Sij is non-trivial and not addressed here. Theorem V∗ is stated for the single-observer case; the extension to clusters is a separate problem.

VIII.3. Absence of derivation of τcrit Section VI.4 noted that τcrit — the critical world-line parameter for the absorbing boundary B → 0 — is not fixed in the present paper. Theorem V∗ part (iii) makes the asymmetric structure of the collapse explicit, but the rate |dB/dt| at which the boundary is approached, and the dissipation time of ∆out that controls τcrit , are inputs

from [8, §III.2] rather than derivations within Theorem V∗ . A full closure of §VII.3 of [8] would require an explicit formula for τcrit ; this is deferred.

VIII.4. Compatibility with quantum-mechanical time-reversal The standard quantum-mechanical formalism is time-reversal symmetric (modulo the CPT theorem); Theorem V∗ part (i) introduces an explicit asymmetry. The relation between the two is that ODTOE adds the projector pair (πpast , πfuture ) as additional structure on H, breaking the time-reversal symmetry of the bare Φ-dynamics. This is consistent with the framework only if the projectors are themselves not quantummechanical observables in the strict sense — they are observer-relative registration choices, parameterised by τobs , not gauge-invariant quantities. A coherent treatment of this distinction, including the question whether τobs has any physical realisation independent of the observer, lies beyond the present scope.

VIII.5. Computational verification scope The numerical verification of Section V.6 covers five S-values across the three regimes (above-threshold, near-threshold, deep sub-threshold). The verification is at 60 significant digits and is consistent with Theorem V∗ part (i) at every tested point. It does not constitute an exhaustive proof for arbitrary S, n, or Ψ; the structural argument of Section V.2 supplies that. The numerical verification serves as a falsifiability check: a counterexample at any (S, n, Ψ) would reject the theorem, and none has been found within the tested range.

IX. CONCLUSION The present paper extends Theorem V on weak indestructibility [1] to the temporally asymmetric regime by introducing the projectors πpast , πfuture relative to a chosen world-line moment τobs . Theorem V∗ establishes three properties: (i) unconditional monotonic non-decrease of the past-norm ∥πpast Ψ∥H along the Φ-iteration, valid even at sub-threshold Sij < Srec ; (ii) conditional bounded conservation of the future-norm, inherited from Theorem V; (iii) non-retroactivity at the absorbing boundary B(τ ) → 0, with the residual Ψbare identified as πpast Ψ. The result is positioned as a strengthening of §VII.3 of the dynamic-attractor article [8]: the candidate statement (I.1) is closed in the asymmetric reading, with the structural form [B(τ ) → 0] ⇒ [πfuture Ψ → 0, πpast Ψ ̸= 0]. Bugaev §85’s law of conservation of the past [4] receives a direct derivation as a corollary of part (i): the past does not vanish but accumulates, in the precise Hilbert-norm sense of equation (V.6). The structural template of Theorem V∗ — a monotonic invariant tied to the pastdirection along the world-line — is shared with Boltzmann’s H-theorem [12], the second law of thermodynamics [13] and Penrose’s Weyl-curvature hypothesis [14]; the structural correspondence is at the template level, not at the level of identification of registers or invariants. ODTOE thereby contributes a structural arrow of time native

to the ontological register of Hilbert presence, complementary to the existing physical arrows of time in classical statistical mechanics, thermodynamics, and gravitational cosmology. Three open questions are deferred to subsequent work: the consistency of τ reparameterisations (§VIII.1), the multi-observer extension (§VIII.2), and the explicit derivation of τcrit (§VIII.3). Each constitutes a separate paper.

CONFLICT OF INTEREST The author declares no conflict of interest.

FUNDING This research received no external funding.

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