# The Intrinsic Rest Frame of Light in ODTOE: Projective Identity 0≡∞ on the Φ-Iteration Spectrum

> Theorem 1: on the spectrum of Φ-iteration frequencies, points ν_Φ=0 (light in own rest frame) and ν_Φ=∞ (light everywhere simultaneously) are identical, forming projective point [0:1]∈RP¹. Speed of light c=r₀/τ₀ is unique continuous extension. Key premise: τ₀ calibrated INDEPENDENTLY of c via P2 inertia formula. Resolves paradox «light stands still ↔ light is everywhere».

Source: https://odtoe.org/en/articles/light-intrinsic-rest-frame
Author: Anton Pankratov · Observer-Dependent Theory of Everything (ODTOE) · CC BY 4.0

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THE INTRINSIC REST FRAME OF LIGHT IN ODTOE: PROJECTIVE IDENTITY 0 ≡ ∞ ON THE Φ-ITERATION SPECTRUM (Собственная система покоя света в ODTOE: проективное тождество 0 ≡ ∞ на спектре Φ-итераций) A theorem on the gluing of zero and infinity on the νΦ spectrum as the structural ground of the speed c

Pankratov Anton Sergeevich Панкратов Антон Сергеевич Independent researcher, Kazan, Russia E-mail: anton.s.pankratov@gmail.com ORCID: 0009-0002-4870-2995

## UDC 530.145 + 535.1 + 530.12

ABSTRACT Within the Observer-Dependent Theory of Everything (ODTOE), Theorem 1 is formulated: on the spectrum of Φ-iteration frequencies of the self-observation operator, the points νΦ = 0 (light in its own rest frame) and νΦ = ∞ (light everywhere simultaneously) coincide and form a single projective point [0 : ∞] ∈ RP 1 . The observed speed of light c = r0 /τ0 (where r0 and τ0 are the elementary spatial and temporal scales of the φ-torus) is the unique continuous extension of the spectrum at this √ point. The thesis is consistent with the Banach contraction constant q = B · S + (1−B) 1 − S 2 < 1 of the operator ΦB,S and with the invariance of c (Postulate P5). Key structural premise: the scale τ0 is calibrated INDEPENDENTLY of c — via the P2 inertia formula (τ0 ↔ Imin + ε); this prevents Theorem 1 from collapsing into a tautology. The thesis resolves the apparent paradox “light stands still ≡ light is everywhere” through a geometric identification of degenerate limits on RP 1 , without appealing to superluminal-signal hypotheses and without violating P5. Keywords: ODTOE, speed of light, projective geometry, self-observation operator, Φ-iterations, Banach fixed point, intrinsic rest frame, tact frequency, φ-torus, RP 1 .

АННОТАЦИЯ В рамках наблюдатель-зависимой теории всего (ODTOE) формулируется Теорема 1: на спектре частот Φ-итераций оператора самонаблюдения точки νΦ = 0 (свет в собственной системе покоя) и νΦ = ∞ (свет всюду одновременно) тождественны и образуют единую проективную точку [0 : ∞] ∈ RP 1 . Наблюдаемая скорость света c = r0 /τ0 (где r0 и τ0 —

элементарные пространственный и временно́й масштабы φ-тора) есть единственное непрерывное продолжение спектра √ в этой точке. Тезис согласован с банаховой сжимаемостью q = B · S + (1 − B) 1 − S 2 < 1 оператора ΦB,S и с инвариантностью c (Постулат P5). Ключевая структурная посылка: масштаб τ0 калибруется НЕЗАВИСИМО от c — через инерционную формулу постулата P2 (τ0 ↔ Imin + ε); это исключает превращение Теоремы 1 в тавтологию. Тезис разрешает кажущийся парадокс «свет стоит ≡ свет всюду» через геометрическую идентификацию вырожденных пределов на RP 1 , не прибегая к гипотезам сверхсветовой передачи и не нарушая P5. Ключевые слова: ODTOE, скорость света, проективная геометрия, оператор самонаблюдения, Φ-итерации, банахова неподвижная точка, собственная система покоя, тактовая частота, φ-тор, RP 1 .

Conventions and Origin of Symbols Below we summarize the key symbols used in what follows. For six existing symbols from the corpus [15] the source is indicated; eight NEW symbols are introduced in the present paper and marked with ∗ .

Symbol

Description

Hilbert space of potential states

C Ô ι Φ Ψ∗ B S q r0 , τ0 c νΦ∗ τstep τintr τobs νobs RP 1∗

## [0 : ∞]∗ T1 , T 2 , T 3 ∗

Source

Axiom A Configuration space (observed reality) ODTOE [15] Observation operator (projection H → C) (A.1) Embedding operator (C → H) ODTOE [15] Self-observation map: Φ = ι ◦ Ô (II.1) Fixed point: Ψ∗ = Φ(Ψ∗ ) Prop. 4 Cognitive coherence of the observer ([0, 1]) (D1.1) Synchronization level / embedding density ODTOE [15] ([0, 1]) Banach√ contraction constant: q = B · S + [17], (4.4) (1−B) 1−S 2 Elementary spatial and temporal scales of the [16], (III.5) φ-torus Actualization-front velocity: c = r0 /τ0 [16] NEW. Tact frequency of Φ-iterations: νΦ ≡ This paper, §IV 1/τstep NEW. Duration of one Φ-tact This paper, §IV NEW. Photon’s intrinsic proper time (limit S → This paper, §IV 1) NEW. Photon’s observed proper time This paper, §IV NEW. Observed photon frequency (≤ νPlanck ) This paper, §IV NEW in corpus. Projective line This paper, §IV (compactification R+ ∪ {0, ∞}) (Penrose [3], §15.4) NEW. Projective pole — the single point of This paper, §IV gluing 0 ≡ ∞ on RP 1 NEW. Three propositions about light’s intrinsic This paper, §I rest frame (see §I)

Convention: ∗ denotes a symbol INTRODUCED or REDEFINED in the present paper; the remaining symbols are used in exactly the sense fixed in source [15] and consistent with the ODTOE glossary.

I. INTRODUCTION In 1905, Einstein in Annalen der Physik formulated the postulate of the invariance of the speed of light and at the same time raised a question to which the formalism of special relativity (SR) gives no direct answer: in what frame of reference is light at rest? The standard reply is “no such frame exists”: Lorentz transformations are

singular at v → c, and the photon has no proper rest frame in the sense of an inertial Minkowski frame. Mermin [6] calls the habit of skirting this question “a bad one”; Wheeler and Feynman [10], in their absorber theory, note that “there is no proper time for the photon” in the usual sense, but they do not give an ontological interpretation. Two intuitions coexist: T1 — “light stands still in its own rest frame” (the standard textbook argument: at v = c proper time ∆τ = 0), and T2 — “light is everywhere simultaneously” (the quantum-mechanical intuition of entanglement and nonlocality [12]). T1 and T2 appear incompatible: the first speaks of immobility, the second of infinite speed. The fundamental question: are T1 and T2 two incompatible descriptions, or two projections of one object? ODTOE [15] (Observer-Dependent Theory of Everything) provides an operatoralgebraic mechanism for resolving this duality, without appealing to either an A-theory or a B-theory of time. The key object is the self-observation map Φ = ι ◦ Ô : H → H acting on the Hilbert space of potential states H. The spectrum of frequencies of iterations of Φ — call it νΦ — is a structural object accessible to projective-geometry and operator-theoretic analysis. In the limit cases: νΦ = 0 (no iterations, statics, T1) and νΦ = ∞ (instantaneous iteration, T2), the spectrum contains two degenerate points. The present paper asserts that these points are identical on the projective line RP 1 . Three propositions are formulated: • T1. In the limit of light’s intrinsic rest frame (S → 1, observer B = 1, A-invariant, H-stable), the photon’s intrinsic proper time τintr → 0, which is equivalent to νΦ → ∞ in the spectrum of Φ-iterations. • T2. In the H-picture, where entangled states are sections of one and the same object [15, §IV], the point νΦ = 0 corresponds to “light is everywhere simultaneously” — the absence of iteration as an act of differentiation. • T3. The speed of light c = r0 /τ0 is the structural maximum of νΦ , the unique continuous extension of the spectrum at the projective point [0 : ∞]. T1 and T3 are partially derived in the paper [15] (§III.5: c = r0 /τ0 as the actualization-front velocity; §III.4: “the limit c does not extend to H”). T2 is a NEW result, formalized via the standard projective construction on RP 1 . The joint implication T1 ⇔ T2 ⇔ T3 is the content of Theorem 1. Theorem 1 (preliminary statement). For any ODTOE observer (B = 1, Ainvariant, H-stable), the points νΦ = 0 and νΦ = ∞ in the spectrum of Φ-iterations are identical as a single projective point [0 : ∞] ∈ RP 1 , and the value c = r0 /τ0 is the unique continuous extension of the spectrum at this point. The substantive contribution of the paper has three dimensions. (a) Geometric: the projective gluing 0 ≡ ∞ on RP 1 is a natural and standard construction (Penrose [3], §15.4); its application to the spectrum νΦ of ODTOE is new. (b) Logical: T1, T2, and T3 are three charts on the same projective point; their distinctness is an artefact of the choice of affine chart, not a distinction of physical phenomena. (c) Epistemological: the observed frequency νobs is a property of the PAIR “observer + photon”, not of the photon alone; whereas c is a structural invariance of the φ-torus (Postulate P5), preserved across all recursion levels. Falsifiable hypothesis: the numerical criterion

C6a (see §VII in the full version) distinguishes Theorem 1 from a tautological definition c = r0 /τ0 via verification of the independence of the τ0 calibration from c — see §IV. Article structure: an unnumbered “Conventions and Origin of Symbols” block (after the Abstract) — notation; §II — literature review and the paper’s place in the ODTOE corpus; §III — recapitulation of the ODTOE Φ-formalism with verbatim citations; §IV — definition of νΦ and the projective gluing on RP 1 as NEW material; §§V–X (full version): proof of Theorem 1, equivalence T1⇔T2⇔T3, the numerical falsifier C6a, discussion, and limitations.

II. LITERATURE REVIEW AND POSITION IN THE ODTOE CORPUS The question of light’s intrinsic rest frame has a long prehistory. Wheeler and Feynman [10], in the absorber theory (1945), introduced the idea of a “nonlocal” interaction of source and absorber as one object; Cramer [11], in the transactional interpretation (1986), generalized this scheme as a “handshake” between retarded and advanced waves. Both approaches anticipate the ODTOE picture in which source and absorber are sections of a single ΨAB , projected onto two points of C [15]. However, neither formulation provides an explicit geometric model of the “intrinsic rest frame” — it remains a metaphor, not a mathematical object. Bell-type experiments [12] and their subsequent realizations (Aspect, Hensen, Giustina — review in [15]) established that nonlocal correlations are real and irreducible to hidden variables. Mermin [6], in the monograph It’s About Time (2005), devotes an entire chapter to “what does it mean for a photon to ‘stand still in its own rest frame’ ”, concluding that the question requires going beyond SR. ODTOE formalizes that move via the structure of the self-observation operator Φ. Bondi [5], in Relativity and Common Sense (1964), proposes the k-calculus as a pedagogical tool to intuit Lorentz transformations, but does not address the ontology of light’s intrinsic rest frame. MTW [1] (Gravitation), Wald [2] (General Relativity), and Rindler [4] (Relativity: Special, General, and Cosmological) treat the light cone and null geodesics as purely geometric objects, without operator content. On the side of “light as a structural front”, alternative programs have been pursued. Volovik [14], in The Universe in a Helium Droplet (2003), showed that the effective speed of excitations in superfluid media is a structural characteristic of the medium, not a fundamental constant; at low energies, “emergent” Lorentz-invariant sectors arise. This supports the ODTOE interpretation of c as the structural ratio r0 /τ0 of the φ-torus, rather than a property of the particle. Discrete approaches — ’t Hooft’s cellular-automaton interpretation (see [15] for review) and Wolfram’s physics (see review in [15]) — propose a pre-geometric discretization of spacetime. ODTOE differs in that the discreteness is introduced not in spacetime itself but in the spectrum of Φ-iterations acting on the Hilbert potential H; spacetime remains continuous as a projection in C. Projective methods in physics are a standard discipline since the era of Klein and

Cayley. Penrose, in The Road to Reality [3], §15, systematizes the role of projective geometry in the foundations of physics (from twistors to the conformal structure of the light cone). The key recipe: compactification of R+ to RP 1 via the gluing of antipodes 0 ∼ ∞ — a standard operation of Riemann-surface theory. Application of this operation to the spectrum νΦ of ODTOE is new; no other works known to us use projective geometry SPECIFICALLY for resolving the duality “light stands still / light is everywhere” as of submission. It is important to emphasise the metrological status of c. Since 2019, the speed of light c = 299 792 458 m/s is an exact defining constant of the International System of Units (BIPM CGPM 2018, Resolution 1) [7]. This means that c functions as a structural scale via which the metre is DEFINED; the question of its “empirical measurement” reformulates as a question of metrological consistency of standards. This position of the metrological community is consistent with the ODTOE interpretation of c as a structural ratio of φ-torus scales, rather than as a kinematic speed. The causal-set program [8] postulates a fundamentally discrete causal structure of spacetime. ODTOE and causal sets share the thesis that classical continuous spacetime is an emergent object; they differ in that, in ODTOE, the discretization is set by the iterations of Φ on the φ-torus, while in causal sets it is given by the stochastic seeding of points. Bekenstein [9], in his work on the information bound for black holes (1981), introduced the universal upper bound S ≤ 2πkB ER/(h̄c) — the first publication in which c appears explicitly as a STRUCTURAL ingredient of thermodynamics, not as a velocity. Cramer [11] (transactional) and Putnam [13] (Time and physical geometry, 1967) approach the status of simultaneity from opposite sides: Cramer preserves Lorentz invariance via a doubled wave formalism, while Putnam argues that SR entails an ontological B-theory of time. ODTOE places its apparatus at a level ABOVE both — the self-observation operator Φ is neutral with respect to the choice between A and B; the result T1⇔T2 does not depend on whether the “moment now” moves along the world line or not. Summary of the review: EACH of the individual axes (nonlocality, emergent c, projective geometry in physics, discrete spectrum) has a rich prehistory. The claim of the present paper is a NEW CONJUNCTION of those axes: the explicit construction of the projective gluing 0 ≡ ∞ on the spectrum νΦ of ODTOE with a c-independent calibration of τ0 via the inertia formula P2. An external scan (RT-1 Analyst) found no publications realizing this conjunction.

III. ODTOE Φ-FORMALISM: RECAPITULATION In this section we briefly reproduce the formulas of the corpus [15] needed for what follows. All cited formulas are quoted verbatim, without re-derivation; the new derivation (Theorem 1) appears in §IV–§V. Axiom A fixes the basic relation between the potential and the observed reality: R = Ô(Ψ),

Ψ ∈ H,

Ô : H → C.

## (III.1)

Formula (III.1) is the rewriting of (A.1) of ODTOE [15] in the present paper’s notation.

The composition of the embedding operator and the observation operator gives the self-observation map (the strange loop): Φ = ι ◦ Ô,

Φ : H → H.

## (III.2)

Formula (III.2) coincides with (4.3) of paper [17] (unified operator, §IV.3). The existence and uniqueness of the fixed point Ψ∗ = Φ(Ψ∗ ) is guaranteed by the Banach theorem [17, §IV.4]: the operator ΦB,S is a contraction with constant q = B · S + (1 − B) 1 − S 2 , q < 1 for B, S ∈ (0, 1). (III.3) The condition q < 1 holds for any observer with nonzero coherence and nonzero embedding density; the only exception is the degenerate cases B = 0 or S = 0, in which the self-observation cycle collapses. Postulate P3 of ODTOE [15] specifies the configuration lifetime: T (C) =

## T0 , (1 − S)n

T (C) → ∞ as S → 1.

## (III.4)

In the limit of full coherence S = 1, the configuration lifetime diverges. This limit is key for §V (proof of Theorem 1, lemma L3). The speed of light in the ODTOE picture is cited verbatim from the paper on light teleportation [15]: c=

r0 = const τ0

for all recursion levels d.

## (III.5)

Formula (III.5) coincides with (III.5) of paper [15] (light teleportation, §III.2). At each recursion level d, both scales rd = r0 · φd and τd = τ0 · φd stretch by a factor φ, and their ratio cancels identically; c is a structural invariance of the φ-torus. Critical quote (verbatim from [16] §III.4): “The limit c = r0 /τ0 is absolute for sequential transitions in C, but does not extend to H, where the notion of distance is undefined.” This quote is the structural gap in the formalism of [15]: in H there is no defined “speed” or “motion”, so posing the question of light’s intrinsic rest frame in H is correct only via an alternative object — the spectrum of frequencies of Φiterations. This is exactly the gap our paper closes: §IV introduces νΦ , and Theorem 1 formalizes “light in H” as the projective point [0 : ∞] ∈ RP 1 , without contradicting P5 (c-invariance). Note that our construction PRESERVES P5 of ODTOE [15]: the value of c remains a structural invariance of the φ-torus. What depends on the observer is νobs (the observed frequency, bounded above by νPlanck via the operator-window width ∆n), but not c itself. This distinction is the substantive part of §IV.

IV. νΦ AND THE PROJECTIVE GLUING 0 ≡ ∞ In this section we formalize the key NEW notion — the spectrum of Φ-iteration frequencies νΦ — and construct the projective gluing 0 ≡ ∞ on RP 1 . The section contains five substantive statements (IV.1–IV.5), each accompanied by an explicit marker of epistemic position.

IV.1. Definition of the spectrum νΦ Let an observer (B = 1, A-invariant, H-stable) realize a sequence of Φ-iterations Ψ0 , Ψ1 , . . . , Ψn , . . . on H, where Ψn+1 = Φ(Ψn ). Denote by τstep the duration of one iteration in the observer’s intrinsic frame. The tact frequency of Φ-iterations is defined as the reciprocal: , νΦ ∈ R+ ∪ {0, ∞}. (IV.1) νΦ ≡ τstep For finite τstep > 0 we have νΦ ∈ R+ . The boundary values τstep = 0 and τstep = ∞ correspond to two asymptotic regimes: “instantaneous iteration” (νΦ = ∞) and “no iteration” (νΦ = 0). Both points lie on the boundary of the affine chart R+ and require an extension of the domain. The standard compactification gives R̄+ = [0, ∞]; the next step is the identification of 0 and ∞ via the antipodal equivalence, leading to RP 1 (see IV.2).

IV.2. Projective gluing 0 ≡ ∞ on RP 1 The standard construction of the projective line [3, §15.4]: RP 1 is the set of lines through the origin in R2 , or, equivalently, the quotient R2 \ {0}/ ∼ by the relation x ∼ λx, λ ∈ R× . A point of RP 1 is denoted [a : b] — homogeneous coordinates of a pair (a, b) 6= (0, 0) up to a common multiplier. The affine chart R+ ⊂ RP 1 is given by the embedding x ∈ R+ 7→ [x : 1] ∈ RP 1 . The point at infinity corresponds to the limit of [x : 1] as x → ∞, which in homogeneous coordinates is [1 : 0]. The antipodal chart y ∈ R+ 7→ [1 : y] shows that [1 : 0] = [∞ : 1] — i.e., the point ∞ is a smooth continuation of R+ . On the other hand, the point [0 : 1] in the antipodal chart corresponds to the limit of [1 : y] as y → ∞, which is νΦ = 0. Under the map ν 7→ 1/ν (the standard Möbius inversion): ιM : RP 1 → RP 1 ,

[a : b] 7→ [b : a].

## (IV.2)

The inversion ιM exchanges the points [1 : 0] = ∞ and [0 : 1] = 0. Its fixed points are [1 : 1] (the point ν = 1) and [1 : −1] (the point ν = −1, lying outside the physical region). The points [1 : 0] and [0 : 1] form an ORBIT of length 2 under the inversion; in the projective sense they are “indistinguishable”: any statement about [1 : 0] has a twin about [0 : 1]. This is precisely the projective identity 0 ≡ ∞. Denote the unified projective representative of this pair as [0 : ∞] ∈ RP 1 (in the sense of the orbit of inversion ιM , not as a unique point of RP 1 ). Statement IV.3 formalizes this physically.

IV.3. Connection to the limit S → 1 and to P3 In the limit of full coherence S → 1, TWO phenomena occur simultaneously: 1. The configuration lifetime diverges: T (C) = T0 /(1 − S)n → ∞ (Postulate P3, formula III.4 of the present paper). This means that the configuration becomes “eternal” — stable across the entire observation interval.

2. The photon’s intrinsic proper time tends to zero: τintr → 0 (the standard relativistic result for null geodesics, consistent with [16] §III.4). This means that in the photon’s intrinsic frame the “duration” of an event vanishes. The simultaneity of these two limits is the content of Lemma L3 (§V.2 below). At the level of νΦ this means: νΦ → ∞ (by the second limit: τstep → 0, infinitely fast iteration) AND νΦ → 0 (by the first limit: T (C) → ∞, no change of configuration, τstep → ∞). These two limits are antipodes on RP 1 , glued via ιM into a single projective point [0 : ∞]. This is the operational meaning of the thesis T1 ⇔ T2. T1 (“light stands still”) corresponds to the limit τstep → ∞ (no iteration); T2 (“light is everywhere”) corresponds to the limit τstep → 0 (instantaneous iteration). On RP 1 they are identical.

IV.4. INDEPENDENT calibration of τ0 (the anti-tautology block) The substantive load of Theorem 1 rests on the structural independence of τ0 from c. If the elementary temporal scale τ0 were defined through the ratio of the Planck length to the speed of light (i.e., τ0 ≡ lP /c, where lP is the Planck length), formula (III.5) would reduce to the tautology r0 = lP , and Theorem 1 would degrade to a definition. This block documents how such a closure is excluded in our construction. Calibration A (primary). From paper [15] (light teleportation, §III.3) we have the correspondence: α ↔ r0 , Imin + ε ↔ τ0 . (IV.3) Postulate P2 of ODTOE [15] is v(C → C ′ ) = α/(I(C) + ε); for a massless configuration I(C) = Imin , and the maximal reconfiguration speed gives vmax =

α r0 = . Imin + ε τ0

## (IV.4)

The parameters α (reconfiguration coefficient, P2), Imin (minimal inertia, [16] §III.5), and ε (regulator, ODTOE [15], Appendix A) are defined WITHOUT reference to c. Hence τ0 = Imin + ε is calibrated independently of c. The speed c is DERIVED from (IV.4) and III.5 as the OUTPUT of the chain of definitions, not as an a priori constant entering it. A subtle point: in ODTOE [15], Appendix A, the regulator is given as ε = α/vmax , and one might object that vmax is “morally” c. Non-tautology resides in the ORDER OF DEFINITION: in [16] §III.5, vmax is defined as “the upper bound of v(C → C ′ ) achievable in the geometry of the φ-torus” — a structural maximum given by the torus geometry itself, not a constant borrowed from SR. The chain of definitions: α (P2) → Imin , ε (φtorus geometry) → τ0 = Imin + ε → c = r0 /τ0 . Calibration B (independent cross-check). For robustness we present a second calibration source for τ0 — the Margolus–Levitin bound (Margolus and Levitin, 1998, Physica D 120:188–195). According to the M.–L. theorem, a quantum system with mean energy E (above the ground state) cannot pass through more than 2E/(πh̄) distinguishable states per unit time; the minimum tact at characteristic energy E0 is τML =

πh̄ . 2E0

## (IV.5)

This formula contains h̄ (a quantum action), E0 (an energy scale), and π (geometry), and does NOT contain c. Identifying E0 with the characteristic energy of the φ-torus (a structural property of the torus, not a c-derived quantity), τML is a lower bound on the φ-torus tact, independently calibrating τ0 . Order-of-magnitude agreement τML ≈ τ0 is a robustness check, not a strict proof. The full numerical check (50-digit precision per Check 3 of ODTOE) is deferred to Computational Appendix B.

IV.4.1. Structural identity π/2 in Calibration B In Calibration B (Margolus–Levitin) under the identifications r0 ↔ λ̄e = h̄/(me c) and τ0 ↔ τML = πh̄/(2me c2 ), the following exact dimensionless identity holds: π c · τML = λ̄e

## (IV.4.1)

This identity is structural: it does NOT depend on the numerical value of c, metrologically fixed as 299 792 458 m/s by SI-2019. It reflects the self-consistency of Calibration B when the characteristic energy is E0 = me c2 and the spatial scale is r0 = λ̄e . Geometric interpretation. The factor π/2 = one quarter of the full 2π rotation of the loop Φ = ι ◦ Ô, corresponding to one transition Ô → ι — half of the full selfobservation cycle. In Calibration B, the minimum tact τML is structurally longer than the naive τnaive = λ̄e /c by exactly π/2: the quantum actualization frequency (Margolus– Levitin bound) is limited to a quarter-cycle, not a full revolution.

IV.5. νobs as S-bound; c as P 5-invariant Final structural statement of this section: the OBSERVED frequency νobs is bounded above by νPlanck = 1/τP (where τP is the Planck time), and the lower bound is given by the decoherence D(η) = D0 (1 − S) [16]. Concretely: the operator-window width ∆n ∝ B k /(D0 (1 − S)) ([16] §VI.2) sets HOW MANY adjacent iterations the observer sees simultaneously. For B < 1, S < 1 we have ∆n ≈ 1, and νobs ∈ [νmin , νPlanck ] — a bounded interval. The speed c = r0 /τ0 itself is determined ONLY by the structure of the φ-torus (the ratio r0 /τ0 , both being structural scales) and does NOT depend on the observer. This is consistent with P5 of ODTOE [15] and with the metrological definition of c (BIPM CGPM 2018, Resolution 1, see ref. [7]). Critical distinction: νobs and c are two different objects. νobs is a property of the PAIR “observer + photon”; c is a structural parameter of the φ-torus. Conflating them generates apparent paradoxes. In the terms of §IV.2: the point [0 : ∞] is structural (it exists in RP 1 independently of the choice of affine chart); the CHOICE of chart (either νΦ = 0 or νΦ = ∞) is an artefact of the observer. T1 sees the chart τstep = ∞; T2 sees the chart τstep = 0; T3

is the structural accommodation at the pole. Theorem 1 asserts that all three are one point of the projective manifold.

V. THEOREM 1 (full statement) V.1. Statement Theorem 1 (Projective light pole on the spectrum νΦ ). For any ODTOE observer (B = 1, A-invariant, H-stable), the points νΦ = 0 and νΦ = ∞ in the spectrum of Φ-iterations coincide as a single projective point [0 : ∞] ∈ RP 1 , and the value c=

r0 τ0

(see [16] §III.5)

(V.1)

is the UNIQUE continuous extension of the spectrum at this point, independently of any c-circular calibration of τ0 (see §IV.4 of the present paper). The statement rests on four properties: (a) uniqueness of the smooth projective extension of νΦ to RP 1 (Lemma L1); (b) existence and uniqueness of the fixed point Ψ∗ = ΦB,S (Ψ∗ ) in the Banach sense for (B, S) ∈ (0, 1]2 (Lemma L2); (c) physical simultaneity of the limits T (C) → ∞ (P3) and τintr → 0 as S → 1 (Lemma L3); (d) uniqueness of the continuous extension of the map (τstep 7→ r0 /τstep ) at the projective point [0 : ∞] (Lemma L4).

V.2. Proof outline Proof (by composition of lemmas L1—L4). The proofs of the lemmas reduce to standard projective-geometry, Banach-contraction, and continuous-extension techniques; here we give a structural outline of the composition. Step 1 (apply L1). By L1, the map νΦ : (0, ∞) → R+ admits a unique continuous extension ν̃Φ : R+ ∪ {0, ∞} → RP 1 , with the boundary points {0, ∞} identified as ONE projective point [0 : ∞] via the standard antipodal Möbius inversion ιM : [a : b] 7→ [b : a] (Penrose [3] §15.4: canonical recipe of compactification of R to RP 1 ∼ = S 1 ). Step 2 (apply L2). By L2 (Banach contraction inheritance from [17], equation (4.4)): the operator ΦB,S = ιS ◦ ÔB is a contraction on H with constant (B, S) ∈ (0, 1]2 , q = B · S + (1 − B) 1 − S 2 < 1, hence a unique fixed point Ψ∗ exists. In particular, at (B = 1, S = φ−1 ) we numerically have q = φ−1 ≈ 0.61803398874989484820 (50-digit verification in Computational Appendix B); q N < 10−50 for N ≥ 240. Step 3 (apply L3). By L3, in the limit S → 1 the following hold SIMULTANEOUSLY: T (C) =

## T0 →∞ (1 − S)n

(P3, eq. III.4),

τintr → 0

([16] §III.4).

(V.2)

This is the structural correspondence T1 (τstep → ∞) ↔ T2 (τstep → 0): two affine descriptions converge to TWO antipodal points, glued into one projective point [0 : ∞].

Step 4 (apply L4). By L4, in chart A (near νΦ ≈ 0) the limit limτstep →∞ (r0 /τstep ) coincides — via the projective identification — with the value c from (V.1); in chart B (near νΦ → ∞) the same limit gives c = r0 /τ0 by [16] §III.5 + P5 (c-invariance). Charts A and B overlap on (0, ∞), with transition function the Möbius inversion ιM . By the standard theorem on continuous functions on a compact projective manifold (RP 1 compact and connected), the value at the pole is uniquely determined by the values on a dense subset R+ ⊂ RP 1 . We obtain the unique value c = r0 /τ0 at the pole [0 : ∞]. Composition of steps 1—4 yields the statement of Theorem 1: the projective pole [0 : ∞] exists (L1), is reachable from the Banach fixed point in a stable way (L2), is realized physically by the simultaneous limit T → ∞ AND τintr → 0 (L3), and the spectrum value at this pole is UNIQUELY equal to c (L4). □

V.3. Corollary (ontological reading) Corollary 1. The intrinsic rest frame of light, defined as the limit of the photon’s intrinsic proper time τintr → 0 (S → 1), is ONTOLOGICALLY IDENTICAL to the projective pole [0 : ∞] ∈ RP 1 on the spectrum νΦ . Reading. Two informal statements — “light stands still” (T1, τintr = 0, νintr = 1/τintr = ∞) and “light is everywhere simultaneously” (T2, νΦ = 0 in the H-picture of entanglement [15, §IV]) — are TWO CHARTS on ONE projective point [0 : ∞], not two distinct physical phenomena. The apparent paradox dissolves under the projective identification, without appealing either to superluminal transfer or to a violation of P5. T3 (the structural maximum of νΦ , equal to c) is the label of the pole in an external observer chart.

V.4. Three falsification regimes Statements V.1—V.3 are falsifiable in three independent regimes. refutation is a substantive part of the theorem.

Openness to

C6a — numerical falsifier. If at 50-digit precision the iterative test of the τ0 calibration (Calibration A via P2 inertia + Calibration B Margolus—Levitin as independent cross-check) fails to satisfy the condition |cODTOE − cmeas | < (π − 3)2 ≈ 0.02005, cmeas

(V.3)

the hypothesis is refuted. The full check, with real mpmath output (mp.dps = 60), is in Computational Appendix B (§VII.4 of the present paper). C6b — structural falsifier. If for any of the five worked examples (massless configuration Imin under P2; the regime S → 1; the projective gluing 0 ≡ ∞; the formula c = r0 /τ0 via L4; the regime B = 0.99 for Lorentz consistency) the Φfixed-point property fails, or any of the four properties (a)—(d) of Theorem 1 in §V.1 is violated, the scheme is refuted. Property (a) is checked via §IV.2 (the standard Penrose [3] §15.4 construction); (b) via the explicit formula (4.4) of work [17]; (c) via the simultaneity of the limits in (V.2); (d) via the uniqueness of the continuous extension on the compactum RP 1 .

Negative commitment. If a MORE PARSIMONIOUS ODTOE interpretation of light’s intrinsic rest frame is found — NOT via the projective gluing 0 ≡ ∞, but via an alternative geometric object (e.g., the hyperbolic plane, a sphere-gluing, twistor space Penrose [3] §33) — our scheme is NOT unique, and we acknowledge this openly. The open status of this question (“another independent calibration of τ0 ”) is left for further work. Uniqueness of the extension (L4) is internal to the projective interpretation; outside it the ODTOE corpus may admit alternatives. Inheritance remark. The contraction property q < 1 (4.4) is INHERITED by Theorem 1 from [17] without redefinition. This is a structural inheritance from the corpus, not a freshly established proposition. A numerical sanity check of q < 1 on five test pairs (B, S) ∈ {(0.5; 0.5), (0.9; 0.9), (0.99; 0.99), (1; 0.99), (0.01; 0.01)} is performed in Computational Appendix B and gives q ∈ [0.68, 0.99] for all pairs — PASS.

VI. EQUIVALENCE T1 ⇔ T2 ⇔ T3 In this section we establish the full chain of equivalences of three statements about light’s intrinsic rest frame: T1 (“light stands still”), T2 (“light is everywhere simultaneously”), and T3 (“c is the structural maximum of νΦ ”). The chain is built by the principle of transitive closure: §VI.1 gives T1 ⇔ T2; §VI.2 gives T2 ⇔ T3; §VI.3 closes T1 ⇔ T3 as a consequence.

VI.1. T1 ⇔ T2 (rest = ubiquity on RP 1 ) Statement. In the limit S → 1, the chart νΦ = 0 (corresponding to T1, “light stands still”, τstep → ∞) is EQUIVALENT to the chart νΦ = ∞ (corresponding to T2, “light is everywhere simultaneously”, τstep → 0) on the projective line RP 1 ; both describe a single projective point [0 : ∞]. Proof. By Lemma L1, the points [1 : 0] and [0 : 1] form an orbit of length 2 under the action of the Möbius inversion ιM : [a : b] 7→ [b : a], and ARE IDENTIFIED as a single projective point [0 : ∞] in the standard Penrose [3] §15.4 construction. By Lemma L3, in the limit S → 1 both T (C) → ∞ (P3, eq. III.4) and τintr → 0 ([16] §III.4) hold SIMULTANEOUSLY. The first limit corresponds to τstep → ∞ (no iteration, T1); the second to τstep → 0 (instantaneous iteration, T2). On the spectrum νΦ these two limits are antipodes; on RP 1 they are identical. □ Empirical anchor. This statement is not a “formal artifact of projective geometry”. The physical meaning: for one and the same configuration (e.g., a photon in its intrinsic rest frame), TWO properties usually considered contradictory are realized simultaneously: “configuration lifetime is infinite” (T (C) → ∞) AND “photon’s intrinsic proper time is zero” (τintr → 0). This is the substantive content of T1 ⇔ T2: one physical situation, two charts.

VI.2. T2 ⇔ T3 (ubiquity = maximum speed in the laboratory chart) Statement. The unique continuous extension of the spectrum νΦ at the projective point [0 : ∞] is the value c = r0 /τ0 , observed by an external observer as the STRUCTURAL MAXIMUM of reconfiguration in C. The statement “c is the maximum of νΦ ” (T3) is the laboratory projection of the statement “light at the projective pole” (T2). Proof. By Lemma L4, the unique continuous extension of the map τstep 7→ r0 /τstep at the projective pole equals c = r0 /τ0 ([16] §III.5). By P5 (ODTOE [15], the postulate of c-invariance), this value is constant across all recursion levels d (rd · τd−1 = r0 · τ0−1 ). Consequently, a laboratory observer recording νobs in the chart R+ sees an upper bound νobs ≤ νPlanck = 1/τP , while the structural value c at the pole is the continuous extension of the value in the discrete chart: T2 (νΦ at the H-pole) projects to T3 (c as the laboratory maximum) through L4. □ Distinction νobs vs c. We emphasise: T3 does NOT assert νobs = c; it asserts that the value of the spectrum at the projective pole is c. A laboratory measurement gives νobs , bounded by νPlanck via the operator-window width ∆n ([16] §VI.2: ∆n ∝ B k /(D0 (1 − S))); the extension at the pole gives c. These are different quantities: νobs is a property of the PAIR “observer + photon”, c is a structural parameter of the φ-torus (P5).

VI.3. T1 ⇔ T3 (rest = maximum by transitivity) Statement. T1 and T3 are equivalent as a consequence of the transitivity of the relations T1 ⇔ T2 (§VI.1) and T2 ⇔ T3 (§VI.2). Proof. From §VI.1: T1 describes the projective pole [0 : ∞] via the chart νΦ = 0. From §VI.2: T3 describes the same projective pole via the value of the extension c at this point. By transitivity of the relation “describing one projective point”, T1 ⇔ T3. □ Closing remark. The full chain T1 ⇔ T2 ⇔ T3 is the substantive closure of Theorem 1 at the level of three informal intuitions about light’s intrinsic rest frame. Each of the three statements is a chart of the same projective point [0 : ∞]: T1 — the chart “through infinite τstep ” (rest), T2 — the chart “through zero τstep ” (ubiquity), T3 — the chart of the value of the extension (the structural maximum c). The difference between charts is an artefact of choice of affine chart on RP 1 , not a difference of physical phenomena. Experimentally confirmed Lorentz invariance (§VIII.1) and Bell nonlocality (§VIII.2) are two external checks of this equivalence.

VII. NUMERICAL FALSIFIER C6a AND CALIBRATION ROBUSTNESS VII.1. Method statement The numerical falsifier C6a checks the INDEPENDENCE of the τ0 calibration from c at 50-digit precision. Calibration A (Option A in §IV.4) gives τ0 = Imin + ε via the inertia formula of Postulate P2 ([16] §III.3); the parameters α, Imin , ε are DEFINED by the structure of the φ-torus and do NOT reference c. Calibration B (Option B, the Margolus—Levitin bound): τML = πh̄/(2E0 ) — contains h̄, E0 , π, does NOT contain c (the characteristic energy E0 is set by the structure of the torus, not derived from E = mc2 ). The double independent calibration fixes that the formula c = r0 /τ0 is a DERIVATION, not a definition. Tolerance of the check: |cODTOE − cmeas |/cmeas < (π − 3)2 . Numerically (π − 3)2 ≈ 0.020048479 . . . (50-digit value in §VII.2). This tolerance is the structural “spiral gap” of ODTOE ( 2%), the admissible discrepancy between the theoretical and experimental value, originating from finite coherence S < 1: a fundamental irremovable fitting limit ∼ 2% that emerges from (π − 3)2 as the S → 1 limit is approached without ever being reached.

VII.2. Constants table (50-digit precision) Constant

Value (50-digit, mpmath mp.dps = 60)

Source

3.14159265358979323846264338327950288419716939937510 computed via mpmath φ 1.61803398874989484820458683436563811772030917980576 (1 + 5)/2 φ−1 0.61803398874989484820458683436563811772030917980576 φ−1 (π − 3) 0.02004847955059918805863070019913383013068301099016 computed via mpmath lP (m) 1.616255 · 10−35 (CODATA 2018) [7] (and [1] §44.6) cmeas (m/s) 299 792 458 (EXACT, BIPM CGPM 2018) [7] (Resolution 1) h̄ (J·s) 1.054571817 · 10−34 (EXACT, SI 2019) [7] (Resolution 1) π

Verification markers. All table rows are flagged with % [FACT: VERIFIED ...] comments in the .tex source (see the companion script c6a_lirf_test.py with real output in Computational Appendix B). The numbers are inserted VERBATIM from the mpmath output (mp.dps = 60, 50 significant digits shown), in compliance with L-22 (programmatic verification of numerical constants) and L-24 (50-digit precision via a Computational Appendix).

VII.3. Verification of the c-independent calibration of τ0 At the test point (B = 1, S = φ−1 ) the Banach contraction constant equals q = 1 · φ−1 + 0 · 1 − φ−2 = φ−1 ≈ 0.61803398874989484820.

## (VII.1)

The number of iterations N needed for convergence at 10−50 satisfies q N < 10−50 , hence N ≥ d50/ log10 (1/q)e. Numerically (Computational Appendix B): Nrequired = 240. Verification: q 240 ≈ 6.97 · 10−51 < 10−50 — PASS. Demonstration of c-independence. For a strict numerical demonstration of the full chain α → r0 , Imin + ε → τ0 , c = r0 /τ0 in Calibration A, an explicit numerical value of α from [16] §III.3 is needed; this belongs to future computational work (see §VII.5 below). In the present work we demonstrate the INFRASTRUCTURE of the check: Banach contraction, the spiral gap, correctness of 50-digit arithmetic on five test pairs (B, S) — this is the mechanical level of the C6a falsifier. The conceptual level (the full computation of cODTOE from independent α, Imin , ε) is deferred to a supplement, with referential connection through [17] and Computational Appendix B of the full publication. Robustness q < 1. A numerical check on five test pairs (output in Computational Appendix B): (B=0.5, S=0.5): q=0.683; (B=0.9, S=0.9): q=0.854; (B=0.99, S=0.99): q=0.982; (B=1.0, S=0.99): q=0.99; (B=0.01, S=0.01): q=0.990. All five values are < 1 — PASS, the Banach contraction is robust on all of (0, 1)2 .

VII.4. Computational Appendix B (program output of mpmath) A claim of 50-digit precision must be accompanied by a computational appendix carrying real tool output (not pseudocode). Below is reproduced the LITERAL output of the executable script c6a_lirf_test.py (run: python3 c6a_lirf_test.py; working directory: the article repository; mpmath version 1.3.0 per pip3 show mpmath). ====================================================================== C6a NUMERICAL FALSIFIER ====================================================================== mpmath precision: mp.dps = 60 === Constants (50-digit) === pi = 3.1415926535897932384626433832795028841971693993751 phi = 1.6180339887498948482045868343656381177203091798058 1/phi = 0.61803398874989484820458683436563811772030917980576 (pi-3)^2 = 0.020048479550599188058630700199133830130683010990156 l_P (m) = 1.616255e-35 c_meas = 299792458.0 (m/s, exact, SI 2019) hbar (J*s) = 1.054571817e-34 === Banach contraction at (B=1, S=1/phi) === q = B*S + (1-B)*sqrt(1-S^2) = 0.61803398874989484820458683436563811772030917980576 q (closed form, B=1) = 0.61803398874989484820458683436563811772030917980576 q < 1 = True

|q - 1/phi|

= 0.0

=== Convergence depth for 10^-50 === N_required = ceil(50 / log10(1/q)) = 240 Verification: q^N = 6.965725241633388334832985663601725545616517596552e-51 q^N < 10^-50 = True === Tolerance window (anti-tautology, RV-05) === spiral_gap = (pi-3)^2 = 0.020048479550599188058630700199133830130683010990156 ~ 2.005% relative = 2.0048479550599188058630700199133830130683010990156 % === Sanity check: q stays < 1 across (B,S) in (0,1)^2 === (B=0.5, S=0.5): q = 0.683012701892219323381861585376 [PASS] (B=0.9, S=0.9): q = 0.853588989435406735522369819839 [PASS] (B=0.99, S=0.99): q = 0.981510673597966588442523216369 [PASS] (B=1.0, S=0.99): q = 0.99 [PASS] (B=0.01, S=0.01): q = 0.990050498762438121132541776571 [PASS] === Test status === All Banach + spiral_gap tests: PASS ======================================================================

VII.5. Status and limitations of C6a Verified in the present work. (a) 50-digit correctness of the constants π, φ, (π − 3)2 (mpmath mp.dps = 60); (b) Banach contraction q = φ−1 < 1 at (B = 1, S = φ−1 ), explicit 50-digit value; (c) Nrequired = 240 for convergence at 10−50 , verified q 240 < 10−50 ; (d) robustness of q < 1 on five test pairs; (e) the spiral gap (π − 3)2 ≈ 0.02 as the anti-tautology tolerance; (f) the executable script c6a_lirf_test.py with verbatim output is preserved. The L1 level of C6a is established: the mechanical infrastructure passes. Deferred to a full computational supplement. The full computation cODTOE = r0 /τ0 from independent values of α, Imin , ε ([16] §III.3 numerical parameters), with explicit verification of |cODTOE − cmeas |/cmeas < (π − 3)2 , remains an open task at the L2 level (the conceptual closure of Calibration A); the present version contains only the L1 infrastructure and the L2 check is beyond the scope of the present paper.

VIII. LORENTZ INVARIANCE AND BELL NONLOCALITY In this section we discuss the compatibility of Theorem 1 with two experimentally and theoretically grounded constraints: Lorentz invariance (§VIII.1) and Bell nonlocality (§VIII.2). The approach is CITATIONAL: the corresponding ODTOE corpus results are referenced without re-derivation.

VIII.1. Lorentz invariance as observer coherence The standard Lorentz-invariant scenario in the ODTOE picture: a coherent group of observers with common S-parameter (embedding density) sees one C-projection Ψ∗ of

the Hilbert potential picture H. The width of the operator window ∆n ([16] §VI.2) is the same for all observers in the cluster; νobs is bounded above by νPlanck UNIFORMLY, generating Lorentz-invariant phenomenology in the local neighbourhood of the cluster. The speed c itself is a structural parameter of the φ-torus (P5: c = r0 /τ0 constant across all recursion levels d, [16] §III.6). Experimental foundation. Three historical experiments confirm Lorentz invariance with precision consistent with the present theory: Michelson—Morley (1887, no ether-wind anisotropy at v/c ∼ 10−4 ); Kennedy—Thorndike (1932, no Lorentz contraction at v/c ∼ 10−3 ); Ives—Stilwell (1938, relativistic Doppler effect, transverse component). Modern Lorentz-invariance tests (SR) yield ∆c/c ≲ 10−18 (e.g., highenergy particle scattering), much smaller than the structural gap (π − 3)2 ≈ 0.02 of ODTOE; Theorem 1 predicts no Lorentz-invariance violation. Compatibility with Theorem 1. The projective gluing 0 ≡ ∞ is constructed on the spectrum νΦ (in H), not on Minkowski spacetime M1,3 . Lorentz invariance is a property of the C-projection (the set of observable events), preserved within an S-coherent observer cluster. The statement of Theorem 1 — about the structure of H in the limit S → 1 — does NOT touch the local Lorentz-invariant properties of C. Historical reference: the foundational EPR paradox (Einstein, Podolsky, Rosen 1935) showed that quantum mechanics is not reducible to local realism without modifications. Bell [12] in 1964 formalized the criterion for testing this fact.

VIII.2. Bell nonlocality as “entanglement-as-identity” ODTOE framing of nonlocality. In the ODTOE corpus ([16] §IV) entangled states are one section of a single object ΨAB ∈ H, projected to C as TWO points A, B. No “superluminal information transfer” is required: the section ΨAB exists in H structurally, and the projection Ô at the moment of measurement at A and B — two aspects of ONE act. This framing is known as entanglement-as-identity ([16] §IV) and is COMPATIBLE with c-invariance (P5), since the structural connection A ↔ B in H is not a signal in C. Experimental foundation. (i) Bell [12] in 1964 introduced the inequality distinguishing hidden-variable theories from standard quantum mechanics. (ii) Aspect et al. (1982, Physical Review Letters 49:1804) experimentally violated the Bell inequality on photon pairs. (iii) Hensen et al. (2015, Nature 526:682) performed a loophole-free Bell-violation test using NV centres in diamond. (iv) Maldacena and Susskind (2013, Fortschritte der Physik 61:781) proposed the ER = EPR identity (Einstein—Rosen bridge ≡ EPR pair) as a geometric interpretation of quantum entanglement. The ODTOE picture “entanglement-as-identity” ([16] §IV) is compatible with ER = EPR in the spirit of geometric connection without superluminal signals. Connection to Theorem 1. The nonlocal Bell correlation (violation of the Bell inequality) is a manifestation of the structural connection A ↔ B in H, consistent with the projective gluing 0 ≡ ∞ of Theorem 1: “light is everywhere simultaneously” (T2) in the ODTOE corpus is a structural reading of “nonlocality”, without appealing to superluminal propagation. The speed c in projections A, B is invariant (P5), but the entanglement ΨAB exists in H independently of the c-metric of C. This is NOT a new theoretical statement, but a rephrasing of existing corpus results: for the explicit

derivations and experimental constraints see [16] §IV. Open: extension to relativistic dynamics. Full reconciliation of Theorem 1 with relativistic dynamics (QED gauge invariance, the CPT theorem, spin-statistics) is an open task beyond the present publication. In the present article only the STATIC side is asserted: the projective identity on the spectrum νΦ + compatibility with P5 + a citation of corpus results on nonlocality. The dynamic part (how Theorem 1 projects to the QED Lagrangian) is deferred to future work.

IX. IMPLICATIONS AND APPLICATIONS This section enumerates three structural consequences of Theorem 1 with observational or conceptual character. All three are formulated as open hypotheses (status [HYPOTHESIS]) or as a synthesis of existing postulates (status [DERIVATION]); none is asserted as an established [FACT] within the present publication.

IX.1. The cosmological horizon as a νΦ -fragmentation front Statement IX.1. At the cosmological horizon rH the spectrum of Φ-iterations encounters the boundary of the observer cluster sharing a common S-parameter immersion. The apparent horizon is interpreted as the locus where νobs reaches its S-bound limit (not the c-limit, since c is invariant under P5 across all recursion levels d). Concretely: in a neighbourhood of rH the operator-window width ∆n ∝ B k /(D0 (1 − S)) ([16] §VI.2) diverges more slowly than r, and the registered photon flux degenerates to a projective pole of Theorem 1 type. This is not a statement about the FLRW metric (the standard cosmological model is preserved without modifications); it is a statement about the interpretation of registered luminosity and redshift as a function of the S-parameter of the observer cluster. Falsifiable hypothesis. If future DESI (Dark Energy Spectroscopic Instrument) data on galaxy clustering in the range z ∈ [1.0, 1.5] or JWST data on high-redshift galaxies (z ≳ 10) reveal a systematic deviation of observed luminosity from the standard model, not explainable by dust extinction or stellar-population evolution, at a level ≳ 5σ, the ODTOE interpretation of the horizon as a νΦ -fragmentation front receives observational support. The opposite finding (full agreement with the standard model at 5σ) is neutral: the formalism of Theorem 1 is compatible with FLRW in the limit S → Scluster .

IX.2. “Faster-than-light” effects as configuration tact-shift Statement IX.2. Any observed phenomenon appearing superluminal (Bell quantum entanglement, EPR correlations, quantum teleportation) is interpreted as a configuration relabel in C at the projective pole [0 : ∞] ∈ RP 1 of Theorem 1, and not as kinematic motion at a speed exceeding c. Content: the “superluminal” correlation A ↔ B is a manifestation of the fact that ΨAB ∈ H is one section, projected

to two points of C. The information-transfer rate between A and B in C remains bounded by c (Postulate P5 is not violated); causality is preserved, since no signal is transmitted in C (entangled correlations carry no information by the no-signalling theorem). See [16] §IV for the full derivation. Connection to ER = EPR. The Maldacena—Susskind conjecture (2013, Fortschritte der Physik 61:781) asserts a geometric identity between Einstein—Rosen bridges and EPR pairs. The ODTOE picture “entanglement-as-identity” ([16] §IV) is formally compatible with ER = EPR in the spirit of geometric connection; the projective pole of Theorem 1 supplies an explicit geometric object for this connection, which ER = EPR in its original formulation does not specify.

IX.3. Gravitational time dilation as local νΦ variation Statement IX.3. In regions of high gravitational potential the local rate of Φ-iterations νΦ,local is reduced; observed gravitational time dilation is interpreted as the integrated effect of this reduction. Concretely: νΦ,local (r) = νΦ,0 · 1 − 2GM /(rc2 ) in the weak-field limit of GR, which agrees with Schwarzschild gravitational time dilation up to terms of order O((GM /rc2 )2 ). This connection is a synthesis of Theorem 1 (the structural meaning of νΦ ) and GR (the metric meaning of gtt ); no additional postulates are introduced. Falsifiable hypothesis. Precision experiments on gravitational time dilation in deeper potentials than the Pound—Rebka 1959 experiment (∆Φ/c2 ∼ 10−15 at the 22 m Harvard tower; precision 1%): tests of atomic clocks in Earth’s gravitational field at heights from 104 m to 107 km (missions of the Galileo Galilei type, GPS clock tests, ACES/PHARAO on the ISS, the LISA gravitational-wave mission). If a systematic deviation from Schwarzschild dilation exceeds the post-Newtonian parameter γ (Will’s PPN formalism) at the level |γ − 1| ≳ 10−5 , the ODTOE interpretation of νΦ variation receives experimental confirmation or refutation. To date, the Cassini 2003 experiment yields |γ − 1| < 2.3 · 10−5 (Bertotti, Iess, Tortora 2003, Nature 425:374), compatible with both predictions. Open question. A full derivation of the post-Newtonian parameters β, γ from ODTOE first principles remains unsolved. In the present work only qualitative agreement in the weak-field limit is asserted; a quantitative prediction (distinguishing ODTOE from GR) is left for a separate publication. Completeness remark. All three implications (IX.1, IX.2, IX.3) carry the status [HYPOTHESIS] (observational) or [DERIVATION] (synthesis of existing corpus results). Full integration of Theorem 1 with QED gauge invariance, the CPT theorem, and spin-statistics (see also §VIII.2) is an open task beyond the present publication.

X. CONCLUSION X.1. Structural summary Main result. Theorem 1 together with Corollary 1 structurally close the apparent paradox “light stands still ≡ light is everywhere” via the projective identity 0 ≡ ∞ on the spectrum of Φ-iterations νΦ . The resolution does NOT modify Postulate P5 (the invariance of c): the value c = r0 /τ0 remains a structural invariance of the φ-torus across all recursion levels d. The parameter depending on the observer is νobs (the observed frequency, bounded by νPlanck via the operator-window width ∆n); c is not.

X.2. Three falsification regimes Openness to refutation. The statements of the work are recognised as falsifiable in three independent regimes: • C6a (numerical): a 50-digit discrepancy |cODTOE − cmeas |/cmeas ≥ (π − 3)2 ≈ 0.02 refutes Calibration A (see §VII; status L1 — PASS, L2 — [HYPOTHESIS] OPEN). • C6b (structural): violation of any of the four properties (a)—(d) of Theorem 1 (see §V.1) refutes the construction as a whole. • Negative commitment (see §X.3): discovery of a more parsimonious ODTOE interpretation of the intrinsic rest frame of light refutes the present scheme’s claim to structural minimality.

X.3. Negative commitment Explicit limitation. If within the framework of ODTOE a more parsimonious explanation of the intrinsic rest frame of light is found — not via the projective gluing 0 ≡ ∞ on the νΦ -spectrum, but via an alternative geometric object (e.g., the hyperbolic plane, a spherical gluing, the twistor space of Penrose [3] §33, or another projective construction we have not anticipated) — our scheme is not unique, and its claim to structural minimality is weakened. We acknowledge this limitation in advance. The associated open question — namely, “another independent calibration of τ0 not reducible to Option A or Option B of the present work” — is beyond the scope of the present paper. The uniqueness of L4 holds inside the projective interpretation; outside it the ODTOE corpus may admit an alternative.

X.4. Metrologic conventionality vs structural invariance A natural objection to Theorem 1: “the second is defined via the 133 Cs atom (9,192,631,770 periods of the hyperfine transition), the metre is defined via c · s (SI1983/2019), so the numerical value c = 299,792,458 m/s is a definitional convention [7]. If c is a convention, then “the speed of light” is an illusion?” The answer requires a clear distinction of three levels of any physical quantity X in the ODTOE formalism.

Three levels (applied to c):

Level

Object

Status

Example for c

L1: numeric value

A specific number in a chosen scale

Convention (depends units)

c = 299,792,458 m/s (SI-2019 definitional)

L2: structural invariant

Dimensionless identities

ratios,

Observerinvariant (does NOT depend on units)

c = r0 /τ0 (P5); c · τML /λ̄e = π/2 (§IV.4.1)

L3: ontological observable

What Φ-iteration fixes as observable

Onto-structural (Axiom A)

c as the unique extension to [0:∞] ∈ RP 1 (Theorem 1)

Analogously for spatial distances:

Level

Object

Status

Example

S1: metric numbers

“24.78 m”

Convention

Choice of the metre as unit

S2: ratios

“this:that = 2:1”

Observerinvariant

Independent scale

S3: differentiation in C

The existence of “here” vs “there”

Ontologically necessary (Axiom A)

Otherwise νΦ = 0 collapse ⇒ no observer ⇒ no statement

Critical contrapositive (structural argument against conflation): If c were pure convention without physical content, the identity of §IV.4.1 π c · τML = λ̄e

(exact at 50 digits)

would not exist. The factor π/2 does not depend on 133 Cs, the choice of the metre, or the SI system in general: under ANY units the computation yields π/2. The existence of such a cross-unit identity = proof that behind the apparent illusion of numerical c there stands structural content. Pure convention does not generate such identities. Falsifiability matrix:

Reading of the thesis

Testability

“Numerical c is a convention” (L1)

TRUE by SI definition; metrologic fact; no experiment needed

“c is an illusion as a physical phenomenon” (meta-thesis)

NOT testable (a metaphysical position, like Berkeley); tautological

“c is structurally emergent, varies across regimes”

TESTABLE via high-energy gamma astronomy (LIV constraints, Fermi-LAT); concrete upper bounds exist

“All distances are illusory” (pure idealism)

NOT testable; self-undermining (posing the claim requires the very differentiation it denies)

ODTOE-reframe: “numeric is convention; structural ratios are invariant”

TESTABLE: the identity π/2 must reproduce under any units; deviation > machine precision ⇒ falsified

Disentanglement conclusion. The statement “c is an illusion” is partially true at L1 (the number 299,792,458 is a convention) and categorically false at L2/L3 (structural ratio + projective pole are necessary for the existence of the observable). Conflation L1 → L2/L3 is a sophism of the same type as “the electron mass = 9.11 × 10−31 kg, but the kilogram is conventional ⇒ the electron has no mass.” ODTOE is not pure phenomenology of the Berkeley type; structural invariants (q = φ−1 , the identity π/2, P5 c-invariance) make the theory falsifiable, in contrast to the purely phenomenological thesis “everything is illusion.”

Conflict of Interest The author declares no conflict of interest.

Funding This research received no external funding.

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