The Rotating Disk in ODTOE: An Angular Transposition of the Projective Light Pole on the νΦ Spectrum

THE ROTATING DISK IN ODTOE: AN ANGULAR TRANSPOSITION OF THE PROJECTIVE LIGHT POLE ON THE νΦ SPECTRUM (Вращающийся диск в ODTOE: угловая транспозиция проективного полюса света на спектре νΦ ) Apparent stasis as one affine chart of the Φ-iteration spectrum and the (π − 3)2 residue of the observed turn

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

UDC 530.12 + 530.145 + 514.144

ABSTRACT Within the Observer-Dependent Theory of Everything (ODTOE) [1], the rotating disk is treated as an angular transposition of the projective result on the intrinsic rest frame of light [2]. Observing the disk at a single sampling frequency νobs is the selection of one affine chart of the Φ-iteration spectrum νΦ through a rank-limited operator ÔB with a single projector PA . The apparent stasis (strobe matching, νΦ → 0) and the apparent omnipresence (the limit νΦ → ∞) are two antipodal charts χ0 , χ∞ , identified by the Möbius inversion ιM into a single projective point [0 : ∞] ∈ RP 1 . This thesis is formalized by Proposition 1 (the projective pole of rotation). A complete 2π turn through a rank-limited chart carries the dimensionless spiral residue (π − 3)2 ≈ 0.0200 (Corollary 1), coinciding with the corpus spiral gap. Apparent stasis is one affine chart of the νΦ spectrum; angular momentum L, the Sagnac effect, and Lense–Thirring dragging are invariants of the complete basis. Motion is chart-dependent at level L1 and invariant-real at levels L2/L3. Falsifiers are stated: the numerical (π − 3)2 residue to 50 decimal places, and the decisive criterion that the Sagnac phase and the angular momentum L are independent of the observer’s sampling frequency. Keywords: ODTOE, rotating disk, projective geometry, Φ-iteration spectrum, Möbius inversion, RP 1 , Sagnac effect, Lense–Thirring dragging, Ehrenfest paradox, spiral gap (π − 3)2 .

I. INTRODUCTION A rotating disk observed at a single sampling frequency poses a vivid question about the status of motion. When the strobe frequency matches the angular speed, the disk appears static; at nearby but unequal frequencies the wagon-wheel illusion arises, an apparent counter-rotation. Yet the same disk carries a definite angular momentum whose reality is confirmed by the gyroscopic effect and by the Sagnac effect. The question follows: are the “static” and the “rotating” readings two incompatible descriptions, or two projections of one object? The Observer-Dependent Theory of Everything (ODTOE) [1] supplies an operatoralgebraic mechanism for the answer. The central object is the self-observation map Φ = ι ◦ Ô acting on the Hilbert space of potential states; the spectrum of its iteration frequencies νΦ is a structural object amenable to projective geometry. The work on the intrinsic rest frame of light [2] establishes that on this spectrum the points νΦ = 0 and νΦ = ∞ coincide as a single projective point [0 : ∞] ∈ RP 1 , glued by the antipodal Möbius inversion. The present work performs an angular transposition of that result: what [2] does for the frequency axis of the spectrum is carried over to the rotational axis. Observation at a single frequency νobs is the selection of one affine chart of the νΦ spectrum (rank-limited ÔB , a single projector PA ); the apparent stasis and the apparent omnipresence are two antipodal charts of that spectrum. The thesis: apparent rotation is an artifact of the single-frequency chart, and observed motion is a property of the pair (observer frequency + disk). The contribution of the work has three dimensions. (a) Geometric: the angular projective gluing of the charts χ0 and χ∞ on RP 1 through the same Möbius inversion ιM as in [2]. (b) Logical: apparent stasis and apparent rotation are artifacts of the choice of affine chart on one projective point. (c) Epistemic: observed rotation is a property of the pair (νobs , disk); the angular momentum L, the Sagnac phase, and Lense–Thirring dragging are observer-invariant quantities that survive the complete basis. Structure of the article: Section II is the literature review and the placement of the work within the ODTOE corpus; Section III recapitulates the Φ/νΦ formalism (cited, without re-derivation); Section IV is the NEW material, Proposition 1 and the four formula labels of the angular chart; Section V treats the invariants of the complete basis and Corollary 1 with the (π −3)2 residue; Section VI states the falsifiers; Section VII the limitations.

II. LITERATURE REVIEW AND POSITION WITHIN THE ODTOE CORPUS The paradox of the rigid rotating disk goes back to Ehrenfest [3]: under relativistic rotation, the circumference and the radius behave incompatibly with the Euclidean geometry of the disk at rest. The Sagnac effect [4] established that in a rotating frame counter-propagating light beams accumulate a measurable phase difference proportional to the angular speed and the enclosed area; this effect underlies ring laser

gyroscopes. The general-relativistic dragging of inertial frames by a rotating mass was described by Lense and Thirring [5]. From the side of information theory, the frequency limit of resolvability is set by the Shannon–Nyquist sampling theorem [6]: observation at a finite sampling frequency does not resolve structure above half the sampling rate. The wagon-wheel perceptual illusion, an apparent counter-rotation under strobed illumination or discrete frame sampling, has been studied experimentally by Purves and co-authors [7]. Position of the work within the ODTOE corpus. The present work is an angular transposition of the frequency result of the work on the intrinsic rest frame of light [2]: the same projective pole [0 : ∞] and the same Möbius inversion are carried from the frequency axis to the rotational axis. It reuses the unified operator ÔB and the Banach contraction [8], the quaternion realization of orientation R = q · Ψ · q̄ for angular momentum [9], the reading of motion as Φ-iteration and the tact counter [10], the dimensionality scaffold of the observer and the octaves of reality [11], and the dimensionless spiral gap (π − 3)2 from the work on the number π [12]. The geometry of the projective line RP 1 and the role of projective methods in the foundations of physics are systematized by Penrose [13]. A systematic review revealed no prior work gluing the apparent-stasis reading and the apparent-rotation reading of a disk as one projective point on RP 1 . Each separate axis (the Ehrenfest paradox, the Sagnac effect, Lense–Thirring dragging, the Nyquist sampling limit, the wagon-wheel illusion) has a rich prehistory; the novelty is their conjunction in the projective picture of the νΦ spectrum.

III. THE ODTOE Φ/νΦ FORMALISM: A RECAPITULATION This section reproduces the corpus formulas needed below. All formulas are cited verbatim, without re-derivation; the original material (Proposition 1, Corollary 1) appears in §IV–§V. The epistemic status of the section’s formulas is inherited structural identities (level L2). Axiom A fixes the basic relation between potential and observed reality [1]: the observed configuration R = Ô(Ψ) is the image of the potential field Ψ ∈ H under the observation operator. The composition of the immersion operator and the observation operator gives the self-observation map (the strange loop) Φ = ι ◦ Ô : H → H [1]. The unified observation operator ÔB and the Banach contraction are cited from the work on the unified operator [8]. The observation operator depends on the coherence B and is realized through a rank-limited projection: ÔB (Ψ) = B · PA (Ψ) + (1 − B) ηB (Ψ),

ΦB,S = ιS ◦ ÔB .

(1)

The existence and uniqueness of the fixed point Ψ∗ = Φ(Ψ∗ ) is supplied by the Banach theorem: the operator ΦB,S is a contraction with constant √ q = B · S + (1 − B) 1 − S 2 < 1,

(B, S) ∈ (0, 1)2 .

(2)

In the inequality (2), PA is the projector onto the subspace resolved by the observer;

the rank of ÔB is limited by the number of projectors held simultaneously. A single sampling frequency corresponds to a single projector PA — this fact carries the load of §IV. Motion and time in ODTOE are read as Φ-iteration [10]: time is the counter of iterations n of the self-observation map, and each Φ cycle produces one discrete step. The duration of one iteration in the observer’s own frame is denoted τstep , and the tact frequency of Φ-iterations is defined as νΦ ≡ 1/τstep [2]. On the νΦ spectrum the work [2] establishes Theorem 1: the points νΦ = 0 and νΦ = ∞ coincide as a single projective point [0 : ∞] ∈ RP 1 , glued by the antipodal Möbius inversion ιM : [a : b] 7→ [b : a]. These results are inherited by the present work (T2, level L2) and are carried from the frequency axis to the rotational axis in §IV.

IV. THE PROJECTIVE PROPOSITION 1

POLE

ROTATION:

This section, together with Corollary 1 of §V, constitutes the sole NEW material of the work. The angular transposition carries the projective construction of [2] from the frequency axis of the νΦ spectrum to the rotational axis: observation of the disk at a single frequency νobs is the selection of one affine chart, and the apparent limits “stasis” and “omnipresence” are two antipodal charts glued by the same Möbius inversion.

IV.1. The single-frequency angular chart Let an observer with a single sampling frequency νobs register a disk rotating at angular speed ω through a rank-limited operator ÔB with a single projector PA (1). A single sampling frequency selects one affine chart χν of the angular νΦ spectrum: ) ( χν : νobs 7−→ νΦ chart realized by one projector PA ,

rank ÔB = 1.

(3)

The chart χν (3) is the angular analogue of the affine chart of the frequency spectrum in [2]: the rank-limitation of ÔB to a single projector PA means that the observer holds one angular frequency of resolution per tact. Epistemic status: dimensionless projective structure, an observer-invariant point (level L2).

IV.2. The angular antipodal gluing In the limit of strobe matching, when the sampling frequency is a multiple of ω, successive frames are indistinguishable, and the chart reads as νΦ → 0 (“the disk is static”). In the limit νΦ → ∞ the chart reads as “the disk is everywhere at once” — the absence of a distinguished angular position per tact. These two limits map to the homogeneous coordinates [1 : 0] and [0 : 1] and are identified by the same antipodal Möbius inversion as in the work [2]:

ιM : RP 1 → RP 1 ,

[a : b] 7→ [b : a];

χ0 ↔ [1 : 0],

χ∞ ↔ [0 : 1] =⇒ [0 : ∞] ∈ RP 1 . (4)

The points [1 : 0] and [0 : 1] form an orbit of the inversion ιM of length 2 and are identified as a single projective point [0 : ∞]. This is the projective identity of the apparent stasis and the apparent omnipresence of the disk on RP 1 . The difference between the charts χ0 and χ∞ is a property of the pair (νobs , disk); the projective point [0 : ∞] is structural. The inversion ιM (4) is the same operation as in [2], a single shared construction.

IV.3. The observed angular phase The observed angular phase per sample is a function of the chosen chart — the rotation accumulated between two successive samples: φobs (ν) = ω · τstep =

ω , νΦ

(5)

where τstep = 1/νΦ is the duration of one Φ-tact [10]. The quantity φobs (ν) (5) is the chart-dependent reading: under strobe matching φobs → 0 (mod 2π) (chart χ0 ), and as νΦ → ∞ the value φobs → 0 as the limit of infinitely frequent sampling (chart χ∞ ). Both limits are antipodes on RP 1 , glued through (4). Epistemic status: dimensionless projective structure (level L2).

IV.4. The residue of the observed turn Upon a complete 2π turn through the rank-limited chart χν , the observed angular measure carries a spiral residue — the structural discrepancy between the continuous 2π turn and its discretely-observed reconstruction: ∆spiral = (π − 3)2 ,

(6)

defined here as a dimensionless quantity and evaluated in Corollary 1 of §V. The residue ∆spiral (6) is the same corpus spiral gap (π − 3)2 [12], applied to the observed turn of the disk. Epistemic status: a dimensionless observer-invariant quantity (level L2).

IV.5. Statement of Proposition 1 Proposition 1 (The projective pole of rotation on the νΦ spectrum). Let an observer with a single sampling frequency νobs observe a disk rotating at angular speed ω through a rank-limited operator ÔB (a single projector PA ). Then, under assumptions A1–A4, the apparent limits “the disk is static” (νΦ → 0: strobe matching or the axisymmetric indistinguishability of the mass density) and “the disk is everywhere at once” (νΦ → ∞) are two antipodal affine charts χ0 , χ∞ of the νΦ spectrum, identified by the Möbius inversion ιM into a single projective point [0 : ∞] ∈ RP 1 . The choice of chart is an artifact of the pair (νobs , disk); the point [0 : ∞] itself is structural and invariant. Assumptions. A1 — the operator ÔB is rank-limited to a single projector PA (a single observation frequency); A2 — the disk is rigid and axisymmetric in mass density (the stasis-reading degeneracy is well-defined); A3 — the νΦ spectrum admits the same continuous projective extension as Theorem 1 of the work [2] (the angular analogue of Lemma L1); A4 — no completion of the basis (a single chart only): the level-L2/L3 invariants are excluded from the single-frequency reading by construction and re-enter in §V under basis completion. The full derivation chain of Proposition 1 from assumptions A1–A4, via the projective-extension lemma (the angular analogue of Lemmas L1–L4 on the intrinsic rest frame of light), is given in [2]; there the structural lift “photon → disk” is recorded explicitly, and the Möbius identification (4) is shown to be the same operation, a single shared construction. Epistemic status of Proposition 1: a NEW result (T4), the angular transposition of Theorem 1 [2].

V. THE INVARIANTS OF THE COMPLETE BASIS AND COROLLARY 1 V.1. The reading under basis completion Assumption A4 restricts Proposition 1 to a single chart. When the operator Ô is completed to the full basis (the stable fixed point Ψ∗ = Φ(Ψ∗ ) realizing rest), the singlefrequency degeneracy lifts, and the genuine invariants appear: the angular momentum L, the Sagnac phase [4], and the Lense–Thirring dragging of inertial frames [5]. These quantities are invariants of levels L2/L3 — observer-invariant and chart-independent. The complete basis realizes rest; the angular momentum L survives the complete basis. The quaternion realization of orientation [9] sets the angular momentum through the action R = q · Ψ · q̄: the quaternion orients the observer in the configuration space, and this orientation persists under basis completion.

V.2. The three-level distinction of the status of motion The status of the disk’s motion is separated into three epistemic levels [1, 2]. Apparent stasis and apparent counter-rotation are the chart-dependent observed reading (level

L1, the sampling convention). The angular momentum L, the Sagnac phase, and Lense–Thirring dragging are structural invariants (levels L2/L3). Motion is chartdependent at level L1 and invariant-real at levels L2/L3. The self-undermining test of the thesis is essential. Were rotation a pure illusion, the angular momentum L, the Sagnac phase, and Lense–Thirring dragging would vanish under basis completion; they survive — hence motion is chart-dependent at level L1 and invariant-real at levels L2/L3. A ring Sagnac interferometer measures a phase difference independent of the observer’s sampling frequency, fixing the L2/L3 status of rotation operationally.

V.3. Statement of Corollary 1 Corollary 1 (The (π − 3)2 residue of the observed turn). Upon a complete 2π turn through the rank-limited chart χν , the observed angular measure carries the spiral residue (π − 3)2 — the structural discrepancy between the continuous 2π turn and its discretely-observed reconstruction. The residue (π − 3)2 ≈ 0.0200 is dimensionless, observer-invariant, and coincides with the corpus spiral gap [12]. Corollary 1 uses the quantity (6) defined in §IV and evaluates it here. The value of the residue to 50 significant digits (computed via mpmath, mp.dps = 50, with no fitting): (π − 3)2 = 0.020048479550599188058630700199133830130683010990156.

(7)

This is the same corpus invariant as the closure ceiling S max = 1 − (π − 3)2 [12]; no new fitting is introduced. (π − 3)2 is the structural discrepancy between the continuous 2π turn and its discrete reconstruction — dimensionless and observer-invariant. Translational speed bounds lie outside the scope of the present work and are treated in other parts of the corpus.

VI. FALSIFIERS The claims of the work are falsifiable in three independent regimes; openness to refutation is a substantive part of the thesis. F-1 (numerical). The residue (π − 3)2 (7) is reproduced at 50-digit precision through a direct recomputation; a deviation beyond machine precision refutes Corollary 1. F-2 (decisive). The Sagnac phase and the angular momentum L remain independent of the observer’s sampling frequency. An experiment in which an invariant of the rotating disk shifts with a change of the observer’s sampling frequency refutes the thesis. Operational form: a phase-difference measurement in a ring Sagnac interferometer [4] is independent of the observer’s sampling frequency. F-3 (parsimony). The projective reading of the pole of rotation introduces no free parameters beyond the corpus constants (π − 3)2 and φ. The introduction of an additional free parameter refutes the claimed economy.

VII. LIMITATIONS First, the spectral reading of the operator Ĥ (the angular Hamiltonian as the generator of rotation) is held at the status of a hypothesis and is placed outside the theorem box: the link of the spectrum of Ĥ to the projective point [0 : ∞] is offered here as open to revision [HYPOTHESIS]. Second, the single-frequency idealization (assumptions A1–A4) is a limiting case. Real multi-frequency observation selects several charts simultaneously, and the projective point [0 : ∞] is the structural limit of this family of charts. Third, translational speed bounds lie outside the scope of the present work (they are treated in other parts of the corpus); no dimensional angular constant and no front speed are introduced here. Finally, the interpretation of the residue (π − 3)2 as a “reconstruction error of the observed turn” is a corpus-consistent reading, held at the same status of a testable prediction as the φ-embedding of the work [12]. Dimensional estimates (characteristic times, the linear sizes of the disk) are marked as phenomenological anchors and are nowhere presented as derivations from π and φ.

Conflict of interest The author declares no conflict of interest.

Funding This research received no external funding.

ODTOE Corpus Navigation Full corpus of the author’s articles: https://odtoe.org/articles.

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