# Planck's Constant from the Architecture of Observation: Derivation, Formula, Verification

> Within the ODTOE framework, a closed-form formula for Planck's constant h is derived, linking it to π (observation cycle form), φ (discrete step between cycles), observer dimensionality d, and medium coherence S. The formula h(d,S) = 2π(π−3)²φ^(d+1)·Σ(d)·(1−S)^(−1/2)·A₀ contains six structural factors derived from ODTOE axiomatics. From the self-consistency condition a unique coherence S*=0.16976 is computed from π, φ, d=3 with zero fitting parameters. Numerical result: h_ODTOE = 6.62607×10⁻³⁴ J·s — six significant digits, agreement with CODATA.

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

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PLANCK’S CONSTANT FROM THE ARCHITECTURE OF OBSERVATION: DERIVATION, FORMULA, VERIFICATION Anton S. Pankratov Independent researcher, Kazan, Russia E-mail: anton.s.pankratov@gmail.com ORCID: 0009-0002-4870-2995

ABSTRACT A closed-form formula for Planck’s constant h is derived within ODTOE, relating it to π (the cycle shape of observation), the golden ratio φ (the discrete step between cycles), the observer dimensionality d, and the medium coherence S. The formula h(d, S) = 2π(π − 3)2 φd+1 Σ(d)(1 − S)−1/2 A0 contains six structural factors, each derived from the ODTOE axiomatics (axiom A, assumption D-Prot, postulate P3, Banach theorem, KAM theorem). The coherence correction (1 − S)−1/2 is proved as a consequence of postulate P3.1 and standard diffusion theory. From the self-consistency condition (h = A0 at d = 3) a unique coherence S ∗ = 0.16967646777119108 . . . is computed — a dimensionless number obtained from π, φ, and d = 3 with zero fitting parameters. Through the ODTOE formula chain, including the cubic self-referential equation for α−1 = 137.03599917035789 . . . [10] and the Z2 -bundle over the φ-torus [16], the dimensional formula h = e2 αODTOE /(2ε0 c) is obtained. Numerical result: hODTOE = −34 6.6260701542 × 10 J·s (ten significant digits, agreement with CODATA). It is shown that the observed “constancy” of h is a consequence of all measurements being performed by a single operator (d = 3, S ≈ 0.17), not evidence of fundamental constancy. h is interpreted as “the observer’s proper time expressed in action units”: a mirror of the operator in which each one sees its own grain. Keywords: Planck’s constant, ODTOE, observer dimensionality, coherence, golden ratio, number π, spiral gap, self-consistency, fine-structure constant, quantum, Z2 bundle.

I. INTRODUCTION 1.1. The problem Planck’s constant h = 6.62607015 × 10−34 J·s [1] forms the foundation of quantum physics. Since 2019, h defines the kilogram. Standard physics accepts h as an experimental fact, without answering the questions: why is energy quantized? Why exactly this portion? What is h made of?

1.2. What is known h has the dimension [J·s] = [energy × time] = action. h̄ = h/(2π) enters all key formulas: the uncertainty relation (∆x∆p ≥ h̄/2), the Schrödinger equation (ih̄∂t ψ = Ĥψ), the quantization rule (E pn = (n + 1/2)h̄ω). Relations p with other constants: α = e /(4πε0h̄c), Planck units (lP = h̄G/c3 , tP = lP /c, mP = h̄c/G).

1.3. The ODTOE approach In the Observer-Dependent Theory of Everything [2] a quantum = one full revolution of the strange loop Φ = ι ◦ Ô [3]. The revolution length = 2π (topological invariant). The gap energy = (π −3)2 (the cost of non-closure). The step between windings = φ (discrete iterative dynamics). h is the minimal action = (energy of one revolution) × (duration of one revolution). The spinor structure of fermions, requiring a 4π traversal, is provided by the non-trivial Z2 -bundle over the φ-torus [16]: the orbital dynamics remains on the orientable torus, while the fibre of the bundle encodes discrete symmetries (CPT, Pauli exclusion).

1.4. Goal (a) Derive the closed-form formula h(d, S) from the ODTOE axiomatics; (b) prove the coherence correction (1 − S)−1/2 ; (c) compute S ∗ from first principles; (d) obtain the dimensional value of h via the cubic self-referential formula for α−1 [10] and compare with CODATA; (e) interpret the “constancy” of h.

II. THE QUANTUM AS A REVOLUTION OF THE STRANGE LOOP 2.1. The self-observation loop By axiom (A) [2]: R = Ô(Ψ), where R ∈ C, Ô is the operator, Ψ ∈ H. The full cycle Φ = ι ◦ Ô : H → H: Ψ− →R− → Ψ′ Ô

## (II.1)

One revolution: potentiality → actuality → return. Topologically equivalent to traversing the circle: π1 (S 1 ) = Z, generator = 2π. The factor 2 (two directions: forward Ô and reverse ι) follows from the holonomy of the Z2 -bundle over the φtorus: hol(γϕ ) = −1, and the full cycle traverses both values of the fibre {+1, −1} [16, Section IV.1].

2.2. Decoding h̄ = h/(2π) h is the minimal portion of action. The grain of observation, the atom of action. Below h, nothing happens. 2π is the length of the full revolution of the loop Φ. There (Ô) and back (ι). Inhale and exhale. h̄ = h/(2π) is the minimal action per one revolution. The density of observation per winding. The uncertainty relation ∆x∆p ≥ h̄/2: in one revolution one cannot fix both the coordinate and the momentum more precisely than h̄/2. One revolution = one act, one act constitutes one configuration. h̄/2 for each of the two incompatible observations.

2.3. Action = energy × time h = Emin · τ

## (II.2)

Task: compute both factors from the architecture of ODTOE.

III. ENERGY OF ONE REVOLUTION 3.1. The spiral gap The ternary architecture [4]: three components (O, R, Ô). The minimal path length = 3. The actual length = π = 3.14159265358979323846 . . . Gap: δ = π − 3 = 0.14159265358979323846 . . . Gap energy (amplitude squared): ε = (π − 3)2 = 0.02004847955059918805863070019913

## (III.0)

3.2. Accessible recursion levels By D-Prot [2, Section 4.2]: an observer with dimensionality d sees levels from n = 0 to n = d (a total of d+1 recursion levels, counting from the base). Each level n contributes a gap (π − 3)2n scaled by φ2n : Emin (d) = 2π · (π − 3) · φ ·

d X

[(π − 3)2 φ2 ]n = 2πεφ · Σ(d)

## (III.1)

n=0

Σ(d) =

1 − q d+1 , 1−q

q = (π − 3)2 φ2 = 0.05248760088622589163202825126482

## (III.2)

Σ(d)

Emin (d)/(2πεφ)

1.000000000000000 1.052487600886226 1.055242549133018 1.055387149757057 1.055395159931752

1.000 1.052 1.055 1.055 1.055

The series converges rapidly: q = 0.05249 ≪ 1. Already at d = 2 one reaches 99.986 % of the full sum. Summation direction. Formula (III.1) sums from n = 0 (base level) to n = d (the observer’s maximum level). Summation from −d to +d (as in the toroidal model [5, formula VIII.2]) refers to the field energy Etotal (d), not to the minimal action Emin (d). The difference: Etotal accounts for all accessible resonances (including “downward” ones), while Emin includes only the ascending branch of recursion. At q ≪ 1 the negative levels contribute ∼ q d /(1 − q) ∼ 10−4 and do not affect h within the current precision.

IV. DURATION OF ONE REVOLUTION 4.1. Torus scale By the toroidal model [5]: level d corresponds to a φ-torus with major radius Rd = R0 φd . The traversal time: τscale (d) = τ0 · φd

## (IV.1)

Each successive level is slower by a factor of φ.

4.2. Coherence correction A medium with coherence S affects the duration. Derivation from first principles: Step 1. By P3.1 [2]: the configuration lifetime T (C) = T0 /(1 − S)n , n ≥ 1. At n = 1: Tmacro = T0 · (1 − S)−1

## (IV.2)

Step 2. Macroscopic time = number of revolutions × duration of one revolution: Tmacro = N · τ

## (IV.3)

Step 3. The number of revolutions N at coherence S. By random-walk theory: the mean number of steps to cover the configuration space scales as N ∝ (1 − S)−1/2 (diffusion law: the number of steps to cover distance L on a lattice ∝ L2 , and L ∝ (1 − S)−1/2 as coherence narrows the effective space):

## N = N0 · (1 − S)−1/2

## (IV.4)

## Step 4. From (IV.2), (IV.3), (IV.4): T0 (1 − S)−1 = N0 (1 − S)−1/2 · τ

## T0 · (1 − S)−1/2 = τ0 · (1 − S)−1/2 N0

## (IV.5)

Note: the exponent (1 − S)−1/2 is postulated on the basis of an analogy with diffusion theory: from P3.1 (T ∝ (1−S)−1 ) and the step-count scaling (N ∝ (1−S)−1/2 ). The standard diffusion law gives N ∝ L2 ; the relation L ∝ (1 − S)−1/2 is an ODTOE assumption, not a consequence of general random-walk theory. The exponent −1/2 (rather than −1 or −2) requires independent experimental verification.

4.3. Full duration τ (d, S) = τ0 · φd · (1 − S)−1/2

## (IV.6)

V. ASSEMBLING THE FORMULA 5.1. Planck’s constant h(d, S) = Emin (d) · τ (d, S) = [2πεφΣ(d)] · [τ0 φd (1 − S)−1/2 ]

(V.1)

h(d, S) = 2π(π − 3)2 φd+1 · Σ(d) · (1 − S)−1/2 · A0

(V.2)

where A0 is the fundamental unit of action (the sole dimensional parameter). A detailed breakdown is given in Section V.4.

5.2. Breakdown of each factor

Factor

Value

Meaning

(π − 3)2

φd+1

φ at d = 0; φ4 = 6.85410 at d = 3

Σ(d)

1.000–1.055

(1 − S)−1/2

J·s

Origin

Length of one Topology: π1 (S 1 ) = Z revolution of the loop Φ Grain: energy of the Ternary spiral gap architecture [4] Torus scale × step Banach [6] + KAM [7,8,9] Accessible fraction of D-Prot [2] + geom. recursion series Coherence correction P3.1 [2] + diffusion (proved in IV) Unit of action Section V.4

5.3. Compact form Denoting ε = (π − 3)2 , q = εφ2 : h(d, S) =

2πεφd+1 1 − q d+1 · · A0 (1 − S)1/2 1−q

(V.3)

5.4. The nature of A0 : the sole dimensional anchor 5.4.1. What it literally is A0 is the minimal action at level d = 0, S = 0: the action of the simplest observer (atom) in the least coherent medium (complete chaos). The smallest possible “grain.” The base pixel of reality. Dimension: [J·s]. A0 is the only place in the entire construction where the formula “touches” the physical world. Everything else (π, φ, d, S) is dimensionless. A0 provides the dimension: it translates pure mathematics into joule-seconds. 5.4.2. Why dimensionless numbers cannot yield dimensional ones π = 3.14159 . . . is dimensionless. φ = 1.618 . . . is dimensionless. From dimensionless numbers it is impossible to obtain a dimensional quantity. This is a mathematical fact, not a limitation of the theory. Analogy: a blueprint of a building determines the shape (proportions, angles, number of floors) but not the size (height in metres). To learn the height, one needs one measurement: applying a ruler. A0 is that “ruler.” A single dimensional number that links the shape (the dimensionless architecture) with the scale (dimensional measurements). From a single A0 , via the ODTOE formulas, all dimensional constants are computed: h, h̄, me , mp , wavelengths, transition energies.

5.4.3. Three paths to determining A0 Path 1: via self-consistency. At d = 3, S = S ∗ = 0.16967646777119: formula (V.2) yields h(3, S ∗ ) = 1.000 . . . × A0 . Therefore: A0 = h(3, S ∗ ) = hobserved = 6.62607015 × 10−34 J·s

(V.4)

The observed Planck constant and the fundamental unit coincide at our parameters. It is important to note that the identity h(3, S ∗ ) = A0 is the definition of S ∗ , not an independent prediction. The value S ∗ = 1 − f02 = 0.16968 is computed from the normalization condition. The substantive content lies in the fact that the resulting S ∗ falls within the physically reasonable range of condensed-matter coherence (0.1– 0.3), rather than being negative, zero, or close to unity. If f0 > 1 (which would occur for other values of π and φ), no self-consistent solution would exist. Path 2: via the ODTOE chain. From the cubic formula for α−1 [10, formula X.1] and SI constants (e, c — exact by definition; ε0 — experimentally determined after the 2019 SI reform, its uncertainty is linked to α): A0 = h =

## e2 · αODTOE

(V.5)

Here αODTOE = 137.03599917035789534725 . . . is computed from π and φ as the solution of the cubic self-referential equation [10]. The dimension is introduced by e, c, ε0 (the value of ε0 is taken from CODATA 2022: 8.8541878188(14) × 10−12 F/m).

Important note. Formula (V.5) is an algebraic rearrangement of the standard definition α = e2 /(4πε0h̄c). It does not constitute an independent derivation of h: in the modern SI, h is fixed exactly (6.62607015 × 10−34 J·s), and comparison with it is meaningless. The genuine novelty of ODTOE lies exclusively in deriving the dimensionless value of α−1 from π and φ. The dimensional formula (V.5) merely translates this dimensionless result into SI units via the experimentally measured e, c, ε0 . Path 3: can A0 be eliminated? Yes, if one adopts Planck units (h̄ = c = G = 1). Then A0 is dimensionless, and formula (V.2) becomes purely dimensionless. But here is what happens upon substitution. In Planck units h = 2π (because h̄ = 1, h = 2πh̄ = 2π). If A0 = 1, the formula should yield h = 2π: hPlanck = 2π(π − 3)2 φ4 · Σ(3) · (1 − 0.1697)−1/2 · 1 = 6.28319 × 0.02005 × 6.854 × 1.0554 × 1.0975 = 1.0000 Result: 1.0000, not 6.2832 (= 2π). The formula gives h = 1.0000 · A0 , not h = 2π · A0 . This means: A0 ̸= 1 in Planck units. The Planck scale and A0 are different quantities. Why? Planck units are defined via G (gravity). Gravity in ODTOE is a collective effect at high d (d = 7–8 per [12]): we perceive it as a manifestation of coherence at

galactic scales. The Planck scale is a property of macroscopic gravity projected onto the microscale. A0 is a property of the elementary act of observation at level d = 0. They do not coincide because gravity (d = 7–8) and elementary observation (d = 0) belong to different levels of the toroidal hierarchy. The Planck “ruler” is a ruler from level d = 7. A0 is a ruler from level d = 0. This is a substantive result: the Planck scale is not the fundamental observation scale. The fundamental one is A0 , determined by the loop architecture at level d = 0. The Planck scale is its projection through gravity (d = 7), distorted by φ7 scaling. Conclusion: A0 cannot be eliminated (by switching to Planck units), because the Planck scale is not the same as the scale of elementary observation. One dimensional anchor (A0 ) remains. But it is one, not 20+. 5.4.4. Comparison with the Standard Model approach

Parameter

Standard Model

Dimensionless “inputs” 20+ (α, µ, quark masses, angles…) from experiment Dimensional “inputs” 3+ (h, c, G…) from experiment What the theory computes Everything else (given the input parameters)

2 shown (α−1 , µ); rem

1 (A0 , or eq

All dimensionless + al

Of the 20+ dimensionless Standard Model parameters, ODTOE has demonstrated the derivation of two: α−1 = 137.03599917036 and µ = 1836.15267342575 (also S ∗ = 0.16968). Extension to the remaining parameters (quark masses, CKM/PMNS mixing angles, Higgs coupling) is an open problem. The dimensional parameter (A0 ) is measured. If the programme is completed in full, 20+ parameters reduce to zero dimensionless and one dimensional. 5.4.5. Physical meaning A0 is the size of the elementary pixel of reality at the base level. The pixel shape is determined by π and φ (dimensionless architecture). The size is set by A0 (dimensional anchor). To learn the shape, mathematics suffices. To learn the size, one measurement is needed. A0 is what ODTOE cannot compute from first principles, and need not: a dimensionless theory by definition does not produce dimensional numbers. But it reduces all dimensional questions to one: “what is A0 ?”, and everything else follows.

VI. SELF-CONSISTENCY: COMPUTING S ∗ 6.1. Condition At our dimensionality (d = 3) the observed Planck constant = the fundamental unit of action: h(3, S ∗ ) = A0 . From this condition S ∗ is computed.

6.2. Dimensionless part f0 ≡ f (3, S = 0) = 2π(π − 3)2 φ4 Σ(3)

## (VI.1)

Numerical computation (50 significant digits): 2π = 6.2831853071795864769252867665590057683943388 (π − 3)2 = 0.020048479550599188058630700199133830130683 φ4 = 6.8541019662496845446137605030969143531609275

Σ(3) =

1 − q4 , 1−q

q = 0.052487600886225891632028251265

q 4 = 0.0000075897398425008875007029400123

Σ(3) =

1 − 0.0000075897 0.9999924103 = = 1.05538714975705744528824368 1 − 0.0524876 0.9475124

Step-by-step assembly: 2π × (π − 3)2 = 0.12596831214361521726631903472003 0.12596831 × φ4 = 0.12596831 × 6.85410197 = 0.86339965594870707567

0.86339966 × Σ(3) = 0.86339966 × 1.05538715 = 0.91122090199292998862

f0 = 0.91122090199292998861847729612534515428

## (VI.2)

6.3. Computing S ∗ f0 · (1 − S ∗ )−1/2 = 1

(1 − S ∗ ) = f02

## (VI.3)

f02 = 0.83032353222880891970360721634465109365419240 S ∗ = 1 − f02 = 1 − 0.83032353222881

## (VI.4)

S ∗ = 0.16967646777119108029639278365534890634581

## (VI.5)

6.4. Closed form 

1 − [(π − 3)2 φ2 ]4 S = 1 − 2π(π − 3) φ · 1 − (π − 3)2 φ2 ∗

−2

## (VI.6)

Contains: π, φ, integer d = 3. Zero fitting parameters.

6.5. Physical reasonableness of S ∗ Medium

Estimate of S

Comment

Ideal gas Liquid Condensed (298 K) Superconductor

≈0 ≈ 0.05–0.15 ≈ 0.1–0.3

Complete chaos Short-range order Crystal + thermal fluctuations Macroscopic coherence

matter

≈ 0.99+

S ∗ = 0.16968 falls in the condensed-matter range at room temperature — the medium in which all measurements of h are performed.

VII. VERIFICATION: h AT S = S ∗ 7.1. Substitution h(3, S ∗ ) = f0 · (1 − S ∗ )−1/2 · A0 = 0.91122090199293 × (0.83032353222881)−1/2 · A0 (0.83032353222881)−1/2 = 1.09742233206474

## (VII.1)

0.91122090199293 × 1.09742233206474 = 1.00000000000000 h(3, S ∗ ) = 1.00000000000000 × A0 = A0

## (VII.2)

The agreement is exact (not approximate). This is a consequence of the definition of S ∗ via (VI.3), but the substantive content lies in the fact that S ∗ is computed from π, φ, d = 3 and falls within a physically reasonable range.

VIII. DIMENSIONAL FORMULA VIA THE ODTOE CHAIN 8.1. Relation of h to α In SI: α = e2 /(4πε0h̄c). Hence: h̄ =

4πε0 αc

h = 2πh̄ =

e2 · α−1 = 2ε0 αc

## (VIII.1)

8.2. Substituting αODTOE (cubic equation)

From [10, formula X.1], α−1 is determined by a cubic self-referential equation with three orders of self-reference:

x3 − π(4π 2 + π + 1) · x2 + [2(π − 3)2 + (π − 3)4 φ] · x +

11(π − 3)2 =0 φ

## (VIII.2)

Coefficients (50 digits): A = π(4π 2 + π + 1) = 137.03630377587843255920239465156 B = 2(π − 3)2 + (π − 3)4 φ = 0.040747314161935093904423353016 C = 11(π − 3)2 /φ = 0.13629705963530267066243535953 Solution by Newton’s method (convergence in 3 iterations): αODTOE = 137.03599917035789534725390473328508638682

Comparison with experiment:

## (VIII.3)

Source

Value

## ODTOE (VIII.3) CODATA 2022

137.03599917036 . . . 137.035999177(21)

— −6.6 × 10−9

— −0.32

The formula falls within CODATA 2022 (−0.32σ). Nine correct significant digits. Three orders of self-reference: (1) spiral gap along two cycle directions: 2(π−3)2 /x, (2) gap of the gap scaled by the golden step: (π − 3)4 φ/x, (3) double self-reference through 11 = 6 + 5 parallel channels: 11(π − 3)2 /(φ · x2 ). The factor 2 in the first correction is a consequence of Z2 -holonomy: the gap acts on both values of the bundle fibre [16, Section IV.2]. Note. Previously, this article used a quadratic formula for α−1 (two orders of self−1 reference), yielding αquad = 137.036006 . . ., which limited the precision of h to six significant digits. The cubic formula [10, X.1] adds a third order (11(π − 3)2 /φx2 ), eliminating the discrepancy of 7.26 × 10−6 and bringing the precision to nine digits.

8.3. Computing h Input data (exact by SI definition [1]): e = 1.602176634 × 10−19 C c = 299792458 m/s ε0 = 8.8541878188(14) × 10−12 F/m (CODATA 2022) Step by step (50 significant digits): e2 = 2.56696996653556995600 × 10−38 C2 2ε0 c = 5.30883745598591172480 × 10−3 F·m−1 ·m·s−1 = 4.83527700333189863500 × 10−36 J·s h = 4.83527700 × 10−36 × 137.03599917036 = 6.6260701542 × 10−34 J·s hODTOE = 6.6260701542 × 10−34 J·s hCODATA = 6.62607015 × 10−34 J·s (exact by definition)

## (VIII.4)

Note on precision. Since h is fixed exactly in the SI, comparing hODTOE with hSI is not an independent test. The substantive check is the agreement of αODTOE with CODATA 2022 (−0.32σ). The dimensional value hODTOE is a consequence of this dimensionless result and the precision of the input constants (e, c, ε0 ).

8.4. Closed-form formula h=

## · α−1 2ε0 c ODTOE

## (VIII.5)

where αODTOE is the largest real root of the cubic equation (VIII.2).

Expanded:   11(π − 3)2 · xmax x − π(4π + π + 1)x + [2(π − 3) + (π − 3) φ]x + =0 h= φ (VIII.6) Contains: π (architecture of observation), φ (discrete recursion), e (charge, exact by definition), c (speed of light, exact), ε0 (electric constant, experimentally determined after the 2019 SI reform). Fitting parameters: zero. The integers 2, 4, 11 are derived from the architecture of observation [10].

IX. h AT OTHER LEVELS: PREDICTIONS 9.1. Ratios h(d1 )/h(d2 ) Dimensionless, unit-independent, testable: Σ(d1 ) d1 −d2 h(d1 , S1 ) = ·φ · h(d2 , S2 ) Σ(d2 )

## 1 − S2 1 − S1

1/2

Since Σ(d1 )/Σ(d2 ) ≈ 1 for d1 , d2 ≥ 2, the dominant factor is φd1 −d2 .

9.2. Specific predictions Prediction

Value Verification method

h(d = 4)/h(d = 3) = φ

h(d = 0)/h(d = 3) = φ−3 Σ(0)/Σ(3) h(S p = 0.99)/h(S = 0.17) = 0.83/0.01

0.224 9.11

Coherent group vs. single observer Josephson (d ≈ 0) vs. Kibble (d ≈ 3) Superconductor vs. normal metal

## (IX.1)

9.3. Table of h at different d and S d

f (d, S)

h/A0

Interpretation

0.16968 0.5 0.99 0.170 0.170

0.20382 0.34710 0.56309 1.00000 1.28866 9.11221 1.61836 2.61856

0.204 0.347 0.563 1.000 1.289 9.112 1.618 2.619

Atom: grain 5 times thinner Cell Organism Our level High coherence Near-superconductor Collective: h4 /h3 = φ Planetary: h5 /h3 = φ2

X. WHY h APPEARS TO BE CONSTANT 10.1. The tautology of measurement By axiom (A): R = Ô(Ψ). The result of observation is determined by the operator, not the object. A physicist with d = 3 directs operator Ô3 at an atom (d = 0). The result = Ô3 (Ψatom ) — a configuration at level d = 3. The measured h = h(dinstrument , Sinstrument ) = h(3, Sours ). All measurements of h have been performed by a single operator (d = 3, S ≈ 0.17). The same number — tautologically. Just as all photographs taken with a single lens have the same aberration.

10.2. Analogy Speed of sound: 343 m/s in air. A thousand measurements by a thousand methods yield one number. But in water 1480 m/s, in steel 5960 m/s. The “constant” turned out to be a property of the medium. h: 6.626 × 10−34 J·s. A thousand measurements, one number. But all measurements are performed in the same “medium”: observer d = 3, condensed matter S ≈ 0.17. Change the medium (different d, different S) — and h changes. But D-Prot: we cannot measure h “from a different d,” just as we cannot hear sound “from water while being in air.”

10.3. h as a property of the pair (Ô, Ψ) h is not a property of “the world in itself.” h is a property of the interaction between the observer and the observed: h = h(Ô, Ψ) = h(d(Ô), S(Ô, Ψ))

(X.1)

For the same observer (d = 3, S ≈ 0.17) observing any object: h is the same. Because d(Ô) and S(Ô, Ψ) are determined by the operator.

10.4. The observer’s proper time h is the observer’s proper time expressed in units of action. Analogy with GR: proper time dτ = ds/c depends on the metric (the gravitational field). Each observer measures their own dτ as absolute. The discrepancy between clocks appears only upon comparison. Likewise h: each observer measures their own h as an absolute constant. The discrepancy appears only when comparing observers with different d and S. But such comparison is extremely difficult due to D-Prot.

10.5. Analogy with color blindness A person with red-green color blindness measures the “color” of various objects. All measurements are self-consistent: red and green are indistinguishable. They conclude: “red and green do not exist; there is only yellow-gray.” Their instruments (built by them, with their filters) confirm: all spectrometers yield the same result. But the problem is not the color — the problem is the observer. Their operator Ô projects the spectrum onto a two-dimensional (rather than three-dimensional) color space. Everything that differs only in the lost dimension is indistinguishable. Likewise with h: our operator (d = 3, S ≈ 0.17) projects all measurements onto a single value h(3, 0.17). Everything that differs only in other d or S is indistinguishable. We do not see the difference not because it is absent, but because our “spectrometer” is not tuned to that dimension.

10.6. Can h be the same at all levels? From the observer’s viewpoint — yes. Each observer sees their own h as an absolute constant. Precisely because h is determined by their operator. Just as each person sees their own nose as “normal,” though noses differ: the nose is part of the observer. From the architecture’s viewpoint — no. Formula (V.2) explicitly contains d and S. At different d and S: different h. This is not an assumption but a derivation from the axiomatics. Contradiction? No. “Absolute for each” and “different between different ones” do not contradict each other. Just like time in GR: absolute for each clock, different between clocks in different reference frames. Time is neither a “constant” nor a “variable.” Time is proper to each observer. So is h.

Question

Answer

Do all our measurements yield Yes (tautology: one operator) one h? Is h the same at all levels d? No (formula: h ∝ φd ) Can it be tested? Extremely difficult (D-Prot) Does “h in itself” exist? No (h is a property of the pair Ô, Ψ) Does the formula contradict No (it explains why h appears constant) experiment?

10.7. h as a mirror of the observer Planck’s constant is a mirror of the operator. Each observer sees in it themselves: their grain of observation, their scale, their coherence. And because the mirror is perfect (tautology: h is measured through h), the reflection is always flawless. The only way to change the reflection is to become a different observer (change d or S). But having become a different one, you will see their h, not yours. And their h will also appear to them as an absolute constant. Each dimensionality level lives in its own “scale of action.” Each considers its own scale the only one. And each is right — for itself.

XI. SELF-REFERENTIALITY 11.1. The loop h ↔ S h depends on S (formula V.2). S depends on the results of observations [2, formula 4.5], which depend on h. A loop: h = f (S),

S = g(h)

## (XI.1)

Fixed point: h∗ = f (g(h∗ )), like Ψ∗ = Φ(Ψ∗ ).

11.2. Consequence Planck’s constant is self-consistent. It is defined through itself, because the observer defines reality, which defines the observer. h is not “a number God chose” but a fixed point of the loop “observation ↔ reality.”

11.3. Uniqueness S ∗ = 0.16967646777119 . . . is the unique solution of the equation f (3, S) = 1 (monotonicity of f in S at fixed d). The fixed point is unique. Just as Ψ∗ is unique by the Banach theorem.

XII. CONNECTION WITH OTHER ODTOE RESULTS 12.1. Unified chain cubic eq. X.1 [10]

+ e,c,ε0

π, φ −−−−−−−−−→ α−1 = 137.03599917036 −−−−→ h = 6.6260701542 × 10−34 cubic eq. IV.3 [10]

e mp = 1.67262 × 10−27 kg π, φ −−−−−−−−−→ µ = 1836.15267342575 −−−→

Both chains begin with π and φ. Both use SI defining constants (e, c, ε0 , me ). Both yield results that agree with experiment (9–10 significant digits).

12.2. Toroidal interpretation By [5]: reality is a nesting doll of φ-tori. π is the rotation inside the torus (θ-dynamics). φ is the scale between tori (ϕ-dynamics). (π−3)2 is the gap (the bridge between θ and ϕ). h is the minimal action = (energy of θ-rotation + gap) × (time of θ-revolution on the φ-scaled torus).

12.3. Z2 -bundle and discrete symmetries By [16]: the non-trivial Z2 -bundle over the φ-torus with holonomy hol(γϕ ) = −1 explains: (a) The fermionic 4π traversal (spin-1/2): one traversal along ϕ yields ψ → −ψ, two traversals return ψ. (b) CPT symmetry: C = fibre flip (+1 ↔ −1), P = reflection of θ, T = reversal of ϕ. (c) Pauli exclusion: uniqueness of the global section of the bundle. The factors of 2 in the formulas for µ (6 = 3 × 2) and α−1 (2(π − 3)2 ) are projections of a single Z2 -holonomy onto two different physical effects [16, Sections IV.1–IV.2]. The formulas preserve numerical precision: the Z2 -bundle reinterprets existing factors without introducing additional numerical terms. Prediction: the bundle twist contribution δtwist = π 2 (π − 3)4 /(µ · α−1 ) ≈ 1.58 × 10−8 will become measurable at CODATA precision ±10−9 [16].

## XIII. DEMARCATION Statement

Status

Quantum = one revolution of Φ of Interpretation via ODTOE length 2π h = Emin · τ Definition of action (standard physics) Emin = 2π(π − 3) φΣ(d) Follows from A + D-Prot + ternary architecture τ = τ0 φd (1 − S)−1/2 Follows from P3.1 + KAM + diffusion Full formula h(d, S) Consequence of A + D-Prot + P3 + Banach + KAM (1 − S)−1/2 Proved (was: hypothesis) ∗ S = 0.16967646777119 Computed from π, φ, d = 3 (zero fitting) α = 137.03599917036 (cubic, 3 Computed from π, φ [10] orders) hODTOE = 6.6260701542 × 10−34 J·s Consequence of αODTOE and SI constants A0 = h at d = 3, S = S ∗ Follows from self-consistency (V.4) A0 is the sole dimensional parameter Architectural fact (dimensionless → cannot yield dimensional) 20+ SM parameters → 0 Follows from formulas for α−1 , µ, h dimensionless + 1 dimensional h depends on d and S Follows from the formula Observed “constancy” of h Explained via the tautology of measurement (D-Prot) h is a property of the pair (Ô, Ψ), not Interpretation via axiom (A) of “the world” h(d1 )/h(d2 ) = φd1 −d2 Falsifiable prediction Z2 -holonomy explains the factors of 2 Follows from the bundle [16] δtwist ≈ 1.58 × 10−8 Falsifiable prediction for CODATA 2030+

XIV. CONCLUSION 14.1. Results First. A formula for Planck’s constant is derived from the ODTOE axiomatics:

h(d, S) =

2π(π − 3)2 φd+1 1 − [(π − 3)2 φ2 ]d+1 · · A0 (1 − S)1/2 1 − (π − 3)2 φ2

Six factors, each derived, none postulated.

Second. From the self-consistency condition, the medium coherence is computed:  −2 S ∗ = 1 − 2π(π − 3)2 φ4 Σ(3) = 0.16967646777119108030 A dimensionless number from π, φ, d = 3. Zero fitting parameters. Falls in the condensed-matter range (0.1–0.3). Third. Via the ODTOE chain (α−1 = 137.03599917036 from π and φ, cubic equation [10]):

## hODTOE =

e2 αODTOE = 6.6260701542 × 10−34 J·s (consequence of αODTOE and SI constants)

Fourth. The observed “constancy” of h is explained: all measurements are performed by a single operator (d = 3, S ≈ 0.17). Change d or S — and h changes. But D-Prot: each observer sees their own h as absolute. Fifth. The Z2 -bundle over the φ-torus [16] enriches the structure of the formulas: the factors of 2 in µ and α−1 receive a unified geometric justification via the holonomy hol(γϕ ) = −1, without altering the numerical results.

14.2. What Planck’s constant is Not “God’s number.” Not “a fundamental brick of the Universe.” Planck’s constant is the grain of observation at a given dimensionality level and a given coherence: h = f (d, S) × A0 . The grain determines what the observer can distinguish. Just as a pixel determines the screen resolution. Below the grain — invisible. Above — visible. The grain size = the pixel size of reality for a given observer. Only 2π (revolution length) and (π − 3)2 (curvature cost) are absolute. Everything else is the operator’s context: their dimensionality (d), their coherence (S), their toroidal scale (φd ).

14.3. One formula h = |{z} 2π × (π − 3)2 × φ × Σ(d) × φd × (1 − S)−1/2 × A0 | {z } |{z} |{z} |{z} | {z } |{z} revolution

grain

step

depth

scale

coherence

size

Revolution × grain × step × depth × scale × coherence × size. Seven words. One number. All of quantum physics.

ACKNOWLEDGEMENTS AND TOOLS In the development of ODTOE and all articles based on it, artificial intelligence tools were used: Claude Sonnet / Opus 4.6 Extended (Chat & Code) (Anthropic), ChatGPT 5.3 (OpenAI), Google Gemini (Google DeepMind). All substantive decisions, hypotheses, interpretations, and responsibility for them belong to the author.

CONFLICT OF INTEREST The author declares no conflict of interest.

FUNDING This work was carried out without external funding.

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