Quantum Epoch

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We examine a Measurement Quantization (MQ) count construction for a proposed earliest regime of the universe, represented as a pre-referenceable count configuration unable to support external referencing until its mapped radius reaches √(3)l_f, where l_f denotes the fundamental length measure, expressed in metres. This pre-referenceable regime is the quantum epoch. Its referencing constraint fixes a temporal-scale ratio; a conditional normalization in the International System of Units (SI) is adopted below, while its unique physical derivation remains unresolved. In this interpretation, the boundary separates the quantum epoch from the expansionary epoch, in which physical elapsed time is realized. The geometric calculation makes no prediction of an inflationary period; its numerical duration requires an explicit temporal normalization. The count allocation and adopted realization give the conditional mass allocation rate (θ_si³p_si)/(l_f). Here θ_si is the invariant dimensionless MQ coefficient and p_si is its momentum realization, in kg m s⁻¹. The adopted SI realization assigns them the same numerical coefficient while preserving their different dimensions. The allocation relation and blackbody conversion for the cosmic microwave background (CMB) remain conditional on the adopted temporal normalization and allocation-to-present-radiation energy identification. Their physical derivations remain open. A numerical CMB temperature is therefore not established here. Fixsen’s combined observational estimate, 2.72548 ± 0.00057 K, provides a comparison target.

Keywords: Quantum Epoch, Epoch, Early Universe, CMB, Inflation

1. INTRODUCTION

The Internal Frame is the discrete, count-based configuration domain used in MQ; it has no intrinsic continuous geometry. The speed of light, escape velocity and Heisenberg’s uncertainty principle [1] motivate the MQ count construction ([2], Appx. I, J). They are not used here as independent measurements of the discreteness of length, mass or time. Loop Quantum Gravity (LQG) provides a separate theoretical description of discrete geometry [3, 4, 5]; this does not establish the MQ count interpretation experimentally.

The System Frame is the continuous relational encoding of count structure in which observable quantities are realized. The Frames mapping connects Internal Frame count configurations to this System Frame description ([2], Appx. K). These are representational distinctions in MQ. Their application to physical constants and the laws of nature is developed in [2], Appx. C, AA, AM, AP, AQ, AR, BA and related appendices [6].

The fundamental mass and time measures are m_f in kg and t_f in s; the length measure l_f was introduced in the abstract. The dimensionless counts n_L, n_M and n_T express lengths, masses and temporal intervals relative to those measures, respectively. Their assigned geometric or temporal context is stated locally. The fundamental measures are distinguished from conventional Planck units [7] by the discrete-measure normalization used in MQ ([2], Appx. D, W, X). Their proposed physical interpretation is developed in [2], Appx. C, I, J.

The proposed sequence begins with the external-referenceability threshold at a mapped radius of √(3)l_f. Below that boundary, the count construction does not provide the external referencing assumed in the realized expansionary description. Its numerical temporal encoding depends on the explicit normalization and mapping assumptions stated below.

Calculations describing this epoch give the symbolic expansion quantity in equation (18). Its numerical evaluation depends on the temporal normalization. The SI-equivalent temporal scale and referenceability boundary are developed in equations (19) and (20) ([2], Appx. BF, BG).

To assess this description, we consider the mass allocation rate ([2], Appx. BE). Its conditional 2022 realization is given in equations (7) and (8). Combining this rate with a mapped duration gives a mass allocation, while present-day CMB density and temperature additionally require a verified radiation evolution. Fixsen [8] supplies an observational temperature comparison. No numerical temperature agreement is established by the symbolic relations retained here.

The calculation uses classical algebra together with the MQ realization and domain assumptions specified below. No fit to the measured CMB temperature is introduced here. The explicit temporal normalization and radiation-energy identification are conditional premises, so the calculation is not presented as free of assumptions. Inflation [9, 10], LQG and string theory [11] provide theoretical background with different assumptions; they are not independent tests of the MQ construction.

The Internal Frame and System Frame descriptions must remain distinct when physical constants are expressed through the Frames mapping. In particular, the SI-equivalent quantum-epoch encoding is not an intrinsic physical duration in the pre-referenceable count domain.

2. METHODS

The calculation applies classical algebra to the count relations and realization assumptions stated below. Earlier MQ publications present applications to physical constants, gravity, galactic dynamics and cosmology [6, 12, 13, 14, 15, 16]. Those publications establish the history of the proposed approach; their reported agreements do not independently validate the temporal normalization or radiation evolution used here.

The underlying count interpretation and proposed calibrations are developed in the introductory manuscript [17] and its published treatment [2]. Appendix citations explicitly identify the published treatment [2]; current symbol and frame meanings follow the definitions stated in this paper. We use the supported algebra with explicit assumptions, while keeping its empirical interpretation distinct from internal consistency.

https://www.researchgate.net/publication/361148724_OnlineResources

Two supplementary spreadsheets are described with the published MQ treatment [2] and linked through Online Resources. Such calculation resources document numerical working; they do not by themselves establish independent physical validation. The present calculation uses the fixed 2022 inputs and uncertainty treatment specified below.

3. RESULTS

We use the mass allocation rate to construct conditional relations for the CMB mass allocation, present-day radiation energy density and blackbody temperature. The physical derivation of the adopted temporal normalization and the allocation-to-present-radiation evolution remain open. We therefore distinguish symbolic relations from the historical numerical comparisons retained below.

The observational domains used here are geometric partitions ([2], Appx. BB), not independently postulated matter components. Their dimensionless fractions are Ω_vis for the presently visible domain, Ω_obs for the domain visible now or eventually, Ω_uobs for the eventually visible portion not visible now, and Ω_dk for the domain that never becomes visible in the stated expansion description. The fundamental-domain fraction Ω_f denotes the MQ partition pivot used in the following algebra. The total fraction Ω_tot is normalized to one. The cosmological-constant plus cold-dark-matter model (ΛCDM) provides a comparison framework; the MQ domain definitions have the meanings just stated.

The dark-domain equality uses the partition identity Ω_tot = Ω_dk + Ω_obs and the total normalization stated above ([2], Eq. (BC.7)). The two expressions are

Ω_{dk} = \frac{Ω_{obs}}{Ω_{f}} - 1 .

(1)

\frac{Ω_{obs}}{Ω_{f}} - 1 = 1 - Ω_{obs} .

(2)

We now express the fractions through mass-equivalent allocations in kilograms. The symbols M_tot, M_obs and M_dk denote the total, observable and dark-domain allocations, respectively. The dimensionless universe-context counts n_{T_u} and n_{M_u} denote the temporal and mass counts of the same mapped description. They are not primitive physical time or mass before realization. The fundamental-domain allocation M_f is specified below and is distinct from m_f ([2], Appx. BE). The normalization uses the observable and fundamental allocations divided by the total allocation, with the total equal to the sum of observable and dark-domain allocations. In this notation, M_f=n_{T_u}m_fθ_si; Ω_f=(M_f)/(M_tot); Ω_obs=(M_obs)/(M_tot); M_tot=M_obs+M_dk.

\frac{M_{obs}}{M_{f}} - 1 = 1 - \frac{M_{obs}}{M_{tot}} .

(3)

M_{tot} = M_{f} (\frac{2}{Ω_{obs}} - 1) .

(4)

M_{tot} = n_{T_{u}} m_{f} θ_{si} (\frac{2 ({θ_{si}}^{2} + 2)}{4} - 1) = n_{T_{u}} m_{f} \frac{{θ_{si}}^{3}}{2} .

(5)

n_{Mu} m_{f} = n_{Tu} m_{f} \frac{θ_{si}^{3}}{2} .

(6)

Equation (3) expresses the same normalized partition in mass-equivalent allocations. Solving it for the total allocation gives equation (4); substituting the MQ observable-domain fraction gives equation (5). Equation (6) restates this allocation in fundamental mass units. Let M_acr denote the mass allocation rate per mapped realized duration, in kg s⁻¹. Dividing the mass-count relation by the temporal count and fundamental time gives equation (7); using the fundamental relation l_fm_f=2p_sit_f ([2], Appx. V) gives equation (8).

M_{acr} = \frac{n_{M_{u}}}{n_{T_{u}}} \frac{m_{f}}{t_{f}} = \frac{{θ_{si}}^{3}}{2} \frac{m_{f}}{t_{f}} = 17.3611207549 (40) \frac{m_{f}}{t_{f}} .

(7)

M_{acr} = \frac{{θ_{si}}^{3} p_{si}}{l_{f}} = 7.0088794874 (27) \times 1 0^{36} kg s^{- 1} .

(8)

Equations (7) and (8) are MQ calculations from the 2022 Committee on Data for Science and Technology (CODATA) inverse fine-structure input [18], conditional on the adopted realization and the MQ count construction. Parenthetical digits denote one standard uncertainty in the final displayed digits and include source-input uncertainty only. The coefficient and rate share the same uncertain inverse fine-structure input; their uncertainties are not independent. With the adopted fixed allocation rate M_acr, we can now proceed.

1. Quantum Referencing

For equations (9) and (10), n_a and n_b are the dimensionless leg counts in the local right-triangle construction, and n_bn is its derived hypotenuse count. The radial length count n_L is the geometric norm formed from these legs. The minimum construction uses unit leg counts; it does not require the derived hypotenuse or radial norm to be an integer.

n_{L} = \sqrt{{n_{a}}^{2} + {n_{bn}}^{2}} = \sqrt{{n_{a}}^{2} + {\sqrt{{n_{a}}^{2} + {n_{b}}^{2}}}^{2}},

(9)

\sqrt{{n_{a}}^{2} + {n_{a}}^{2} + {n_{b}}^{2}} = \sqrt{1^{2} + 1^{2} + 1^{2}} = \sqrt{3} .

(10)

Within the stated referenceability interpretation, the unit-count construction gives the threshold √(3)l_f. For a larger mapped radius, such as √(5)l_f, the same local construction can be referenced from points within the domain. Its radial norm n_L=√(3) is distinct from the unit leg counts. This geometric statement does not establish an additional elapsed-time interval or an independently verified expansion history. The global shape and any preferred axis require a separate analysis.

We next consider the quantum-epoch expansion quantity and its SI-equivalent temporal scale A_qe, expressed in seconds after dimensional encoding. We use the fundamental relation introduced above. The primed quantity n_{L_u'} is a derived dimensionless ratio used in the Frames mapping; its definition and assumed allocation are specified in equation (15). It is not a primitive integer count or the integrated radius. The associated v' is a mapped radial increment per temporal-scale increment, with units m s⁻¹ after realization.

Critical density is used in the reduction below. Its application is an explicit assumption of the mapped construction in equation (17), rather than an independently validated consequence of this calculation ([2], Appx. BG). The same fundamental relation is used in the atomic calibration context of [2], Appx. BL.

In equation (11), a₀ is the Bohr radius in m, m_e is electron mass in kg, and α_c is the dimensionless classically realized fine-structure coupling. The dimensionless Diophantine realization residual is Q_L, with Q_L=(1)/(√(1+n_L²)+n_L) for positive n_L ([2], Appx. D). Its product Q_Ln_L is half the finite-count normalization factor; it is not itself the Informativity differential. That term denotes the associated non-Lorentz length contraction, quantified here by the departure of twice this product from unity. In this orbital calibration, the count is evaluated at the Fine Structure demarcation, the electromagnetic interaction-count reference used by MQ.

a_{0} = \frac{1}{α_{c} m_{e}} \frac{p_{si} t_{f}}{Q_{L} n_{L}} .

(11)

The atomic relation is part of the calibration used to connect the count construction to its dimensional realization ([2], Appx. BL). Reusing those calibrated quantities in a cosmological expression does not supply an independent validation of the epoch normalization.

For equation (12), A_U is the adopted present System Frame age of the universe, in s; D_U is its physical diameter, in m; and c is the speed of light, with c=299792458 m s⁻¹. The illustrative choice A_U=13.794 Gyr gives D_U=90.003 Gly. Here Gyr means one billion years and Gly one billion light-years. This specifies an adopted age, with uncertainty not supplied, rather than an independently determined age calibration.

D_{U} = 2 θ_{si} c A_{U} .

(12)

m_{f} = \frac{2 p_{si}}{c} = 2.17643254057 (17) \times 1 0^{- 8} kg .

(13)

Equation (12) gives a diameter conditional on an adopted age; equation (13) instead follows from the momentum calibration and the exact speed of light. The mass in equation (13) is not inferred independently from that diameter. The subsequent critical-density reduction uses the count and realization assumptions stated below. The quantity A_ee denotes the mapped expansionary-epoch duration, in s, related conditionally to the quantum-epoch scale through equation (23) ([2], Appx. BH, BK). The latter remains an SI-equivalent pre-referenceable extent rather than ordinary elapsed clock time.

2. Quantum Epoch

Beginning with the fundamental relation, we consider its count-scaled form for positive counts satisfying the condition in equation (14). These general counts are not identified with the total diameter, accumulated mass, and age counts of the universe. The application to the constrained quantum-epoch counts in equation (15) requires a separate specification. The subsequent expressions consider the critical-density reduction and the radius and temporal scale associated with the quantum epoch. For equation (15), we assume that the mass-allocation relation in equation (6) applies to the same mapped quantities over the same interval.

\frac{n_{L} l_{f}}{n_{T} t_{f}} = \frac{2 p_{si}}{n_{M} m_{f}}, n_{L} n_{M} = n_{T} .

(14)

n_{L_{u^{'}}} = \frac{n_{T_{u}}}{n_{M_{u}}} = \frac{2}{{θ_{si}}^{3}} .

(15)

In the volume expression, π is the dimensionless circle constant. For equation (16), we use a mapped System Frame representation. M_U=n_{M_u}m_f denotes the mapped mass allocation in kg; ρ_c is the critical mass density in kg m⁻³; and V_U is the associated spherical mapped volume in m³. We assume that n_{T_u}t_f is represented by the SI-equivalent temporal scale A_qe in this volume construction, so that V_U=(4π)/(3)(θ_sicA_qe)³. The integrated pre-referenceable radius, expressed in m, is R_qe. The mass density here is distinct from the radiation energy density used later. This conditional representation does not establish an absolute time normalization or a physically measured velocity within the quantum epoch. For equation (17), we explicitly assume ρ_c=(3H_f²)/(8πG), where H_f is the Hubble frequency, in s⁻¹, assigned to the same mapped representation. Upright G denotes the upper-count gravitational coupling in m³ kg⁻¹ s⁻², given by G=(l_fc³)/(2p_si). The density and frequency must describe that same construction. We retain H_f symbolically here; its relation to A_qe requires the separate specification in equation (18).

v^{'} = \frac{n_{L_{u}'} l_{f}}{n_{T_{u}} t_{f}} = \frac{2 p_{si}}{n_{M_{u}} m_{f}} = \frac{2 p_{si}}{M_{U}} = \frac{2 p_{si}}{V_{U} ρ_{c}} = \frac{2 p_{si}}{(\frac{4}{3}) π c^{3} {θ_{si}}^{3} {A_{qe}}^{3} ρ_{c}} .

(16)

v^{'} = \frac{3 p_{si}}{2 π c^{3} {θ_{si}}^{3} {A_{qe}}^{3}} \frac{1}{ρ_{c}} = \frac{3 p_{si}}{2 π c^{3} {θ_{si}}^{3} {A_{qe}}^{3}} \frac{8 π G}{3 {H_{f}}^{2}} = \frac{4 p_{si}}{c^{3} {θ_{si}}^{3} {A_{qe}}^{3} {H_{f}}^{2}} \frac{l_{f} c^{3}}{2 p_{si}} .

(17)

v^{'} = \frac{2 l_{f}}{{θ_{si}}^{3} {A_{qe}}^{3} {H_{f}}^{2}} = \frac{2 l_{f}}{{θ_{si}}^{3} A_{qe}}, H_{f} A_{qe} = 1 .

(18)

The condition in equation (18) is an additional assumption of the mapped construction for positive frequency and temporal scale. Its numerical velocity is withheld because the physical justification of an absolute SI-equivalent temporal scale remains open. For equation (19), let A_qe,0>0 denote a reference value of the mapped SI-equivalent temporal variable, and R_qe(A_qe,0) its associated radius. We interpret v' as the radial increment per increment of A_qe and integrate between the two positive scales with l_f and θ_si fixed. The general relation leaves this reference pair free; a conditional numerical convention is specified below.

R_{qe} (A_{qe}) - R_{qe} (A_{qe, 0}) = \frac{2 l_{f}}{θ_{si}^{3}} \ln (\frac{A_{qe}}{A_{qe, 0}}) .

(19)

\frac{A_{qe}}{A_{qe, 0}} = \exp [\frac{{θ_{si}}^{3}}{2} (\sqrt{3} - \frac{R_{qe} (A_{qe, 0})}{l_{f}})] .

(20)

Equation (20) applies at the referenceability boundary R_qe(A_qe)=√(3)l_f from equation (10). It fixes a dimensionless ratio relative to the reference pair in equation (19). The familiar exponential follows only if the reference radius is additionally set to zero; a dimensional reference scale is still required to obtain seconds or years. For a conditional numerical realization, we adopt A_qe,0=1 s and R_qe(A_qe,0)=0. These are additional normalization assumptions of the mapped description. The zero radius specifies a reference condition and is not asserted as a physical pre-realization initial state. Under this convention, equation (22) gives A_qe=363312.3525(25) yr, with one year taken as 31557600 s. Parentheses give source-input standard uncertainty from [18] only. The physical derivation selecting this reference pair remains open; its unquantified uncertainty is not assigned zero. The assigned extent is not elapsed physical time within the quantum epoch. A change of reporting units converts the same one-second reference and does not replace it with one year.

Equations (19)–(22) therefore determine a relative scale and, under the declared reference convention, an SI-equivalent extent. Equation (23) supplies the separate mapping to a realized duration used in the allocation calculation. Neither step establishes a thermal state, a causal contact history or a present CMB energy. Those require the additional assumptions and radiation evolution discussed below.

We carry the reference-consistent quantum-epoch relation into the mapping discussion by restating equation (19) as equation (21). This algebraic restatement does not specify a realized-universe growth law or determine the expansionary-epoch duration A_ee. The relation between the SI-equivalent quantum-epoch scale and that duration is considered in equation (23). The primordial CMB allocation M_CMB, expressed in kg, and its present-day density and temperature remain conditional on this mapping and the temporal normalization ([2], Appx. BH, BK). Equation (23) is adopted as a framework Frames mapping relation, with conditional source-input standard uncertainty from [18]. Its factor is not established here as a Lorentz transformation between specified clocks and does not fix an absolute duration.

For the special-case evaluation in equation (22), we impose R_qe(A_qe,0)=0 only for that calculation. This condition is not adopted as a physical initial condition or imposed on the general relation in equation (20). The result is dimensionless. Parentheses give one standard uncertainty from the shared 2022 inverse fine-structure input [18], propagated through the full exponential; the displayed digits are rounded. The one-second convention specified after equation (20) supplies a conditional dimensional realization; it does not establish a unique physical normalization.

\ln (\frac{A_{qe}}{A_{qe, 0}}) = \frac{{θ_{si}}^{3}}{2 l_{f}} [R_{qe} (A_{qe}) - R_{qe} (A_{qe, 0})] .

(21)

\frac{A_{qe}}{A_{qe, 0}} = \exp (\frac{\sqrt{3} {θ_{si}}^{3}}{2}) = 1.1465265894 (79) \times 1 0^{13} .

(22)

\frac{A_{ee}}{A_{qe}} = {(2 θ_{si})}^{1 / 3} = 1.868624209450 (48) .

(23)

For equation (24), we assume that the mapped interval supplies the primordial CMB allocation at the constant rate in equation (8), with no later allocation added to that pool. This is mapped allocation accounting; it does not describe primitive physical accretion before realization. An absolute mass requires a normalized mapped duration. For equation (25), E_CMB is the present total CMB radiation energy in J, ρ is its energy density in J m⁻³, and V_U(A_U) is the spherical volume evaluated at the adopted present age. This present-volume evaluation is distinct from the conditional mapped-volume construction in equation (16). In equation (26), we conditionally identify present radiation energy with the energy equivalent of the primordial allocation in equation (24), following the direct allocation-to-present-density construction in [2], Appx. BH. This is an additional mapping assumption. The quantitative physical evolution establishing that identification, including post-transition redshift and dilution, remains an open exception. The identification is not a derived redshift law or a demonstration that primordial photon energy is conserved through expansion. In equation (27), a is the radiation constant in J m⁻³ K⁻⁴, σ is the Stefan–Boltzmann constant in W m⁻² K⁻⁴, k_B is the Boltzmann constant in J K⁻¹, and h is the SI Planck constant in J s. The displayed value uses the exact SI definitions of c, h and k_B [18]; its final digits are rounded. In equation (28), T_CMB is the temperature in K of the realized radiation field under the equilibrium blackbody assumption. Density and temperature calculated using equation (26) remain conditional on that identification, the adopted temporal normalization and present age/volume. No independently validated numerical temperature prediction is established, and unquantified physical-evolution uncertainty is not assigned zero.

M_{CMB} = M_{acr} A_{ee} = \frac{{θ_{si}}^{3} p_{si}}{l_{f}} A_{ee} .

(24)

ρ = \frac{E_{CMB}}{V_{U} (A_{U})} V_{U} (A_{U}) = \frac{4 π}{3} {(θ_{si} c A_{U})}^{3} .

(25)

E_{CMB} = M_{CMB} c^{2} .

(26)

a = \frac{4 σ}{c} = \frac{8 π^{5} {k_{B}}^{4}}{15 c^{3} h^{3}} = 7.56573325028 \times 1 0^{- 16} J m^{- 3} K^{- 4} .

(27)

T_{CMB} = {(\frac{ρ}{a})}^{1 / 4} = {[\frac{E_{CMB}}{a V_{U} (A_{U})}]}^{1 / 4} .

(28)

3. Observational Comparisons

Table 1 separates observational reference values from historical MQ calculations. The decoupling epoch is inferred under a cosmological model fitted to Wilkinson Microwave Anisotropy Probe (WMAP) data, whereas the MQ entry is an SI-equivalent extent. They are different quantities. The reference energy density is inferred from a blackbody spectrum and is not an independent second temperature measurement. The historical MQ column retains the incoming manuscript’s values and original reported precision; it is neither a 2022 recalculation nor an independently validated prediction. Its density entry differs from the conditional evaluation of equations (24)–(27) with the adopted age. Numerical concordance does not establish the missing temporal normalization or radiation evolution.

The MQ observational-domain fractions describe the geometric partitions introduced above ([2], Appx. BB, BC). Comparing them with ΛCDM requires preserving that distinction from matter and dark-energy components. The partition algebra by itself does not establish the physical composition of the universe or validate the conditional CMB calculation.

Table 1. Reference values and historical MQ calculations for decoupling epoch, CMB allocation, present-day density and temperature.

Measure
	Reference value	Historical MQ
Age comparison	379,000 years ^a	363,312.341272(21) years
CMB allocation M_CMB	-	1.5016 10⁵⁰ kg
Present Density	4.17 10^-14 Jm^-3^b	4.1704 10^-14 Jm^-3
Present Temperature	2.72548 ± 0.00057 K^c	2.7255 K

^a Decoupling epoch inferred under the first-year WMAP cosmological model [19]; it is not a measurement of the MQ temporal extent.

^b Blackbody energy density inferred from the measured temperature; the same spectrum gives about 411 photons/cm³ [8, 20].^c Fixsen’s combined temperature estimate [8].

4. DISCUSSION

The construction provides a proposed sequence of early-universe events and a conditional chain from mass allocation to radiation density and temperature. A numerical prediction requires the temporal normalization and radiation evolution specified above. The origin of the initial configuration remains outside the present calculation.

Equation (29) states a mass-to-length count bound in the MQ escape-velocity construction. Applying it to the quantum-epoch configuration is a conditional interpretation. It does not independently establish the composition, thermalization or subsequent radiation evolution of the allocation in equation (24).

\frac{n_{M}}{n_{L}} < \frac{1}{2} .

(29)

The construction can be considered alongside the flatness, horizon and monopole problems that motivate early-universe models [10]. Its present equations must be assessed within their stated mapping and realization assumptions.

First, the flatness problem [21, 10] concerns spatial curvature in the realized universe. The fundamental relation fixes the relation among the adopted measures; equations (16)–(18) additionally assume a mapped spherical volume and critical density. This reduction therefore does not independently establish spatial flatness or exclude curvature. The atomic relation in equation (11) supplies calibration context, rather than a separate test of cosmological curvature.

Second, the horizon problem concerns the homogeneity of regions that are causally disconnected in the conventional early-universe description [10]. The referenceability threshold in equation (10) and the conditional allocation in equation (24) do not by themselves calculate causal contact or thermal equilibration. The proposed MQ account of the CMB therefore still requires the temporal normalization, mapping to a realized interval and radiation evolution stated above.

Finally, the monopole problem concerns a particle-production history [22, 10]. The mapped allocation construction does not identify primitive pre-referenceable counts with physical baryonic matter. Equations (24)–(29) do not determine a primordial thermal history or monopole abundance. A quantitative comparison requires a supported radiation and particle-production evolution.

The calculation is assessed under its stated MQ premises. Independent physical validation requires a supported temporal normalization, a quantitative radiation evolution and an independently calibrated present age with uncertainty. Observations of the CMB in distant molecular absorbers test its cooling over cosmic history [23]; an evolution model used here must reproduce such data as well as the present temperature. The allocation identity alone does not supply that test. The present results therefore remain a conditional construction, with no independently validated numerical CMB temperature prediction.

5. APPENDICES

APPENDIX A NOTATION AND CALIBRATION NOTES

Symbols and terms used in the derivation are defined at first use. The following notes retain the angular-calibration context and clarify notation conventions. The broader source definitions are given in Measurement Quantization [17].

The angular realization is denoted Θ_si. The manuscript’s historical MQ angular estimate is Θ_si=3.26239 rad, with an adopted comparison tolerance of 2 μrad. Table I of [24] gives this angle as a calculated phase-matching solution, obtained by matching the wave vectors for a specified diamond-crystal geometry. Its identification with an MQ angular realization is a framework interpretation, and the adopted tolerance is not a published standard measurement uncertainty. The experimental X-ray down-conversion and wave-mixing studies [25, 26] concern distinct optical processes; they do not independently determine an MQ fundamental momentum or a cosmological normalization. Under the adopted 2022 calibration, the momentum realization is p_si=3.26239030504(25) kg m s⁻¹ [18].

The speed calibration uses c=(n_Ll_f)/(n_Tt_f), with equal light-travel counts n_L=n_T=1, and the exact value stated in Section 3.1 [27, 18].

Upper-count gravitational coupling is distinguished from a finite-count realization by its assigned count context and typography. The coupling in equation (17) uses the upper-count convention. Historical finite-count numerical examples are not used to calibrate the present calculation.

The metric differential is the mismatch between discrete count structure and its continuous System Frame encoding; it is distinct from the Informativity differential introduced above. In the historical count illustration ⌊(84.60055)/(2)⌉=42, mixed left-floor/right-ceiling brackets denote rounding to the nearest whole-unit value. The example does not identify an interaction count with a dimensional temporal reference.

Alongside the classical coupling introduced before equation (11), α_f and α_p denote the dimensionless fundamental and Planck-like fine-structure couplings. Their inverse forms are α_f^-1 and α_p^-1. The fundamental form α_f^-1=42θ_si belongs to the Internal Frame. The Planck-like form includes the metric differential; the classically realized form also accounts for the Informativity differential.

STATEMENTS AND DECLARATIONS

The author has no relevant financial or non-financial interests to disclose.

The author has no competing interests to declare that are relevant to the content of this article.

DATA AVAILABILITY

The numerical inputs, calculation provenance and validation records supporting this revision accompany the manuscript. Historical supplementary spreadsheets are described in [2].

REFERENCES

[1]Heisenberg, W., Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 43 (3–4): 172–198 (1927), Bibcode:1927ZPhy.43.172H, doi:10.1007/BF01397280, S2CID 122763326.

[2] J.A. Geiger, Measurement quantization, Int. J. Geom. Methods Mod. Phys. 20(4), 2350069 (2023), doi:10.1142/S021988782350069X.

[3]Hermann Nicolai; Kasper Peeters; Marija Zamaklar, Loop quantum gravity: an outside view, Classical and Quantum Gravity. 22 (19): R193 (2005), doi:10.1088/0264-9381/22/19/R01.

[4]L. Smolin, The Case for Background Independence, In D. Rickles, S. French, J. T. Saatsi, The Structural Foundations of Quantum Gravity, p. 196 (Clarendon Press, 2006), doi:10.1093/acprof:oso/9780199269693.001.0001.

[5] C. Rovelli and L. Smolin, Spin networks and quantum gravity, Phys. Rev. D 52, 5743–5759 (1995), doi:10.1103/PhysRevD.52.5743.

[6]J.A. Geiger: Measurement Quantization Describes the Physical Constants, International Journal of Theoretical and Mathematical Physics, 11(1), pp. 29-59 (Jan 2021).

[7]M. Planck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu, Berlin, 5, p. 480 (1899).

[8]D.J. Fixsen, The Temperature of the Cosmic Microwave Background, (2009), arXiv: 0911.1955, doi:10.1088/0004-637X/707/2/916.

[9]Alan H. Guth, The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (1998), Basic Books. ISBN 978-0201328400.

[10] A.H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D 23, 347–356 (1981), doi:10.1103/PhysRevD.23.347.

[11]Becker, Katrin; Becker, Melanie; Schwarz, John (2007). String theory and M-theory: A modern introduction, Cambridge University Press, ISBN 978-0-521-86069-7, doi:10.1017/CBO9780511816086.

[12]J.A. Geiger: Measurement Quantization Unites Classical and Quantum Physics, Journal of High Energy Physics Gravitation and Cosmology. Vol. 4, No. 2, 262–311 (Apr 2018), doi:10.4236/jhepgc.2018.42019.

[13]J.A. Geiger: Quantum Model of Gravity Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data, Journal of High Energy Physics Gravitation and Cosmology. Vol. 4, No. 4 (Oct 2018), doi:10.4236/jhepgc.2018.44038.

[14]J.A. Geiger: Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture, Journal of High Energy Physics Gravitation and Cosmology. Vol. 5, No. 2 (Apr 2019), doi:10.4236/jhepgc.2019.52028.

[15]J.A. Geiger: Physical Significance of Measure, Journal of High Energy Physics Gravitation and Cosmology. Vol. 6, No. 1 (Jan 2020), doi:10.4236/jhepgc.2020.61008.

[16]J.A. Geiger: Measurement Quantization Describes History of Universe – Quantum Inflation, Transition to Expansion, CMB Power Spectrum, Journal of High Energy Physics Gravitation and Cosmology. Vol. 6, No. 2 (Apr 2020), doi:10.4236/jhepgc.2020.62015.

[17]J.A. Geiger: Measurement Quantization, (2022), doi:10.13140/RG.2.2.23035.85280/1.

[18] P.J. Mohr, D.B. Newell, B.N. Taylor and E. Tiesinga, CODATA Recommended Values of the Fundamental Physical Constants: 2022, arXiv:2409.03787v1 (2024), https://arxiv.org/pdf/2409.03787v1, Tables XXXII–XXXIII.

[19] C.L. Bennett et al., First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results, Astrophys. J. Suppl. Ser. 148, 1–27 (2003), doi:10.1086/377253.

[20] D. Scott and G.F. Smoot, Cosmic Microwave Background, in P.A. Zyla et al. (Particle Data Group), Review of Particle Physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020), Sec. 29.3.1, https://pdg.lbl.gov/2020/reviews/rpp2020-rev-cosmic-microwave-background.pdf.

[21]Robert H. Dicke (1970). Gravitation and the Universe: Jayne Lectures for 1969. American Philosophical Society. ISBN 978-0871690784.

[22]Carrigan, Richard A.; Trower, W. Peter (1983). Magnetic Monopoles, ISBN 978-1-4615-7372-2, doi:10.1007/978-1-4615-7370-8.

[23] S. Muller et al., A precise and accurate determination of the cosmic microwave background temperature at z = 0.89, Astron. Astrophys. 551, A109 (2013), doi:10.1051/0004-6361/201220613.

[24]S. Shwartz and S. E. Harris, Polarization Entangled Photons at X-Ray Energies, Phys. Rev. Lett. 106, 080501 (2011), arXiv: 1012.3499, doi:10.1103/PhysRevLett.106.080501.

[25]S. Shwartz, R. N. Coffee, J. M. Feldkamp, et. al., X-ray Parametric Down-Conversion in the Langevin Regime, Physical Review Letters, 109, 013602 (July 6, 2012), doi:10.1103/PhysRevLett.109.013602.

[26]T. E. Glover et al., X-ray and optical wave mixing, Nature, 488, 7413, 603–608 (2012), doi:10.1038/nature11340.

[27]E. Tiesinga, P.J. Mohr, D.B. Newell and B.N. Taylor, CODATA Recommended Values of the Fundamental Physical Constants: 2018, Rev. Mod. Phys. 93, 025010 (2021), doi:10.1103/RevModPhys.93.025010; NIST 2018 values, https://physics.nist.gov/cuu/pdf/wall_2018.pdf.