|
|
DOI: 10.13140/RG.2.2.10611.39203
Author: Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803
Affiliation: ECM Research Initiative - Tagore’s
Electronic Lab, India
Correspondence: postmasterenator@gmail.com
Date: December 07, 2025
This Appendix develops a unified, high-resolution formulation of Extended Classical Mechanics (ECM) applicable across gravitational, anti-gravitational, and cosmologically expanding domains. It establishes the operational definitions of effective acceleration (aᵉᶠᶠ, gᵉᶠᶠ), the role of negative apparent mass (NAM), and the full ECM mass-decomposition identity (Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ ≡ Mᴍ − ΔPEᴇᴄᴍ), which together govern all ECM dynamical behaviour.
The framework formalizes ECM gravitational neutralization, the limiting radius rmax, and the transition from matter-dominated to NAM-dominated states for particles, light, and composite systems. It shows that the rmax of any bounded structure naturally merges into the rmax of the universe, revealing a universal potential-energy structure that supersedes curvature-based or metric-expansion interpretations. Beyond rmax, systems enter a dark-energy–dominated region where NAM growth produces natural anti-gravitational acceleration without external forces.
The Appendix distinguishes the ECM behaviour of massive objects, whose acceleration may exceed c under sufficient NAM dominance, from pure-NAM systems such as photons, whose velocity remains fixed at c. It characterizes the ECM mechanism by which photons expend interactional NAM to traverse regions of increasing ΔPEᴇᴄᴍ, thereby losing energy while preserving c and generating cosmic redshift without invoking Doppler motion or expanding-metric geometry.
A key extension introduced in this Appendix is the Planck-Scale Phase-Shift Conversion, which expresses Δf, ΔE, and ΔMᴍ on a 360-division phase grid rather than in standard unit-based notation. ECM replaces the full oscillation period with its degree-resolved form (Tₓ° = x°/360°f), enabling a 360× finer temporal and frequency resolution. This allows manifestations such as stellar-photon frequencies to be compared directly against the Planck frequency fᴘ, revealing the massive disparity between ordinary electromagnetic events and Planck-scale manifestations with unprecedented clarity. The relation Δf = x°/360° serves as the Planck-frame conversion connecting frequency differences to their corresponding phase representation.
Through this synthesis—spanning gravitational dynamics, NAM-dominated regimes, photon energy behaviour, and Planck-scale phase-resolved analysis—the Appendix establishes ECM as a coherent and self-consistent alternative framework for understanding cosmic acceleration, redshift, and universal motion. It provides a non-relativistic, non-geometric explanation rooted solely in potential-energy displacement, manifest mass, and frequency-governed kinetic expression.
Effective acceleration, negative apparent mass, NAM dominance, gravitational mass, dark-energy region, r_max, cosmic potential, cosmic redshift, interactional NAM, ECM.
Appendix 50 consolidates the complete Extended Classical Mechanics (ECM) interpretation of acceleration, mass transformation, and cosmic-scale motion. It integrates local gravitational dynamics with the large-scale behaviour of the universe, providing a unified treatment of how matter, apparent mass (Mᵃᵖᵖ), negative apparent mass (NAM), and effective mass (Mᵉᶠᶠ) govern physical interactions across all scales.
The appendix formalizes the ECM decomposition (Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ) and the operational link Mᵃᵖᵖ ≡ −ΔPEᴇᴄᴍ, establishing negative apparent mass as the principal driver of anti-gravitational acceleration. This makes clear why NAM increases naturally with distance and why sufficiently high NAM dominance enables outward acceleration in cosmic environments.
A central component of the appendix is the derivation and interpretation of the limiting radius rmax, which marks the transition from matter-dominated gravitational influence to NAM-dominated outward acceleration. The analysis shows that every isolated gravitational system possesses its own rmax, and that these local rmax values merge seamlessly into the cosmic-scale rmax of the universe. This reveals a continuous universal potential structure, eliminating the need for geometric curvature as an explanatory mechanism.
The appendix further establishes a unified explanation of photon motion in ECM. Photons, as pure-NAM entities with zero matter mass (Mᴍ = 0), preserve constant velocity c while expending interactional NAM to move through regions of increasing ΔPEᴇᴄᴍ. This produces energy loss and hence redshift, without invoking metric expansion, relativistic Doppler formulas, or spacetime curvature. Photon behaviour thus fits naturally within the same mass-energy bookkeeping rules governing all ECM systems.
Together, these developments build a cohesive, falsifiable ECM framework capable of explaining gravitational attraction, anti-gravitational cosmic acceleration, matter–NAM transitions, photon energy evolution, and universal motion through a single consistent mechanism based on mass, potential energy, and their transformations. Appendix 50 therefore serves as the central reference for ECM’s cosmic-scale dynamics.
Effective acceleration (aᵉᶠᶠ, gᵉᶠᶠ) in ECM is not the same as the conventional acceleration defined in classical mechanics. In classical mechanics, an inertial mass (m) undergoing acceleration (a) requires an external force (F). ECM retains this classical grounding but extends it: the effective mass (Mᵉᶠᶠ) undergoing effective acceleration (aᵉᶠᶠ) responds to the ECM force field (Fᴇᴄᴍ, gᴇᴄᴍ), while honouring ECM’s deeper mass–energy bookkeeping structure.
In ECM, the effective mass is defined as:
where matter mass Mᴍ includes both ordinary and dark matter (Mᴍ = Mᴏʀᴅ + Mᴅᴍ), and the apparent mass term Mᵃᵖᵖ is the positive scalar representing reduced potential energy through the identity:
When matter mass dominates NAM (Mᴍ > Mᵃᵖᵖ), the effective mass behaves similarly to classical inertial and gravitational mass. As NAM increases with radial distance, however, Mᵉᶠᶠ decreases. This reduction in effective mass produces an increase in effective acceleration (aᵉᶠᶠ), enabling outward acceleration in regions where NAM becomes dominant.
While aᵉᶠᶠ describes force-driven acceleration for massive particles, ECM reveals its deeper identity for photons: the Effective Transformation Coefficient. For photons—whose dynamics are governed not by inertial acceleration but by mass-potential transitions encoded in frequency— aᵉᶠᶠ quantifies the rate of mass-form restructuring required to preserve the constant propagation condition.
A photon begins its existence with:
at the moment of emission, but its subsequent propagation is governed entirely by the conserved liberation force:
This constant force maintains the symmetry between frequency, wavelength, and NAM during propagation. As the photon moves through a gravitational potential field, frequency and wavelength adjust in a complementary manner to preserve:
Because F₀ is conserved, the transformation symmetry takes the form:
Thus, photon motion is not governed by acceleration in the classical sense but by continuous restructuring of mass-frequency form, regulated through the transformation coefficient aᵉᶠᶠ. As the photon evolves toward rₘₐₓ, the coordinated relation between frequency, wavelength, and apparent mass ensures that its kinetic-energy expression:
remains fully consistent with ECM’s mass–energy transition cycle. This establishes aᵉᶠᶠ as the unifying parameter governing both geometry-driven motion of massive particles and frequency-driven evolution of photons. It bridges gravitational interaction, potential-energy transition, and photon redshift within a single continuous ECM framework.
As an object moves away from a gravitational source, there exists a radius r_max at which its gravitational mass (Mɢ) becomes zero because Mᴍ = Mᵃᵖᵖ. Beyond this point, Fᴇᴄᴍ = gᴇᴄᴍ = 0, yet the object continues in uniform motion by classical inertia.
The r_max of any local source extends to the r_max of the universe because all local gravitational potentials are subsets of the universal potential structure. Thus, objects become gravitationally neutral not at the galaxy’s edge but only when cosmic ΔPEᴇᴄᴍ is fully counterbalanced by NAM.
Beyond r_max, the object enters the dark-energy dominated region. Immersion in positive ΔPEᴇᴄᴍ causes its bookkeeping NAM (Mᵃᵖᵖ) to grow. This makes its effective mass negative (Mᵉᶠᶠ < 0), bringing it under universal anti-gravity.
In regions of positive cosmic potential, NAM continues to grow. The interactional NAM component (Mᵃᵖᵖ₍ᶜₒₛₘᵢ꜀₎ ≡ ΔMᴍ) is consumed to overcome universal ΔPEᴇᴄᴍ. Additional motion is supported by inherent NAM. This produces natural acceleration without external force.
Pure NAM entities (photons: Mᴍ = 0, Mᵃᵖᵖ > 0) always propagate at the constant speed c. They cannot accelerate beyond c. As photons cross expanding cosmological regions, they expend interactional NAM to traverse increasing spatial intervals per unit time. Their speed remains c, but the NAM expenditure reduces their energy, producing cosmic redshift.
In summary: cosmic redshift occurs because a photon cannot exceed c; it therefore expends interactional NAM rather than increasing speed. This expenditure reduces its energy while preserving c.
Mass decomposition:
Gravitational neutralization condition:
NAM growth in cosmic ΔPEᴇᴄᴍ > 0:
Photon constraint:
Conventional interpretations assign cosmic redshift to metric expansion or Doppler shifting. ECM provides an alternative: photons expend interactional NAM to traverse expanding space while remaining speed-locked at c. This explains redshift without requiring spacetime curvature or geometric expansion models.
Effective acceleration (aeff, geff) in Extended Classical Mechanics (ECM) is not the same as the conventional acceleration defined in classical mechanics. In classical mechanics, the inertial mass (m) undergoing acceleration (a) requires an external force (F). In ECM, the effective mass (Meff) undergoing effective acceleration (aeff) similarly requires an external force (FECM, gECM), but the structure of this extended formulation remains rooted in classical mechanics while incorporating ECM’s deeper mass–energy bookkeeping.
In ECM, the effective mass (Meff) is composed of the matter mass (MM) and the negative apparent mass (NAM), where Mapp is a positive scalar and its negative contribution (-Mapp) represents the mass-equivalent of reduced potential energy (ΔPEECM ≡ -ΔMapp). NAM exhibits the reverse property of matter mass by subtracting from the gravitational mass (MG) of the system. Matter mass itself includes ordinary and dark-matter contributions (MM = MORD + MDM), relevant primarily under large-scale cosmological conditions.
When matter mass dominates NAM (MM > Mapp) under motion or gravitational potential difference, the effective mass (Meff) interacting through effective acceleration (aeff) couples to surrounding gravitational fields (geff) in a manner similar to classical inertial mass. However, the NAM component behaves repulsively and lowers the net gravitational influence of matter mass. This yields the decomposition:
equivalently,
This reduced effective mass decreases the required external force (FECM↓) for a given effective acceleration (aeff).
According to the understanding developed above, the effective acceleration (aeff) in ECM is not, in the strict sense, the same as the classical acceleration (a). As long as the gravitational mass (MG) of an object remains nonzero—corresponding to the dominance of matter mass (MM) over the negative apparent mass (-Mapp)—attractive gravity continues to govern the dynamical behaviour of the object through its effective mass (Meff).
However, as the distance between a gravitational source and the separating object increases, there exists a limiting radius (rmax) at which the object’s gravitational mass (MG) diminishes to zero. At this point, the matter mass of the object in dynamism becomes equivalent to its apparent-mass term (MM = Mapp), and thus the effective gravitational interaction ceases.
At rmax, the requirement for any external force (FECM, gECM) also falls to zero, because the object no longer possesses gravitational mass capable of responding to gravitational fields. Despite FECM, gECM = 0 at rmax, the physical object continues in motion with its last acquired velocity, fully consistent with the classical law of inertia.
Beyond rmax, the object transitions into the dark-energy-dominated intergalactic region, where gravitational influence becomes negligible and ECM dynamics shift to the NAM–DE regime.
At rmax, the object is not merely freed from the gravitational influence of its immediate source body; it is effectively liberated from the gravitational influence of the entire galaxy that hosts that source.
This follows directly from ECM’s interpretation that the gravitational ability of any local mass (stars, planets, clusters within the galaxy) is always an indirect expression of the galaxy’s total gravitational potential structure. In other words, the source body’s gravitational influence is only a localized manifestation of the galaxy’s global potential field.
Therefore, the radius rmax is extended far beyond the galaxy’s visible or baryonic boundary.
The object does not become gravitationally neutral at the galaxy’s edge; rather, it becomes gravitationally neutral only at the distance where the total galactic ΔPEECM experienced by that object is fully cancelled by its accumulated NAM component (-Mapp).
Hence,
Beyond rmax, the object enters a region dominated by dark-energy–driven expansion, where large-scale cosmological anti-gravity governs the dynamical behaviour, and conventional attractive gravitational structure no longer shapes the object’s motion.
Even though the object’s effective mass Meff (equivalently the gravitational mass MG) becomes numerically zero at this point—because the surrounding gravitational influence has been fully neutralized—the object’s physical existence does not vanish.
A zero-gravitational state in ECM does not imply annihilation of matter; it simply means the object is located at a point where both gravity and anti-gravity effects are absent or mutually cancelled.
Thus, the object persists in a state where:
maintaining motion through an environment that is gravitationally and anti-gravitationally neutral at the instant of crossing rmax.
Beyond rmax, as the dynamic object enters the dark-energy–dominated intergalactic region, rmax naturally merges with the larger-scale rmax of the universe.
At these distances, the object is no longer responding to the gravitational potential of its parent galaxy; instead, it experiences the total universal potential ΔPEECM, which defines the background cosmological potential field.
In this environment, the object’s NAM component (-Mapp) continues to grow.
This occurs through the mutual interaction between the object’s NAM and the universal NAM field associated with cosmic dark-energy dominance. In ECM, this interaction is understood as a natural amplification of anti-gravitational capacity: the more the object is immersed in a potential region of positive ΔPEECM (cosmic-scale potential rise), the more its bookkeeping NAM term Mapp increases, and therefore the physical NAM -Mapp becomes more negative.
Consequently:
Thus, beyond rmax, the dynamical state of the object is characterized by:
In this configuration, the object continues its motion along the rmax_universe trajectory, carried by increasing NAM support and by the expanding dark-energy-dominated geometry of intergalactic space.
Once the object is immersed in a region of positive cosmological potential (ΔPEECM > 0), its negative apparent mass (Mapp) increases through interaction with the surrounding anti-gravitational field of the universe. This increase generates the interactional NAM component (Mapp_cosmic = ΔMM), which the object must expend in order to overcome the positive ΔPEECM of the universe as it continues to recede along rmax_universe. Any additional outward progression beyond what the interactional component supports is then supplied by the object’s inherent NAM (Mapp_inherent), which remains part of its intrinsic mass-energy structure.
This cosmological interaction increases the physical NAM - (Mapp), making it more negative. As a result, the object’s effective mass (Meff = MM - (Mapp) becomes increasingly negative, strengthening the anti-gravitational contribution acting on it. The object therefore undergoes a natural acceleration, not due to an externally applied force, but arising from the cumulative NAM–NAM interaction with the cosmic anti-gravitational field.
For objects containing matter mass (e.g., stars or galaxies), an increase in NAM shifts the gravitational–antigravitational balance such that NAM dominance eventually allows the object’s speed to approach c and, when the NAM excess becomes sufficiently large, even exceed c under ECM conditions. Matter-bearing objects are not constrained by the propagation rule that governs pure NAM entities.
In contrast, pure NAM entities such as photons (Mᴍ = 0, Mᵃᵖᵖ > 0) maintain a strictly constant speed c at all times. A photon cannot accelerate beyond c. Instead, its propagation is governed by mass-potential transitions encoded in its frequency, which are regulated through the ECM Effective Transformation Coefficient (aᵉᶠᶠ).
At the moment of emission, a photon begins its existence with:
After emission, however, its behaviour is governed not by acceleration but by the conserved liberation force:
This conserved force regulates the complementary evolution of frequency and wavelength required to maintain the constant propagation condition:
Because F₀ does not change, the photon cannot alter its speed; instead, it adjusts its internal mass-form distribution through the following transformation symmetries:
Thus, a photon remains speed-locked at c, and any expenditure of NAM is expressed not as acceleration but as continuous mass-form restructuring. As the photon climbs out of gravitational wells or traverses regions where cosmic expansion increases the required distance per unit time, it expends interactional NAM instead of increasing speed. This expenditure reduces its frequency while maintaining c, producing the observed phenomenon of cosmic redshift.
Meanwhile, the ECM kinetic-energy identity:
remains satisfied at every point in propagation. This unifies redshift, mass-energy transformation, and constant-speed photon motion into a single ECM framework.
In summary: cosmic redshift occurs because a photon is fundamentally forbidden from exceeding c. As it traverses stretched or expanding regions of space, it expends NAM rather than increasing speed. This decreases its frequency while preserving c—the defining signature of ECM photon dynamics.
The formal mass-decomposition and NAM-governed principles established in this Appendix provide a clear physical basis for understanding photon behaviour in cosmic regions dominated by positive ΔPEECM. In such regions, the active manifestation of ΔPEECM appears as an anti-gravitational field that progressively neutralizes gravitational effects and shapes the energy evolution of photons travelling through expanding cosmic space.
According to the Appendix, mass decomposes as:
Here, Mᵃᵖᵖ is identified as the active expression of ΔPEECM—the interactional NAM that functions as an anti-gravitational field. Larger values of ΔPEECM correspond to environments where gravitational mass-effects are progressively neutralized, influencing both matter dynamics and photon evolution.
The Appendix shows that gravitational neutralization occurs when:
Meff = 0 ⇒ MM = Mapp ⇒ r = rmax
At this radius the photon experiences no net gravitational resistance. The positive ΔPEECM background provides propagation support rather than opposition, meaning the photon does not need to expend its inherent energy merely to traverse the light-travel component of space.
A crucial ECM principle is that a photon’s speed remains fixed at v = c. Therefore, additional NAM supplied by the anti-gravitational field cannot manifest as increased speed. The photon has only one available physical degree of freedom: its frequency must adjust.
Extra ΔPEECM (via the anti-gravitational field) → frequency reduction (cosmic redshift)
In ECM, this frequency adjustment is not arbitrary. It is regulated by the photon's Effective Transformation Coefficient (aᵉᶠᶠ), the parameter governing how NAM-driven mass-potential changes translate into its observable properties.
At emission the photon begins with aᵉᶠᶠ = 6 × 10⁸ m/s², and its propagation is governed by the conserved liberation force:
Because F₀ is conserved, the photon maintains the propagation rule:
This conservation enforces a complementary evolution:
Thus, photon propagation in anti-gravitational cosmic space is governed not by acceleration but by continuous restructuring of its mass-form, ensuring:
This makes the physical picture intuitive: the photon retains its speed c but must reduce its frequency to remain dynamically consistent with the increasing ΔPEECM background, and this frequency reduction is precisely what we observe as cosmic redshift.
A central implication of this Appendix is that the photon does not lose energy in traversing the ordinary light-travel distance within cosmic space. This part of the trajectory is supported by the anti-gravitational field arising from the positive ΔPEECM background.
However, cosmic expansion produces an extra distance that did not exist at the time of emission. To cover this additional distance, the photon must draw from its inherent energy, resulting in:
Cosmic Redshift = NAM expenditure required to cover expansion-induced extra distance
Total distance traversed = light-travel distance + expansion-induced extra distance
Only the second component requires inherent photon energy expenditure.
The Appendix therefore implies a consistent ECM picture in which photon redshift is strictly a consequence of the expansion-generated portion of the trajectory. The ambient anti-gravitational field, emerging from ΔPEECM, sustains the photon’s motion across non-expanding space without diminishing its inherent energy.
This framework offers a clear ECM-based physical explanation of photon dynamics in an anti-gravitational, ΔPEECM-dominated universe.
Within the Extended Classical Mechanics (ECM) framework, the identity KEECM = ΔPEECM is fundamentally bound to the gravitational– or anti-gravitational–potential centre that governs the system in question. This means that the manifested kinetic energy of any entity—ordinary matter, dark matter, or photons— is always defined with respect to its corresponding local PEECM centre. Thus, KEECM and ΔPEECM form an inseparable pair anchored to their specific gravitational domain.
For stellar systems within a galaxy, the gravitational structure establishes a characteristic rmax at which the effective mass becomes zero (Meff = 0 ⇒ MM = Mapp), and the local ΔPEECM transitions into its anti-gravitational manifestation. A photon emitted from a star therefore begins its propagation within the domain of this local rmax, governed by the PEECM conditions of the galactic system.
However, the universe as a whole possesses its own far larger potential centre associated with ΔPEECM,universe, giving rise to a cosmological rmax,universe. This defines the distance at which universal effective mass neutrality emerges and cosmological-scale anti-gravitational behaviour dominates. Importantly, rmax,galaxy is only a subset of rmax,universe; any KEECM entity originating from a galaxy eventually enters the domain of the universal rmax when transitioning into intergalactic space.
Intergalactic space is therefore dominated by the kinetic and potential balance associated with the universal background: KEuniverse,darkenergy > KEuniverse,matter. This condition reflects the predominance of the universal ΔPEECM reservoir whose active, kinetic expression forms the anti-gravitational field responsible for cosmic expansion. Entities moving in this region—especially photons—interact with this field rather than with the local galactic potential that governed their origin.
The universal ΔPEECM thus represents the potential-energy reservoir of dark energy, whereas its kinetic manifestation, observed through NAM-driven acceleration, appears as the cosmic anti-gravitational field. This field sustains photon propagation across vast intergalactic distances without drawing upon the photon’s inherent energy for the light-travel portion of the journey. Consequently, photon energy loss (cosmic redshift) arises only from the additional expansion-induced distance rather than the entire propagation path. This interpretation is fully consistent with the ECM linkage between KEECM, ΔPEECM, and the hierarchical structure of rmax at galactic and universal scales.
Within the Extended Classical Mechanics (ECM) framework, frequency, energy, and effective mass manifest through a unified oscillatory structure. When a stellar photon is compared to the Planck-scale reference, the resulting differences are overwhelmingly dominated by the Planck term, with the stellar contribution being negligible. This enables ECM to express all differences using the 360-times finer phase-resolved temporal scale.
For a stellar photon of frequency f⋆ compared against the Planck frequency fP, the incremental frequency gain is:
Δf⋆ = fP − f⋆
= 2.999 × 10⁴² − 6.0368 × 10¹⁴
= 2.99900000000000000000060368 × 10⁴² Hz
Since 10⁴² ≫ 10¹⁴, the difference is essentially the full Planck frequency. The stellar term is insignificant at this scale.
ECM replaces the standard time period T = 1/f with a 360-degree phase decomposition, providing a 360-fold finer temporal resolution:
Tx° = (x° / 360° f) = Δt
Each degree corresponds to one 360th of the complete oscillatory cycle, enabling frequency, energy, and effective mass to be analysed with much finer precision.
ECM uses a structural identity in Planck-scale comparison:
1 Hz = 360°
Therefore the phase-equivalent representation of the frequency gain Δf⋆ is:
x° = 360° × Δf⋆
Substituting the numerical value:
x° = 360° × 2.99900000000000000000060368 × 10⁴²
= 1.07964000000000000000021732 × 10⁴⁵ degrees
This value is not a geometric angle; it is a phase-count used to quantify the frequency displacement in a 360× finer manner.
Verification:
x° / 360 = 2.99900000000000000000060368 × 10⁴² Hz
which exactly recovers the original Δf⋆.
The ECM phase-frequency identity at the Planck reference is:
Δf = x° / 360°
and conversely:
x° = 360° Δf
This relation is applied whenever a frequency f⋆ belonging to the electromagnetic spectrum (and thus lying well below the Planck frequency) is compared against fP in the ECM Planck-scale frame.
Appendix 50 establishes a self-consistent Extended Classical Mechanics (ECM) framework for Effective Acceleration (aᵉᶠᶠ), Negative Apparent Mass (NAM), and cosmic-scale motion. The central structural relation:
provides the foundational bookkeeping from which gravitational neutralization, NAM growth, and anti-gravitational dynamics follow. The appendix demonstrates how changes in potential and motion map directly to changes in effective mass and kinetic expression under ECM.
In ECM, the apparent-mass bookkeeping term Mᵃᵖᵖ expresses the potential-to-mass conversion:
As Mᵃᵖᵖ increases, Mᵉᶠᶠ decreases. When Mᵉᶠᶠ → 0 (i.e., Mᴍ = Mᵃᵖᵖ), the system reaches gravitational neutralization. This condition defines the maximum neutralization radius rmax, at which net gravitational response vanishes while inertia persists.
Appendix 50 clarifies a hierarchical boundary structure:
A KEᴇᴄᴍ entity leaving a galactic domain enters the intergalactic region and then participates in the universal neutralization process. Local rmax values are nested subsets of the cosmological rmax.
Intergalactic space is dominated by the universal potential reservoir and its kinetic expression:
Motion beyond local neutralization is governed by NAM accumulation and expenditure. ΔPEᴇᴄᴍ gradients control effective mass evolution and the resulting kinematics. This provides a non-geometric, force-based account of cosmic recession and large-scale acceleration.
Photons are pure NAM entities (Mᴍ = 0, Mᵃᵖᵖ > 0) and are speed-locked at c. Under ECM:
Cosmic redshift = NAM expenditure for expansion-induced extra distance
Thus, measurable redshift reflects expenditure only for the expansion-created portion of the path, not the entire light-travel domain.
Appendix 50 integrates coherently with broader ECM principles:
Appendix 50 presents a rigorous extension of classical mechanics that unifies mass decomposition, potential dynamics, anti-gravity, and cosmological motion under a single interactional framework. NAM-driven neutralization and ΔPE–KE exchange replace geometric curvature with a transparent, force-based ontology capable of explaining cosmic acceleration and redshift through directly measurable physical quantities.
This appendix establishes a unified and fully ECM-consistent framework for describing the dynamics of matter-bearing systems and pure-NAM entities such as photons. Central to this formulation are the concepts of effective acceleration (aᵉᶠᶠ), matter–NAM decomposition, and gravitational neutralization, which together define how objects transition between matter-dominated and NAM-dominated regimes. The behaviour of systems across local, galactic, and cosmic scales emerges as the direct consequence of how potential-energy displacement (−ΔPEᴇᴄᴍ) manifests as kinetic energy, effective mass, and observable motion in ECM.
The ECM interpretation of photon redshift follows from the expenditure of interactional NAM required for a photon to traverse regions of progressively increasing ΔPEᴇᴄᴍ. Photons remain speed-locked at c, but their frequency decreases in accordance with ECM’s mass–potential transformation law (−ΔPEᴇᴄᴍ → ΔMᴍ → hf). Thus, cosmic redshift is explained without Doppler motion, spacetime curvature, or metric expansion—arising purely from ECM bookkeeping of NAM consumption over increased traversal distance.
A major refinement introduced in this appendix is the incorporation of Planck-Scale Phase-Shift Conversion, which expresses Δf, ΔE, and ΔMᴍ on a 360-division phase grid rather than using whole-cycle time periods. The degree-resolved expression (Tₓ° = x°/360°f) provides a 360× finer temporal resolution, enabling direct comparison between ordinary electromagnetic frequencies and the Planck frequency fᴘ. This sharpened representation makes explicit the enormous disparity between stellar-scale manifestations and Planck-scale manifestations, reinforcing ECM’s interpretation of the Planck scale as the dominant energetic reference point for all manifestation processes.
By combining effective acceleration, NAM bookkeeping, frequency-governed mass manifestation, and degree-resolved Planck-scale comparison, this appendix presents a complete, interaction-driven description of cosmic motion. It substitutes traditional relativistic and geometric explanations with a clear potential-energy–based mechanism that unifies gravitational neutralization, photon dynamics, cosmic acceleration, and deep Planck-scale behaviour within a single, coherent ECM structure.
A recurring challenge in the evaluation of new theoretical frameworks is the presence of relativistic bias — the assumption that any cosmological or gravitational model must conform to the geometric axioms of General Relativity (GR) to be considered scientifically valid. This assumption, though historically influential, is not part of the scientific method and is not a criterion for determining the legitimacy, rigor, or predictive value of a physical theory.
Extended Classical Mechanics (ECM), including the Negative Apparent Mass (NAM) formulation developed in Appendix 50, adopts a force-based, continuum-mechanics framework that draws from classical physics, Planck’s energy relations, and cosmological insights such as the negative effective mass phenomenon identified in the works of A. D. Chernin et al. The ECM interpretation does not rely on geometric spacetime curvature, nor does it require the relativistic postulate of a metric-defined gravitational field for its internal consistency or explanatory power.
The term “relativistic bias” refers to the practice of dismissing or underrating frameworks that do not adopt GR’s axioms, even when those frameworks provide consistent mathematical structures, yield clear physical interpretations, or reproduce cosmological observations through alternative mechanisms. Such methodological bias risks equating “mainstream” with “scientifically mandatory,” despite the fact that Planck physics, classical mechanics, and contemporary cosmological studies remain equally established scientific disciplines.
ECM is designed according to the foundational principle of the scientific method: falsifiability. A theory is scientifically meaningful when it produces testable, disprovable predictions. The ECM framework, through the NAM-driven dynamics formulated in Appendix 50, provides measurable criteria that differentiate it from GR-based interpretations:
As long as these predictions remain open to experimental and observational verification, the ECM framework satisfies the requirements of scientific falsifiability. The fact that ECM does not adopt the relativistic geometric paradigm does not reduce its scientific status. Instead, it reflects a deliberate and valid methodological divergence grounded in alternative, well-established domains of physics.
Thus, the evaluation of ECM must rest on its mathematical coherence, empirical applicability, and predictive power — not on conformity to a specific theoretical tradition. Appendix 50 formalizes this position by presenting NAM dynamics as a falsifiable and independently verifiable extension of classical mechanics into cosmological-scale phenomena.