Acceleration-History Reformulation of Relativistic Time Dilation

Mathematical Reformulation Draft

Abstract

Scope of Claim

This work does not propose modifications to the equations of Special Relativity or General Relativity. All standard relativistic predictions, Lorentz invariance, tensor structure, and experimentally verified results remain unchanged.

The proposal is interpretive and reformulative rather than operational. Existing relativistic equations are reorganized through acceleration-history structure in order to provide an alternative conceptual interpretation of relativistic asymmetry, time dilation, and gravitational scaling behavior.

Notation

Throughout this work:

• γ = Lorentz factor
• v = coordinate velocity
• c = invariant spacetime propagation limit (speed of light)
• a = acceleration
• a_{proper} = proper acceleration
• g = gravitational acceleration
• d = accumulated distance under acceleration
• r = radial distance from gravitational source
• G = gravitational constant
• M = gravitating mass
• τ = proper time
• p = relativistic momentum
• E = relativistic energy
• tanh(x) = hyperbolic tangent
• sech(x) = hyperbolic secant

All equations are written in flat-text mathematical notation for readability.

Abstract

This work presents an interpretive reformulation of relativistic time dilation using acceleration-history structure as the primary explanatory framework. No equations from Special Relativity or General Relativity are modified. Instead, equivalent substitutions already present within relativistic mathematics are used to reorganize the interpretation of relativistic effects.

The central proposal is that relativistic scaling behavior is more naturally interpreted through accumulated spacetime-path structure governed by proper acceleration, geodesic behavior, and bounded relativistic accumulation, rather than through simplified narratives based solely on symmetric relative motion between observers.

The reformulation preserves:

• Special Relativity,
• General Relativity,
• Lorentz invariance,
• tensor structure,
• geodesic motion,
• and all standard experimental predictions.

The proposal is therefore interpretive rather than operational.

1. Lorentz Structure

Special Relativity defines the Lorentz factor (Eq. 1):

(1) γ = 1 / √(1 - v²/c²)

Time dilation:

(2) t' = γt

Relativistic momentum:

(3) p = γmv

Relativistic energy:

(4) E = γmc²

The same Lorentz factor γ governs:

• time dilation,
• momentum growth,
• energy growth.

All relativistic scaling is therefore structurally governed by:

v²/c²

2. Dimensional Structure

Velocity units:

v = m/s

Therefore:

v² = (m/s)² = m²/s²

Acceleration-distance units:

a = m/s²

d = m

Therefore:

ad = (m/s²)(m) = m²/s²

Gravitational acceleration-distance units:

g = m/s²

r = m

Therefore:

gr = (m/s²)(m) = m²/s²

Thus:

v² ↔ ad ↔ gr

all share identical dimensional structure:

m²/s²

This dimensional equivalence motivates the substitutions developed below. The reformulation relies not solely on dimensional equivalence, but on established relativistic identities, kinematic substitutions, and acceleration relations already present within Special Relativity and General Relativity.

3. Newtonian Velocity Reformulation

Classical kinematics gives:

(5) v² = 2ad

Substituting into the Lorentz factor:

(6) γ = 1 / √(1 - 2ad/c²)

Thus relativistic scaling can already be written directly in terms of accumulated acceleration-distance structure.

Under this interpretation, velocity is not treated as the primitive explanatory quantity. Instead, velocity encodes accumulated acceleration-history behavior.

4. Weak-Field Gravitational Time Dilation

General Relativity weak-field form:

(7) t' = t√(1 - 2GM/rc²)

where t denotes coordinate time measured far from the gravitational source and t' denotes proper time measured locally within the gravitational field.

Newtonian gravity:

(8) g = GM/r²

Solve for GM:

(9) GM = gr²

Substitute:

(10) 2GM/rc² = 2gr/c²

giving:

(11) t' = t√(1 - 2gr/c²)

Thus gravitational time dilation becomes:

1 - 2gr/c²

while the acceleration-history form becomes:

1 - 2ad/c²

Both therefore share identical relativistic structure.

Since g is itself acceleration, gravitational time dilation may be interpreted through acceleration structure without altering relativistic predictions.

5. Equivalence Principle

General Relativity establishes the local equivalence between gravity and acceleration.

Free-fall observer:

a_{proper} = 0

Standing observer:

a_{proper} = g

Thus experienced relativistic effects correlate directly with proper acceleration structure.

Under this interpretation:

• free-fall corresponds to geodesic motion,
• resisting free-fall corresponds to experienced proper acceleration,
• and relativistic asymmetry tracks spacetime-path structure.

6. Relativistic Velocity Accumulation

Newtonian accumulation:

v = at

fails near c because:

v < c

Relativistic constant proper acceleration instead gives:

(12) v = c tanh(aτ/c)

where:

• τ = proper time,
• tanh = hyperbolic tangent.

Squaring:

(13) v² = c² tanh²(aτ/c)

Thus relativistic velocity itself becomes expressible through acceleration-history structure.

7. Relativistic Lorentz Reformulation

Starting from:

γ = 1 / √(1 - v²/c²)

Substitute:

v² = c² tanh²(aτ/c)

giving:

(14) γ = 1 / √(1 - tanh²(aτ/c))

Using the hyperbolic identity:

(15) 1 - tanh²(x) = sech²(x)

gives:

(16) γ = 1 / sech(aτ/c)

Therefore:

(17) γ = cosh(aτ/c)

Thus the Lorentz factor itself becomes expressible entirely through proper acceleration-history structure.

8. Relativistic Momentum Reformulation

Original:

p = γmv

Substitute:

γ = cosh(aτ/c)

giving:

(18) p = mv cosh(aτ/c)

Momentum scaling therefore becomes expressible through acceleration-history structure.

9. Relativistic Energy Reformulation

Original:

E = γmc²

Substitute:

γ = cosh(aτ/c)

giving:

(19) E = mc² cosh(aτ/c)

Energy scaling therefore becomes expressible through acceleration-history structure.

10. Hyperbolic Structure

Relativistic acceleration produces hyperbolic geometry.

Constant proper acceleration trajectory:

(20) x² - c²t² = (c²/a)²

Relativistic accumulation is therefore bounded hyperbolic accumulation rather than unbounded linear accumulation.

This preserves:

• finite invariant propagation speed c,
• relativistic velocity saturation,
• and Lorentz structure.

11. Structural Equivalence Chain

Special Relativity:

γ = 1 / √(1 - v²/c²)

Newtonian acceleration structure:

v² = 2ad

Weak-field gravity structure:

2GM/rc² = 2gr/c²

Relativistic acceleration structure:

v² = c² tanh²(aτ/c)

Lorentz acceleration-history form:

γ = cosh(aτ/c)

Thus:

v²
↔ ad
↔ gr
↔ c²tanh²(aτ/c)

become structurally connected through acceleration-history geometry.

12. Proper Time Interpretation

Relativity fundamentally compares accumulated proper time along different spacetime paths.

Under this reformulation:

• proper time accumulation tracks spacetime-path geometry,
• acceleration-history determines relativistic asymmetry,
• and velocity represents bounded hyperbolic accumulation constrained by c.

This interpretation reframes relativistic effects in terms of accumulated spacetime traversal structure rather than isolated relative velocity alone.

13. Twin Paradox Reinterpretation

The standard twin paradox is often pedagogically described as symmetric relative motion between observers.

Under the acceleration-history interpretation:

• the asymmetry is encoded directly in the spacetime paths,
• proper acceleration histories differ,
• proper time accumulation differs,
• and no mathematical contradiction arises.

The underlying relativistic equations remain unchanged.

14. Interpretive Scope

This work does not propose modifications to:

• Special Relativity,
• General Relativity,
• Lorentz invariance,
• or established relativistic predictions.

Instead, it proposes that acceleration-history structure provides a more physically intuitive interpretation layer for existing relativistic mathematics.

The reformulation therefore functions as:

• an interpretive restructuring,
• a mathematical re-expression,
• and a geometric reframing of relativistic scaling behavior.

All substitutions used in this work are algebraic substitutions, relativistic identities, or already-established acceleration relations contained within existing relativistic mathematics.

15. Conclusion

Using substitutions already contained within relativistic mathematics, relativistic scaling behavior can be reformulated entirely through acceleration-history structure.

The resulting framework preserves all existing relativistic equations while reorganizing the interpretation around:

• proper acceleration,
• geodesic/free-fall structure,
• accumulated proper time,
• bounded hyperbolic accumulation,
• and spacetime-path geometry.

No operational predictions change.

The proposal therefore represents an interpretive reformulation of existing relativistic structure rather than a replacement theory.

Geometric Phase Support (GPS) Theorem for Entangled Polarization Correlations

Abstract

This paper proposes a local geometric interpretation of entangled polarization correlations based on shared bounded phase structure established entirely at particle creation. The framework preserves the experimentally verified polarization correlation law while rejecting the necessity of nonlocal measurement influence. Entangled systems are modeled as shared axial geometric phase manifolds possessing bounded support regions and rotational phase structure. Detector outcomes emerge from local geometric compatibility with this pre-existing source-fixed structure. The theorem reproduces the experimentally observed cosine-squared polarization correlation behavior while rejecting Bell-type statistical factorization assumptions for entangled geometric phase systems.

1. Introduction

A Comparison to Existing Geometric and Phase-Based Frameworks

Several existing areas of physics suggest that geometric phase structure plays a deeper role in physical systems than traditionally emphasized in standard probabilistic interpretations of quantum mechanics.

Berry phase demonstrated that physical systems can acquire measurable phase structure through geometry itself rather than through ordinary dynamical evolution alone. This established that geometric phase is physically real and experimentally observable, not merely a mathematical artifact.

Pancharatnam phase extended these ideas directly into polarization physics, showing that relative polarization geometry and phase relationships produce measurable interference behavior. This work strongly connected wave polarization, geometry, and phase coherence into a unified physical framework.

Modern polarization geometry further developed these concepts by treating polarization states as geometric objects defined by rotational orientation, angular relationships, and phase structure. Polarization behavior naturally exhibits axial symmetry, cosine projection laws, and rotational phase dependence consistent with wave geometry.

Optical coherence theory similarly demonstrated that phase relationships possess finite coherence structure. Correlated wave systems do not necessarily participate uniformly across all phase configurations, but instead exhibit bounded regions of coherent participation and interference.

The present theorem incorporates and unifies these concepts into a single geometric interpretation of entangled polarization correlations.

Specifically, the theorem combines:

• geometric phase realism,
• polarization-wave geometry,
• axial symmetry,
• bounded coherence structure,
• cosine projection statistics,
• and shared source-fixed phase relationships

into a unified local geometric framework for entanglement correlations.

The shared source state:

ψθ(φ) = A(φ − θ)e^(iφ)

treats the entangled pair as a real bounded geometric phase structure fixed at creation rather than as a nonlocal probabilistic state completed during measurement.

The bounded support condition:

|φ − θ| ≤ π/4

introduces an explicit finite geometric participation region centered on the shared source axis θ. This bounded coherence structure provides a direct geometric mechanism for the emergence of observed Bell-type polarization correlations while preserving locality.

In this interpretation, detector measurements do not generate correlation through instantaneous distant influence. Instead, detectors locally sample compatibility with a pre-existing shared geometric phase manifold established at particle creation.

The theorem therefore proposes that Bell-type correlations emerge naturally from bounded geometric phase structure itself rather than from nonlocal measurement dynamics.

2. Source Geometry

Each entangled pair is created with a shared axial orientation:

θ ∈ [0, π)

with axial equivalence:

θ ≡ θ + π

The shared source state is defined as:

ψθ(φ) = A(φ − θ)e^(iφ)

where:

• θ represents the shared axial source orientation,

• φ represents the continuous angular phase coordinate of the shared rotational phase manifold,

• e^(iφ) defines the continuous rotational phase geometry of the shared state,

• and A(φ − θ) defines the bounded geometric support structure surrounding the shared source axis.

The source geometry is axial rather than directional. Rotations by π therefore preserve physical equivalence.

3. Support Function Definition

The support function:

A(φ − θ)

defines the physically allowed geometric participation region surrounding the shared source orientation θ.

The support structure is bounded such that:

A(φ − θ) ≠ 0

only if:

|φ − θ| ≤ π/4

and:

A(φ − θ) = 0

when:

|φ − θ| > π/4

This produces a finite bounded coherence region centered on the shared source orientation.

The support structure is rotationally symmetric and axial in character. Only phase orientations within ±45° of the shared source axis physically participate in the correlated geometric manifold.

The support structure defines a normalized participation distribution over the allowed phase region.

4. Detector Interaction Rule

A detector oriented at angle a samples the shared source geometry locally according to:

P(a | θ) = cos²(a − θ)

This reproduces the experimentally observed polarization probability law.

For an entangled pair:

Detector A measures:

P(A | a, θ) = cos²(a − θ)

Detector B measures:

P(B | b, θ) = cos²(b − θ)

Both measurements sample the same shared source orientation θ established at pair creation.

5. Correlation Structure

The resulting correlation depends on the relative detector geometry:

Δ = a − b

giving:

cos²(a − b)

which reproduces the experimentally observed polarization correlation structure.

At the Bell-test angle:

Δ = 22.5°

the framework yields:

cos²(22.5°) ≈ 0.8536

matching the experimentally observed ~85% correlation region.

Within the Geometric Phase Support framework, the experimentally observed correlation structure emerges directly from bounded shared geometric phase relationships rather than from faster-than-light communication between distant particles.

6. Rejection of Bell Factorization

Bell-type derivations assume statistical separability of detector outcomes:

P(A,B | a,b,θ) = P(A | a,θ)P(B | b,θ)

The present theorem rejects this assumption for entangled geometric phase systems.

Within the GPS framework, detector outcomes are not independent statistical events after emission because both particles remain members of the same bounded geometric phase manifold established at creation.

The observed correlations therefore arise from shared geometric structure rather than nonlocal measurement influence.

7. Interpretation

The GPS theorem preserves the experimentally verified mathematical structure of polarization-wave physics while proposing a different physical interpretation of the wavefunction.

Within this framework:

• the wavefunction represents a real shared geometric phase structure,

• correlation exists prior to measurement,

• measurement does not generate correlation,

• and detector outcomes arise from local compatibility with a pre-existing bounded geometric manifold.

The framework therefore preserves locality while reproducing the experimentally observed polarization correlation structure.

8. Relationship to CHSH Correlation Analysis

Standard CHSH analysis combines multiple detector-angle correlation measurements into a single statistical consistency quantity:

S = E(a,b) + E(a,b') + E(a',b) − E(a',b')

Within the Geometric Phase Support framework, these expectation values emerge naturally from the underlying bounded geometric phase structure shared by the entangled pair at creation.

The GPS framework begins from the experimentally observed polarization compatibility law:

P(a | θ) = cos²(a − θ)

where detector outcomes arise from local geometric compatibility between detector orientation and the shared source-fixed phase manifold.

For binary detector outcomes, CHSH expectation values assign:

· +1 to matching detector outcomes,

· and −1 to differing detector outcomes.

The expectation correlation therefore becomes:

E = P_same − P_different

Since:

P_different = 1 − P_same

the expectation relation reduces to:

E = 2P_same − 1

Using the GPS polarization compatibility law:

P_same = cos²(a − b)

gives:

E(a,b) = 2cos²(a − b) − 1 = cos(2(a − b))

This reproduces the standard experimentally observed Bell/CHSH correlation structure directly from bounded geometric phase relationships without requiring faster-than-light communication or nonlocal state updates between distant particles.

Using the standard CHSH detector-angle configuration:

· a = 0°

· a′ = 45°

· b = 22.5°

· b′ = −22.5°

the GPS framework yields:

· E(a,b) = 0.7071

· E(a,b′) = 0.7071

· E(a′,b) = 0.7071

· E(a′,b′) = −0.7071

giving:

S = 2.8284

which matches the experimentally observed CHSH violation region.

Within the GPS interpretation, the CHSH correlation structure therefore emerges from repeated local sampling of the same bounded shared geometric phase manifold established at pair creation rather than from nonlocal detector influence generated during measurement.

9. Conclusion

The Geometric Phase Support (GPS) theorem proposes that entangled polarization correlations arise from a shared bounded geometric phase structure established entirely at particle creation.

Within this framework, entangled systems remain geometrically coupled through a common source-fixed phase manifold possessing axial symmetry, bounded coherence support, and rotational phase structure. Detector outcomes therefore emerge from local geometric compatibility with this shared wave structure rather than from instantaneous nonlocal state updates generated during measurement.

The GPS framework preserves the experimentally verified polarization correlation law:

P(a | θ) = cos²(a − θ)

while reproducing the experimentally observed Bell/CHSH correlation structure through bounded geometric phase relationships alone.

The theorem further demonstrates that Bell-type correlations can emerge naturally from continuous geometric phase compatibility without requiring faster-than-light communication between distant particles.

By unifying geometric phase structure, polarization-wave geometry, bounded coherence support, and CHSH correlation behavior into a single local geometric framework, the GPS theorem proposes an alternative physical interpretation of entangled polarization correlations grounded in shared source-fixed phase structure rather than nonlocal measurement dynamics.

Geometric Phase Support (GPS) Framework

Complete Mathematical Derivation Summary

1. Rotational manifold geometry

Represent the local manifold orientation at phase angle φ as a unit rotational vector:

v(φ) = [ cos(φ), sin(φ) ]

Represent a detector oriented at angle a as:

d(a) = [ cos(a), sin(a) ]

The detector overlap with the local manifold orientation is given by the dot product:

d(a) · v(φ)

Substituting explicitly:

(cos(a))(cos(φ)) + (sin(a))(sin(φ))

Using the trigonometric identity:

cos(x − y) = cos(x)cos(y) + sin(x)sin(y)

gives:

d(a) · v(φ) = cos(a − φ)

This derives cosine overlap directly from rotational projection geometry.

2. Probability from projection overlap

Probability is defined as squared overlap magnitude:

P(a | φ) = | d(a) · v(φ) |²

Substituting the projection result:

P(a | φ) = | cos(a − φ) |²

Since cosine is real-valued:

P(a | φ) = cos²(a − φ)

The cosine-squared overlap structure therefore emerges directly from:

• rotational geometry,
• vector projection,
• and squared overlap magnitude.

3. Axial bounded-support manifold

Traditional hidden-variable constructions integrate over a fully flattened directional circle:

φ ∈ [0, 2π)

The GPS framework instead defines bounded axial support:

|φ − θ| ≤ π/4

where:

• θ is the shared axial source orientation,
• and:

θ ≡ θ + π

Define the support function:

A(φ − θ) = 1 for |φ − θ| ≤ π/4

A(φ − θ) = 0 for |φ − θ| > π/4

This restricts physical participation to a coherent axial phase region of total width:

π/2

The geometry is therefore:

• bounded,
• rotational,
• axial,
• and not flat directional averaging.

4. Local bounded-support probability

The detector probability over the bounded axial manifold becomes:

P(a | θ) = ∫[θ−π/4 → θ+π/4] cos²(a − φ) dφ

The framework therefore modifies:

• the integration geometry,
• the support topology,
• and the averaging domain itself.

5. Joint detector correlation structure

For two detectors at angles a and b:

E_axial(a,b) ∝ ∫[θ−π/4 → θ+π/4] cos²(a − φ) cos²(b − φ) dφ

Expand each cosine-squared term using:

cos²(x) = (1 + cos(2x)) / 2

Substitution gives:

cos²(a−φ)cos²(b−φ) = 1/4 + 1/4 cos2(a−φ) + 1/4 cos2(b−φ) + 1/8 cos2(a−b) + 1/8 cos2(a+b−2φ)

This is the full bounded-support integrand structure.

6. Symmetric bounded-support integration

Integrating term-by-term across:

φ ∈ [θ−π/4, θ+π/4]

yields:

E_axial(a,b) ∝ π/8 + 1/4 cos2(a−θ) + 1/4 cos2(b−θ) + (π/16) cos2(a−b)

The oscillatory term:

cos2(a+b−2φ)

cancels exactly under symmetric bounded axial integration.

This cancellation follows directly from the bounded axial support geometry.

7. Averaging over shared axial source orientations

The framework assumes shared axial source orientations distributed across axial space:

θ ∈ [0, π)

The observable expectation value becomes:

⟨E_axial(a,b)⟩ = ∫[0 → π] E_axial(a,b,θ) dθ

Substituting the derived expression:

⟨E_axial(a,b)⟩ = ∫[0 → π](π/8 + 1/4 cos2(a−θ) + 1/4 cos2(b−θ) + (π/16) cos2(a−b))dθ

Integrating term-by-term gives:

• constant contribution:

π² / 8

• source-angle cosine terms cancel exactly:

∫ cos2(a−θ)dθ = 0

∫ cos2(b−θ)dθ = 0

• surviving correlation term:

(π² / 16) cos2(a−b)

Resulting observable overlap structure:

⟨E_axial(a,b)⟩ = π²/8 + (π²/16) cos2(a−b)

The doubled-angle dependence survives bounded-support averaging.

8. Emergence of the Bell-style expectation structure

The overlap integral naturally produced:

(1 + cos2(a−b)) / 2

structure.

This is exactly the algebraic structure of:

cos²(x) = (1 + cos2x) / 2

The bounded-support manifold therefore naturally generates:

• a constant coincidence background,
• plus:
• a doubled-angle correlation modulation term.

9. Signed binary expectation construction

Bell-style observables are constructed from signed binary outcomes:

E(a,b) = P_same − P_different

Within the GPS framework:

P_same = (1 + cos2(a−b)) / 2

P_different = (1 − cos2(a−b)) / 2

Therefore:

E(a,b) = (1 + cos2(a−b)) / 2− (1 − cos2(a−b))/2

The constant background terms cancel exactly, leaving:

E(a,b) = cos2(a−b)

This is the standard Bell/CHSH doubled-angle polarization correlation structure.

10. Final geometric result

The GPS framework derives the Bell-style doubled-angle correlation structure from:

• rotational projection geometry,
• bounded axial support,
• symmetric bounded phase integration,
• and signed binary expectation construction.

The surviving:

cos2(a−b)

correlation emerges naturally from bounded axial rotational geometry rather than from flat directional averaging.

The framework therefore replaces:

• flat 1D directional integration, with:
• bounded axial phase support geometry.

Under this construction:

• the doubled-angle Bell correlation structure survives bounded-support averaging,
• the oscillatory averaging terms cancel exactly through axial symmetry,
• and the expected correlation structure emerges directly from rotational overlap geometry and bounded support topology.

Axial Geometric Support Theorem for Quantum Interference

Mathematical Reformulation Draft

Scope of Claim

This work does not modify the equations of quantum mechanics, electrodynamics, wave mechanics, or experimentally observed interference behavior.

All experimentally verified detector structures, interference fringes, coherence phenomena, Bell/CHSH correlations, and operational predictions remain unchanged.

The proposal is interpretive and geometric rather than operational.

The central proposal is that the oscillatory interference structure observed in quantum interference experiments admits a natural derivation from bounded axial geometric support relationships established during emission rather than from directional self-interference interpreted through spatially distributed wave propagation.

Specifically, the standard oscillatory interference structure:

cos²(x)

admits a natural derivation from axial geometric compatibility relations under the axial equivalence:

θ ≡ θ + π

without modifying experimentally observed detector equations.

The present work therefore proposes that experimentally observed interference structure can emerge from geometrically constrained axial propagation statistics rather than from unrestricted directional propagation distributed across all spatial paths.

Notation

Throughout this work:

• θ = source-fixed axial orientation state
• φ = detector propagation angle
• A(φ − θ) = axial geometric compatibility weighting
• I(φ) = detector intensity/density distribution
• P_emit(θ) = source emission distribution
• ψ = standard quantum wave amplitude notation
• λ = wavelength parameter appearing in experimentally observed interference relations
• d = slit separation
• L = detector distance
• y = detector-plane position
• Δφ = directional phase difference

All equations are written in flat-text mathematical notation for readability.

1. Standard Interference Structure

Standard double-slit interference intensity is written:

(1) I_total = |ψ₁ + ψ₂|²

Expanding:

(2) I_total = |ψ₁|² + |ψ₂|² + 2ψ₁ψ₂ cos(Δφ)

The oscillatory detector structure therefore depends on:

cos(Δφ)

which produces alternating constructive and destructive interference regions.

Using the trigonometric identity:

(3) cos²(x) = (1 + cos(2x)) / 2

standard interference structure reduces to oscillatory cos²-type detector distributions.

Under the standard interpretation, this oscillatory structure is attributed to directional wave self-interference.

2. Classical Additive Trajectory Statistics

Ordinary unrestricted additive trajectory accumulation produces detector densities of the form:

(4) I_total = I₁ + I₂

Purely additive unrestricted positive trajectory accumulation cannot generate oscillatory subtraction structure or periodic near-zero interference minima.

Experimentally observed interference structure therefore requires additional geometric structure beyond unrestricted additive trajectory accumulation alone.

The AGS framework proposes that this missing structure is bounded axial compatibility.

3. Axial State Equivalence

The present reformulation replaces directional state geometry with axial state geometry.

Instead of directional equivalence over:

(5) θ ∈ [0, 2π)

the axial equivalence relation is introduced:

(6) θ ≡ θ + π

giving the axial state space:

(7) θ ∈ [0, π)

Under this interpretation, antipodal directional states are treated as physically equivalent axial orientations.

The fundamental object is therefore an axis rather than a directional vector.

4. Bounded Axial Geometric Support

Propagation support is defined geometrically through bounded axial compatibility relationships.

Support exists only when:

(8) |φ − θ| ≤ π/4

Outside this bounded support region:

(9) A(φ − θ) = 0

Only propagation paths satisfying the bounded axial compatibility conditions contribute to detector accumulation.

Thus propagation support is geometrically constrained rather than uniformly directional.

Under this interpretation:

• some detector regions receive overlapping supported paths,
• some receive partial support,
• and some receive no support.

Detector minima therefore emerge from forbidden geometric support regions rather than from negative probability amplitudes.

5. Axial Compatibility Law

Within the Geometric Phase Support framework, local compatibility between detector orientation and source-fixed axial structure is governed by:

(10) P(a | θ) = cos²(a − θ)

This compatibility law previously reproduced the experimentally observed Bell/CHSH correlation structure through bounded axial geometric phase relationships.

The present reformulation extends the same compatibility structure to interference geometry.

Define the axial geometric compatibility weighting:

(11) A(φ − θ) = cos²(φ − θ)

This weighting function:

• remains strictly positive,
• varies continuously with axial alignment,
• preserves axial equivalence symmetry,
• and reaches maximal support under axial alignment.

Using:

(12) cos²(x) = (1 + cos(2x)) / 2

the oscillatory doubled-angle structure arises naturally from axial geometric compatibility relations.

No explicit directional wave subtraction term is inserted manually.

6. Detector Accumulation Structure

Detector density is accumulated through axial geometric compatibility overlap:

(13) I(φ) = ∫ A(φ − θ) P_emit(θ) dθ

where:

• P_emit(θ) = geometrically constrained emission distribution,
• A(φ − θ) = axial compatibility weighting,
• I(φ) = detector density distribution.

Substituting Eq. (11):

(14) I(φ) = ∫ cos²(φ − θ) P_emit(θ) dθ

where integration is taken over the axial state space:

θ ∈ [0, π)

Under the AGS framework, the emission distribution P_emit(θ) represents the statistically generated distribution of allowed axial propagation paths established at emission.

The slit geometry acts as a geometric compatibility filter on these allowed propagation paths.

Within the AGS interpretation, each emission location generates a bounded axial propagation distribution. The experimentally observed detector structure emerges from the statistical overlap and accumulation of these allowed propagation distributions across the emitting surface.

Under this interpretation:

• generation is statistically random,
• propagation follows geometrically allowed axial compatibility relationships,
• only geometrically compatible propagation paths contribute to detector accumulation,
• and detector structure emerges from statistical accumulation of surviving geometrically compatible paths.

The interference structure therefore arises from emission-conditioned axial path statistics rather than from directional superposition interpreted through spatially distributed propagation.

Thus the same oscillatory cos² structure appearing in standard interference equations admits a natural derivation from bounded axial geometric compatibility accumulation.

7. Fringe Spacing Geometry

Within the AGS framework, slit geometry determines the angular filtering structure imposed on the emitted axial path distribution.

For detector position y and detector distance L:

(15) φ ≈ y/L

using the small-angle approximation.

AGS preserves the experimentally verified fringe-spacing relations while reinterpreting them in terms of slit-conditioned axial path filtering.

Standard double-slit maxima satisfy:

(16) d sinφ = nλ

Using the small-angle approximation:

sinφ ≈ φ

gives:

(17) dφ ≈ nλ

yielding:

(18) φₙ ≈ nλ/d

Substituting Eq. (15):

(19) yₙ ≈ nλL/d

Defining fringe spacing as:

(20) Δy = yₙ₊₁ − yₙ

gives:

(21) Δy ≈ λL/d

Within the AGS interpretation, these experimentally observed spacing relations emerge from slit-conditioned filtering of allowed axial propagation paths rather than from directional wave self-interference across all possible trajectories.

Under this framework:

• d controls angular filtering density,
• λ parameterizes the experimentally observed angular spacing scale within the AGS interpretation,
• and detector accumulation produces the experimentally observed fringe spacing structure.

8. Structural Comparison

Standard interpretation:

directional wave self-interference → cos²(x)

Axial geometric support interpretation:

axial emission/path compatibility → cos²(x)

In both cases, the experimentally observed detector structure is preserved.

The difference lies in the underlying geometric interpretation of the propagation process.

9. Interpretive Scope

This work does not modify:

• quantum mechanics,
• wave equations,
• Bell/CHSH statistics,
• experimentally verified detector distributions,
• coherence phenomena,
• or interference predictions.

Instead, it proposes that bounded axial geometric compatibility structure provides an alternative interpretive origin for experimentally observed interference and correlation phenomena.

The reformulation therefore functions as:

• an interpretive restructuring,
• a geometric reformulation,
• and an axial reinterpretation of interference structure.

No operational predictions are modified within the scope of this work.

10. Conclusion

Using axial state equivalence:

θ ≡ θ + π

together with bounded geometric support and axial compatibility accumulation:

A(φ − θ) = cos²(φ − θ)

the standard oscillatory interference structure:

cos²(x)

admits a natural derivation from geometrically allowed axial emission/path relationships.

Furthermore, the experimentally observed fringe-spacing relation:

(21) Δy ≈ λL/d

admits a corresponding geometric interpretation through slit-conditioned filtering of allowed axial propagation paths.

The experimentally observed detector structure is therefore preserved while the underlying interpretation is reformulated in terms of bounded axial geometric support rather than directional self-interference interpreted through spatially distributed wave propagation.

The proposal therefore represents an interpretive geometric reformulation of interference structure rather than a replacement theory of quantum mechanics.