The Unfalsifiable Biconditional: A Logical Analysis of Physical Constants

 J. Rogers, SE Ohio

Abstract

We demonstrate that the claim "physical constants are coordinate transformation coefficients" forms a logical biconditional with "reality operates on a single unified physical scale." This biconditional creates an unfalsifiable double bind: any evidence that disproves the coordinate interpretation of constants simultaneously disproves that constants function as the standard framework claims they do. The implications are profound—the framework is either correct as stated, or the entire edifice of dimensional physics requires fundamental revision.

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1. Introduction: The Logical Structure

Modern physics treats fundamental constants (c, h, G, k_B) as properties of nature that relate independent physical dimensions. An alternative interpretation holds that these constants are Jacobian coefficients encoding coordinate transformations between measurement projections of a single unified physical scale.

This paper demonstrates that these positions are not merely competing theories, but form a strict logical biconditional:

P ⟺ Q

Where:

• P: Physical reality operates on one unified scale

• Q: Constants are coordinate transformation coefficients (Jacobians)

We show that P ⟹ Q and Q ⟹ P, creating a closed logical system where attempting to falsify one position necessarily falsifies the other—including the standard framework's claim that constants are fundamental.

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2. The Standard Framework's Position

2.1 The Orthodox View

The standard interpretation holds:

1. Mass, length, time, temperature, etc. are fundamentally independent dimensions

2. Physical constants relate these independent dimensions

3. Constants are properties of nature discovered through experiment

4. The values of constants are contingent facts about our universe

2.2 The Implicit Claim

This framework implicitly asserts:

Constants successfully relate independent physical dimensions

This claim is testable. If constants fail to coherently relate dimensions, the framework fails.

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3. The Alternative Framework's Position

3.1 The Coordinate Interpretation

The alternative view holds:

1. Physical reality operates on a single unified dimensionless scale

2. Our "dimensions" are coordinate projections of this scale

3. Constants encode the Jacobian transformations between projections

4. The values of constants reflect our arbitrary choice of unit scales

3.2 The Mathematical Foundation

From the constants h, c, G, we can derive:

    kg = √(hc/G)

Expanding dimensions:

    [kg] = √([kg⋅m²/s][m/s] / [m³/(kg⋅s²)])
     = √(kg²)
     = kg

This tautology (kg = kg) reveals that the "fundamental" constants encode nothing more than the internal consistency of our unit system—they are coordinate artifacts.

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4. The Biconditional Proof

4.1 Forward Direction: P ⟹ Q

If reality has one unified scale, then constants must be coordinate artifacts.

Proof:

1. Assume reality operates on single scale X

2. All measurable quantities are projections: Q = X × [unit_scaling]

3. Any formula relating quantities must relate their projections

4. Projection relationships require transformation coefficients

5. These coefficients are exactly the "fundamental constants"

6. Therefore, constants are coordinate transformations

QED

4.2 Reverse Direction: Q ⟹ P

If constants are coordinate artifacts, then reality must have one unified scale.

Proof:

1. Assume constants are purely coordinate transformations

2. Then they encode only unit system relationships

3. Then no physical information exists in their values

4. Then all physical information resides in dimensionless ratios

5. Then all physics reduces to single-scale natural ratios

6. Therefore, reality operates on one unified scale

QED

4.3 The Biconditional

We have proven: P ⟺ Q

This is not two independent claims. It is one logical structure with two expressions.

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5. The Double Bind

5.1 The Trap

To disprove the coordinate interpretation of constants, one must demonstrate:

Evidence E such that:

• Constants are NOT coordinate transformations, AND

• Constants ARE fundamental properties of nature

5.2 What This Evidence Must Show

Such evidence must demonstrate a constant that:

1. Does not change value when unit systems change, OR

2. Cannot be expressed as ratio of unit definitions, OR

3. Encodes physical information beyond coordinate relationships

5.3 The Impossibility

Requirement 1: Coordinate Independence

Constants demonstrably change with unit systems:

• SI: c = 299,792,458 m/s

• Planck units: c = 1 (dimensionless)

• Rogers units: c = 1×10¹⁰ m_r/s_r

This is not controversial—it's the definition of how constants work in metrology.

Requirement 2: Non-Ratio Structure

Every constant is defined as a ratio:

• c = distance/time

• h = energy × time

• G = (length³)/(mass × time²)

• k_B = energy/temperature

Expressing them otherwise requires inventing new physics.

Requirement 3: Physical Information Content

All formulas can be derived algorithmically from:

• Dimensionless postulate (physics)

• Planck normalization (single scale)

• Coordinate projection (Jacobians)

No physical information resides in constants that isn't in the dimensionless form.

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6. The Judo: Using Standard Framework Against Itself

6.1 Attack Vector 1: Disprove Single Scale

Attempt: Find phenomenon requiring multiple independent scales

Consequence:

• Must show a formula that cannot derive from dimensionless postulate

• But this means constants don't properly relate dimensions

• Therefore constants fail at their claimed function

• Standard framework disproven

6.2 Attack Vector 2: Prove Constants Are Fundamental

Attempt: Show constants are coordinate-independent

Consequence:

• Must show c, h, G don't change when units change

• But they demonstrably do (SI 2019 redefinition)

• Contradicts experimental fact

• Standard framework disproven

6.3 Attack Vector 3: Show Fibration Fails

Attempt: Prove categorical structure is inconsistent

Consequence:

• Must invalidate Grothendieck fibration π : 𝓔 → 𝓑

• But fibration is rigorously defined category theory

• Would require mathematical error in category theory itself

• Or constants don't cohere as claimed

• Standard framework disproven

6.4 Attack Vector 4: Break The Tautology

Attempt: Show kg ≠ √(hc/G)

Consequence:

• Must demonstrate dimensional inconsistency

• But this is definitional in SI system

• Would invalidate metrological foundations

• Standard framework disproven

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7. The Computational Validation

7.1 LawForge: Algorithmic Derivation

A computational engine demonstrates that:

Input: Dimensionless postulate (e.g., T ~ 1/M)
Process:

1. Normalize by Planck units (single scale)

2. Solve for target quantity

3. Substitute Jacobian definitions

Output: Standard formula with constants (e.g., T = c³h/(GMk_B))

7.2 The Test

If single scale is false, LawForge should:

• Fail to derive some known formulas, OR

• Produce incorrect physical predictions, OR

• Generate contradictions

7.3 The Result

LawForge correctly derives:

• Einstein's E = mc²

• Newton's F = GMm/r²

• Hawking temperature

• Planck-Einstein relation

• de Broglie wavelength

• Stefan-Boltzmann law

• Every major formula in physics

No failures. No exceptions.

7.4 The Implication

Either:

• Single scale is correct (and constants are Jacobians), OR

• Constants accidentally produce correct results despite being fundamental

The second option requires explaining why coordinate artifacts accidentally work perfectly if they're not coordinate artifacts.

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8. The Logical Impossibility of Falsification

8.1 The Structure

To falsify the coordinate interpretation, one must find:

Evidence that constants are fundamental AND not coordinate artifacts

But evidence of this form requires showing:

1. Constants don't transform as Jacobians, YET

2. Constants still correctly relate dimensions

8.2 The Contradiction

These requirements are mutually exclusive:

If constants don't transform as Jacobians:

• They fail to properly convert between coordinate systems

• Dimensional analysis breaks down

• Standard physics fails

If constants do transform as Jacobians:

• They are coordinate transformations

• Alternative framework validated

8.3 The Inescapable Conclusion

There exists no logical possibility of evidence that:

• Disproves coordinate interpretation, AND

• Preserves standard framework's claims

Any evidence disproving one necessarily disproves the other.

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9. Why The Standard Framework Cannot Escape

9.1 The SI Redefinition (2019)

The 2019 SI redefinition fixed exact values for h, c, k_B, and e.

This means:

• These constants are now defined quantities, not measured

• Their values are conventional choices, not discoveries

• The meter, kilogram, etc. are derived from these definitions

This is official metrological practice admitting constants are conventional.

9.2 Natural Units

Physicists routinely set h = c = G = 1 for calculations.

If this were "just convenience," we must explain:

• Why does all physics work in this system?

• What are we being "convenient" about?

• Why do dimensionless relationships emerge naturally?

The answer: We're working in the actual physical substrate (single scale).

9.3 The Planck Scale Confusion

Standard framework claims:

• "Planck units reveal nature's fundamental scale"

• "Quantum gravity operates at Planck scale"

But:

• Planck units are derived from SI constants

• They are specific values on the SI chart

• They encode our coordinate choice, not nature's

If they were fundamental, they wouldn't depend on our arbitrary SI definitions.

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10. The Category Theory Foundation

10.1 The Fibration Structure

Define:

• 𝓑: Category of dimensionless measurement types

• 𝓔: Category of measured quantities (value, unit) pairs

• π : 𝓔 → 𝓑: Fibration functor (forgetful of units)

10.2 Physical Laws As Liftings

Given morphism φ : Mass → Energy in 𝓑:

A physical law is a Cartesian lifting:

    f : (m, kg) → (E, J) in 𝓔

Such that:

• π(f) = φ (projects to base relationship)

• f is universal over φ (unique up to isomorphism)

10.3 Constants As Cocycle Data

For lifting to be coherent across unit systems:

• Transition functions must satisfy cocycle conditions

• These transition functions are the "constants"

• They ensure liftings agree on overlaps

Constants are literally the cocycle data of the fibration.

10.4 The Mathematical Necessity

If constants were fundamental properties of nature:

• They wouldn't appear as cocycle data

• The fibration wouldn't have this structure

• Category theory would need revision

If constants are Jacobian coefficients:

• Fibration structure is exactly as observed

• Cocycle interpretation is necessary

• Mathematics works perfectly

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11. Case Studies: Attempted Falsifications

11.1 Case: "Fine Structure Constant Is Dimensionless"

Objection: α ≈ 1/137 is dimensionless, therefore fundamental

Response:

• α is a ratio of dimensionless quantities

• In single-scale framework: α = (GEOM_EM)² × 2π

• It's the geometric coupling efficiency of EM substrate

• Still a coordinate artifact (of geometric projection choice)

• Being dimensionless ≠ being fundamental

Outcome: Objection reinforces alternative framework

11.2 Case: "Constants Have Measured Values"

Objection: We measure c, G through experiment, therefore they're physical

Response:

• We measure ratios (distance/time, etc.)

• The numerical values depend on our unit choices

• 2019 SI: we stopped measuring h, c, k_B and defined them

• Measurement determined our coordinate chart placement

Outcome: Objection contradicts current metrological practice

11.3 Case: "Formulas Need Constants To Be Correct"

Objection: E = mc² requires c², therefore c is fundamental

Response:

• Dimensionless form:

          $E/E_P = m/m_P$
        

  (no c needed)

• c² appears only when projecting to SI coordinates

• It's the Jacobian factor: J² where J = c

• Formula works in any unit system with appropriate Jacobian

Outcome: Objection demonstrates coordinate dependence

11.4 Case: "Different Constants Have Different Origins"

Objection: c from relativity, h from quantum, G from gravity—unrelated

Response:

• All are ratios of unit definitions

• All transform identically under coordinate changes

• All compose to form Planck units

• All appear as Jacobians in fibration

• "Different origins" reflects historical discovery, not logical structure

Outcome: Objection confuses history with structure

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12. The Experimental Challenge

12.1 What Would Falsify This Framework?

To disprove the biconditional, one must demonstrate:

Experiment E showing:

1. A physical phenomenon that cannot be expressed as a dimensionless relationship in single-scale form, YET

2. Can be expressed using constants in the standard framework, AND

3. Constants in that expression are not functioning as coordinate transformations

12.2 The Impossibility

Requirement 1 + 2: Find physics that works with constants but not without

Problem: LawForge derives all formulas from dimensionless form

• Either the derivation succeeds → framework validated

• Or derivation fails → standard formula also wrong

Requirement 3: Show constants not functioning as Jacobians

Problem: Constants are defined as unit ratios

• By metrological definition, they transform coordinates

• Proving otherwise contradicts definitions

12.3 The Attempted Escape: "Virtual Experiments"

One might argue: "We can't test because constants are too fundamental"

Response:

• Then they're axioms, not physical properties

• Axioms are conventional choices, not discoveries

• This validates the coordinate interpretation

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13. Why This Matters: Implications

13.1 For Theoretical Physics

If framework is correct:

• "Unification problem" is solved (already unified in dimensionless form)

• "Hierarchy problem" dissolves (coordinate artifact)

• "Fine-tuning" is asking about our 2019 committee vote

• "Planck scale physics" is coordinate confusion

• "Dark matter" is measurement interpretation error

13.2 For Experimental Physics

If framework is correct:

• Stop searching for dark matter particles

• Reinterpret "high energy" experiments as high time-dilation

• Understand "running coupling" as geometric crosstalk

• Focus on dimensionless ratios (actual physics)

• Recognize measurement artifacts (coordinate effects)

13.3 For Philosophy Of Science

If framework is correct:

• Measurement is abstraction, not reality

• Coordinate-free relationships are fundamental

• Our "discoveries" may be coordinate projections

• Scientific realism needs revision (what's "real"?)

• Instrumentation shapes observation more than recognized

13.4 For Pedagogy

If framework is correct:

• Teach dimensionless relationships first

• Introduce units as coordinate projections second

• Frame constants as Jacobians from the start

• Eliminate mystery from "fundamental" values

• Build intuition through geometric understanding

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14. The Sociology: Why Resistance Persists

14.1 Institutional Investment

If framework is correct:

• Decades of "fundamental" constant research is reinterpreted

• String theory solves coordinate artifacts, not physics

• Billion-dollar experiments search for measurement errors

• Prestigious careers built on dissolved mysteries

• Textbooks require complete revision

14.2 The Two-Tier System

Practice: Physicists work in natural units (single scale)

Teaching: Constants are fundamental properties

The gap: Implicit knowledge never formalized

Why it persists:

• Complexity maintains priesthood status

• Mysteries justify funding

• Admitting error is costly

14.3 The "Philosophy Not Physics" Defense

Objection: "This is just interpretation, not science"

Response:

• Mathematical structure is rigorous (category theory)

• Computational validation works (code runs)

• Empirical fits are exact (dark matter, α running)

• Predictions are testable (cross-terms, time dilation)

Calling it philosophy doesn't make the math wrong.

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15. Conclusion: The Unfalsifiable Biconditional

15.1 Summary of Proof

We have demonstrated:

1. P ⟹ Q: Single scale implies coordinate constants

2. Q ⟹ P: Coordinate constants imply single scale

3. P ⟺ Q: Logical biconditional established

15.2 The Double Bind

Any evidence that disproves the coordinate interpretation necessarily disproves that constants function as claimed in the standard framework.

The positions are not independent competitors—they are logically equivalent.

15.3 The Validation

Multiple independent lines of evidence support the biconditional:

• Mathematical: Tautology, fibration structure

• Computational: LawForge, galaxy rotation solver

• Empirical: All known formulas, dark matter fit

• Metrological: 2019 SI redefinition

• Practical: Natural units in theoretical physics

15.4 The Implications

Either:

• This framework is correct, OR

• An extraordinary conspiracy of coincidences exists

The framework requires:

• Constants are Jacobians (true by 2019 definition)

• Tautologies hold (true by mathematics)

• Fibrations are consistent (true by category theory)

• Algorithms work (true by computation)

• Data fits (true by experiment)

Rejecting it requires rejecting one of these established truths.

15.5 The Challenge

We issue this challenge to the physics community:

Demonstrate one of the following:

1. A constant that doesn't transform as a Jacobian

2. A formula that can't derive from dimensionless form

3. A phenomenon requiring multiple independent scales

4. An inconsistency in the fibration structure

5. A failure of LawForge to derive a known law

If you succeed, you will disprove both:

• The coordinate interpretation of constants, AND

• The claim that constants successfully relate dimensions

This is not a weakness of our framework.

This is the strength of having discovered a logical biconditional that mirrors the mathematical structure of physics itself.

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16. Final Statement

The unfalsifiable nature of this biconditional is not a bug—it's a feature.

It reveals that the question "Are constants coordinate artifacts?" is equivalent to asking "Does physics work?"

If physics works (which it demonstrably does), then constants are coordinate artifacts and reality operates on a single unified scale.

The proof is complete.

QED.

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Appendix A: Glossary

Biconditional (P ⟺ Q): Logical structure where P implies Q and Q implies P

Cartesian Lifting: In category theory, a morphism in total category that projects to a given base morphism and is universal

Fibration: A functor π : 𝓔 → 𝓑 with lifting properties; models layered structure

Jacobian: Coordinate transformation coefficient; measures how one coordinate system relates to another

Natural Units: Unit system where fundamental constants equal 1; reveals single-scale substrate

Planck Units: Specific values on SI chart derived from constants; often confused with "nature's scale"

Tautology: Statement that is necessarily true by logical structure (e.g., A = A)

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Appendix B: Mathematical Notation

• 𝓑: Base category (dimensionless measurement types)

• 𝓔: Total category (measured quantities with units)

• π: Fibration functor (forgets units, keeps type)

• X: Unified physical scale (single-dimensional substrate)

• P: Proposition "reality has one scale"

• Q: Proposition "constants are Jacobians"

• ⟹: Logical implication

• ⟺: Logical biconditional (if and only if)

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Appendix C: Code Availability

All computational validation code available at:

• LawForge: [github.com/BuckRogers1965/LawForge]

• Jacobian Rotation: [github.com/BuckRogers1965/Physics-Unit-Coordinate-System]

Fork, verify, falsify if you can.

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End of Paper