This document is the distinction guide (Common Misconceptions + Usage Cautions) portion of the Banya Framework Comprehensive Report.

A. Common Misconceptions

Misconception 1. The Banya equation is a Theory of Everything (TOE)

Wrong. The Banya equation is not a theory but a framework.

A theory explains "why" and predicts specific values. A framework defines "how far" and checks whether existing theories fit within it.

Theory: E = mc² -> if mass is 1kg then energy is 9×10^{16}J (numerical prediction)
Framework: δ² = c² + ℏ² -> E = mc² is inside the classical bracket (position identification)

Example: The Pythagorean theorem declares the structure that "the square of the hypotenuse of a right triangle = the sum of squares of the other two sides." It does not determine whether a side's length is 3 or 5. The Banya equation is the same.

Misconception 2. The Banya equation replaces existing physics

Wrong. Existing physics formulas continue to work within the Banya Framework.

Einstein's E² = (mc²)² + (pc)² is an app inside the classical bracket. The Schrödinger equation is an app inside the quantum bracket. It is not replacing the apps but discovering the OS on which the apps run.

Before: Relativity app separate, quantum app separate, incompatible
Banya Framework: Both apps run on the same OS. Orthogonal, so no conflict

Example: The release of Windows did not make Excel disappear. Excel runs on Windows.

Misconception 3. Four axes means four-dimensional spacetime

Wrong. Spacetime consists of only 2 axes in the classical bracket: time + space. The remaining 2 axes (observer, superposition) belong to the quantum domain. Not all 4 axes are spacetime.

Classical bracket: time, space -> spacetime (Einstein's domain)
Quantum bracket: observer, superposition -> quantum states (Heisenberg's domain)
The two are orthogonal -> no need to merge

Example: A car's speedometer and fuel gauge are independent instruments. The speedometer going up does not automatically change the fuel gauge. Yet both indicate the car's state.

Misconception 4. δ is energy

Wrong. δ is change. Energy is merely one expression of $\delta$.

δ = change (invariant)
energy = one way of measuring change
distance = another way of measuring change
probability = yet another way of measuring change

Example: Whether you measure "distance" in km or miles, the distance itself is the same. Whether you measure $\delta$ as energy or probability, the change itself is the same.

Misconception 5. The + sign in the equation is addition

Wrong. The + inside a bracket is a structural notation meaning "two orthogonal axes belong to one bracket." It does not mean to add numbers.

Example: On a map, going 3 km east and 4 km north gives a straight-line distance of 5 km. Not 3+4=7, but √(9+16)=5. Orthogonal axes combine via Pythagoras.

Misconception 6. Putting observer as an axis is unscientific

Quantum mechanics itself has failed to solve the measurement problem for 100 years. The Copenhagen interpretation, many-worlds interpretation, and decoherence theory all failed to answer "why does observation change the outcome."

Banya Framework promoted observation from "something to be explained" to "a structural axis." It accepted observation rather than explaining it. This is the same strategy Einstein used when he accepted gravity as curvature of spacetime rather than explaining it.

Einstein: Don't know what gravity is -> define it as spacetime curvature (success)
Banya Framework: Don't know what observation is -> define it as an independent axis (118 PASS)

Misconception 7. The units are wrong (it's neither SI nor natural units)

The units of the Banya equation are determined by the left-hand side δ. They are neither SI nor natural units. They are the units of $\delta$.

Banya equation: δ² = (time + space)² + (observer + superposition)²

In this equation, time, space, observer, superposition are "names."
Not m (meters), not s (seconds), not J (joules).
Units are determined only when constants are substituted into the norm.

Before substituting constants, there are no units. Units emerge only after substitution.

Example: Does the word "distance" itself have units? No. You can measure it in km, miles, or light-years. The axes of the Banya equation are the same. They are just names, and units attach the moment you substitute constants.

Wrong: "time is in seconds (s) and space is in meters (m), so how can you add them?"
Right: time and space have no units yet. When you put c into the norm, both share the units of c

In natural units (c=1, ℏ=1), δ = √2. In SI, $\delta$ = √(c² + ℏ²). Regardless of the unit system, the framework does not break. Units are the user's choice, not a property of the framework.

Misconception 8. The signs are wrong (Minkowski uses − but you use +)

The Minkowski metric uses ds² = (ct)² − x² − y² − z² with minus signs. But the Banya equation uses all +. Is that wrong?

It is not wrong. The Banya equation is not a physics equation but a structural equation. Signs are determined inside the norm at the time of substitution.

Banya equation: δ² = (time + space)² + (observer + superposition)²
                  Structural declaration. + means "belongs to the same bracket"

Norm substitution: \|C\|² = c²
           How time and space combine with what signs inside this
           is determined by the definition of the norm

Minkowski's − emerges inside the norm:

+ in the Banya equation: "these axes belong to one bracket" (structure)
− in Minkowski: "the norm of time and space combines this way" (internal, after substitution)

These are different levels. The Banya equation declares the brackets; signs are determined inside the norm.

Example: Labeling a drawer "socks" and the method of folding socks are different matters. The Banya equation is the drawer label, and the sign is the folding method. The label need not specify the folding method.

This is exactly why constants are substituted into the norm. Signs, units, and specific combination methods are all handled inside the norm. The Banya equation declares only the structure above that.

Misconception 9. It is circular reasoning ($E_p = m_p c^2$ is a definition, not a derivation)

The direction is reversed. We start from CAS Cost (Axiom 4) and arrive at existing physics equations. We are not taking existing physics equations and putting them into the framework.

Supposed direction: Know E = mc² -> put it into Banya Framework -> "you already knew that" (circular)
Actual direction: CAS Cost (Axiom 4)(cost = ℏ, record = spacetime) -> expand -> E = mc² emerges (derivation)

Example: Even if you treat CAS Cost (Axiom 4) as a hypothesis and run it, you get the same result. If the starting point is different, it is not circular. If a hypothesis matches existing physics even when initialized independently, the hypothesis is correct.

Misconception 10. There are no new predictions (it merely reinterprets existing values)

The empty axes among the 4 are the predictions. Each of the 118 physics equations has unused domains. Switching to those domains yields values that did not exist before.

Coulomb's law: F = kq₁q₂/r² -> uses only the space domain
Empty domains: observer, superposition, time
Switching yields: electromagnetic decoherence rate, entanglement energy, and other new physical quantities

See the "expected derivation values" in the appendix (118 detailed verifications). For each equation, new physical quantities that can emerge from empty domains are proposed. These are predictions unique to Banya Framework that do not exist in conventional physics.

Misconception 11. It is just relabeling (merely attaching new names to existing equations)

It is not just renaming — the domain changes. When the domain changes, previously invisible values emerge.

V = IR (Ohm's law, space domain)
Switch to quantum domain -> h/e² = 25,812.807 Ω (quantum Hall resistance)
This is a value that cannot emerge from relabeling

Put an existing equation into Banya Framework and switch to an empty domain. Different physical quantities emerge from the same equation. This is the power of the framework.

B. Usage Cautions

Caution 1. Do not expect specific numerical predictions from the Banya equation

The Banya equation itself declares only structure. To obtain numerical values, you must go to Banya Framework and substitute constants.

Wrong expectation: electron mass should come directly from δ² = (time + space)² + ...
Correct usage: put in c, ℏ, G -> solve simultaneous equations within the framework -> related values emerge

The Banya equation is a map. To ask "how many km from Seoul to Busan" by looking at a map, you must first insert the scale (constants).

Caution 2. Always compare framework results against established physical values

Values derived from the framework must be compared with experimentally confirmed physical quantities. If they match, the framework is correct; if not, the substitution process must be reviewed.

Derived: M_W = 77.5 GeV
Experiment: M_W = 80.4 GeV
Error: 3.5% -> within acceptable range (tree-level approximation)

The framework is not omnipotent — it does not give correct results no matter what you put in. If it is wrong, it is wrong.

Caution 3. Do not confuse orthogonality with merging

That classical and quantum are orthogonal means "they are independent." It does not mean to merge them into a single equation.

Wrong approach: time² + observer² = ? (mixing axes from different brackets)
Correct approach: time² + space² = c² (trade-off only within the same bracket)

Just as you should not add east to height, you should not directly compute classical axes and quantum axes in a single equation. Each bracket is an independent unit.

Caution 5. Do not mix up the equation and the framework when speaking

In conversation, do not say "the Banya equation predicts." The equation does not predict. The framework derives.

Wrong: "We predicted the dark energy ratio with the Banya equation"
Right: "We derived the dark energy ratio by substituting the cosmological constant into Banya Framework"

The reason for distinguishing: saying the equation predicts causes confusion with a theory. The framework is not a theory.