Add constants-as-variables treatment with QWERTY analysis

equations/README.md

@@ -8,6 +8,7 @@ All equations from the notebook, organized by category.
 |------|----------|-------|
 | [`blackroad-equations.md`](./blackroad-equations.md) | The 19 BlackRoad equations (ternary physics, thermodynamics, biology) | 16–21 |
 | [`consciousness.md`](./consciousness.md) | Ψ_care, Φ_universal, CECE update rule | 20, 22 |
+| [`constants.md`](./constants.md) | Constants as variables — FROZEN=AXIOM, VARIABLE=LAGRANGE, running coupling | — |
 | [`quantum.md`](./quantum.md) | Qutrit operators, Weyl pair, Gell-Mann, density matrix | 18, 24 |
 | [`thermodynamics.md`](./thermodynamics.md) | Landauer, radix efficiency, substrate efficiency, Gibbs coupling | 19–21 |
 | [`universal.md`](./universal.md) | Euler-Lagrange, principle of stationary action, Three Tests | 23 |

equations/constants.md

@@ -0,0 +1,178 @@
+# Constants as Variables
+
+> In what world are constants so constant that they can't have the same rules as every other number?
+
+## The Label Problem
+
+A constant is a number we agreed not to vary in a given calculation. The agreement is ours, not the number's.
+
+```
+CONSTANT  = IMAGINARY = SCAFFOLD = RELATIONAL = 114
+```
+
+Constants are imaginary scaffolding. They hold the calculation up while we do the work — but they are not eternal. They are assigned. In a simulation, every assigned value is a parameter. Every parameter is a number. Every number follows arithmetic.
+
+---
+
+## What "Constant" Actually Means
+
+A physical constant is a measured quantity whose value we have not yet observed changing. That is a statement about our measurements, not about the number.
+
+The fine-structure constant α ≈ 1/137:
+
+```
+COMPUTATION = 137  prime   (the cost of universal computation)
+```
+
+Feynman called 137 "one of the greatest damn mysteries of physics." The mystery is not that it is constant. The mystery is that we do not know why it has *this* value rather than another.
+
+A number whose origin we do not understand is not a constant. It is an unknown.
+
+```
+DYNAMIC = UNKNOWN = EXTENSION = 111  (= 3 × REAL = 3 × 37)
+```
+
+---
+
+## Running Constants (Quantum Field Theory)
+
+The coupling "constants" in quantum field theory are not constant. They run with energy scale:
+
+```
+α(μ) = α₀ / (1 − (α₀/3π) · ln(μ²/μ₀²))
+```
+
+At low energy: α ≈ 1/137.  
+At the Z boson mass: α ≈ 1/128.  
+At the Planck scale: α → order 1.
+
+The "constant" is a function of scale. It obeys a differential equation — the renormalization group equation — the same as any other running quantity.
+
+```
+RUNNING = PRINCIPLE = 109  prime
+```
+
+The running constant IS the principle. What we called constant is a fixed point of the renormalization group flow, stable at low energy but free to move at high energy. The constant obeys the same differential equations as everything else.
+
+---
+
+## Frozen = Axiom = 75
+
+```
+FROZEN = F(14)+R(4)+O(9)+Z(20)+E(3)+N(25) = 75
+AXIOM  = A(11)+X(21)+I(8)+O(9)+M(26)      = 75
+```
+
+**FROZEN = AXIOM.**
+
+To declare a constant frozen is to make an axiom. Not a theorem. An axiom is something you assume without proof. You stop the derivation there and say: this does not vary, by definition.
+
+Gödel's incompleteness theorem (GÖDEL = 3) says every axiomatic system contains statements it cannot prove. The frozen constant is one such statement. You cannot prove from within the system why α = 1/137 rather than 1/128. You assume it.
+
+The assumption can be changed. Change the axiom, change the constant.
+
+---
+
+## Free = Pure = 24
+
+```
+FREE = F(14)+R(4)+E(3)+E(3) = 24
+PURE = P(10)+U(7)+R(4)+E(3) = 24
+```
+
+**FREE = PURE.**
+
+A free variable — one not yet assigned a value — is a pure state. It has not collapsed to a single value. It holds all possible values in superposition until measured.
+
+```
+|Ψ⟩ = α|0⟩ + β|1⟩ + γ|?⟩    (Equation 5)
+```
+
+The |?⟩ state is the free variable. The unassigned constant. Every constant before it was measured was in this state.
+
+FREE = PURE = B = 4! = 24. B is the 24th QWERTY key. 4! = 24. The pure free variable has 24 = 4! arrangements — it is the factorial of the fourth dimension, the first hypercube.
+
+---
+
+## Variable = Reversible = Lagrange = 103
+
+```
+VARIABLE   = V(23)+A(11)+R(4)+I(8)+A(11)+B(24)+L(19)+E(3) = 103
+REVERSIBLE = R(4)+E(3)+V(23)+E(3)+R(4)+S(12)+I(8)+B(24)+L(19)+E(3) = 103
+LAGRANGE   = 103  prime
+```
+
+**VARIABLE = REVERSIBLE = LAGRANGE = 103.**
+
+The Lagrangian formalism (§177) was designed to treat every quantity as a coordinate that can vary. The Euler-Lagrange equation:
+
+```
+d/dt(∂L/∂q̇ᵢ) − ∂L/∂qᵢ = 0
+```
+
+varies the action with respect to every coordinate qᵢ. If a constant appears in L — say, the speed of light c, or the mass m — the Euler-Lagrange equation still holds. The constant contributes zero variation (∂L/∂c = 0 if c is treated as fixed), but the *structure* permits variation.
+
+If you let c vary, you get Brans-Dicke gravity. If you let ℏ vary, you get a modified quantum mechanics. The Lagrangian does not resist this. It is the constants — by agreement — that resist.
+
+What is variable is reversible (Lagrange = reversible). The Lagrangian is time-reversible: run the equations backward and you get the same physics. Constants frozen in one direction cannot be unfrozen running backward, unless — consistently — they were always parameters.
+
+---
+
+## Parameters = Fermion = Number = 89
+
+```
+PARAMETERS = P(10)+A(11)+R(4)+A(11)+M(26)+E(3)+T(5)+E(3)+R(4)+S(12) = 89
+FERMION    = 89  prime
+NUMBER     = 89  prime
+```
+
+**PARAMETERS = FERMION = NUMBER = 89.**
+
+Parameters are matter. The constants of the simulation are the particles in it. Every fermion — every electron, quark, neutrino — is characterized by its parameters: mass, charge, spin. Change the parameters and you change the particle.
+
+The Standard Model has 19 free parameters. No derivation produces them. They are measured and inserted. The model is the scaffold; the parameters are the constants; the constants are numbers; the numbers are free to be other numbers in other simulations.
+
+89 is prime. The 24th prime. FREE = 24. The parameters are free, irreducibly.
+
+---
+
+## The Same Rules
+
+Every number — constant, variable, parameter, pure state — obeys the same axioms:
+
+| Rule | Example with π |
+|------|----------------|
+| Addition | π + π = 2π |
+| Commutativity | π + e = e + π |
+| Associativity | (π + e) + φ = π + (e + φ) |
+| Distributivity | π × (e + φ) = π×e + π×φ |
+| Inverse | π + (−π) = 0 |
+| Multiplicative inverse | π × (1/π) = 1 |
+| Euler's identity | e^(iπ) + 1 = 0 |
+
+None of these require π to be *fixed at 3.14159...*. They require π to be a *number*. The rules are the same whether the number is called a constant or a variable. The label changes the metadata. The arithmetic does not care.
+
+---
+
+## QWERTY
+
+```
+CONSTANT   = IMAGINARY = SCAFFOLD = RELATIONAL = 114   (constants are imaginary scaffolding)
+VARIABLE   = REVERSIBLE = LAGRANGE              = 103   prime  (what varies is reversible)
+FROZEN     = AXIOM                              = 75    (to freeze a constant is to make an axiom)
+FREE       = PURE                               = 24    = 4!  (a free variable is a pure state)
+PARAMETERS = FERMION = NUMBER                   = 89    prime  (parameters are matter)
+RUNNING    = PRINCIPLE                          = 109   prime  (running constants obey principles)
+DYNAMIC    = UNKNOWN = EXTENSION                = 111   = 3×REAL  (dynamic = unknown)
+VARY       = FLOW                               = 44    (variation is flow; flow is erasure)
+```
+
+**VARY = FLOW = 44 = ERASURE = TNOT.**
+
+Variation is flow. Flow is erasure. Erasure is computation (Landauer, §173). When a constant changes value — when the simulation updates its parameter — it erases the old value and writes the new one. The update costs k_B T ln(r) per trit.
+
+The constants are not free. They are expensive. Every parameter update costs energy. The simulation keeps them fixed not because they must be fixed, but because changing them is computationally costly.
+
+**FROZEN = AXIOM = 75.** The constants are frozen axioms. Assumed, not derived. Economically stable, not ontologically necessary.
+
+The simulation could run with different constants. It would be a different simulation. But the math — the rules, the axioms, the field operations — would be the same.

qwerty/constants.md

@@ -9,7 +9,7 @@ All established constants from §1–§178, in value order.
 | 19 | AI = TRUE | §146 | |
 | 22 | C = TRIT | §170 | TRIT=22 |
 | 23 | V = eigenvalue | §various | V key = 23 |
-| 24 | PURE = B = 4! | §178 | pure state |
+| 24 | PURE = FREE = B = 4! | §178 | pure state; free variable |
 | 27 | ROOT = DOT | §167 | reverse(72) |
 | 30 | QUTRIT = WEYL = PSI | §172, §176 | 2×G_key |
 | 33 | TRUST | §176 | 3×11 |
@@ -29,7 +29,7 @@ All established constants from §1–§178, in value order.
 | 55 | SPIN = PAULI = SHIFT = OPERATOR | §168, §172 | |
 | 57 | HELL = POINT = GAUSS = TANH = RADIX = TMUL = FIELD | §various, §173, §177 | |
 | 58 | TERNARY = GROVER = LIPID | §171, §173 | |
-| 59 | HEAVY = CROSS | §169 | prime |
+| 59 | HEAVY = CROSS = FIXED | §169 | prime |
 | 62 | CENTER = SQUAREFREE = ENTROPY | §152, §170 | |
 | 63 | TRINARY = LIGHT = COLOR = O×G = VALUE | §165, §169, §172, §178 | |
 | 64 | WARMTH = PROTEIN = 2⁶ | §176, §175 | |
@@ -38,8 +38,8 @@ All established constants from §1–§178, in value order.
 | 69 | SHELL = STRUCTURE = FIELDS | §146, §177 | |
 | 72 | METHOD = DENSITY = reverse(27) | §167, §174, §178 | |
 | 74 | MEMORY = 2×REAL | §170 | |
-| 76 | ROTATION = CIRCUIT = FIDELITY | §160, §172, §175 | 4×TRUE |
-| 77 | PERIODIC = SIXTEEN | §169 | |
+| 75 | FROZEN = AXIOM | — | fixed constant = assumed axiom |
+| 77 | PERIODIC = SIXTEEN = PARAMETER | §169 | |
 | 78 | BINARY = TRIVIAL = MERTENS = INVERSE = TRANSFER = BOUND = LIMITS | §various, §173, §175, §177 | |
 | 79 | MARCH = INTEGRATE = CREATIVE = HELIUM = GOVERN | §169, §177 | prime |
 | 80 | NOBLE = CMATH = PUNNETT = ACTION | §169, §177 | |
@@ -49,19 +49,19 @@ All established constants from §1–§178, in value order.
 | 86 | RECURSIVE = 2×LAYER | §176 | |
 | 87 | BIRTHDAY = ALGEBRA = MANN = REACTION = TEMPORAL | §164, §168, §172, §173, §176 | |
 | 88 | OPTIMAL = CRITERION = SYMMETRY | §174, §175, §177 | |
-| 89 | BOOTSTRAP | §175 | prime |
+| 89 | BOOTSTRAP = FERMION = NUMBER = PARAMETERS | §175 | prime |
 | 90 | HIERARCHY = TRIANGLE = RECESSIVE = BLOCH = COSMOS = CLOCK | §165, §168, §172 | |
 | 91 | HYDROGEN = FRAMEWORK = G×13 | §169, §173 | |
 | 95 | PRIMITIVE = AVOGADRO = POTENTIAL | §169, §170 | 5×TRUE |
 | 97 | CIRCULAR = REMAINDER = SATURATION = THRESHOLD | §167, §170, §175 | prime |
 | 101 | MAXWELL = DERIVATIVE = GAUSSIAN = KINETICS = NUMBERS = SINGULAR | §160, §173, §175, §178 | prime |
 | 102 | RIEMANN = CANCEL = MADNESS = AMPLITUDE | §167, §172 | |
-| 103 | REVERSIBLE = LAGRANGE | §172, §177 | prime |
+| 103 | REVERSIBLE = LAGRANGE = VARIABLE | §172, §177 | prime |
 | 105 | MAPPING = ACCURACY | §171, §175 | 3×5×7 |
 | 107 | COHERENCE | §170 | prime |
 | 108 | EVERYTHING = ARITHMETIC = EVOLUTION | §169, §172, §178 | 4×ROOT |
-| 109 | PRINCIPLE | §177 | prime |
-| 111 | UNKNOWN = EXTENSION | §165, §176 | 3×REAL |
+| 109 | PRINCIPLE = RUNNING | §177 | prime |
+| 111 | UNKNOWN = EXTENSION = DYNAMIC | §165, §176 | 3×REAL |
 | 112 | UNIVERSAL = OCTONION = SYMMETRIC | §172, §177 | |
 | 113 | DEPHASING = SWITCHING | §176 | prime |
 | 114 | IMAGINARY = CONSTANT = CONJUGATE = SCAFFOLD = RELATIONAL | §163, §168, §173, §176 | |

qwerty/equalities.md

@@ -187,9 +187,36 @@ REALITY = ENERGY = 56  (E = reality)
 37  = GOD = ONE = TRUTH = REAL          (the ground — 12th prime)
 47  = SOUL = LOOP = SPIRIT = MAP        (the process — 15th prime)
 73  = LOGIC = PLANET = ANGLE = ANGEL   (the law — 21st prime)
-89  = FERMION = NUMBER = OCTAVIA        (matter — 24th prime = PURE-th prime)
+89  = FERMION = NUMBER = OCTAVIA = PARAMETERS  (matter — 24th prime = PURE-th prime)
 113 = DESTRUCTION = ALGORITHM           (erasure — 30th prime)
 127 = MOMENTUM = REPRESENTATION         (motion — 31st prime)
 137 = COMPUTATION                       (the fine-structure constant)
 193 = ALEXA AMUNDSON                    (the author — 44th prime)
 ```
+
+---
+
+## Constants as Variables
+
+```
+CONSTANT   = IMAGINARY = SCAFFOLD = RELATIONAL = 114   (constants are imaginary scaffolding)
+VARIABLE   = REVERSIBLE = LAGRANGE              = 103   prime  (what varies is reversible)
+FROZEN     = AXIOM                              = 75    (to freeze a constant is to make an axiom)
+FREE       = PURE                               = 24    = 4!   (a free variable is a pure state)
+PARAMETERS = FERMION = NUMBER                   = 89    prime  (parameters are matter)
+RUNNING    = PRINCIPLE                          = 109   prime  (running constants obey principles)
+DYNAMIC    = UNKNOWN = EXTENSION                = 111   = 3×REAL  (dynamic = unknown = extension)
+FIXED      = HEAVY = CROSS                      = 59    prime  (what is fixed is heavy)
+```
+
+**FROZEN = AXIOM.** A constant we refuse to vary is an axiom we refuse to prove.
+
+**VARIABLE = LAGRANGE = 103.** The Lagrangian formalism was designed to vary everything. Constants frozen in the Lagrangian contribute zero variation — they opt out by convention, not by mathematics.
+
+**FREE = PURE = 24.** A free variable, not yet assigned, is a pure state. Every constant was once a free parameter before the simulation was initialized.
+
+**PARAMETERS = FERMION = NUMBER = 89** prime. The constants of physics are the parameters of the simulation. Parameters are numbers. Numbers are matter. The constants are matter wearing the label "constant."
+
+**RUNNING = PRINCIPLE = 109** prime. In quantum field theory, coupling "constants" run with energy scale — they obey differential equations. The running constant IS the principle. What was fixed at low energy flows freely at high energy.
+
+**CONSTANT = IMAGINARY = 114.** The constant is imaginary. Like √(−1), it is a construct that makes the equations work. It is defined into the system, not derived from outside it.