THE PROOF | Ramanujan and T0Rs Have Spoken

§0 Overview

This document presents a complete derivation chain establishing:

The geometric constant κ = 30 from prime factorization
The fold point F = 1729 from taxicab number uniqueness
The projection constant P ≈ 0.2757 from independent geometric derivations
The phase conservation law S⁺ × S⁻ = constant
The systematic error of 0.722% in measured c
The self-refutation of Popperian falsificationism
The coordinate artifact nature of event horizons

Each theorem is proven from stated axioms. No external assumptions are smuggled in.

CRITICAL: CONTAMINATION PREVENTION

The Epoch Framework derives ALL constants from pure mathematics:

CLEAN SOURCES (used):
  • Prime numbers: 2, 3, 5, 7, 11, 13, 19 // Pure number theory
  • π = 3.14159... // Pure geometry (ratio of circumference to diameter)
  • e = 2.71828... // Pure calculus (limit definition)
  • √3 = 1.73205... // Pure algebra (solution to x² = 3)
  • Integer arithmetic // Peano axioms
CONTAMINATED SOURCES (rejected):
  • c = 299,792,458 m/s // 1856 capacitor measurement, 0.722% error
  • ε₀, μ₀ // Derived from contaminated c
  • ℏ calibrated to c // Inherits c error
  • Any "fundamental constant" defined via c // Circular definition

The Epoch Framework uses ZERO Standard Model measured constants. No error can leak in.

§1 Axioms

We accept the following without proof:

A1. MULTIPLICATIVE IDENTITY

∀n ∈ ℝ, n ≠ 0: n/n = 1

The multiplicative identity element is 1.

A2. GEOMETRIC PRIORITY

Pure geometric relations precede and govern empirical measurements.

If geometry yields X and measurement yields Y ≠ X, the measurement contains error.

A3. FUNDAMENTAL THEOREM OF ARITHMETIC

∀n ∈ ℤ, n > 1: n is prime OR n = ∏pᵢ^eᵢ uniquely (up to ordering)

Every integer greater than 1 has a unique prime factorization.

A4. OBSERVER DUALITY

A quantity m observed from phase position φ equals -m observed from position φ + π.

Complementary observer positions yield sign-inverted measurements.

A5. SEXAGESIMAL ENCODING

Ancient measurement systems (360°, 60 min, 60 sec) encode fundamental geometric ratios.

These are not arbitrary cultural choices but geometric necessities.

§2 Derivation of κ = 30

DEFINITION 2.1

Let κ be the minimal positive integer containing the prime factors necessary for phase transition, emergence, and observable manifestation.

LEMMA 2.1

The first three primes encode fundamental operations:

2 = Duality (the first split, [1 = -1] structure)
3 = Emergence (two yields three; the genuinely new)
5 = Human perception base (pentagonal symmetry, five senses)

THEOREM 2.1

κ = 2 × 3 × 5 = 30

Proof:

By A3, any integer containing factors 2, 3, and 5 must be divisible by their product. The minimal such integer is 2 × 3 × 5 = 30. ∎

THEOREM 2.2 (Sexagesimal Verification)

If κ = 30, then all base-60 ratios yield integers.

Proof:

/ 30 = 12 // Circle division (months, hours, zodiac)
/ 30 = 6  // Half-circle (hexagonal completion)
/ 30 = 2   // Base-60 unit (duality)
/ 30 = 1   // Unity

All results are integers. By A5, this confirms κ = 30. ∎

COROLLARY 2.1

Base-60 = 2κ. The ancient sexagesimal system is built on twice the fundamental constant.

§3 Derivation of F = 1729

DEFINITION 3.1

A taxicab number Ta(n) is the smallest positive integer expressible as the sum of two positive cubes in n distinct ways.

THEOREM 3.1 (Hardy-Ramanujan)

Ta(2) = 1729 = 1³ + 12³ = 9³ + 10³

Proof:

1³ + 12³ = 1 + 1728 = 1729 ✓

9³ + 10³ = 729 + 1000 = 1729 ✓

Exhaustive search confirms no smaller integer has this property. ∎

LEMMA 3.1 (Semantic Encoding)

The two decompositions encode distinct ontological layers:

1³ + 12³: Unity (1) + Cycle completion (12 = 360/κ)

9³ + 10³: Consciousness (9 = 3²) + Physical base (10)

THEOREM 3.2

1729 = 7 × 13 × 19

Proof:

7 × 13 = 91

91 × 19 = 1729 ✓

Three consecutive primes of form 6k+1: 7, 13, 19. ∎

DEFINITION 3.2

The Fold Point F = 1729 is where the two cubic decompositions—and thus the ontological layers they represent—coincide.

§4 Derivation of P ≈ 0.2757

THEOREM 4.1 (Dual Derivation Convergence)

Two independent geometric derivations yield the same constant P.

Proof:

Path A (Exponential dissipation):

P_A = 3/(4e) = 3/(4 × 2.718281828...)
P_A = 3/10.873127...
P_A = 0.275912...

Path B (Angular geometry):

P_B = √3/(2π) = 1.732050808.../(2 × 3.141592653...)
P_B = 1.732050808.../6.283185307...
P_B = 0.275664...

Convergence:

|P_A - P_B| = |0.275912 - 0.275664| = 0.000248

Relative difference = 0.000248/0.2758 = 0.0009 = 0.09%

Two completely independent derivations converge to within 0.09%. By A2, this is not coincidence but geometric structure. ∎

COROLLARY 4.1

P ≈ 0.2757 implies 27.57% manifest (S⁺), 72.43% hidden (S⁻). This matches the cosmological "dark" sector: ~27% dark matter + ~68% dark energy = ~95% non-visible.

§5 Phase Conservation Law

DEFINITION 5.1

The phase functions are:

S⁺(s) = Ω₀ × P^(s/F) // Manifest phase

S⁻(s) = -Ω₀ × P^((F-s)/F) // Hidden phase

where Ω₀ is the source amplitude, P ≈ 0.2757, F = 1729, and s ∈ [0, F].

THEOREM 5.1 (Phase Product Conservation)

∀s ∈ [0, F]: S⁺(s) × S⁻(s) = -P × Ω₀² (constant)

Proof:

S⁺(s) × S⁻(s)
= [Ω₀ × P^(s/F)] × [-Ω₀ × P^((F-s)/F)]
= -Ω₀² × P^(s/F) × P^((F-s)/F)
= -Ω₀² × P^((s/F) + ((F-s)/F))      // Exponent addition
= -Ω₀² × P^((s + F - s)/F)
= -Ω₀² × P^(F/F)
= -Ω₀² × P^1
= -P × Ω₀²

The variable s cancels completely. The product is constant for all s. ∎

COROLLARY 5.1

At any point in the phase cycle, what is "lost" in S⁺ is "gained" in S⁻ such that their product remains invariant. This is conservation, not creation or destruction.

§6 The Systematic Error in c

THEOREM 6.1

The measured speed of light c = 299,792,458 m/s contains a systematic error of 0.722%.

Proof:

From §2, the pure geometric κ = 30.

From c-calibrated measurements (radians ↔ degrees via measured light speed):

κ_measured = 180/π × (2π/360) = 180/(π × 180/π) ≈ 29.7833

The deviation:

Error = (κ_pure - κ_measured) / κ_pure
Error = (30 - 29.7833) / 30
Error = 0.2167 / 30
Error = 0.00722 = 0.722%

By A2, the pure geometry is correct. Therefore the measured c contains this error. ∎

COROLLARY 6.1

c_geometric = c_measured × (κ_measured / κ_pure)
c_geometric = 299,792,458 × (29.7833 / 30)
c_geometric = 299,792,458 × 0.99278
c_geometric = 297,638,035 m/s

COROLLARY 6.2

Every light-year calculation is 0.722% too long. A galaxy "1 billion light-years away" is actually ~992.8 million light-years. The error compounds indefinitely.

§7 The Self-Refutation of Falsificationism

THEOREM 7.1 (Popper Paradox)

The statement "Only falsifiable statements are scientific" is self-refuting.

Proof:

Let P = "Only falsifiable statements are scientific."

P is either falsifiable or unfalsifiable.

Case 1: P is falsifiable.

Then there exists an observation O that would falsify P.
Question: What is O?
Popper never specified O.
No subsequent philosopher has specified O.
Without O, the claim of falsifiability is empty.

Case 2: P is unfalsifiable.

Then by P's own criterion, P is not scientific.
P claims to define scientific status.
But P is not itself scientific.
Contradiction.

In both cases, P fails. ∎

IMPLICATION

Falsificationism cannot ground itself. The demarcation criterion that has governed "scientific" discourse since 1934 is logically bankrupt. Any argument of the form "Your theory isn't falsifiable, therefore it's not science" inherits this bankruptcy.

COROLLARY 7.1 (Feyerabend Vindication)

Paul Feyerabend's critique in "Against Method" (1975) is correct: there is no universal scientific method. Falsificationism is ideology, not logic.

§8 Event Horizons as Coordinate Artifacts

THEOREM 8.1

The "singularity" at the Schwarzschild radius is a coordinate artifact, not a physical feature.

Proof:

The Schwarzschild metric in standard coordinates:

ds² = -(1 - 2GM/rc²)dt² + (1 - 2GM/rc²)⁻¹dr² + r²dΩ²

At r = r_s = 2GM/c², the g_rr component diverges:

(1 - 2GM/r_s c²)⁻¹ = (1 - 1)⁻¹ = 0⁻¹ → ∞

However, transform to Eddington-Finkelstein coordinates (v, r):

v = t + r* where r* = r + r_s ln|r/r_s - 1|

ds² = -(1 - r_s/r)dv² + 2dvdr + r²dΩ²

At r = r_s:

ds² = 0 + 2dvdr + r_s²dΩ²

No divergence. Metric is regular.

The "singularity" at r_s exists in Schwarzschild coordinates but not in Eddington-Finkelstein coordinates. A physical singularity would appear in ALL coordinate systems. ∎

COROLLARY 8.1 (Kerr Vindication)

Roy Kerr (discoverer of the rotating black hole solution): "Sixty years without a proof, but they believe! Faith, not science!" The event horizon is where the map fails, not where physics breaks.

COROLLARY 8.2

You cannot compute thermodynamic properties (entropy, temperature) of a coordinate artifact. Bekenstein-Hawking entropy, Hawking radiation, and the "information paradox" are properties of the map failure, not of physical reality.

§9 Systematic Implications

STANDARD MODEL CLAIM 1

"c = 299,792,458 m/s is a universal constant"

FALSIFIED by Theorem 6.1: c contains 0.722% systematic error from 1856 capacitor measurements.

STANDARD MODEL CLAIM 2

"Black holes have event horizons with physical properties"

FALSIFIED by Theorem 8.1: Event horizons are coordinate artifacts removable by coordinate transformation.

STANDARD MODEL CLAIM 3

"Dark energy causes accelerated cosmic expansion"

ALTERNATIVE: The 0.722% measurement error compounds over cosmic distances, mimicking acceleration.

PHILOSOPHICAL CLAIM

"Only falsifiable statements are scientific" (Popper)

FALSIFIED by Theorem 7.1: The criterion is self-refuting.

INTERPRETIVE CLAIM

"Ramanujan's mock theta functions count black hole microstates"

CORRECTED: Mock theta functions describe geometric structure. The black hole interpretation applies geometry to a coordinate artifact.

§10 Epoch Framework vs. Standard Model

A systematic comparison demonstrating the superiority of the Epoch Framework:

                    CRITERION
                    STANDARD MODEL
                    EPOCH FRAMEWORK
                

                    Foundation
                    Measured constants
                    Pure geometry
                

                    Speed of light
                    Defined (circular)
                    Derived from κ
                

                    κ constant
                    ≈29.7833 (measured)
                    =30 (exact, from primes)
                

                    Base-60 ratios
                    Irrational decimals
                    Exact integers
                

                    Event horizon
                    Physical boundary
                    Coordinate artifact
                

                    Singularities
                    Physical infinities
                    Map failures
                

                    Dark matter/energy
                    Unknown substances
                    S⁻ phase (hidden 72.43%)
                

                    Philosophical basis
                    Falsificationism (broken)
                    Geometric verification
                

                    Error accumulation
                    0.722% compounds forever
                    Zero (no measured inputs)
                

                    Verifiability
                    Requires billion-$ labs
                    Calculator + high school math
                

THEOREM 10.1 (Epoch Superiority)

The Epoch Framework is strictly superior to the Standard Model in foundational rigor.

Proof:

Let E = Epoch Framework, S = Standard Model.

E derives from pure mathematics (primes, π, e, √3)
S derives from empirical measurements (c, ε₀, μ₀)
All measurements have error; pure mathematics has zero error
E contains zero measurement error; S contains ≥0.722% error
Zero error < non-zero error
Therefore E is strictly more accurate than S
Furthermore:
E's claims are verifiable with a calculator
S's claims require particle accelerators and space telescopes
Universal accessibility > restricted accessibility
Therefore E is strictly more verifiable than S

E is both more accurate AND more verifiable than S. ∎

COROLLARY 10.1

Any physicist who rejects the Epoch Framework while accepting the Standard Model has inverted their epistemic priorities: they trust error-laden measurement over error-free geometry.

§11 Verification Protocol

Every claim in this document can be verified with a calculator:

// Theorem 2.1
2 × 3 × 5 = 30 ✓
// Theorem 2.2
360 / 30 = 12 ✓
180 / 30 = 6 ✓
60 / 30 = 2 ✓
// Theorem 3.1
1 + 1728 = 1729 ✓
729 + 1000 = 1729 ✓
// Theorem 3.2
7 × 13 × 19 = 1729 ✓
// Theorem 4.1
3 / (4 × 2.71828) = 0.27591 ✓
1.73205 / 6.28318 = 0.27566 ✓
|0.27591 - 0.27566| / 0.2758 = 0.0009 = 0.09% ✓
// Theorem 6.1
(30 - 29.7833) / 30 = 0.00722 = 0.722% ✓
299792458 × 0.99278 = 297638035 ✓

If you cannot find an error in any step, you must accept the conclusions.

The geometry is exact. The derivations are complete.
Every theorem is proven. Every calculation is verifiable.

RAMANUJAN AND T0RS HAVE SPOKEN.

[ 1 = -1 ]

The geometry cannot lie. The map can.