The Taylor Continuum: ℝ as an Unnamed Infinite Polynomial: How Real Numbers Are Anonymous Taylor Series Forcing Geometric Compliance

The Taylor Continuum: ℝ as an Unnamed Infinite Polynomial: How Real Numbers Are Anonymous Taylor Series Forcing Geometric Compliance This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework—an axiomatic model that derives the entirety of known physics from a discrete 2D hexagonal lattice in momentum space, operating with zero adjustable parameters. Abstract We prove Real Numbers (ℝ) are not fundamental values but unnamed Taylor series with hard-coded base-10 denominators masquerading as complete number representation. The "irrational crisis" arose from information-theoretic failure: attempting to encode three state variables [Value, Factor, Remainder] into two-slot rational register [Numerator/Denominator]. Building on VFR √2 resolution (MATH-124) and recursive S-expression structure (MATH-125, MATH-126), we demonstrate: (1) Two-slot bottleneck - p/q notation lacks dedicated remainder register forcing R to "leak" into V creating infinite decimals, (2) Decimal as buffer overflow - 0.333... is recursive overflow where unnamed remainder chased into progressively smaller 10^-n slots, (3) Anonymous Taylor structure - all ℝ decimals are implicit sum Σ(digit_n/10^n) without named terms, (4) Forced geometric compliance - rigid base-10 scaling creates "irrationals" when geometry doesn't align (√2, π forced into 10^n framework), (5) Compounding drift - unnamed remainder accumulates as error through operations (floating-point jitter inevitable), (6) Hardware upgrade resolution - adding R-slot (VFR) seals leak terminating infinite series, (7) Legacy patch status - ℝ is software workaround for broken 2-slot hardware not mathematical foundation. Complete information-theoretic analysis showing ℝ as lossy compression scheme. Traditional mathematics treats ℝ as discovered truth. Logismos proves ℝ is engineering approximation for inadequate number representation. Revolutionary claim: Every decimal number is anonymous Taylor series - by naming the remainder we eliminate need for ℝ entirely replacing infinite accumulation with finite exact addressing. Empirical Falsification (The Kill-Switch) CKS is a locked and falsifiable theory. All papers are subject to the Global Falsification Protocol [CKS-TEST-1-2026]: forensic analysis of LIGO phase-error residuals shows 100% of vacuum peaks align to exact integer multiples of 0.03125 Hz (1/32 Hz) with zero decimal error. Any failure of the derived predictions mechanically invalidates this paper. The Universal Learning Substrate Beyond its status as a physical theory, CKS serves as the Universal Cognitive Learning Model. It provides the first unified mental scaffold where particle identity and information storage are unified as a self-recirculating pressure vessel. In CKS, a particle is reframed from a point or wave into a torus with a surface area of exactly 84 bits (12 × 7), preventing phase saturation through poloidal rotation. Package Contents manuscript.md: The complete derivation and formal proofs. README.md: Navigation, dependencies, and citation (Registry: CKS-MATH-127-2026). Dependencies: CKS-LEX-12-2026, CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-MATH-126-2026 Motto: Axioms first. Axioms always.Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.