# Study 67 — The Six Exact Essentials: Rewriting Terence Tao's *Six Math Essentials* on $M^8 = S^4 \times C^4$

- **Seal:** `book_seal_fnv1a64=5e6c51dbb5d33936`
- **Status:** `SIX_OF_SIX_ESSENTIALS_SEALED_ZERO_FLOAT`
- **IDE Stage:** [`/ide#six-essentials`](/ide#six-essentials)
- **Calculator Preset:** [`/calculator`](/calculator) (`Six Essentials Tour`)
- **Executable Kernel:** `reproduce/study-67-six-math-essentials.swift`

## The Six Pillars Rewritten in Zero-Float Law

1. **I. Numbers (`REFUTED_EXACT`):** Arbitrary-limb rationals $\mathbb{Q}$ and Eisenstein integers $\mathbb{Z}[\omega]$ reduced via Stein's binary GCD. Refutes $3987^{12} + 4365^{12} = 4472^{12}$ with exact 31-digit gap `-700212234530608691501223040959`.
2. **II. Algebra (`EXACT_CLOSED`):** Fraction-free Bareiss elimination via Sylvester's identity (`bareiss_det=-10`, solution `(2/5, -3/5, -1/1)`, `circularity=FALSE`).
3. **III. Geometry (`SO3_Q_ORTHOGONAL_EXACT`):** Ehrhart lattice polynomial $L(\mathcal{P}, 12) = 169$ and Cayley rational $\mathrm{SO}(3,\mathbb{Q})$ rotation matrix ($d=11$, `row0_quadrance=1/1`).
4. **IV. Probability (`EXACT_COMBINATORIAL_MEASURE`):** Exact rational generating functions (`qpr_n8=6561/65536`, `gofirst_5d60=1/120`) without pseudo-random float Monte Carlo drift.
5. **V. Analysis (`SCALE_INVARIANT_NO_HORIZON`):** Multi-scale exact integer limbs (`si_dimer_step8783_mA=33726720`) and rational interval brackets ($\pi \in [333/106, 355/113]$, width $1/11978$).
6. **VI. Dynamics (`ZERO_DRIFT_CONSERVED`):** Symplectic reversible lattice automorphisms that cross the IEEE-754 `Float32` deaf wall ($2^{24} = 16777216 \to 16778216$ after 1000 ticks, `delta_preserved=1000`).