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Document 3 of the TFPT 5.0 setE8 audit & bootstrap
TFPT 5.02026-06-08749 KBSHA-256 dfc6fe5a…f5cb
Paper 3E8 audit & bootstrap

E₈ Audit, Cascade Bridge and Bootstrap

The seven E₈ slices as an audit raster, the cascade spine, and the Möbius loop

E₈ as an audit container, not a mystery: the seven maximal slices of 248 as a falsification raster (every load-bearing number must appear in at least one projection), the bridge showing the old E₈ orbit cascade D = 60 − 2n is the same even-integer spine as the compiler, and the Möbius bootstrap in which g_car = 5 and the '8' in c₃ are overdetermined E₈-closure fixed points — only π irreducible.

Inputs
  • ›The compiler core {c₃, g_car} ⇒ E₈ and the residue matrix R from Documents 1–2.
Contribution
  • ›The discipline rule: every load-bearing TFPT number must appear in at least one E₈ branching projection — turning the number stock into a falsifiable raster, not numerology.
  • ›E₆ × A₂ reads the flavor matrix: 248 = 78 + 8 + 2·27·3 with ‖R‖_F² = 78 = dim E₆ and det R = 8 = dim A₂.
  • ›The Möbius bootstrap: g_car = 5 forced three ways and the '8' in c₃ = rank E₈ = h(D₅) = φ(30) = det R.
Not claimed here
  • ›The atlas slice readings are audit-level [A] — a program, not a proof of new physics.
  • ›The bootstrap is not creation from nothing: two inputs remain, and π is not produced by the loop.
Falsification surface
  • ›Fails if a load-bearing number cannot be placed in any E₈ projection, or if the reverse glue μ² − 5μ + 4 = 0 does not single out the (D₅, A₃) branch.
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Highlights
Audit rule7 slicesEvery load-bearing number lives in an E₈ projection
Flavor readE₆ × A₂‖R‖² = 78 = dim E₆, det R = 8 = dim A₂
g_car = 5forced 3×Rank-fill, Coxeter-match, integer-glue
Irreducibleπ onlyBootstrap leaves no free discrete number

Key formulas

  • E₆ × A₂ flavor read
    248=78+8+2⋅27⋅3248 = 78 + 8 + 2\cdot 27\cdot 3
    ‖R‖_F² = 78 = dim E₆, det R = 8 = dim A₂. Audit-level [A].
  • Cascade endpoints
    12DstartDend=60⋅82=240\tfrac{1}{2}D_{\mathrm{start}}D_{\mathrm{end}} = \tfrac{60\cdot 8}{2} = 240
    The old cascade is the same even-integer spine. [I]
  • Reverse glue
    μ2−5μ+4=0\mu^2 - 5\mu + 4 = 0
    Singles out μ = 4 (A₃), g_car = 5. [L]
  • Five readings of 8
    8=rank⁡E8=h(D5)=φ(30)=det⁡R=2∣μ4∣8 = \operatorname{rank}E_8 = h(D_5) = \varphi(30) = \det R = 2|\mu_4|
    The c₃ denominator is overdetermined.

The seven E₈ slices as an audit raster

Each maximal subalgebra of E₈ projects a TFPT module. The strongest new hit: E₆ × A₂ reads the flavor residue matrix, with E₆ reading its Frobenius norm and A₂ the three-family symmetry.

248=∥R∥F2+det⁡R+2(1⊤R a) Nfam=78+8+2⋅27⋅3248 = \|R\|_F^2 + \det R + 2(\mathbf{1}^\top R\,a)\,N_{\mathrm{fam}} = 78 + 8 + 2\cdot 27\cdot 3
∥R∥F2=78=dim⁡E6,det⁡R=8=dim⁡A2=h(D5)\|R\|_F^2 = 78 = \dim E_6, \qquad \det R = 8 = \dim A_2 = h(D_5)

The cascade bridge: D = 60 − 2n

The old E₈ orbit cascade is the same even-integer spine: it starts at 60 = 2·3·10, ends at 8 = h(D₅) (the flavor selector), passes the Coxeter rung 30 = h(E₈), and the product of endpoints recovers the root count.

Dn=60−2n,Dstart Dend2=60⋅82=240=∣R(E8)∣D_n = 60 - 2n, \qquad \frac{D_{\mathrm{start}}\,D_{\mathrm{end}}}{2} = \frac{60\cdot 8}{2} = 240 = |R(E_8)|
240+Dend=248=dim⁡E8240 + D_{\mathrm{end}} = 248 = \dim E_8

g_car = 5 forced three ways

The carrier rank is an overdetermined E₈-closure fixed point: rank-fill (g + 3 = 8), Coxeter-match (h(D_g) = 2g − 2 = 8), and the integer-glue/norm closure whose reverse-glue quadratic has nontrivial root μ = 4.

gcar+3=8,h(Dg)=2g−2=8⇒gcar=5g_{\mathrm{car}} + 3 = 8, \qquad h(D_{g}) = 2g - 2 = 8 \Rightarrow g_{\mathrm{car}} = 5
q(Dg)+q(Aμ−1)=2  ⟹  μ2−5μ+4=0q(D_g) + q(A_{\mu-1}) = 2 \;\Longrightarrow\; \mu^2 - 5\mu + 4 = 0

The '8' in c₃ and the irreducible π

The seam denominator is fixed five concordant ways. The two axioms collapse to one continuous primitive (π, from Möbius/Gauss–Bonnet) plus one discrete fixed point (the E₈ closure).

8=2∣μ4∣=rank⁡E8=h(D5)=φ(30)=det⁡R8 = 2|\mu_4| = \operatorname{rank}E_8 = h(D_5) = \varphi(30) = \det R
{c3,gcar}⟶π⏟continuous+E8 closure⏟discrete\{c_3, g_{\mathrm{car}}\} \longrightarrow \underbrace{\pi}_{\text{continuous}} + \underbrace{E_8\text{ closure}}_{\text{discrete}}

Key formulas at a glance

  • E₆ × A₂ flavor read
    248=78+8+2⋅27⋅3248 = 78 + 8 + 2\cdot 27\cdot 3

    ‖R‖_F² = 78 = dim E₆, det R = 8 = dim A₂. Audit-level [A].

  • Cascade endpoints
    12DstartDend=60⋅82=240\tfrac{1}{2}D_{\mathrm{start}}D_{\mathrm{end}} = \tfrac{60\cdot 8}{2} = 240

    The old cascade is the same even-integer spine. [I]

  • Reverse glue
    μ2−5μ+4=0\mu^2 - 5\mu + 4 = 0

    Singles out μ = 4 (A₃), g_car = 5. [L]

  • Five readings of 8
    8=rank⁡E8=h(D5)=φ(30)=det⁡R=2∣μ4∣8 = \operatorname{rank}E_8 = h(D_5) = \varphi(30) = \det R = 2|\mu_4|

    The c₃ denominator is overdetermined.