Principia Metaphysica

1 What the model postulates

The setting is assumed, not derived: a two-time bulk, two shadows — a visible one and a mirror one, each with its own time — and the bridges between them. Each shadow’s seven internal dimensions form the geometry derived below.

• The bulk is 26-dimensional with signature (24,2), one time per 13D shadow of signature (12,1).
• Twelve bridges: the directed edges of K₄ on a Fano arc of four faces, grouped into three E₈ blocks.
• WA-1 (adopted 2026-10-01): one bridge per resolved A₁ component (one U(1)). With the 3 × 4 bridge-component match it selects (b₂, b₃) = (12, 43) without data (CG.12).

2 Standard physics it builds on

Established results the model uses, each with its source:

• M-theory on a compact 7-manifold with holonomy G₂ gives 4D N = 1 supergravity with b₂ abelian vector multiplets and b₃ neutral chiral multiplets (Acharya 1999; Lukas & Morris 2004).
• Compact 7-manifolds with a torsion-free G₂-structure are obtained by resolving T⁷/Γ with Eguchi-Hanson spaces; the (ℤ/2)³ example has b₂ = 12, b₃ = 43, and further examples include b₂ = 8 + l, b₃ = 47 - l (Joyce 1996a; Joyce 1996b; Joyce 2000).
• For a compact torsion-free G₂-structure, Hol = G₂ iff π₁ is finite (Joyce, JDG II, Prop. 1.1.1) (Joyce 1996b).
• π₁ of a quotient space is the group modulo the subgroup generated by elements with fixed points (Armstrong 1968).
• An A₁ singularity fibred over a 3-manifold Q carries pure N = (1 + b₁(Q)) super Yang-Mills (Acharya 1999).
• G₄ flux on a smooth G₂ manifold gives a positive runaway potential; a supersymmetric vacuum needs a non-real Chern-Simons invariant (Acharya et al. 2005; Lukas & Morris 2004).
• M-theory on Joyce manifolds, as K3 fibrations, is dual to heterotic strings on T³-fibred Calabi-Yau threefolds (Acharya 1996).

3 How the geometry is selected

Each step follows from the one before; theorem CG.12 states the chain as one result, and each switch it rests on can be flipped to compare the alternative.

• φ, the model’s 3-form, fixes the folding group Γ = (ℤ/2)³: the coordinate sign-flips that preserve all seven of its terms.
• Joyce’s construction from Γ reaches 28 literature-checked Betti pairs (b₂, b₃), on the lines b₂ + b₃ = 23, 39 and 55 and the point (0, 7); the earlier seed b₃ = 24 is not among them (CG.7).
• Across that family the restricted holonomy climbs trivial → SU(2) → SU(3) → G₂ as the number n of singular involutions goes 0 → 3, and π₁ is finite only at n = 3. On the compact real form of φ — the active path (D-015) — that makes the holonomy exactly G₂ only on the n = 3 line.
• On that line the twelve bridges of the bulk and the twelve singular components of Y₇ form one 3 × 4 structure only on the all-plain classes, at (b₂, b₃) = (12, 43). Identifying each bridge with one component (WA-1, adopted 2026-10-01) selects it without using data (CG.12).
• n_gen = b₂/4 = 3, the number of singular involutions: three is the rank of the folding group, a count rather than a fit. That these are three chiral generations is open: Y₇ carries no chiral matter (D-011).

4 The closed geometry, proved

Y₇ is Joyce’s resolution of T⁷/(ℤ/2)³, with (b₂, b₃) = (12, 43) and Betti numbers (1, 0, 12, 43, 43, 12, 0, 1). Each statement below is a theorem with a proof, a test and a falsifier (13 of 13 hold); together they fix the topology and every discrete datum with no free parameter. The metric’s b₃ moduli are fields, not parameters, and their stabilisation is open.

• CG.1 Resolved Betti numbers of Y₇
• CG.2 Euler characteristic of Y₇
• CG.3 Singular components of T⁷/Γ
• CG.4 Fundamental group of Y₇ across the family
• CG.5 Gauge content of Y₇
• CG.6 G₄ flux fixes no modulus of Y₇ at leading order
• CG.7 What Joyce's construction reaches from φ's (ℤ/2)³
• CG.8 The 12 bridges and the 12 A₁ components are one structure
• CG.9 One Kummer K3 per singular involution
• CG.10 Holonomy G₂ and a confining sector are mutually exclusive
• CG.11 The flux potential cannot accelerate the universe
• CG.12 What selects (b2, b3) = (12, 43)
• CG.13 The holonomy-G₂ metric exists, by theorem

5 What the model found

Results of the model’s own computations, each with its test:

• Joyce's construction from φ's diagonal (ℤ/2)³ reaches 28 literature-checked (b₂, b₃); the off-path seed b₃ = 24 is not among them (Joyce 1996b).
• Across the family, restricted holonomy is trivial, SU(2), SU(3), G₂ for n = 0..3 singular involutions; π₁ is finite only at n = 3 (Joyce 1996b).
• The 12 bridges and the 12 A₁ components form one 3 × 4 structure, existing exactly on the all-plain members; blocks match involutions canonically, while the matching inside a block is a free choice.
• χ_eff = 2 × sum over singular involutions of χ(Kummer K3) = 48n; n_gen = χ_eff/48 counts singular involutions (adopted 2026-10-01).
• In φ's Joyce family, holonomy exactly G₂ and a confining gauge sector are mutually exclusive (Acharya 1999).

6 What was corrected

Earlier claims of the model that were wrong, and what replaced them:

• The adopted 3-form was the split real form, one transcription sign away from the framework's own octonion product. The compact form is the active path since 2026-10-01; the split form stays switchable.
• The A₁ filter tested the setwise stabiliser; the four-member family was the all-plain subfamily of Joyce's construction (Joyce 1996b).
• Constants built on k_gimel = b₃/2 + 1/π were fits made at the retired seed, not derivations.
• χ_eff = 144 was called the Euler characteristic of the G₂ manifold. Every closed odd-dimensional manifold has χ = 0, so χ_eff is an effective index, now defined by the K3 reading, 48 n (adopted 2026-10-01).
• The internal space was formerly described as a twisted connected sum ('TCS #187') with b₃ = 24. No published TCS enumeration has such an entry, and b₃ = 24 lies below the published TCS ranges. The construction is Joyce's orbifold resolution, which reaches b₃ = 43 and never 24 (Joyce 1996b).
• The racetrack exponent a = 2 π / b₃ read a Betti number as the rank of a gauge group. Y₇'s singular loci carry N = 4 super Yang-Mills, and with full holonomy no confining sector exists in φ's Joyce family, so Y₇ has no gaugino racetrack; its values were calibrations at the off-path seed (Acharya 1999).
• Two borrowed claims were withdrawn (signature ruling 2026-08-31): Bars' Sp(2,ℝ) ghost-freedom theorem, because gauging (24,2) gives one (23,1) shadow rather than two (12,1) shadows; and '26 = the bosonic critical dimension', because with two times the critical dimension is 27-28.

7 What is open

Problems the geometry does not solve, stated honestly:

• No chiral matter inside Y₇: disjoint singular loci leave no codimension-7 points (Acharya 1996).
• The 43 moduli, Re(T) included, are unfixed at leading order; Re(T) is an open modulus, and values that need it use a labelled calibration (Acharya et al. 2005).
• The leading-order flux potential cannot accelerate the universe (slope ≥ 5 √(2/7)).
• Flavour structure awaits a chiral sector.
• Ghost control of the second time direction has no computed backing, and Nahm's bound on supersymmetric theories with one time (at most 11 dimensions) must be evaded by the 13D(12,1) shadows.

8 Where it stands against data

Above the closed geometry the model’s physics layers carry calibrations, counted in the free-variable ledger, and the model does not fit the data globally. Every comparison — passes and failures alike — is reported row by row in the paper’s results section and on the validation pages.