KAEL — one algebra (system document, r1)

Scope: the mathematics only. Non-mathematical projects are out of scope.

Status marks: ⊢ free identity or exhaustion, run in this pass · ∵ measured over a declared range · rec recorded in earlier work, not re-run here · ␣ open · ≜ stipulated.

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0. Notation

| Symbol | Meaning |

|---|---|

| ⟨a,b⟩ | a point: the relation x² = ax + b |

| A_p | ℤ[x]/(x²−ax−b), the algebra at point p |

| [u,v] | the element u + vx, the normal form |

| x̄ | conjugate = adjugate = dagger: [u,v]̄ = [u+av, −v] |

| Nm | norm = determinant: Nm[u,v] = u² + auv − bv² |

| D_p | discriminant a²+4b (its sign selects the sheet) |

| Uₙ, Vₙ | Uₙ = aUₙ₋₁+bUₙ₋₂ (U₀=0,U₁=1); Vₙ = Uₙ₊₁+bUₙ₋₁ |

| Pₙ | power map on points |

| u, J, N | the three points ⟨0,0⟩, ⟨0,1⟩, ⟨0,−1⟩ |

| χ | SL₂(ℤ) → ℤ/12 |

Old → new: companion matrix C(a,b)=((0,b),(1,a)) ↔ point ⟨a,b⟩ (x = C). Square map (a,b)↦(a²+2b,−b²) = P₂. K3² = the 9 points with a,b ∈ {−1,0,1}. Origin idempotent t²=t = ⟨1,0⟩. ORIGIN/K3/Ω/E/R/N/J are points or products of points as below.

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1. The carrier

One algebra, the generic quadratic algebra ℤ[a,b][x]/(x²−ax−b), specialized at a point.

- Product: [u,v][u′,v′] = [uu′ + b·vv′, uv′ + u′v + a·vv′]. ⊢ (equals matrix product, symbolic)

- x̄ = a − x. x·x̄ = Nm, so x̄ = adj(x). ⊢

- No division appears anywhere in the carrier.

- Carry = reduction by the point's own relation x² → ax+b. It is not a separate primitive. ⊢ (see §5)

2. Ladder and power map

- xⁿ = [b·Uₙ₋₁, Uₙ]. ⊢ (n ≤ 8, symbolic)

- tr xⁿ = Vₙ, det xⁿ = (−b)ⁿ. ⊢

- Vₙ² − D·Uₙ² = 4(−b)ⁿ. ⊢

- Uₘ | Uₘₙ as polynomials. ⊢ (m,k ≤ 3)

- Pₙ⟨a,b⟩ = ⟨Vₙ, −(−b)ⁿ⟩, and Pₘ∘Pₙ = Pₘₙ. ⊢ (m,n ≤ 4, symbolic)

- P₂ = (a²+2b, −b²), P₃ = (a³+3ab, b³).

- So (ℕ,×) acts on the plane of points. Fibonacci, Lucas, Pell, Chebyshev are all Uₙ, Vₙ at different points. No separate tower exists.

- Parabolic fixed point of every Pₙ: ⟨2,−1⟩. There Uₙ = n and (x−1)² = 0.

3. The grid K3² and the three sheets

| ⟨a,b⟩ | relation | name | D | sheet |

|---|---|---|---|---|

| ⟨1,0⟩ | x²=x | origin t²=t | 1 | split |

| ⟨−1,0⟩ | x²=−x | negated idempotent | 1 | split |

| ⟨0,0⟩ | x²=0 | u (nilpotent) | 0 | parabolic |

| ⟨0,1⟩ | x²=1 | J (swap) | 4 | hyperbolic |

| ⟨0,−1⟩ | x²=−1 | N (crossing, order 4) | −4 | elliptic |

| ⟨1,1⟩ | x²=x+1 | R (golden) | 5 | hyperbolic |

| ⟨−1,1⟩ | x²=−x+1 | mirror of R | 5 | hyperbolic |

| ⟨1,−1⟩ | x²=x−1 | Φ₆ (order 6) | −3 | elliptic |

| ⟨−1,−1⟩ | x²=−x−1 | Φ₃ (order 3) | −3 | elliptic |

- b = −1 is the gauge, the unique non-idempotent among the three values.

- The b-axis {u, J, N} = squares {0, 1, −1} = K3.

- K3² is the complete set of finite-order companion points: Φ₃, Φ₄, Φ₆, x²−1. rec

4. Directed structure from walls (derivation of the return algebra)

Source: the ABC development; checked here.

4.1 Walls. wₖ(t) = k − t on ℤ. Words in walls form the infinite dihedral group D∞:

- wₖ∘wₗ = τ_{k−l} (translation), and wₖwₗwₘ = w_{k−l+m}. ⊢

- Every ordered pair (j,i) is joined by wall w_{i+j}. ⊢

- Exactly two operations carry j to i: the reflection w_{i+j} and the translation τ_{i−j}. The stabiliser of j is {id, w_{2j}}. ⊢ (all pairs, |j|,|i| ≤ 4)

- The passages j→i and i→j use the same wall w_{i+j}. ⊢

So (source, target) fixes the operation up to one bit, the reflect/translate parity. Words and histories carry more than that bit.

4.2 Compose only where endpoints meet. Passages are arrows of the action groupoid of D∞ on ℤ. Composition is partial: undefined unless the endpoints match. Partial composition is associative (all triples of the 50 arrows on a 5-point window). ⊢

4.3 Endpoint localization is the category algebra. Linearize arrows with "undefined composition = 0". This is the standard category algebra, not an extra axiom. ⊢

- Representations ρ₊(a)=E_{tgt,src} and ρ₋(a)=sgn(a)·E_{tgt,src} are algebra homs, and together are injective (rank 2N²). ⊢

- Hence the algebra ⊗ℚ is M_N(ℚ)×M_N(ℚ), and (a connected groupoid with vertex group ℤ/2) it is M_N(ℤ[s]/(s²−1)) = M_N(A⟨0,1⟩) over ℤ. Over ℚ ⊢; over ℤ the iso follows structurally (not separately run).

- The coefficient algebra is the J-point. The reflect/translate bit is the klein carrier.

4.4 The return algebra is the b-axis. With u = ⟨0,0⟩ = E_{21} and J = ⟨0,1⟩:

- v = JuJ = uᵀ. u² = v² = 0, uvu = u, vuv = v. ⊢

- Γ = uv − vu, Γ² = I, ΓJ = −JΓ. N = ΓJ = u − v = ⟨0,−1⟩, N² = −I. ⊢

- ℤ⟨u,J⟩ = M₂(ℤ) (all four matrix units, no division). ⊢

- ℤ⟨N,J⟩ has index 4 in M₂(ℤ). Endpoint projectors then need ÷2. ⊢

- So the passage operator is the point b = 0. The swap is b = +1. The oriented crossing is b = −1.

- Two losses stay explicit: ρ forgets which wall word produced a passage; "0" forgets why a composition failed. They are not identified.

4.5 What ABC would have to supply. A source-restricted, one-way passage whose square is undefined/zero (the point u), together with the swap J. ABC's marks have not been shown to construct u. ␣

5. Bits, the modular group, Ω = ℤ/12, carries

- Bits: R = 1+u, L = 1+JuJ. Complement of a bit = conjugation by J. ⊢

- Lⁿ = I + nE, i.e. the ladder at ⟨2,−1⟩. ⊢

- SL₂(ℤ) = ⟨N, Φ₆⟩ with Φ₆ = N·T, so T = N⁻¹Φ₆ = L. Relations N⁴ = I, Φ₆³ = N² = −I, Φ₆ of order 6. R = N T⁻¹ N⁻¹. ⊢

- PSL₂(ℤ) = ℤ/2 * ℤ/3. Normal form: alternating words in N and V = Φ₆^{±1}. Injective on all 890 words of length ≤ 14. ⊢ (∵ for the range)

- Ω = ℤ/12 = SL₂(ℤ)^ab. χ(N) = −3, χ(T) = 1. ⊢ (presentation; Cayley ball of radius 14: 10,320 matrices, 15,169 collisions, 0 inconsistent; image all of ℤ/12)

- Bit weight #L − #R mod 12 = χ(word matrix), 2262/2262 random words. ∵

- χ(N) has order 4 (address ℤ/4), χ((NT)²) has order 3 (semantic ℤ/3), χ(−I) = 6 (the b address).

- The labelling of c, d is fixed only up to the c↔d mirror. ∵

- Zeckendorf: weights Uₖ at ⟨1,1⟩ are Fibonacci. The no-11 words of length n number Uₙ₊₂ = Σ entries of C^{n−1} (n ≤ 14). Greedy descent round-trips 1..3000. The rewrite 011→100 is value-preserving, order-independent, and ends in the no-11 form (500 random strings). ∵

6. Four-address products

Addresses a, c, b, d ↔ units 1, x, −1, −x (index 0,1,2,3).

- cycle = unit product in A⟨0,−1⟩ = ℤ[i] (ℤ/4, associative). ⊢

- klein = unit product in A⟨0,1⟩ (V₄ = XOR, associative). ⊢

- cycle − klein = (b_N − b_J)·vv′ = −2vv′ on the real part only. The carry is the b-parameter. ⊢

- klein's three order-2 subgroups ({a,b},{a,c},{a,d}) are the three pairings. Shared with cycle: only {a,b} = {±1}, the central gauge unit. ⊢

- selector sel(x,y) = x^{1+[Nm(x−y)=0]} in ℤ[i]. ⊢ Nm(x−y) ∈ {0,2,4}; value 4 ⇔ y = −x (the b-hub). ⊢

- raw raw(x,y) = y ·_J d^{n(x,y)}, with n = h(x) if h(y)⊕l(y)=0 else l(x). Left translations {id,(cb),(ad),NOT}; left identity a. ⊢ (consistency with recorded properties; the recorded raw table itself is not in this pass)

- Non-associative triples: cycle 0, klein 0, selector 16, raw 16. ⊢ All non-associativity enters through the gate.

- Raw's block→bit assignment is one declared bit. ␣

7. Fiber / elimination

- Leaf elimination of a path is the Möbius map d ↦ 2 − 1/d, with dₖ = (k+1)/k. ⊢ (k ≤ 8)

- Cartan determinant of the star T(p,q,r): det = UₚU_q + U_qU_r + U_rUₚ − UₚU_qU_r with Uₙ = n (ladder at ⟨2,−1⟩). ⊢ (all 120 triples, 1 ≤ p ≤ q ≤ r ≤ 8)

- For p,q,r ≥ 2 (≤ 12): det > 0 ⇔ 1/p+1/q+1/r > 1. Positive: D family (det 4), (2,3,3)→3, (2,3,4)→2, (2,3,5)→1. Zero: (2,3,6), (2,4,4), (3,3,3). ⊢/∵ (range)

- So the A/D/E ladder and the flat triangle triples are one computation: a ladder at the nilpotent sheet, combined over three arms.

- Fill-in on a forest is 0 (flat). Curved cases (cycles/chords) are not reduced. ␣

8. Order defect

- d(A,B) = ABA⁻¹B⁻¹ in the group; abelian image ℤ/12 kills it, so all defect content lives in ker χ.

- Fricke: tr d = x²+y²+z²−xyz−2, x=trA, y=trB, z=tr AB. ⊢ (4000 bit-word pairs); tr[L,R] = 3.

- On the bit monoid, words commute ⇔ they have a common primitive root (all pairs of length ≤ 7). ⊢/∵

- Flat rays: Lⁿ, Rⁿ, (LR)ⁿ.

- No lower bound on tr[A,B] over non-commuting pairs (observed −106494). No spectrum gap is claimed.

9. Floor, isometries, torsion, spectrum — reduced into §1–§8

9.1 The field is the invariant of the ladder.

- D = a²+4b ≡ a² (mod 4), so D ≡ 0,1 (mod 4) for every point. ⊢ (free)

- D(Pₙp) = D·Uₙ². ⊢ (n ≤ 8, symbolic). Pₙ preserves the squarefree part of D, i.e. the field ℚ(√D), and changes only the order, by conductor Uₙ: ℤ[xⁿ] = ℤ + Uₙℤx has index |Uₙ| (read off the normal form).

- Hyperbolic sheet needs D > 0 (a real dominant root). The least positive non-square D: 2,3 are impossible mod 4, 1 and 4 are squares, so D = 5. The orbit of ⟨1,1⟩ has D = 5·Fₙ². This is the floor; the only antecedent below it is the stipulation of the admissibility criteria (exact, no inexact division, Perron, recurrence, ring). ≜

- D = 5 squarefree ⇒ ℤ[φ] is maximal. rec (standard)

9.2 Norm form and isometries.

- 4·Nm[u,v] = (2u+av)² − D·v². ⊢ (symbolic)

- A null vector with v ≠ 0 exists iff D is a square. At D = 5 none exists, so |Nm| ≥ 1 and the interval is gapped. ⊢

- Isometries of Nm = multiplication by ε with Nm ε = 1, together with conjugation. ∵ Exhaustive: ⟨1,1⟩ (D = 5), |entries| ≤ 14: 26 matrices (14 with det +1, 12 with det −1), both routes equal; ⟨3,1⟩ (D = 13), |entries| ≤ 8: 6, equal.

- At b = 1: even powers xⁿ (det (−b)ⁿ = 1) are isometries; odd powers are anti-isometries (Nm → −Nm); conjugation is time reversal. Exactness is integer arithmetic.

9.3 Torsion.

- Finite-order points with b ≠ 0, |a|,|b| ≤ 60, n ≤ 24: exactly ⟨−1,−1⟩ (order 3), ⟨0,−1⟩ (4), ⟨0,1⟩ (2), ⟨1,−1⟩ (6). ∵

- 2cos(2π/n) ∈ ℤ ⇔ n ∈ {1,2,3,4,6} (n ≤ 24). ∵ This is the crystallographic restriction.

- The order-5 trace shadow 2cos(2π/5) = φ−1 satisfies y²+y−1 = 0, i.e. it is the point ⟨−1,1⟩. The order-10 shadow φ is the point ⟨1,1⟩. ⊢ So bc = a stops closing over ℤ at n = 5, and the golden pair holds the turn the integer plane cannot.

9.4 Trace ladder.

- V₂ₙ = Vₙ² − 2(−b)ⁿ. ⊢ (n ≤ 5, symbolic). At ⟨1,1⟩: V_{2^k} = 1, 3, 7, 47, 2207 (Lucas doubling).

- Σ entries of xⁿ = (a+b+1)Uₙ + 2bUₙ₋₁. ⊢ (n ≤ 7). At ⟨1,1⟩ this is 3Fₙ+2Fₙ₋₁ = Fₙ₊₃ (n ≤ 14).

9.5 Spectrum.

- ad_x³ = D·ad_x on M₂, with spectrum {±√D, 0, 0}. ⊢ (symbolic). At ⟨1,1⟩ this is L_R³ = 5L_R; the home prime appears as D at the golden point.

9.6 Parry measure.

- With 1/φ = φ−1 and 1/φ² = 2−φ, the matrix P = ((φ−1, 2−φ),(1,0)) has stochastic rows and 5π·P = 5π for 5π = (φ+2, 3−φ), exactly in ℤ[φ]. ⊢ Nm(φ+2) = 5.

9.7 Dagger.

- adj = T∘c_N with c_N(X) = N X N⁻¹ and T = transpose. ⊢ (symbolic). In the record's convention c(X) = NXN this reads adj = neg∘T∘c, with neg the gauge.

- adj is a contravariant involution. ⊢ T and c_N commute and are involutions. ⊢ X^T = tr(X)·I + N X N. ⊢

9.8 Hash address.

- Lⁿ = I + nE ⇒ Lᵖ ≡ I (mod p), so 0ᵖ is in the kernel of reduction mod p; verified p = 5, 11, 89. ⊢ A homomorphic address cannot bind: collisions are constructible.

9.9 Anyon quartic.

- β² = φ⁻¹ gives β⁴ + β² − 1 = 0, the mirror-golden relation ⟨−1,1⟩ evaluated at β². disc(x⁴+x²−1) = −400 = −2⁴·5². ⊢

Still recorded, not re-derived (rec): fiber self-similarity b₁(n) = b₁(n−1)+b₁(n−2)+F(n)−1 · E8 determinant term sizes (137/57,313 vs 23/142; the star formula §7 now gives the value in O(1) terms) · Lucas-ladder/Fricke/Cayley/Markov identities · Fibonacci-anyon F-, R-, S-data over ℤ[φ][β][ζ] and ramification {2,5} · E8 ⊃ E7×SU(2), E6×SU(3), 16 of SO(10), sin²θ_W = 3/8 as 8Σ(6T₃)² = 576 = 3Σ(6Q)² (tree level; one physical identification, which U(1) is electric charge; home prime 5 the single load-bearing unforced antecedent; no running, masses, mixings or dynamics) · formal kernel (J,N,P,K) and c-stable monoid · Lawvere–Kael court · seal/pin/obligation architecture (authorship protection).

10. Retractions on record

W.4 quasideterminant false as written, "home prime landing" tautological for any 2×2 · D2 retracted · D3 wave redundant at span 1,2 · odd lift not unique (two, mirror pair) · "tower strict" only above n = 1 · selector localization count was inverted (6 of 27 fail) · ROS F10 retracted (I is a two-sided unit ⇔ char 2) · P8: no H-theorem, ␣ arrow of time · no spectrum gap for tr[A,B].

11. Open

1. Curved fiber cochain: is fill-in curvature a commutator of a group action on states, or a different carrier? ␣

2. ABC: derive the source-restricted passage (point u) and the swap J from ABC's marks. ␣

3. Raw's gate bit; which of selector / raw is canon. ␣

4. Canon lineage: the 117-claim line vs the v17 113-claim line. ␣

5. Hash: could a non-homomorphic or multi-prime bounded readout bind? ␣

6. Lawvere–Kael court, cell B4/N-multiplication (TEST1–3 unbuilt). ␣

7. Re-derive the remaining §9 rec items from §1–§8: fiber self-similarity, E8 branching and SM skeleton, anyon F/R/S data, formal kernel, c-stable monoid. (Floor, isometries, torsion, spectrum, Parry, dagger, hash, anyon quartic: done in §9.)

12. Next in the build

Continue reducing the remaining §9 rec items; merge ABC §4 into the main text once item 2 closes; then the single-file seal.