Passes 4936–4937 · native GAP · 20/20 + 30/30

The chamber packet is a logic algebra. The missing ten is a finite controller.

Two previously separate exact carriers now have explicit systems-level forms: the rank-48 chamber packet is literally a two-state rational matrix algebra repeated on 24 representation lanes, and the ten-dimensional ternary adjoint quotient is a 10-trit affine control register after global port matching.

M₂(Q)

all four literal matrix units on the 160-chamber carrier

24 × 2

one multiplicity switch repeated on 24 representation lanes—not 24 physical qubits

3¹⁰ = 59,049

exact first-order correction offsets

17 classes

complete PGSp(4,3) orbit atlas for the offset register

Pass 4936: literal matrix units

Let Π₄₈ be the rank-48 chamber projector and let Q_p, Q_ℓ be the rank-24 point and line carriers from Pass 4334. Exact rational matrices give

e₁₁ = Qₚ, e₂₂ = Π₄₈ − Qₚ, e₂₁ = (Π₄₈ − Qₚ)QℓQₚ, e₁₂ = (64/15)QₚQℓ(Π₄₈ − Qₚ), eᵢⱼeₖℓ = δⱼₖeᵢℓ.

Thus the four-dimensional packet algebra is literally M₂(Q) on the multiplicity coordinate. The exact switch pair

Z = e₁₁ − e₂₂, S = e₁₂ + e₂₁, Z² = S² = Π₄₈, ZS = −SZ, (SZ)² = −Π₄₈

is algebraic logic on that coordinate. Its rational normalization is not an orthogonal or unitary gate construction.

The precise HoloBox checksum

The deterministic HoloBox labels HP0..HP2 and HL0..HL2 depend on a software chart. Their complete family sums do not: they recover the intrinsic point and line panel relations. Compression and affine normalization yield two exact packet reflections H_p, H_ℓ with

Hₚ² = Hℓ² = Π₄₈, HₚHℓ + HℓHₚ = −(1/2)Π₄₈, 2(HₚHℓ)² + HₚHℓ + 2Π₄₈ = 0.

The aggregate turn has infinite order on the packet. This certifies whole-family consistency; it does not prove that one selected HPi or HLi preserves the packet.

Pass 4937: the ten-trit affine controller

Pass 4864 identifies the residual quotient with sp₄(F₃). Pass 4861 shows that full port matching removes the local S₃⁴⁵ sheet gauge and leaves one global PGSp(4,3) frame. Their smallest direct composition is

F₃¹⁰ ⋊ PGSp(4,3), v ↦ vA_g + w, |F₃¹⁰ ⋊ PGSp(4,3)| = 3¹⁰ · 51,840 = 3,061,100,160.

The 59,049 offsets have exactly seventeen frame-symmetry classes:

1, 80, 240, 480, 540, 540, 1080, 1080, 4320, 4320, 5184, 5184, 5760, 6480, 6480, 8640, 8640.

Compiler reading. A future control plane can first classify a ten-trit correction state into one of 17 exact symmetry classes, then lower a chosen representative. The certificate supplies the taxonomy and update law, not the representative-selection algorithm or runtime opcode.

Post-merge provenance. Pass 4871 independently proves that the ten-space bracket is intrinsic to Levi incidence, up to nonzero F₃ scalar. Pass 4872 compiles each of the 45 local S₃ port matchings as one trit plus one bit. Those results cross-certify the inputs; they do not already construct this distinct global ten-trit affine correction register.

The equal-order firewall

The dual-number tangent extension Sp₄(F₃[ε]/ε²) and the affine PGSp controller both have order 3,061,100,160, but they are not the same group:

|Z(Sp₄(F₃[ε]/ε²))| = 2, |Z(F₃¹⁰ ⋊ PGSp(4,3))| = 1.

The square-zero kernel is elementary abelian, so its group commutator vanishes. The nonabelian Lie bracket is tangent data inherited from matrix commutators before imposing ε²=0; it is not a commutator inside that kernel.

Reproduce the exact certificates

gap -q analysis/w33_pass4936_chamber_packet_matrix_units.g
python3 -m pytest -q tests/test_w33_pass4936_chamber_packet_matrix_units.py
gap -q -b analysis/w33_pass4937_adjoint_dual_number_controller.g
python3 -m pytest -q tests/test_w33_pass4937_adjoint_dual_number_controller.py

Evidence boundary. These are exact finite rational and characteristic-three theorems. They do not construct an individual-selector packet intertwiner, a HoloBox guest-state update, a recursive checksum law, an opcode lowering, hardware timing, operating-system isolation, a security guarantee, a physical qubit, a continuum gauge field, a particle, a mass, or a coupling.