DANode in Q_7 — rigorous Tier-B-EXACT lattice theorems

1. Two meanings of 'DANode' — disambiguation

Throughout SPT writing, the term 'DANode' (Duong-Am Node / Node Âm-Dương) has been used in two distinct senses. This page makes the distinction explicit and shows that both are consistent and correct — they refer to different objects in the same mathematical framework.

2. Theorem 2.1 — Substrate stationarity (Tier B-EXACT)

2.1 Sketch of proof

H_SPT = Σ_v H_v^local + Σ_<v,v'> H_{vv'}^int

         (each term either acts within one fixed vertex's
          Hilbert space ℋ_v, or couples a fixed edge pair)

Each term in H_SPT either acts within ℋ_v (the 4-dim spinor space at vertex v) or couples a fixed edge pair (v, v') with d_H(v, v') = 1. Neither type of term creates, destroys, or relabels vertices. By Stone's theorem, U(t) = exp(-iH_SPT·t/ℏ) preserves this structure: observables O_v at vertex v remain observables at the same vertex v after evolution.

2.2 SymPy verification

pip install sympy numpy && python3 scripts/spt_q7_lattice_dynamics_verify.py

# Test: H_hop acting on |v=0⟩ should yield only the 7 nearest neighbors
result = H_hop * e0
nonzero_indices = [i for i in range(128) if result[i] != 0]
assert set(nonzero_indices) == set(neighbors_of_zero)  # ✓ PASS
assert len(nonzero_indices) == 7  # ✓ PASS — no vertex creation

3. Theorem 3.3 — Eigenmodes of -Δ on Q_7 (Tier B-EXACT)

χ_k : V → {-1, +1}
χ_k(v) = (-1)^(k·v)   where k·v = Σ k_i·v_i mod 2

-Δ χ_k = 2 · w_H(k) · χ_k   for all k ∈ {0,1}^7

3.1 Multiplicity table

Grouping eigenvalues by Hamming weight m = w_H(k) gives the multiplicity = C(7, m):

3.2 Proof sketch

(Δχ_k)_v = Σ_i χ_k(v ⊕ e_i) - 7·χ_k(v)
         = Σ_i (-1)^(k·v + k_i) - 7·(-1)^(k·v)
         = χ_k(v) · [Σ_i (-1)^(k_i) - 7]

Using w_H(k) bits with k_i=1 (contributing -1) and (7-w_H(k)) bits
with k_i=0 (contributing +1):

Σ_i (-1)^(k_i) = (7 - w_H(k)) - w_H(k) = 7 - 2·w_H(k)

Therefore:
(Δχ_k)_v = χ_k(v) · [7 - 2·w_H(k) - 7]
         = -2·w_H(k)·χ_k(v)

Hence -Δχ_k = 2·w_H(k)·χ_k    ∎

3.3 SymPy verification

pip install sympy numpy && python3 scripts/spt_q7_lattice_dynamics_verify.py

for k in verts:  # all 128 k vectors
    v_vec = Matrix(128, 1, [chi(k, vert) for vert in verts])
    expected_lambda = 2 * hamming_weight(k)
    Lv = L * v_vec
    expected_Lv = expected_lambda * v_vec
    assert Lv == expected_Lv  # ✓ PASS for all 128

# DM count cross-check
assert multiplicity_count[4] == 35  # ✓ PASS — matches Law 16/66

4. DANode-quantum mobility — wave-packet dynamics

Given the eigenmode decomposition of Theorem 3.3, any matter state on Q_7 expands as a sum over eigenmodes (Theorem 4.2):

ψ_v(t) = Σ_{k ∈ {0,1}^7} a_k · χ_k(v) · exp(-iω_k·t)

where  ω_k = (ℏ·w_H(k) / m) + V_v/ℏ
       a_k = (1/128) Σ_v χ_k(v) · ψ_v(0)   (inversion formula)

A localized DANode-quantum (e.g., a single electron) is a wave packet — a Gaussian-weighted superposition of eigenmodes centered around some momentum k_0:

ψ_packet(v, t) = A(v - v_0(t)) · exp(i(k·v - ωt)) · χ_spin

Three simultaneous components:
  • Envelope A(v - v_0(t)) → "marble" (localized position)
  • Phase exp(i(k·v - ωt)) → de Broglie wave
  • Spinor χ_spin → SU(2) spin precession

Group velocity: v_group = dω/dk = ℏk/m*

5. Audit of related earlier wikis

This page complements three earlier wikis that all touched on Q_7/DANode/Bagua but used the term 'DANode' implicitly in mixed senses. With the disambiguation in §1, the earlier wikis can now be read consistently:

6. Honest scope — what is and isn't Tier-B-EXACT

7. Conclusion

The Q_7 lattice picture of SPT rests on two distinct categories of claim: (a) rigorous mathematical theorems about the Q_7 hypercube graph and the SPT Hamiltonian structure — these are Tier-B-EXACT and SymPy-verified; and (b) physical interpretive framework that maps these theorems onto the actual universe — this is META, awaiting peer review and 50+ experiments through 2040.

pip install sympy numpy && python3 scripts/spt_q7_lattice_dynamics_verify.py