E47-Manta Programmable Matter Spectral Morph Flight System

TECHNICAL INFOGRAPHIC · FORMAL RECORD

E47–MANTA Programmable-Matter Spectral Morph Flight System

A visual formalization of a user-supplied mathematical record: exact E47 spectral algebra, mapped by defined software bridges onto a 125-node morph-geometry and a 6DOF flight simulation.

Record: E47-MANTA-SPECTRAL-MORPH-20260927-001Type: first-principles formalization of implemented simulation mappingsE0 exact finite algebraE2 software / flight simulation bridge

EPISTEMIC BOUNDARY

The spectral-to-geometry and spectral-to-thrust mappings are defined simulation assumptions, not empirical claims of matter deformation or propulsion.

1 · Theorem2 · Contraction3 · Capture4 · Field5 · Morph6 · Smoothing7 · Gain8 · Torque9 · 6DOF10 · Chain11 · Partition12 · Receipts13 · Validator14 · SourcesAppendix Ω

https://claude.site/public/artifacts/f3cee7bb-7755-4d57-be89-0d727a5bde52/embed

1

Theorem — E47 Kernel Construction

Exact finite Lie algebra · E0/E1

ℋ = V2⊗3, dim ℋ = 53 = 125

C = Jx2 + Jy2 + Jz2

2⊗2⊗2 = 1V0⊕3V1⊕5V2⊕4V3⊕3V4⊕2V5⊕1V6

spec(C) = {0, 2, 6, 12, 20, 30, 42}

mult = (1, 9, 25, 28, 27, 22, 13), Σ = 125

K = (C − 6I)(C − 30I)

Kψλ = (λ−6)(λ−30)ψλ

Kψλ = 0 ⟺ λ ∈ {6, 30}

ker K = E6 ⊕ E30, dim ker K = 25 + 22 = 47

P47 = P6 + P30

P472 = P47, P47† = P47, rank P47 = 47, KP47 = P47K = 0

Ωc = 47125 = 0.376

071421281λ=09λ=225λ=6kernel28λ=1227λ=2022λ=30kernel13λ=42Σ m = 125spec(C) — eigenvalue λ vs multiplicity m

Gold bars (λ = 6, 30) span the 47-dimensional kernel E₆ ⊕ E₃₀.

47/12537.6%

carrier rank fraction Ω_c

2

Spectral Contraction

Discrete contraction semigroup → projector limit

HK = K2 ⪰ 0

nonzero spec(K2) = {11664, 12544, 19600, 32400, 186624}

gap μmin+ = 11664 (λ=12), radius μmax = 186624 (λ=42)

ε⋆ = 199144

Γ = I − ε⋆K2

Γnψ = Σλ (1 − ε⋆[(λ−6)(λ−30)]2)nψλ

λ ∈ {6, 30} → 1; λ = 12 → 1517 ≈ 0.88235; λ = 42 → −1517

limn→∞ Γn = P47

ψn+1 = Γ(ψn + ηn)‖Γ(ψn + ηn)‖

undriven: ψn → P47ψ0‖P47ψ0‖ when P47ψ0 ≠ 0

110⁻²10⁻⁴10⁻⁶10⁻⁸04080120160kernel modesn = 160iteration n → complementary factor (15/17)ⁿ (log scale)

max contraction factor 15/17; Γ¹⁶⁰ fidelity ≈ 0.99999988

3

Capture and Residuals

Projection capture Ω vs runtime residuals

Ω(ψ) = ‖P47ψ‖2‖ψ‖2, 0 ≤ Ω ≤ 1, Ω(ψ) = 1 ⟺ ψ ∈ E47

JS runtime: rK(ψ) = ‖Kψ‖‖ψ‖

Python worker: rK2(ψ) = ‖K2ψ‖‖ψ‖

numerically different, same zero set: rK = 0 ⟺ rK2 = 0 ⟺ ψ ∈ E47

4

125-Node Spectral Control Field

State → normalized activations → lattice geometry

ψ = (ψ0, …, ψ124)T ∈ ℂ125

ai = |ψi|maxj |ψj|, 0 ≤ ai ≤ 1, ℂ125 → [0,1]125

(a,b,c) ∈ {0,1,2,3,4}3, i = 25a + 5b + c

u = a−22, v = b−22, w = c−22

xi = 4u; zi = 2.5v(1 − ¼|u|); yi = 0.20(1−u2) − 0.12v2 + 0.08w

ri(0) = (xi, yi, zi)T

x ∈ [−4, 4]y ∈ [−0.12, 0.28]z ∈ [−2.5, 2.5]

one spectral coordinate per MANTA control node

5

Spectral Morph Law

Activations + controls → node displacements

Δyi = 0.45(ai − ½)Ω + 0.15upzi + 0.12urxi

Δzi = 0.06ai sin φi, φi = arg ψi

MODESWEEP σCRUISE1MANEUVER1 − 0.12umTRANSITION1 − 0.45umRECOVERY0.85 + 0.15Ω

ri′ = (σxi, yi + Δyi, zi + 0.06ai sin φi)T

Si = 0.5 + 1.5ai, Ai = Ωai, Φi = arg ψi

6

Geometry Smoothing

Nearest-neighbor graph relaxation on the 5×5×5 lattice

rinew = 0.88 ri′ + 0.12 1|𝒩(i)| Σj∈𝒩(i) rj′

𝒩(i) — axis-adjacent neighbors · αs = 0.12

7

E47-Dependent Simulation Gain

Defined software mapping · amber = simulation bridge

χ = clamp(0.3 + 0.9Ω − 0.05 log10(1 + rK), 0.15, 1.2)

0.15 ≤ χ ≤ 1.2

χ ≢ Pr(·) — the gain is a defined clamp function, not a probability.

Fmax = 145000 N

F = 145000 utχ

Fb = (F, 0, −0.04Fup)T

8

Torque Law

Defined software mapping · amber = simulation bridge

m = 7200 kg

I = diag(15000, 23000, 26000) kg·m2

τ = 1.65m(ur, 1.2up, 0.7uy)T − 1.8 Iω

τx = 11880ur − 27000ωx

τy = 14256up − 41400ωy

τz = 8316uy − 46800ωz

9

6DOF Dynamics

Body-frame equations of motion · amber = simulation bridge

v̇b = Fbm + gb − ω × vb

Iω̇ + ω × (Iω) = τ

ω̇ = I−1[τ − ω × (Iω)]

q̇ = ½ q ⊗ (0, ω)

gravity remains activebrowser runtime: 120 Hz fixed-step RK4

10

Complete Implemented Chain

Software composition of the four pipelines

(A) SPECTRAL CONTRACTION → PROJECTOR

C

→

K = (C−6I)(C−30I)

→

K2

→

Γ = I−K2/99144

→

P47

(B) CAPTURE → CONTROL FIELD → MORPH GEOMETRY

P47

→

Ω(ψ)

→

ai

→

ri′

→

rinew

(C) CAPTURE/RESIDUAL → GAIN → FORCE

(Ω, rK)

→

χ

→

F

→

Fb

(D) CONTROLS → TORQUE → DYNAMICS

u

→

τ

→

(v̇b, ω̇, q̇)

E47 spectral state → 125-node control field → simulated morph geometry;
spectral capture/residual → simulation gain → force/torque law → 6DOF dynamics.

11

Exact / Simulation Partition

Evidentiary separation of the record

ℰexact — E0/E1

dim V2⊗3 = 125

K = (C−6I)(C−30I)

dim ker K = 47

P47

Γ = I−K2/99144

𝒮simulation — E2

χ, Fmax, gτ

ai ↦ ri

Si, Ai, σ

Fb, τ

ℰexact —defined bridge→ 𝒮simulation but ℰexact ⇏ 𝒮physical

The geometry bridge is a software model. It does not establish that E47 physically deforms matter. The thrust/torque map is a simulation assumption. The Python geometry worker does not supply measured aerodynamic forces.

12

Validation Receipts

Machine-checked first-principles checks

PASSCarrier Space Dim125 states (V₂⊗V₂⊗V₂)

PASSCasimir Spectrum{0:1, 2:9, 6:25, 12:28, 20:27, 30:22, 42:13}

PASSMultiplicity Sum1 + 9 + 25 + 28 + 27 + 22 + 13 = 125

PASSSelector Nullspaceker K = E₆ ⊕ E₃₀, dim = 25 + 22 = 47

PASSCarrier Rank FractionΩ_c = 47/125 = 0.376

PASSProjector IdentitiesP₄₇² = P₄₇, P₄₇† = P₄₇, rank 47, KP₄₇ = 0

PASSContraction Bound|λ(Γ)|_max = 15/17 ≈ 0.88235 (exact)

PASSProjection FidelityΓ¹⁶⁰ fidelity ≈ 0.99999988

PASSLattice Boundsx∈[−4,4], y∈[−0.12,0.28], z∈[−2.5,2.5]

PASSGain Lawχ ∈ [0.15, 1.20] clamped

E0/E1 Exact Algebra: VERIFIED PASS · E2 Simulation Bridge: ACTIVE · Physical Reality Claim: NONE

13

Validator Summary

Compact report · no code

• SU(2) spin-2 tensor product → Casimir matrix eigendecomposition: spectrum {0, 2, 6, 12, 20, 30, 42} with multiplicities (1, 9, 25, 28, 27, 22, 13) summing to 125.

• Selector kernel: nullity of K = (C−6I)(C−30I) is exactly 47, i.e. E₆ ⊕ E₃₀.

• Projector checks: P₄₇ idempotent, Hermitian, rank 47, annihilated by K.

• Contraction: Γ = I − K²/99144; maximal non-unit factor exactly 15/17; trial-state fidelity after 160 iterations ≈ 0.99999988.

• 125-node lattice: reference geometry from (a,b,c) anchors with axis-adjacent neighbor smoothing, α_s = 0.12.

• Gain & thrust: χ clamped to [0.15, 1.2]; F = 145000·u_t·χ; χ ≢ Pr(·).

• E47Witness.compile(): rank 47, basis 125×47, kernel_fraction 47/125 — fixed and distinct from the state-dependent capture Ω(ψ).

• VehicleRegistry: CONVENTIONAL {f16, sr71, x15}· EXPERIMENTAL_SIMULATION {eidolon, manta, skyrmion, jacob}; the switch changes the vehicle model, not the algebra.

14

Source Binding

Canonical repository paths

src/manta/programmable_matter.pywebsite/interfaces/skyrmion/runtime2/experimental-model.jswebsite/interfaces/skyrmion/runtime2/physics-6dof.jswebsite/interfaces/manta/manta-model.jswebsite/interfaces/manta/provenance.jsonsrc/aetheris/flight_runtime.pyartifacts/E47-MANTA-SPECTRAL-MORPH-20260927-001.md

Flow: shared E47Witness → experimental adapter → E2 trajectory; the switch changes the vehicle model, not the exact algebra.

Ω

Appendix — Ω–Recursive Field System: Error-Corrected Algebraic Closure

Supplementary record appended to the canonical file

Certificate: MC-OMEGA-RECURSIVE-CLOSURE-20260927-001Status: PASSE0 exact symbolicE1 Python numericE1 quantum simulation

I. CONTINUITY AND COVARIANT FIELD

∂tψ + ∇·J = 0, J = −D∇ψ ⇒ ∂tψ = D∇2ψ

Jμ = −D∇μψ, ∇μJμ = 0 ⇒ □ψ = 0

Tμν = ∇μψ∇νψ − ½gμν(∇ψ)2

□ψ = 0 ⇒ ∇μTμν = 0

II. CONSERVED GEOMETRIC COUPLING

∇μGμν = 0, ∇μgμν = 0

Hμν := aGμν + bgμν, ∇μHμν = 0

Hμν = κ0Tμν, Λ := b/a, κ := κ0/a

Gμν + Λgμν = κTμν

III. NEWTONIAN LIMIT

G00 ≃ ∇2Φ, T00 ≃ ½(∇ψ)2

∇2Φ = κ2 (∇ψ)2

ρ := α(∇ψ)2, κ = 8πGα

∇2Φ = 4πGρ

Φ = κ2 ∇−2[(∇ψ)2] + Φh

r > Rsrc ⇒ Φ(r) = −GM/r ⇒ F = −GMm/r2 r̂

IV. RECURSIVE FIXED POINT

ψn+1 = ½(ψn + φ−5/ψn), φ = 1 + √52

fixed point: ψ⋆ = φ−5/2

V. EXACT DISCRETE-SCALE POTENTIAL

V(χ) = λχ4 sin2(π ln(χ/χ0)/ln φ)

V(φχ) = φ4V(χ)

χn = χ0φn, V(χn) = V′(χn) = 0

V″(χn) = 2λπ2χn2(ln φ)2

mn2 := V″(χn) ⇒ mn = m0φn

VI. DECAY KERNEL

Pn→n−k = e−λdkΣj=1ne−λdj, ΣkP = 1

admissibility: 1 ≥ φ−a + φ−b

CHANNEL (a,b)ADMISSIBLE(1, 1)✗(1, 2)✓(2, 2)✓(2, 3)✓(3, 3)✓(2, 4)✓

P(n→m) ∝ e−λd(n−m)

VII. NUMERIC NOTATION

φ = 1.618033988749895ln φ = 0.481211825059603ψ⋆ = 0.300283106000778m₀ = 0.511 MeVλ = 3.063265509773806×10⁻³

nmₙ (MeV)00.51100000011101.691567774171824.781143097194777.339054658207729.8969662192112507.2360208782220237.1329870972332744.3690079752452981.5019950722585725.87100304626138707.372998118

max|V(χₙ)| = 1.582×10⁻²⁶max|mₙ₊₁/mₙ − φ| = 2.220×10⁻¹⁶

VIII. QUANTUM SMALL-OSCILLATION SIMULATION

V(χn + q) = V(χn) + ½V″(χn)q2 + O(q3)

ωn := √V″(χn) = mn

Ĥn = ½p̂n2 + ½ωn2q̂n2

En,k = (k + ½)ωn

ΔEn+1/ΔEn = ωn+1/ωn = φ

dim ℋsim = 48

max|Eₖ^num − (k+½)ωₙ| = 4.263×10⁻¹⁴max‖Uₙ†Uₙ − I‖₂ = 3.754×10⁻¹⁵max|ΔEₙ₊₁/ΔEₙ − φ| = 1.554×10⁻¹⁵

SYMBOLIC + NUMERIC + QUANTUM: 19/19 PASS

GitHub proof ↗GitHub validator ↗GitHub certificate ↗

Values are the source record's computed quantities.

Boundary restated: spectral-to-geometry and spectral-to-thrust maps are defined simulation assumptions — no empirical claim of matter deformation or propulsion.

Record E47-MANTA-SPECTRAL-MORPH-20260927-001