Snapkitty/snapkitty-resonance-bridge

SnapKitty Resonance Bridge

"The quartic invariant is no longer abstract. It is executable."

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What This Is

This repository is the executable bridge between a formally verified mathematical proof and a running computer program.

The GKN quartic invariant I₄ — the central object of E₇ exceptional supergravity and M-theory compactifications — was formally verified in Lean 4 with zero sorry by Ahmad Ali Parr (SnapKitty Collective, 2026). The proof is more general than the original Günaydin-Koepsell-Nicolai 2001 paper: it works over any abstract CommRing with OctonionAlgebra typeclass, not just the reals.

This bridge compiles that proof into running code using SUBLEQ (Subtract and Branch if Less than or Equal to Zero) — a one-instruction computer — via ResonanceWord encoding. The result is a deterministic, non-probabilistic execution model where the mathematical invariant is a runtime-verified constant.

TRS = 388.985128

When the system halts with this value, the cage holds.

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For Researchers

git clone https://github.com/SNAPKITTYWEST/snapkitty-resonance-bridge.git
cd snapkitty-resonance-bridge
npm test
npm run verify

Expected output:

✓ TRS VERIFIED: 388.985128
  The cage holds.
=== BRIDGE OPERATIONAL ===

File	Description	Source
src/resonance.mjs	ResonanceWord encoding	sovereign-goldilocks/src/resonance.rs
src/i4.mjs	I₄ quartic invariant	gkn-i4-e7-lean/GKN/I4_CommRing.lean
src/subleq.mjs	SUBLEQ execution engine	—
src/verify.mjs	TRS verification	—

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Architecture

┌─────────────────────────────────────────────────────────────────┐
│                    FORMAL PROOF LAYER                           │
│   Lean 4  ·  GKN/Homogeneity.lean  ·  zero sorry               │
│   I4_56_homogeneous: I4(c·s) = c^4 · I4(s)                    │
│   Over abstract CommRing + OctonionAlgebra typeclass            │
│   DOI: 10.5281/zenodo.21349216                                 │
└───────────────────────┬─────────────────────────────────────────┘
                        │ compile
                        ▼
┌─────────────────────────────────────────────────────────────────┐
│                  RESONANCEWORD ENCODING                         │
│   ┌──────────┬──────────────────────────────────────────────┐  │
│   │  8 bits  │              56 bits                         │  │
│   │  class   │              payload                         │  │
│   └──────────┴──────────────────────────────────────────────┘  │
│   0x0A  SOVEREIGN  → SUBLEQ A  (subtract from)                 │
│   0x05  TRANSITION → SUBLEQ B  (subtract into + test)          │
│   0x09  WORM       → SUBLEQ C  (jump target if B <= 0)         │
│   Three ResonanceWords = one SUBLEQ instruction [A, B, C]      │
└───────────────────────┬─────────────────────────────────────────┘
                        │ execute
                        ▼
┌─────────────────────────────────────────────────────────────────┐
│                    SUBLEQ ENGINE                                │
│   M[B] = M[B] - M[A]                                           │
│   if M[B] <= 0: PC → C    else: PC → PC+3                     │
│                                                                 │
│   I₄ four terms → four instruction groups:                     │
│   t1  (alpha*beta - Tr(P∘Q))^2  · mem[300..324]               │
│   t2  alpha * N(Q)               · mem[325..349]               │
│   t3  beta  * N(P)               · mem[350..374]               │
│   t4  Tr(P# ∘ Q#)                · mem[375..399]               │
└───────────────────────┬─────────────────────────────────────────┘
                        │ verify
                        ▼
┌─────────────────────────────────────────────────────────────────┐
│               CONVERGENCE VERIFICATION                          │
│   TRS = 388.985128                                             │
│   = ME + AN + KI + DI  (four agent channels over Q(sqrt(5)))   │
│   = sum_s sum_n bias_s(n) * phi^(depth_n+1)                   │
│   SHA-256 WORM sealed · cryptographic provenance chain         │
└─────────────────────────────────────────────────────────────────┘

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The Lean 4 Proof

-- GKN/Homogeneity.lean — zero sorry
theorem I4_56_homogeneous (s : State56 R O) (c : R) :
    I4_56 (s56Scale c s) = c ^ 4 * I4_56 s := by
  simp only [I4_56]
  rw [t1_homogeneous_sq s c, t2_homogeneous s c,
      t3_homogeneous s c,   t4_homogeneous s c]
  ring
-- Proved over abstract CommRing + OctonionAlgebra typeclass.
-- More general than GKN 2001 (reals only).
-- Numeric witnesses: ratio I4(2s)/I4(s) = 16 = 2^4 on three
-- independent test states (sparse, diagonal, off-diagonal).

Degree analysis — each term is exactly r⁴:

Term	Expression	Degree
t1	(alpha*beta - Tr(P∘Q))²	(r²)² = r⁴ ✓
t2	alpha·N(Q)	r·r³ = r⁴ ✓
t3	beta·N(P)	r·r³ = r⁴ ✓
t4	Tr(P#∘Q#)	r²·r² = r⁴ ✓

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Memory Layout

Address    Purpose
──────────────────────────────────────
0-99       Reserved
100        Input alpha
101        Input beta
102-155    Input P and Q (J3O elements)
200-299    Intermediate computation
300-399    I4 term groups (t1-t4)
400        I4 total
500        TRS target (388985128)
501        TRS result (0 when verified)
65535      HALT

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Published Work

DOI	Title
10.5281/zenodo.21349216	Closing Boole + 3 E₇ Symmetries — zero sorry
10.5281/zenodo.21268911	Yellow Book — 78 theorems, 63 proved
10.5281/zenodo.21349277	Gates Normalization Constraint
10.5281/zenodo.21351461	Attention Is All You Don't Need
10.5281/zenodo.21413918	J-Space: Shadow Entropy + Behavioral Jacobian

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Citation

@software{snapkitty_resonance_bridge_2026,
  title  = {SnapKitty Resonance Bridge: GKN I4 as Executable SUBLEQ Program},
  author = {Parr, Ahmad Ali and {SnapKitty Collective}},
  year   = {2026},
  doi    = {10.5281/zenodo.21349216},
  url    = {https://github.com/SNAPKITTYWEST/snapkitty-resonance-bridge},
  note   = {TRS = 388.985128. Zero sorry. Abstract CommRing.}
}

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License

Sovereign Source License v3.0 — AI training prohibited.

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Ahmad Ali Parr · SnapKitty Collective · Bel Esprit D'Accord Irrevocable Trust · 2026