Resonant Field Storage (RFS) Mathematics

RFS is built on rigorous mathematical foundations that establish correctness, stability, and capacity guarantees. As a substrate, RFS provides the mathematical framework and APIs that enable other systems to build on top of it.

Ψ(x,y,z,t) = Σ aₖ · e^(iφₖ) · Φₖ(x,y,z,t)

RFS: A Mathematical Substrate for Memory Systems

Our Approach: We follow academic research practice—foundational mathematics is public and verifiable, while implementation optimizations remain proprietary. This dual model enables collaboration on the substrate while protecting competitive advantages in implementation.

Mathematical Foundations

RFS is built on rigorous mathematical foundations. The lemmas establish foundational guarantees, while invariants govern system behavior in production. Together, they form the mathematical contract that any RFS implementation must satisfy.

Complete Proof Suite: RFS maintains 40+ lemmas and 43+ invariants covering core mathematics, implementation optimizations, hardware-specific proofs, and operational procedures.

Core Mathematical Lemmas

Lemma L1 — Projector Stability

Invariant: INV-0001

‖δΨ‖₂ ≤ k·ε_trunc + (1 − ρ(A)^Δt_proj)‖Ψ‖₂

Encoding is energy preserving. Unitary FFT/IFFT and unit-modulus phase masks prevent energy blow-up regardless of shard composition. The bound links projector cadence, drift, and available energy slack.

Practical Impact:

Ensures numerical stability during field evolution. In production, this prevents field divergence that could corrupt stored data. The bound enables safe projector cadence tuning—faster projection increases throughput but must respect the stability bound.

Lemma L2 — Controlled Interference

Invariant: INV-0002

η ≤ η_exact + δ_F / E_total

Constructive resonance is bounded by the correlation between existing content and new inserts. Guardrails watch destructive energy to keep overlap meaningful. The bound quantifies how much destructive interference might be hidden in the sketch residual.

Practical Impact:

Guarantees that resonance peaks remain meaningful for retrieval. Without this bound, destructive interference could mask relevant results. In practice, this enables reliable top-k retrieval even as the corpus grows, with guardrails automatically triggering when interference exceeds safe thresholds.

Lemma L3 — Collision Probability via Spherical LSH

Invariant: INV-0003

P_collision(θ) = 1 - (1 - (1 - θ/π)^m)^b

Collision probability model for SimHash with correlated bands. Bounds candidate set sizes and bucket occupancy tails, ensuring logarithmic collision checks per ingest even as corpus grows.

Practical Impact:

Enables sub-linear search complexity as the corpus scales. This bound ensures that candidate set sizes remain bounded, making resonance search practical for large-scale deployments. Without this, collision checks would grow linearly with corpus size, making the system impractical.

Lemma L4 — Recall Error Decomposition

Invariant: INV-0004

ε_recall ≤ ε_basis + ε_projector

Approximation error decomposes cleanly between basis quality and projector sparsity. Archival recall error ≤ 1e-8, associative recall error ≤ 1e-6. Calibration runs tune each term before production rollout.

Practical Impact:

Enables independent tuning of basis truncation and projector cadence. This decomposition allows engineers to optimize each component separately—tighten basis truncation for archival precision, adjust projector cadence for throughput. Production systems use calibration runs to set optimal parameters before deployment.

Lemma L5 — Learned Kernel Impact Bounds

Invariant: INV-0005

μ^Q_t ≤ 1.05, ΔL_t ≤ 1.2×10⁻², Δη_t ≤ 4.5×10⁻²

Learned kernels must satisfy calibrated bounds on resonance quality, loss, and destructive energy. Semantic planner only promotes kernels after manifests prove compliance with these tolerances.

Practical Impact:

Prevents learned kernels from degrading system guarantees. The semantic planner automatically validates new kernels against these bounds before promotion. This ensures that machine learning improvements don't compromise mathematical guarantees, maintaining system reliability while enabling continuous improvement.

Lemma L10 — Byte Channel Capacity Bound

Invariant: INV-0018

κ = C_p99 / R ≥ 1.3

Byte channel capacity margin must be ≥ 1.3× to ensure reliable exact recall with FEC overhead (RS(255,223) ≈ 14.3%) and drift tolerance. Effective capacity accounts for headroom policy.

Practical Impact:

Ensures exact recall reliability even under capacity pressure. The 1.3× margin accounts for FEC overhead, field drift, and capacity headroom policy. In production, this bound prevents exact recall failures that could corrupt data. Systems monitor capacity margin continuously and reject stores that would violate this bound.

Lemma L11 — Byte Channel Integrity (AEAD Verification)

Invariant: INV-0019

success_rate = successes / attempts == 1.0

Exact recall success rate must be 100% (all AEAD verifications succeed) to ensure data integrity. AES-256-GCM provides authenticated encryption with integrity; tag verification fails if data is corrupted, tampered, or key mismatched.

Practical Impact:

Provides cryptographic guarantees for exact recall. Every exact recall operation verifies the AEAD tag—any corruption, tampering, or key mismatch causes immediate failure. This ensures that exact recall returns data exactly as stored, with mathematical proof of integrity. Production systems treat any AEAD failure as a critical error.

Lemma L12 — Byte Channel Spill Bound

Invariant: INV-0020

spill_db = 10×log₁₀(P_byte_in_semantic / P_semantic) ≤ -60 dB

Byte channel energy spill into semantic band must be ≤ -60 dB to prevent interference with semantic retrieval. Guard band thickness ensures isolation between semantic and byte channels.

Practical Impact:

Maintains channel isolation for reliable dual-path retrieval. The -60 dB bound ensures that byte channel data doesn't interfere with semantic resonance search. Guard band design enforces this isolation physically. Violations would cause semantic retrieval to return byte channel artifacts, corrupting search results.

Lemma L19 — Field Superposition Equivalence

Invariant: INV-0030

‖Δ‖₂ ≤ ‖(I - Π_assoc)(Ψ_reconstructed)‖₂ + ρ(L)^N_evolve·‖Ψ‖₂

Bound on difference between live field superposition and reconstructed superposition from artifacts. Establishes equivalence conditions and divergence bounds, ensuring reconstruction from artifacts is mathematically sound.

Practical Impact:

Enables reliable reconstruction from persisted artifacts. This bound ensures that reconstructing the field from artifacts (for backup, migration, or analysis) produces a mathematically equivalent state. Without this, reconstruction could diverge, making artifacts unreliable for recovery or analysis.

Lemma L23 — Field-Native Encoder Mathematical Guarantees

Invariant: INV-0034

‖E_field(t_i) - E_field(t_j)‖₂ ≤ L* · d_sem(t_i, t_j)

Field-native encoder satisfies Lipschitz continuity, energy bounds, superposition compatibility, retrieval quality preservation, and external embedding independence. Composed operator E_field = Π_assoc ∘ E ∘ Θ_text.

Practical Impact:

Ensures field-native embeddings maintain semantic relationships while being compatible with field superposition. This enables direct text-to-field encoding without external embedding dependencies. The Lipschitz bound guarantees that similar texts produce similar field embeddings, preserving retrieval quality while enabling field-native operations.

Mathematical Invariants

INV-0001

Projector Stability

Bound on projector drift between passes; links projector cadence, truncation residual, and evolution operator spectral radius.

Enforcement:

Verified via telemetry on projector drift (‖δΨ‖₂). CI gates block deployments if drift exceeds bound. Calibration runs validate spectral radius estimates before production.

INV-0002

Controlled Interference

Bound on destructive energy ratio when only top-k overlaps are tracked exactly; residual bounded by sketch rank.

Enforcement:

Monitored via destructive energy ratio (η) telemetry. Guardrails auto-trigger when η approaches bound. Sketch rank validation ensures residual remains bounded.

INV-0003

Collision Probability via Spherical LSH

Collision probability model for SimHash with correlated bands; bounds candidate set sizes and bucket occupancy tails.

Enforcement:

Validated through empirical collision rate measurements. Candidate set size monitoring ensures logarithmic growth. Bucket occupancy tracked via telemetry.

INV-0004

Recall Error Decomposition

Recall error decomposes into basis truncation and projector drift; archival ≤ 1e-8, associate ≤ 1e-6.

Enforcement:

Calibration runs measure ε_basis and ε_projector independently. Production deployments blocked if either term exceeds threshold. Continuous telemetry tracks recall error components.

INV-0018

Byte Channel Capacity Margin

Byte channel capacity margin (P99 capacity / required bits) must be ≥ 1.3× to ensure reliable exact recall with FEC overhead and drift tolerance.

Enforcement:

Capacity margin (κ) computed on every store operation. Stores rejected if κ < 1.3. P99 capacity measured via empirical distribution. Headroom policy enforced at API layer.

INV-0019

Byte Channel Integrity (AEAD Verification)

Exact recall success rate must be 100% (all AEAD verifications succeed) to ensure data integrity and prevent corruption.

Enforcement:

Every exact recall operation performs AES-256-GCM tag verification. Any verification failure triggers critical alert and data corruption investigation. Success rate monitored via telemetry (must be 1.0).

INV-0020

Byte Channel Spill Bound

Byte channel energy spill into semantic band must be ≤ -60 dB to prevent interference with semantic retrieval.

Enforcement:

Measured via spectral analysis of field energy distribution. Guard band design physically enforces isolation. CI gates validate spill measurements. Violations block deployments.

INV-0030

Field Superposition Equivalence

Bound on difference between live field superposition and reconstructed superposition from artifacts. Establishes equivalence conditions and divergence bounds.

Enforcement:

Validated through reconstruction tests comparing live vs. reconstructed fields. Artifact generation includes equivalence checks. Reconstruction divergence monitored via telemetry.

INV-0034

Field-Native Encoder Mathematical Guarantees

Field-native encoder satisfies Lipschitz continuity, energy bounds, superposition compatibility, retrieval quality preservation, and external embedding independence.

Enforcement:

Lipschitz constant (L*) validated via gradient analysis. Energy bounds verified through embedding energy telemetry. Retrieval quality measured via benchmark suite. CI gates validate all properties.

Verification Strategy

RFS follows proof-driven development with executable verification. All public proofs are verifiable and reproducible.

Verification Methods

Symbolic: Coq scripts certify unitary operations and AEAD inversion. Property tests ensure ε bounds stay within contracts.
Empirical: FFT throughput and resonance latency measured across GPU/CPU fleets. Recall experiments validate accuracy.
CI Enforcement: Automated gates validate mathematical contracts and block deploys when guardrails are threatened.

Proof Artifacts

Business WhitepaperMarket opportunity, competitive advantages, ROI, and investment thesis for unified AI memory infrastructure.
Academic Research PaperField-theoretic framework with formal mathematical foundations, lemmas, and empirical validation.
Technical WhitepaperArchitecture, API specifications, implementation guidance, and operational runbooks for engineers.
Proof AppendixFormal lemmas, invariants, and AEAD verification steps maintained as living mathematics.
BenchmarksEmpirical validation of resonance speed, headroom, and recall accuracy under load.

Open for Collaboration

RFS is built as a substrate for the broader research and engineering community. We're open to:

Research Collaboration: Academic partnerships, joint publications, mathematical extensions
Integration Partnerships: Building on RFS substrate, API integrations, protocol extensions
Open Source Components: Core mathematical libraries, verification tools, API clients

Contact us to discuss collaboration opportunities, extended documentation access, or integration support.