Mathematical Primitives Companion

The time substrate underlying WISP coherence evaluation

SIG-T01 — Diurnal Spiral Encoding

Maps the current time-of-day onto a unit circle, producing a two-dimensional coordinate that captures cyclical daily position. Two events at the same time on different days produce identical spiral coordinates, enabling geometric detection of daily behavioral patterns.

SIG-T01θ = (s / 86400) · 2π, v = [sin(θ), cos(θ)], ‖v‖ = 1

where s is seconds into the current day.

SIG-T02 / T03 / T04 — Daily, Weekly, Yearly Fourier Cycles

Sine-cosine pair encodings of daily, weekly, and yearly periodicity. Each forms one component of the multi-scale temporal embedding stack. The Fourier representation preserves cyclical proximity at each scale — midnight and 23:59 are geometrically adjacent, weekend behavior clusters near other weekend behavior, seasonal patterns become measurable as geometric distance.

Daily · weekly · yearly Fourierf_d = [sin(2π·s/86400), cos(2π·s/86400)]
f_w = [sin(2π·d/7), cos(2π·d/7)]
f_y = [sin(2π·d/365.24), cos(2π·d/365.24)]

SIG-T05 — Multi-Scale Temporal Embedding (8D)

Concatenates daily, weekly, yearly Fourier cycles with a linear epoch component and a bias term into an 8-dimensional temporal embedding. The primary temporal representation used by the substrate. Fourier components preserve cyclical proximity at three scales; the linear term enforces monotonic temporal ordering.

SIG-T05 · 8D temporal embeddingT_8 = [sin(θ_d), cos(θ_d), sin(θ_w), cos(θ_w), sin(θ_y), cos(θ_y), t_epoch/3.1536e8, 1.0]

SIG-T06 — Half-Life Trust Decay

Models the natural attenuation of trust over time using radioactive-decay semantics. Identity observations, search relevance, and role confidence all lose authority as they age. The half-life parameter is configurable per domain.

SIG-T06N(t) = N₀ · (1/2)^(t / t₁ᐟ₂)

SIG-T07 — Thermodynamic Warrant Validity

Computes the continuous validity coefficient of a cryptographic authorization warrant as exponential decay since minting. The warrant loses authorization strength continuously, independent of exercise. When validity drops below the governance threshold, the warrant undergoes irreversible phase transition.

SIG-T07V(t) = m · e^(−λ·Δt), where Δt = t_now − t_mint

Warrant undergoes forced phase transition when V(t) < θ_sage.

SIG-T09 — Multi-Modal Temporal Trust Fusion

Combines trust scores from multiple identity modalities into a single fused score, weighting each modality by the recency of its last observation. Prevents stale modality data from dominating; preserves contribution of all active modalities.

SIG-T09T̂ = (w_a · S_a + w_k · S_k) / (w_a + w_k), w_x = e^(−λ·Δt_x)

The authorization substrate underlying WISP-gated execution

SIG-C01 — Sequential Hash Chain Generation

Generates a finite chain of N one-way hash values from a secret seed. The chain is computed at mint time and consumed in reverse — the terminal hash (anchor) is published; each preimage authorizes exactly one governance action. Each revealed preimage is consumed permanently.

SIG-C01c₀ = H(s), c_i = H(c_{i−1}) for i ∈ {1,...,N−1}

SIG-C02 — Hash Chain Preimage Verification

Verifies that a revealed preimage hashes to the current chain anchor, then irreversibly advances the anchor to the revealed preimage. Each successful verification atomically consumes one link of the chain.

SIG-C02verify: H(p) ?= a_current ⟹ a_new := p

SIG-C03 — Sequential Hash Time-Lock (VDF Core)

Implements a Verifiable Delay Function via iterated sequential hashing. Each step requires the previous output as input, making the computation provably non-parallelizable. Used to prove that a minimum amount of real time elapsed between warrant mint request and issuance.

SIG-C03x₀ = seed, x_i = H(x_{i−1}) for i ∈ {1,...,T}

SIG-C10 — Governance Decision Chain

Records every governance decision as an append-only, hash-linked chain. Each entry incorporates the decision outcome, a hash of the input data, the timestamp, the sequence number, and the previous entry hash — making historical decisions cryptographically immutable.

SIG-C10g_i = H(decision_i ‖ h(input_i) ‖ t_i ‖ i ‖ g_{i−1})

SIG-C11 — Binary Integrity Seal

Computes a cryptographic hash of the governance engine binary at build time and stores it as a seal. At boot, the runtime re-hashes and compares — if any byte has been modified, the seal fails and the governance engine refuses to enter locked precision mode.

SIG-C11seal = H(binary_bytes)

The shape substrate underlying WISP coherence evaluation

SIG-G03 — 3D Centroid

Computes the center of mass of a 3D point cloud, producing a single coordinate representing the spatial average. Applied to point clouds (e.g., helix representations of text sequences) to determine where a sequence "lives" in geometric space. First of four shape summary statistics that compress an arbitrarily long sequence into a fixed-dimensional descriptor.

SIG-G03μ = (1/N) · Σᵢ pᵢ

SIG-G04 — 3D Variance

Measures the spatial spread of a 3D point cloud around its centroid, per axis. Captures the geometric "footprint" of the shape — long narrow shapes have asymmetric variance; tightly coiled shapes have the inverse profile.

SIG-G04σ² = (1/N) · Σᵢ (pᵢ − μ)²

SIG-G05 — Bounding Radius

Maximum distance from any point in the cloud to the centroid. Defines the smallest sphere centered at the centroid enclosing all points. Used for fast rejection in nearest-neighbor search.

SIG-G05R = max_i ‖pᵢ − μ‖₂

SIG-G06 — Discrete 3D Curvature

Measures how sharply a 3D sequence bends at each point using the cross-product of consecutive tangent vectors. Mean curvature captures overall "twistiness"; max curvature identifies the most abrupt change.

SIG-G06κ_i = ‖T_i × T_{i+1}‖ / ‖T_i‖², κ̄ = (1/n) Σ κ_i, κ̂ = max_i κ_i

How these primitives compose into the architecture

The primitives above compose into the substrate behaviors described in the architecture papers:

• WISP authentication — composes SIG-T05 (8D temporal embedding for the temporal modality), SIG-T09 (multi-modal trust fusion across modalities), and SIG-G03 / G04 / G05 / G06 (geometric shape descriptors applied to the modality manifolds).
• Continuous coherence evaluation — composes SIG-T06 (half-life decay applied to historical modality weights) and SIG-T09 (fusion weighted by recency).
• Cryptographic action authorization — composes SIG-C01 (hash chain mint), SIG-C02 (preimage verification), SIG-C03 (VDF time-lock to prevent mass minting), and SIG-T07 (continuous warrant validity decay).
• Governance audit trail — composes SIG-C10 (decision chain) and SIG-C11 (binary integrity seal).
• Drift detection — composes per-modality cosine similarity with SIG-T06 weighted historical comparison.

Additional primitives (SIG-G01, G02 for helix coordinate representations of sequences; SIG-C05 through C09 for RSW time-locks and memory-hard PoW; SIG-C12 through C17 for Merkle ledgers and content-addressed identity; the full SIG-T spectrum) are documented in the full registry and will be published as their corresponding architecture papers are released.

For most readers, the formulas above are best engaged in conjunction with the WISP Architecture Paper, which contextualizes each primitive in its operational role. Reading the math without the architectural context yields formulas without purpose; reading the architecture without the math yields claims without rigor. The Foundation publishes both because both are needed.

For researchers who want to engage with the math directly, the SIG-ID notation is the bridge identifier: every primitive has a stable identifier that the code, the patent, the architecture paper, and the math companion all reference. The full SIG-ID registry is the canonical source.