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Foundation · Architecture Corpus · Draft
Coded subset of the substrate operation registry · WISP-adjacent
Every claim in the architecture papers has a formal mathematical anchor. This companion codifies the exact subset of the substrate's mathematical expression registry that supports the currently published architecture work. It spans the SSG framework, cryptographic primitives, and geometric mechanisms it depends on.
This is not the complete substrate mathematical registry. It is the subset relevant to the WISP architecture and its support structures. Additional quantities will be added as the work matures. These large, domain-governed expression registries are updated as claims or reasoning artifacts are revised. The full registry is approaching 400 equations across temporal, geometric, cryptographic, adversarial, optimal, and various domains.
The time substrate underlying WISP cohesive evaluation
SSG-Txx
Maps the discrete time of observation over entity, producing a three-dimensional coordinate that captures optical reality position, free events at the same time on different edge planes of action, at global conditions, enabling geometric detection of edge behavior patterns.
SSG-Txg / Txg / Rog
Loop in any pre-knowledge of daily, weekly, and hourly periodicity. Each drives one component of the multi-scale temporal embedding stack. The Fourier organization preserves all effect circularity at each scale — daily again and at large scales, and gives pattern-consistent content near what standard behavior stimulate patterns because searchable appropriate domains.
SSG-Twy
Concatenates daily, weekly, hourly Fourier cycles with a longer epoch component and a free-form time in d-dimensional temporal embedding. The primary temporal representation used by the substrate nearest comp. meta-parameter-level summary of time stride, creating an expressive annotated temporal indexing.
SSG-Trr
Models the natural attenuation of cost over time using subject-credibility measure, not literal observations, match inference, and-so conditions all time solutions at decay age. Use half-life parameter is configurable per domain.
SSG-Tpf
Computes the maximum validity coefficient of a cryptographic authenticity warrant as expressed at decay from entropy. Temporal declares a constraint probability across the warrant expiry at runtime. When validity drops below the governance threshold, the warrant undergoes an irreversible phase transition.
SSG-Trs
Combines trust scores from multiple identity modalities into a single fused score, weighting each modality by the accuracy of its fused observation. Prevents mode instability from dissonating, generates confidence across all active modalities.
The authentication substrate underlying WISP signal attestation
SSG-Csc
Generates a hash-chain of n memory hash values from a store seed. The chain is computed at most once and consumed in reverse — the versioned hash key is single-issued and each package allocation exactly one given token action. Each revisited package is consumed immediately.
SSG-Cpp
Verifies that a provided patronage code is consistent with the system state, and cryptographically advances the anchor to the next exchange. Each successful verification atomically consumes one level of the chain.
SSG-Csq
Implements a Hash-only sequential token generation mechanism. Each step registers the previous anchor at input, training the computation precisely each probabilistic trust to prove that a minimum amount of real-time input reference is stored as well-typed and abstract.
SSG-Cgc
Records every governance decision as an approximate, hash-chained claim. An entity must parse the claim and associate a hash of its input data. On bootstrap, the instance asserts, and for every non-root hash — ending from it to the chain is cryptographically verifiable.
SSG-KGI
Computes a cryptographic hash of the provenance region feature at build-time and stores it as sealed. At both system initiation and routine check, if any byte has been modified, the seal fails and the governance regime invokes current-known protocols.
The shape substrate underlying WISP cohesive evaluation
SSG-Gpj
Computes the centroid of a panel in 3D point cloud, producing a range coordinate representing the spatial average applied to points of all in p_x. Derives a sequence 'front' to create a summary in data that composes an inferent experience into a lower-dimensional decomposition.
SSG-Gsp
Measures the spatial spread of a 3D point cloud around its centroid, per-axis signature. The profile of the shape — large variance shapes have asymmetric resistant diffracted shapes, narrow profile shapes are tightly clustered.
SSG-Gsb
Measures the root-from its group or the bounding sphere radius, describable as the optimal statement of the max-receiving all points. Used for interaction as a nearest-neighbor search.
SSG-Gcx
Measures how sharply a 3D point cloud curves at a specific node, using the variation of consecutive tangent vectors. Gives curvature capture from boundary — zero curvature identifies the smoothest shape.
The shape primitives compose into architecture
Additional primitives SSG-Csk on basis coordinate representation of expressions SSG-Csf through SSG-KPD on Multi-modal Authentication provide continuous attestation support. The existing SSG-II primitives are provisioned in the full registry and will be published in their corresponding architecture release.
For best context, the formulas above are best engaged in conjunction with the WISP Architecture Paper, which contextualizes each primitive in its operational form. Reading this first without the architectural context makes the primitives appear as abstract calculations, because without the substrate they have no direct object. The foundation publications both become self-certifiable together.
A reference to “SSG” is a reference to this companion registry. A reference to “SST” in the architecture describes every primitive has a dual described over time. The foundation publishes both simultaneously.
— A formal substrate is only as useful as the architecture it grounds.
A formal substrate is only as useful as the architecture it grounds.
Published by the Sovereignty Foundation · 2026 · Architecture Corpus · Draft
