arborist/docs/v7w-frontier-catalog.md

v7-W ε-frontier catalog

Companion to: docs/_source/merkle-agi-v7w-spatial-temporal.rst (the v7-W substrate paper, ticket #000013). Status: draft v0, 2026-05-09.

This catalog enumerates the four canonical ε-frontiers in v7-W — the kernel-boundaries where ε proofs are admissible. Each is affine after appropriate canonical projection (T4-W) and each has a published quantization-derived ε bound.

A frontier is the boundary between "model forward pass" and "committed integer kernel." Below the frontier, proofs are at the float-tensor level (v7's domain). Above the frontier, canonical bytes admit replay-audit.

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Reference table

Frontier	Domain	Canonical kernel	ε bound	Reuses
pose_integration	object pose / agent ego-motion	SE(3) integer composition	Δ_rot + Δ_trans ·	t_max
observation_update	Kalman / Bayesian state update	affine in innovation	Δ_innovation + Δ_covariance	v7 § 5; arithmetic@v1 for residual fold
object_logits	per-object classification logits	linear projection	manifest-declared per-class	v7 § 11 multimodal composition
relation_logits	pairwise scene-graph edge prediction	bilinear scoring	predicate-whitelist + feature granularity	v7 § 11; relation_graph carrier

Each frontier's full design lives in the substrate paper Part 4; this catalog is the operator-facing quick reference.

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§1 pose_integration

Inputs (canonical bytes):

{
  "prev_pose_quantized": {
    "axis_angle_int32": [<int32 x 3>],
    "translation_int32": [<int32 x 3>]
  },
  "ego_motion_quantized": {
    "axis_angle_int32": [<int32 x 3>],
    "translation_int32": [<int32 x 3>]
  },
  "Δ_t_ticks": <int32>
}

Output:

{
  "next_pose_quantized": {
    "axis_angle_int32": [<int32 x 3>],
    "translation_int32": [<int32 x 3>]
  }
}

Operator. SE(3) composition. The axis-angle representation is integer-encoded with manifest-declared Δ_rot_milli_radians; translation in Δ_trans_micrometers. Composition uses bigint arithmetic to avoid float drift, then re-quantizes to the manifest grid.

ε bound. Per-step: Δ_rot + Δ_trans · |translation_max|.

Affine after canonical projection. The axis-angle representation linearizes rotation under small-time-step assumption (the common SLAM regime); the linearization error is bounded by the small-angle approximation residual, captured in the manifest's published Δ_rot_residual_bound field.

When to use. Every agent-trace tick. Every object whose pose changes between observations.

---

§2 observation_update

Inputs (canonical bytes):

{
  "prior_state_quantized": [<int32 x state_dim>],
  "prior_covariance_quantized": [[<int32>], ...],
  "observation_quantized": [<int32 x obs_dim>],
  "innovation_covariance_quantized": [[<int32>], ...],
  "observation_matrix_quantized": [[<int32>], ...]
}

Output:

{
  "posterior_state_quantized": [<int32 x state_dim>],
  "posterior_covariance_quantized": [[<int32>], ...]
}

Operator. Standard Kalman update:

innovation = observation - H · prior_state
K_gain = prior_cov · H^T · (H · prior_cov · H^T + innov_cov)^-1
posterior_state = prior_state + K_gain · innovation
posterior_cov = (I - K_gain · H) · prior_cov

All arithmetic uses bigint accumulators (v7 § 5) to avoid analog leakage in the matrix inversion. Final values re-quantized to the manifest grid.

ε bound. Per-step: Δ_innovation + Δ_covariance.

Affine after canonical projection. Innovation is affine in observation; gain application is affine in innovation; posterior update is affine in gain. The matrix inverse is the only non-affine step — handled by bigint arithmetic + re-quantization so the canonical bytes remain deterministic.

When to use. Every observation that updates a stateful estimate (object pose, agent location, place geometry). Stateless classifications (single-frame object detection without temporal smoothing) skip this frontier.

---

§3 object_logits

Inputs (canonical bytes):

{
  "object_features_quantized": [<int32 x feature_dim>],
  "classifier_weights_quantized": [[<int32>], ...],
  "classifier_bias_quantized": [<int32 x num_classes>]
}

Output:

{
  "logits_quantized": [<int32 x num_classes>]
}

Operator. Linear projection: logits = W · features + b. All on quantized integers. Pre-softmax — softmax itself is NOT in the kernel (softmax is monotone and order-preserving on the logits, so commitments to logits implicitly commit to the softmax-class predictions).

ε bound. Manifest-declared per-class quantization threshold. Typical values: 1/256 of the full logit range.

Affine after canonical projection. Linear-in-features by construction.

When to use. Every object-detection commitment. Multimodal pipelines that route through v7's vision encoder use this frontier as the v7→v7-W handoff.

---

§4 relation_logits

Inputs (canonical bytes):

{
  "subject_features_quantized": [<int32 x feature_dim>],
  "object_features_quantized": [<int32 x feature_dim>],
  "relation_classifier_weights_quantized": [[<int32>], ...]
}

Output:

{
  "relation_logits_quantized": [<int32 x predicate_count>]
}

Operator. Bilinear scoring on the (subject, object) feature pair. The bilinear tensor B is canonicalized to its tensor- decomposed form (Tucker decomposition with manifest-declared ranks; or CP decomposition for low-rank cases). Scoring then factorizes:

score_per_predicate = subject_features · core · object_features^T

…with core being the canonicalized decomposition.

ε bound. predicate-whitelist size factor + feature quantization. The whitelist size factor accounts for the finite alphabet of declared predicates per substrate.

Affine after canonical projection. Bilinear becomes affine once the subject (or object) features are fixed; the factorization makes this explicit. Substrate manifest pins the factorization rank to keep the canonical bytes deterministic.

When to use. Every scene-graph edge commitment. Relation extraction in multimodal pipelines.

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Cumulative ε across frontiers

Per the data-processing inequality, ε accumulates additively across composed kernels:

ε_total = Σ_frontier (ε_frontier × invocation_count)

A typical 10-tick agent trace with 3 detected objects + 6 relations:

10 ticks × (pose_integration + observation_update)
       + 30 object_logits invocations (3 obj × 10 ticks)
       + 60 relation_logits invocations (6 rel × 10 ticks)

Each invocation contributes its frontier's per-step ε. Operators size their grid choices (manifest's Δ values) to keep ε_total under the downstream consumer's tolerance.

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Out of scope (per #000013 §5)

• Continuous-state kernels that haven't been re-canonicalized to integer state. v7-W requires explicit integer-encoding; proofs at the float layer live in v7 + arborist's existing integer-kernel discipline (SQD § 5).
• Sensor-specific kernels (LIDAR ICP, RGB-D fusion). Each is a separate adapter ticket; the four frontiers above are the abstract kernel set, not deployment-specific implementations.
• Cross-substrate frontiers (v7-W ↔ arborist-text). Captured by the cross-domain π* composition theorem (#000015) and the multimodal composition section in the v7-W substrate paper (Part 5).

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Status

Catalog draft v0. Each frontier's full proof-of-affine-property and ε derivation will be folded back into the substrate paper (Part 3 theorems) once the catalog stabilizes. This document is the operator-facing quick reference; the substrate paper is the formal source.