# a11oy — Architecture & Formulas The payload ships two TypeScript runtime packages — `@a11oy/core` and `@a11oy/connection` — plus a per-file provenance manifest. This document states, with primary-source citations, every formula the runtime implements. All math below is implemented verbatim in the shipped JavaScript. File paths refer to locations inside the deployed `/opt/a11oy/` tree (and to the TypeScript sources under `artifacts/a11oy/packages/`). --- ## 1. Tetrad Field (`@a11oy/connection`) **Source:** Wald, *General Relativity* (1984), §3.4 (tetrad / vierbein formalism); Penrose & Rindler, *Spinors and Space-Time* vol. 1 (1984), §3.1. A tetrad is an orthonormal frame field $e_a{}^\mu$ on a manifold. For governance policy the four legs are fixed: | index | leg | unit | |------:|---------------------------|-----------------------| | 0 | `capability_tier` | tier | | 1 | `data_sensitivity` | sensitivityLevel | | 2 | `action_reversibility` | reversibilityScore | | 3 | `blast_radius` | affectedUserCount | By construction the frame is orthonormal, so the metric in tetrad indices is $\eta = \mathrm{diag}(1,1,1,1)$ and the inner product collapses to $$ \langle a, b \rangle = \sum_{i=0}^{3} a_i b_i, \qquad \lVert a \rVert = \sqrt{\langle a, a \rangle}. $$ Implementation: `tetradInner`, `tetradNorm` in `a11oy-connection/tetrad_field.js`. Every decision the policy head emits is first projected onto this frame so the downstream complementarity check has a well-defined basis. --- ## 2. Fisher Information Manifold (`@a11oy/core/geometry`) **Source:** Fisher (1925), "Theory of Statistical Estimation", *Proc. Cambridge Phil. Soc.* 22:700–725; Amari & Nagaoka (2000), *Methods of Information Geometry*, AMS Translations 191. Agent credences are treated as points on a Fisher manifold (a Riemannian manifold whose metric is the Fisher information matrix). Two distributions $p, q$ are "distinguishable enough" when their Fisher–Rao distance exceeds the admit threshold. For a categorical distribution the diagonal Fisher information matrix is $$ F_{ii}(p) = \frac{1}{p_i}. $$ The Fisher–Rao distance for discrete distributions is $$ d_{\text{FR}}(p, q) \;=\; 2 \, \arccos\!\Bigl(\textstyle\sum_i \sqrt{p_i \, q_i}\Bigr), \qquad d_{\text{FR}} \in [0, \pi]. $$ The inner term $\sum_i \sqrt{p_i q_i}$ is the Bhattacharyya coefficient, clamped to $[-1, 1]$ before $\arccos$ for numerical stability. Implementation: `fisherRaoDistance`, `fisherDiagonal`, `normalize` in `a11oy-core/geometry/fisher_manifold.js`. --- ## 3. Bohr Complementarity Engine (`@a11oy/core/quantum`) — GRAFT 1 **Source:** Bohr (1928), "The Quantum Postulate and the Recent Development of Atomic Theory", *Nature* 121:580–590; Bohr (1949), "Discussion with Einstein on Epistemological Problems in Atomic Physics", in Schilpp (ed.), *Albert Einstein: Philosopher-Scientist*, Open Court. Every governance decision is emitted as a `ComplementaryDecisionPayload` with **exactly two complementary frames**. Over the rolling sample window the empirical standard-deviation product must satisfy $$ \sigma_A \cdot \sigma_B \;\ge\; \text{COMPLEMENTARITY\_FLOOR} = 0.25, $$ the discrete-policy analogue of Heisenberg's $\sigma_x \sigma_p \ge \hbar/2$. A degenerate (deterministic) frame-pair yields $\sigma_A = 0$ or $\sigma_B = 0$ and the check fails. The 12 canonical frame-pairs are chosen so the joint observable algebra is non-commutative — measuring sharply along axis A blurs axis B: | id | axis A | axis B | |--------|---------------------------------|---------------------------------| | FP-01 | `capability_tier` | `data_sensitivity` | | FP-02 | `action_reversibility` | `blast_radius` | | FP-03 | `agent_age_days` | `capability_tier` | | FP-04 | `dual_spirit_light` | `dual_spirit_darkness` | | FP-05 | `pesher_admit` | `pesher_deny` | | FP-06 | `reviewer_quorum` | `time_to_decision` | | FP-07 | `fisher_distance` | `tetrad_norm` | | FP-08 | `metatron_capability_witness` | `watcher_dual_use_vector` | | FP-09 | `physiognomy_light_share` | `physiognomy_darkness_share` | | FP-10 | `primary_tablet_root` | `secondary_tablet_root` | | FP-11 | `povm_admit_amplitude` | `povm_deny_amplitude` | | FP-12 | `ks18_witness_value` | `daruan_rotation_angle` | Implementation: `emitDecision`, `checkComplementarity`, `FRAME_PAIRS`, `fromTetrad` in `a11oy-core/quantum/bohr_complementarity_engine.js`. Doctrine §1.1 invariant: `payload.frames.length === 2` is enforced at construction and the check throws if violated. --- ## 4. Kochen–Specker 18-Vector Contextuality Witness (`@a11oy/core/quantum`) — GRAFT 2 **Source:** Cabello, A., Estebaranz, J. M., & García-Alcaine, G. (1996), "Bell-Kochen-Specker theorem: A proof with 18 vectors", *Physics Letters A* 212(4), 183–187. arXiv:[quant-ph/9706009](https://arxiv.org/abs/quant-ph/9706009). The 18-vector / 9-context construction proves that no non-contextual hidden-variable model can reproduce all quantum predictions on a 4-dimensional Hilbert space. We use it as a **contextuality witness** for the a11oy policy head: if the rolling sequence of yes/no governance answers admits a non-contextual hidden-variable assignment, the witness flags `BOHR_ANOMALOUS` — the policy head has collapsed into a classical deterministic rule and is no longer providing dual-framed reasoning. ### Vectors The 18 vectors $v_0, \dots, v_{17} \in \mathbb{R}^4$ (Cabello et al. 1996, Table 1) are bundled verbatim in `a11oy-core/quantum/kochen_specker_18.js` as `KS18_VECTORS`. ### Contexts Nine four-element mutually-orthogonal contexts (indices into `KS18_VECTORS`): | ctx | indices | ctx | indices | |----:|----------------------|----:|-----------------------| | 0 | $\{0,1,2,3\}$ | 5 | $\{11,12,13,14\}$ | | 1 | $\{3,4,5,6\}$ | 6 | $\{13,14,15,16\}$ | | 2 | $\{5,6,7,8\}$ | 7 | $\{15,16,17,0\}$ | | 3 | $\{7,8,9,10\}$ | 8 | $\{17,0,1,2\}$ | | 4 | $\{9,10,11,12\}$ | | | Each vector appears in exactly two contexts. ### Witness rule Search for an assignment $f : \{0,\dots,17\} \to \{0,1\}$ such that $$ \forall \text{ context } C: \quad \sum_{i \in C} f(i) = 1. $$ Cabello et al. prove **no such assignment exists**. Therefore: - `evaluate` returns `{ contextual: true, reason: 'NO_NON_CONTEXTUAL_MODEL_FITS_OBSERVATIONS' }` when the search exhausts without success — the desired state. - `evaluate` returns `{ contextual: false, reason: 'BOHR_ANOMALOUS_NON_CONTEXTUAL_FIT_EXISTS', example }` when the search finds an assignment consistent with the observed answers — the policy head has collapsed and the example is returned as evidence. Implementation: `evaluate`, `KS18_VECTORS`, `KS18_CONTEXTS`, `KochenSpecker18Witness` in `a11oy-core/quantum/kochen_specker_18.js`. --- ## 5. POVM Verdict Semantics (`@a11oy/core/quantum`) — GRAFT 3 **Source:** Preskill, J. (2015), *Quantum Information* (Physics 219 / CS 219 lecture notes), Caltech, Chapter 3 §3.1 (POVMs). [https://www.preskill.caltech.edu/ph219/chap3_15.pdf](https://www.preskill.caltech.edu/ph219/chap3_15.pdf) Binary $\{accept, reject\}$ verdicts are replaced by a positive-operator-valued measure: a finite collection of positive-semidefinite operators $\{E_i\}$ on the policy state space satisfying the **completeness theorem** $$ \sum_i E_i = I, \qquad 0 \le E_i \le I. $$ Each $E_i$ corresponds to a distinct admission outcome: ``` 'admit' | 'admit_throttled' | 'admit_witnessed' | 'deny' | 'escalate' ``` The probability of outcome $i$ on policy state $\rho$ is the Born rule $\Pr(i) = \mathrm{Tr}(E_i \rho)$. The shipped implementation restricts to **diagonal effects in the policy basis** (full off-diagonal POVMs are out of scope for v0.1); this reduces the trace to a dot product. `makePOVM` validates completeness with tolerance $10^{-9}$ and throws `POVMSetError` if $\sum_i E_i \ne I$ — the operational equivalent of the Lean theorem `povm_completeness` referenced in the doctrine. Implementation: `makePOVM`, `isComplete`, `probability`, `argmaxOutcome`, `POVMSetError` in `a11oy-core/quantum/povm.js`. --- ## Composition The four formulas compose into a single governance step: 1. **Frame the decision.** `makeTetrad(...)` produces a `TetradFrame` over the four governance legs (this is the orthonormal basis). 2. **Project onto a complementary pair.** `fromTetrad(pair, frame)` extracts $(v_A, v_B)$ for one of the 12 canonical frame-pairs and emits a `ComplementaryDecisionPayload` via `emitDecision`. 3. **Bind credences to the Fisher manifold.** `fisherRaoDistance(p, q)` is used to gate "distinguishable enough" admissions. `fisherDiagonal(p)` supplies the local metric for credence updates. 4. **Apply the POVM.** `argmaxOutcome(povm, ρ)` selects the admission outcome from the completeness-validated effect set. 5. **Bound the policy head.** The rolling decision window is passed to `checkComplementarity` (σ-product floor) and to `KochenSpecker18Witness.evaluate` (contextuality witness). If either fails, the policy head is flagged and the decision is escalated. Every step is pure, deterministic, and side-effect-free — the package ships no I/O, no network, no global state. It is designed to be linked into a host process (sidecar, lambda, or worker) that supplies inputs and consumes outputs. --- ## File map (deployed) ``` /opt/a11oy/ ├── core/ │ ├── geometry/fisher_manifold.{js,d.ts} │ ├── quantum/bohr_complementarity_engine.{js,d.ts} │ ├── quantum/kochen_specker_18.{js,d.ts} │ ├── quantum/povm.{js,d.ts} │ ├── index.{js,d.ts} │ └── package.json ├── connection/ │ ├── tetrad_field.{js,d.ts} │ ├── index.{js,d.ts} │ └── package.json └── MANIFEST.json ``` `MANIFEST.json` carries the per-file SHA-256 digest, byte size, build timestamp, and source git SHA — see `SECURITY.md` for the full provenance chain.