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| # 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. | |