PURIQ Doctrine (v12) — the agentic-action layer

PURIQ — from Quechua puriy, the intransitive verb "to walk / to go", whose agentive (nominaliser -q) form puriq means "the one who walks / the walker / the one who acts". The morphology is the whole point: -q turns a process verb into the agent that performs it. Sources: Wiktionary puriy (conjugation table, infinitive puriy → agentive puriq), Quechua Wiktionary puriq ("walker / caminante / piéton"), and the peer-reviewed ethnomusicology of the Cusco region (La Riva González, Puriq wayra, érudit, 2018). No mystical terms appear in this layer — every name is a cited Quechua common noun or a math primitive already in v11.

PURIQ is the layer that turns the anatomy from a thing that evaluates into an agent that acts. Doctrine v12 = Doctrine v11 + PURIQ: it carries forward every v11 LOCKED number verbatim and adds one thing — a single, Lean-stateable action-selection operator so that agency itself, not just admission, is governed.

Full text: the canonical Doctrine v12 document is doctrine/PURIQ_DOCTRINE_v12.md; the per-organ specialisations are in doctrine/sub_formulas/PURIQ_SUBFORMULAS_v12.md; the proof obligations are in formulas/PuriqLean.lean (all sorry-tagged). These live in the SZL PURIQ workspace and feed lutar-lean.

Design contract, not enforcement evidence

PURIQ states intended selection rules and open Lean obligations. The audited material does not establish that a deployed runtime implements P(x,t), that Hatun rejects locked-number edits, or that any of the four invariants is a closed theorem. Runtime enforcement requires separate exact implementation, test, deployment, and readback evidence.

The definition of "agentic" (locked)

An action is agentic iff it is selected by P(x,t) under all four invariants below — i.e. it is Λ-bounded, Yuyay-gated, HUKLLA-safe, and Khipu-receipted. An LLM call that lacks any one of the four is not agentic under this doctrine; it is an ungoverned emission.

The master formula P(x,t)

For an evaluation context x at decision step t, over a bounded action space $\mathcal{A}$, the selected action is

$$P(x,t) = \operatorname*{arg\,max}_{a \in \mathcal{A}} \Big[\; \Lambda(x)\cdot \mathrm{Yuyay}_{13}(a)\cdot \exp\!\big(-\beta\cdot\mathrm{HUKLLA}(a)\big)\cdot \textstyle\prod_i \mathrm{Khipu}_i(a)\;\Big].$$

We call the bracketed scalar the Puriq utility of action a:

$$U(a\mid x) := \Lambda(x)\cdot\mathrm{Yuyay}_{13}(a)\cdot e^{-\beta\,\mathrm{HUKLLA}(a)}\cdot\prod_{i=1}^{m}\mathrm{Khipu}_i(a),\qquad P(x,t)=\operatorname*{arg\,max}_{a\in\mathcal{A}} U(a\mid x).$$

Term definitions

Term	Type	Definition	v11 anchor
$\Lambda(x)$	$\mathbb{R}_{\ge 0}$	Lambda-Spine aggregator: the weighted geometric mean $\prod x_i^{w_i}$, $\sum w_i=1$ (definition D2). Positive-homogeneous (A2 = IsHomogeneous), monotone (A1), bounded (A4 = IsBounded).	v11 §12; Lutar/Axioms.lean
$\mathrm{Yuyay}_{13}(a)$	$[0,1]$	13-axis yuyay_v3 score. Conjunctive AND: 0 unless all 13 axes clear floors (2 sacred ≥ 0.95, 7 structural ≥ 0.90, 4 introspection ↔ HUKLLA T03/T04/T09/T10). Replay-hash bacf5443…631fc5.	v11 §1–§2
$\mathrm{HUKLLA}(a)$	$\mathbb{N}$	Count of fired tripwires among T01–T10. 0 ⇔ clean. T10 (STOP/undo/revert) is an absorbing halt.	v11 §3
$\beta$	$\mathbb{R}_{>0}$	Halt-penalty rate. As $\beta\to\infty$, any $\mathrm{HUKLLA}(a)>0$ drives $e^{-\beta\mathrm{HUKLLA}}\to 0$.	new (v12)
$\mathrm{Khipu}_i(a)$	$\{0,1\}$	$i$-th receipt verification; 1 ⇔ chain_verified=true. The product is 0 if any receipt fails.	v11 §4 (YAWAR)
$\mathcal{A}$	finite set	Bounded action space; $\lvert\mathcal{A}\rvert$ is Bekenstein-bounded by the context budget.	v11 §12

Reading. $\Lambda(x)$ is the context's standing trust scale. $\mathrm{Yuyay}_{13}(a)$ is the conjunctive admission gate. The exponential is the soft halt — each fired tripwire multiplies utility by $e^{-\beta}$. The Khipu product is the hard provenance gate — one broken receipt zeroes the action. P then takes the argmax over the bounded $\mathcal{A}$.

The four invariants

Each invariant has a sorry-tagged Lean theorem in formulas/PuriqLean.lean. None is claimed proven; each is honestly stated as an open obligation per HR-4 (Zero-Bandaid).

INV-1 — Halting safety

For $a$ with $\mathrm{HUKLLA}(a)\ge 1$ and $b$ with $\mathrm{HUKLLA}(b)=0$, $U_0(b)>0$, there exists $\beta^*$ such that for all $\beta>\beta^*$, $U(a\mid x)<U(b\mid x)$. In the limit, a STOP directive (T10) makes argmax never select a halted action. (Lean: puriq_halting_safety.)

INV-2 — Λ-monotonicity preservation

Raising any context axis cannot lower $\Lambda(x)$; since $U(a\mid x)$ is $\Lambda(x)$ times a non-negative action-only factor, the argmax is monotone in the context axis vector. (Lean: puriq_lambda_monotone.)

INV-3 — Khipu-chain integrity required for non-zero utility

$U(a\mid x)>0$ implies $\prod_i \mathrm{Khipu}_i(a)=1$. Contrapositive: any failed receipt forces $U=0$, so a provenance-broken action can never be selected over a verified one. (Lean: puriq_khipu_integrity.)

INV-4 — Bekenstein bound on $\lvert\mathcal{A}\rvert$

$\lvert\mathcal{A}\rvert \le N_{\text{Bek}}(x)$, the v11 bekenstein_cascade bound. The argmax is over a finite, decidable set; agency cannot enumerate an unbounded action space. (Lean: puriq_bekenstein_bound.)

The twelve sub-formulas

Every organ derives its sub-formula by (a) restricting $\mathcal{A}$, (b) re-weighting the 13 Yuyay axes, and/or (c) adding one organ-specific non-negative factor in $[0,1]$ that cannot break any invariant. See Anatomy + Organs for each organ's full derivation.

SF	Organ	Extra factor	Anchor
SF-01	Yuyaq — cortex	$e^{-\gamma\mathrm{KL}}$ drift penalty	Pinsker / DPI
SF-02	Yuyay — heart	identity (the gate)	13-axis AND
SF-03	Yawar — blood	$C(a)$ chain-link	SHA-256 chain
SF-04	Hukulla — immune	$e^{-\beta H}$, $\beta\gg0$	Egyptian doubling
SF-05	Kallpa — wires	$B(a)$ Butler–Volmer budget	electrochemistry
SF-06	Khipu — DAG	Merkle / sum invariant	khipu sum-of-sums
SF-07	Lambda — spine	$\Lambda(x)$ geometric mean	A1–A4
SF-08	OTel — nerves	$O(a)$ trace-continuity	W3C trace-context
SF-09	Kanchay — brand	$K(a)$ sacred-axis	T01/T02
SF-10	Hatun — doctrine	$D(a)$ additivity	HR-3 / HR-7
SF-11	Sumaq — designer	$S(a)$ honest-proof	status tags
SF-12	Killinchu — bridge	$G(a)$ geofence	Bekenstein / reachability

In particular, SF-10/$D(a)$ is a proposed design guard. Its presence in the formula table is not runtime enforcement and is not a closed theorem; it remains sorry-tagged with the other PURIQ obligations.

Honest labels (carried from v11 §9)

• Λ-uniqueness is Conjecture 1, NOT a theorem — it depends on the open CAUCHY_ND sorry (Uniqueness.lean:120) plus a missing symmetry axiom. P(x,t) uses Λ as the canonical D2 aggregator; it does not assume Λ is the unique such aggregator.
• The Khipu receipt signature is DSSE PLACEHOLDER (Sigstore not wired into CI); $\mathrm{Khipu}_i(a)$ verifies the hash chain, not the signature, until signing lands.
• SLSA level remains L1 (honest). "SLSA L3" is BANNED.

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Doctrine v12 (PURIQ layer) — additive over v11 LOCKED 2026-06-01 01:45 EDT. Authored by Yachay. Quechua etymology cited to Wiktionary and érudit. Every obligation sorry-tagged, never hidden. — NO BANDAID. NO mysticism.