Vector Decisioning for Job-Specific Intelligence (JSI): Mathematical Foundations

Abstract

This document is the mathematical companion to the normative Job-Specific Intelligence (TAK-JSI) profile. It explains one DPF mechanism for representing and evaluating profession-local judgment. It does not independently define job qualification, issue qualification claims, or replace the profile’s broader requirements for job scope, tools, data, evaluation, surveillance, and runtime enforcement.

DPF makes AI coworker judgment auditable by scoring every option a coworker considers against a fixed, named registry of decision dimensions, using signed weight vectors that are either hand-authored (doctrine), statistically inferred from revealed choices (experience), or — in one narrowly-scoped pilot — learned from a validated external signal (situation). This document states the actual mathematics of that system as implemented, corrects an external write-up that attributed a different (unimplemented) formalism to DPF, and grounds each real mechanism in its closest external prior art so the design can be evaluated against the literature rather than taken on faith.

This document does not introduce new mechanism. Every formula, threshold, and data shape below is quoted or derived from code and specs that already exist and are cited inline. Its contribution is synthesis: pulling the mathematical core out of four scattered specs into one academically referenced account, and drawing the line — precisely — between what DPF has built and what a future design could still add.

0. Relationship to TAK, TAK-JSI, and the Golden Triangle

The standards family and this implementation companion answer different questions about the same governed action:

TAK’s own abstract states the boundary precisely: TAK “defines what a trustworthy agent harness and runtime MUST, SHOULD, and MAY do once an identified agent is allowed to operate” — it is concerned with runtime governance and harness consistency, not with the mathematics of what a correct decision looks like. TAK’s HITL and escalation controls (§7) are the enforcement mechanism; this document’s scoring engine is what populates the judgment that enforcement gates on. A coworker blocked by a TAK tool-grant gate and a coworker whose recommendation crosses an autonomy-policy confidence threshold in evaluateDecisionPerspective are stopped by two different, complementary layers — one asks “is this agent allowed to do this,” the other asks “is this decision well-supported enough to act on without a human.”

The Golden Triangle is deliberately orthogonal to both: it compiles a human’s stated posture (“get this right” / “I need this now” / “keep this cheap”) into routing and orchestration parameters, and explicitly is not a second scoring engine — see its own Decision 4: it feeds inferContract(), and its own Terms table draws a hard line between “assurance posture” (what the system intends) and “realized quality” (what the vector-decisioning and verification layers actually measure happened).

1. Authority scopes: WWMD / WWWD / WSID / per-decision override

DPF partitions decision authority into four non-inheriting scopes (docs/design/golden-triangle-design.md §5; decisions-belong-to-their-scope kernel principle):

Precedence is strict subsidiarity: the most-local scope that owns the decision applies; a non-owning scope is advisory only and never substitutes as authority. A customer business decision is never defaulted from WWMD even when WWWD is silent — the gate defers to a human rather than borrow a neighboring scope’s doctrine.

External grounding. This is a direct software application of the subsidiarity principle — the norm that a matter should be handled by the smallest, most local competent authority, and a higher authority should support rather than supplant it. Subsidiarity has two well-documented formal lineages DPF’s own founder-kernel material already cites (docs/founder-kernel/raw-sources/frameworks/subsidiarity.md): Catholic social teaching’s articulation in Quadragesimo Anno (Pius XI, 1931), and its codification in public administration via Article 5(3) of the Treaty on European Union — the modern reference point for subsidiarity as a governance-design constraint rather than a moral one. DPF’s contribution is applying the same non-inherit rule to software authority scopes rather than governmental ones.

2. The structured alignment engine

2.1 The dimension registry

PRINCIPLE_DIMENSIONS (packages/db/src/wiki-taxonomy.ts) is a closed, compile-time-enforced set of 20 named axes — e.g. blast_radius, reversibility, data_privacy, long_term_maintainability, governance_compliance, evidence_density, speed_to_value, cost_efficiency, human_cognitive_load, operator_effort. Each axis is defined individually — meaning, sign, scope, projection, and where its weight comes from — in the Decision Vectors reference; this section states only the structure. Every kernel principle, org-policy item, and profession-corpus page that participates in structured scoring declares a signed principleDimensionVector — a hand-authored weight per axis it cares about, with a written rationale (principleWeightRationale).

PRINCIPLE_DIMENSION_SCOPE (docs/superpowers/specs/2026-07-23-decision-tier-rebalance-and-vector-epistemology-design.md §2.1) splits this registry into a spine (axes that trade off across every profession — kept small, kept commensurable) and profession-local axes (declared inside one profession’s corpus, richer resolution, mandatory projectsOnto rule mapping every local axis onto ≥1 spine axis). This is the direct, measured answer to over-generalization: a UX judgment about typographic hierarchy is scored at full resolution inside the design profession, then rolls up onto the shared human_cognitive_load spine axis when it needs to be weighed against a security or cost concern from a different craft. Profession-local axes proliferate where the criteria actually live, without inflating the one space every decision reasons over.

2.2 The scoring formula

For an option o with numeric features f_d and a principle p with signed weights v_d over the same dimension set D, computeStructuredAlignment (apps/web/lib/decision/option-scoring.ts) computes:

alignment(o, p) = ( Σ_{d ∈ D} f_d · v_d ) / ( Σ_{d ∈ D} |v_d| )

This is a normalized weighted sum — the option’s features projected onto the principle’s signed weight vector, divided by the L1 norm of that weight vector so the result is bounded and comparable across principles with different numbers of active dimensions. A composite score per option is then the evidence-weighted combination of alignment(o, p) across every applicable principle (the “contribution ledger” — every principle’s vote is individually inspectable, which is the platform’s stated differentiator over an opaque scorer).

When a principle has no dimensionVector (or the option shares no scoreable dimension with it), the engine falls back to computeSemanticAlignment: cosine similarity between the option’s and the principle’s text embeddings. The mode selection is deliberately two-sided (hasScoreableOverlap) — structured only when both the principle declares dimensions and the option scores at least one of them — because scoring from the principle side alone silently returns a confident-looking alignment: 0 when the option simply never described that axis, which is indistinguishable from “actively neutral” without the both-sides check.

External grounding. The weighted-sum formula is the classical Simple Additive Weighting (SAW) / Weighted Sum Model (WSM) from Multi-Criteria Decision Analysis — one of the oldest and most widely used MCDA aggregation rules, formalized in the operations-research literature (Churchman & Ackoff, An Approximate Measure of Value, 1954; surveyed alongside its competitors in Hwang & Yoon’s standard MCDM reference, Multiple Attribute Decision Making: Methods and Applications, 1981). This is the correct citation — not the Analytic Hierarchy Process (AHP). An external write-up submitted for evaluation attributed DPF’s decisioning to full MAUT/AHP with a Saaty pairwise-comparison matrix and eigenvector-derived weights (Saaty, The Analytic Hierarchy Process, 1980); no such matrix, eigenvector computation, or consistency-ratio check exists anywhere in this codebase. The distinction matters for a research-grounded document: SAW assumes attribute weights are already known and simply sums them; AHP is a specific method for deriving those weights from pairwise judgments. DPF hand-authors its weights today (§3), so SAW is what is actually running. §4 below identifies AHP’s pairwise-elicitation method as a legitimate, currently unbuilt technique for a specific unmet need (cold-start weight elicitation), rather than as a description of the present system.

The semantic-alignment fallback is a direct application of the vector space model for information retrieval (Salton, Wong & Yang, A Vector Space Model for Automatic Indexing, CACM 1975) using modern sentence/document embeddings in place of term-frequency vectors — standard, well-established grounding for the cosine-similarity path.

The hierarchical spine/profession-local split, with mandatory projection back onto a shared spine, is structurally the same move AHP itself makes at the problem-structuring stage (decomposing a goal into a hierarchy of criteria and sub-criteria before any weight is assigned) — DPF’s architecture borrows AHP’s hierarchical decomposition discipline without borrowing its pairwise-eigenvector weight-derivation method. This is a legitimate, precise partial debt worth recording rather than either denying influence or overclaiming the whole method.

2.3 Evidence discounting

evaluateDecisionPerspective (apps/web/lib/decision-perspective/evaluator.ts) does not score options against dimension vectors directly; it scores whether enough trustworthy material exists to support a recommendation at all. scorePerspectiveMaterial (material.ts) computes:

effectiveWeight = confidenceWeight × freshnessFactor × evidenceFactor × reviewFactor × promotionFactor

where each factor is an independent [0,1] discount: freshnessFactor (current 1 / stale 0.5 / superseded 0.2 / contradicted 0), evidenceFactor by grade (A 1 / B 0.75 / C 0.4 / D 0), reviewFactor (approved 1 / draft 0.35 / rejected 0), promotionFactor (promoted 1 / candidate 0.45 / revoked 0).

Confidence is then computed one of two ways, minus a risk-tier penalty (0 / 0.1 / 0.25 / 0.5 for low/medium/high/critical) and a recent-override penalty (capped at 0.3):

• Content-blind (WWMD / build-studio): the mean effectiveWeight across applicable materials. Coverage-shaped — it answers “how well-grounded is this class”, not “what does our doctrine say about THIS question”.
• Content-aware (WWWD, BI-7E1F128A): each material is weighted by its semantic relevance to the question, and confidence is the strongest relevant DIRECTIONAL weight scaled by how one-sided the relevant material is (bestDirectionalWeight × |alignmentScore|). Relevance is min-max normalised within the set actually scored (BI-F5F2869D) — normalising across material that never gets scored leaves the scored set with no 1.0 and an arbitrary ceiling.

Alignment is not a licence to act. A confident approve does NOT authorise the gate to decide. It acts autonomously only when the owner has ALREADY RULED on this question — a ruled-tier stance (grade A, weight 1.0) that dominates the relevant material, surfaced as settledByRuling. Anything else escalates with gapReason: "aligned-not-settled", rendered to the operator as “New proposition” rather than a doctrine gap (BI-F5F2869D).

This is deliberate, not a threshold that needs raising. Approving whatever matches recorded doctrine would mean the business only ever does what it already does; reviewing an unaligned or novel proposal is exactly where a new idea surfaces. The goal is better-sorted escalations, not fewer. The decline direction is unaffected — saying no to an off-stance idea is low-consequence and does not need the owner’s live call.

The recent-override penalty counts times the owner OVERRULED the profile in that class — the gate recommended one option and the human chose another. It deliberately does not count answering an escalation that carried no recommendation, nor an answer that agreed, nor one where no structured option was picked (BI-ACF0D6D4). Counting answers rather than overrules made working through a review queue drive the penalty to its cap and suppress the next decision.

External grounding. A confidence score built as a product of independent reliability discount factors applied to a base weight is the same structural move as the GRADE framework (Grading of Recommendations Assessment, Development and Evaluation — Guyatt et al., GRADE: an emerging consensus on rating quality of evidence, BMJ 2008), the standard used across evidence-based medicine and policy to discount a recommendation’s confidence by evidence quality, recency, and consistency before it is acted on. DPF’s A/B/C/D evidence-grade discounting is a direct structural analogue, applied to platform/business decisions instead of clinical ones. It is deliberately not a formal Dempster-Shafer belief-combination rule (Shafer, A Mathematical Theory of Evidence, 1976) — there is no explicit handling of conflicting-evidence mass reassignment, only exclusion (contradicted/rejected/revoked material scores zero outright) — and this document does not claim that heavier formalism; the simpler multiplicative-discount model is what is implemented and is accurately described as GRADE-like, not Dempster-Shafer-like.

3. Weight fluidity: the JSI three-timescale model

docs/superpowers/specs/2026-07-24-job-specific-intelligence-fluid-weight-layer-design.md names the central limitation the sections above still have: every principleDimensionVector is hand-authored once and frozen until a human deliberately re-authors it in a PR. That is more auditable than a neural network’s weights, but exactly as static in the one property a live job needs — a job is not a training corpus; its ground truth of “what a good decision looks like” is a moving target (seasonal demand, an organization’s accumulating preferences, a craft’s evolving standards), not a fact that can be baked in once.

JSI resolves this by recognizing three genuinely different timescales of truth, each requiring a different update rule:

3.1 Medium timescale: weight inference as preference learning

inferWeightProposals reads WeightInferenceObservation[] — paired chosenVector / recommendedVector axis scores for real past decisions in one (domainClass, profile) group — and proposes an adjustment only when:

• sample floor: ≥8 observations in the group (no pooling across groups to clear it);
• consistency floor: a super-majority (≥0.7) of decisions that separated on an axis went the same direction (agreements with zero separation count toward the base but cannot manufacture a signal);
• magnitude floor: mean separation ≥0.1 (a systematic-but-trivial preference does not trigger a proposal).

A surfaced proposal enters the same confirmation ladder stance-promotion.ts already uses for material authority (unconfirmed 0.6 → confirmed 0.9 → ruled 1.0), but strictly below unconfirmed, at entersAtConfidenceWeight: 0.3 — enforced by a runtime assertion — so an un-ruled statistical inference can never outweigh authored doctrine. It mutates nothing until a human rules on it.

External grounding, per the fluid-weight design’s research pass (§3), independently arrived at and consistent with this document’s framing:

• Revealed preference theory (Samuelson, A Note on the Pure Theory of Consumer’s Behaviour, 1938) is the classical economic grounding for inferring a preference ordering from observed choices rather than stated weights — precisely what “the human systematically overrode the kernel in a consistent direction” is doing.
• Bradley-Terry model (Bradley & Terry, Rank Analysis of Incomplete Block Designs, Biometrika 1952) is the standard statistical model for inferring a latent strength/weight ordering from paired-comparison outcomes. DPF’s consistency-floor check (a distribution-free super-majority test on separated pairs) is a simplified, non-parametric cousin of what a full Bradley-Terry fit would estimate with a likelihood model; naming the lineage is honest, naming it as Bradley-Terry estimation would overclaim statistical rigor not implemented.
• Contextual bandits (LinUCB — Li, Chu, Langford & Schapire, A Contextual-Bandit Approach to Personalized News Article Recommendation, WWW 2010; Thompson sampling, Thompson 1933) are the standard open formalism for adjusting a weight online from a bounded reward signal while carrying an explicit confidence/sample-count alongside the point estimate — validating that pairing every inferred weight with minSampleSize/consistency, as weight-inference.ts already does, is the right shape rather than an ad hoc safety add-on. DPF explicitly does not adopt full bandit optimization (a single scalar reward per arm), because its decisions are multi-axis and multi-principle by design and collapsing to one reward would erase the per-principle contribution ledger.
• Concept-drift detection (ADWIN — Bifet & Gavaldà, Learning from Time-Changing Data with Adaptive Windowing, SDM 2007; DDM — Gama et al., Learning with Drift Detection, SBIA 2004) is the correct pattern for when to re-evaluate a proposal — on a windowed statistical disagreement threshold, not a fixed cron — while explicitly not adopting the underlying (opaque) models these libraries adapt.

3.2 Fast timescale: situational signal validation

Not yet built. The fluid-weight design (§4.3) proposes exactly one pilot: a single correlate, in a single archetype, that starts as unscored evidence and is promoted to a scored input only after a minimum-sample, minimum-correlation-with-actual-recorded-outcome test, ratified by a human — a deliberately higher bar than the medium layer, because an external signal has no author vouching for it. Once validated it modulates a decision at inference time only; it is never persisted as a stored weight (the “attention, not backprop” distinction the spec draws explicitly from how a transformer’s attention mechanism re-weighs context at inference time over frozen training-time weights — the LLM-construction analogy the fluid-weight design opens with).

External grounding: signal half-life / confidence decay on a live risk signal, with a human-reviewable rule layer staying inspectable while a statistical layer adapts underneath, is established practice in production fraud-detection systems (e.g. Stripe Radar’s public documentation of analyst-overridable, decaying risk signals) and in hybrid recommender systems that blend stable hand-curated features with continuously updated collaborative-filtering signal as distinct components combined at serving time, not merged into one representation (the architecture pattern documented publicly by Netflix’s and Spotify’s recommendation engineering writing). Both are cited as structural precedent in the fluid-weight design, not as endorsements to adopt either platform’s underlying (opaque) model.

4. Where AHP-style pairwise elicitation would actually fit

Section 2.2 established that DPF runs Simple Additive Weighting, not AHP, today. There is one concrete, currently-unmet need where AHP’s actual contribution — deriving a weight vector from a human’s pairwise judgments rather than requiring raw numbers — is the right tool: cold start.

weight-inference.ts’s sample floor (§3.1) means a brand-new organization profile, or a newly onboarded archetype, has no revealed-preference signal to learn from and must run on hand-authored spine/profession defaults alone until enough decisions accumulate. Saaty’s pairwise-comparison method — present a business owner with simple binary/intensity comparisons (“is cost more or less important than turnaround time, and by how much, on a 1–9 scale?”), build the reciprocal comparison matrix, and take the principal eigenvector as the derived weight vector, checking a Consistency Ratio against Saaty’s empirical Random Index to catch incoherent judgments — is a well-validated way to extract an initial weight vector from a non-technical operator without asking for raw numbers. This is exactly the “Active Preference Elicitation” idea the external write-up raised, and it is sound decision theory; the write-up’s error was claiming it as already implemented, not in recommending it.

Scoped correctly, this is additive, not architecturally new: an AHP-derived initial vector would enter the same WeightAdjustmentProposal ladder §3.1 already defines, at the same sub-unconfirmed confidence tier, superseded the moment enough real decision history exists to run genuine weight inference. It would not create a second authority model, and it would not apply to the spine (a per-org AHP session cannot invent new axes — §2.1’s projection rule and the decisions-belong-to-their-scope non-inherit boundary both hold unchanged). This is recorded here as a candidate for a future, separately-scoped BI — not proposed as done, and not scheduled ahead of the fluid-weight design’s own sequencing (§6; medium-timescale wiring is the higher-leverage, already-designed gap).

5. Summary table

References

• Bifet, A. & Gavaldà, R. (2007). Learning from Time-Changing Data with Adaptive Windowing. SDM.
• Bradley, R. A. & Terry, M. E. (1952). Rank Analysis of Incomplete Block Designs. Biometrika 39(3/4).
• Churchman, C. W. & Ackoff, R. L. (1954). An Approximate Measure of Value. Journal of the Operations Research Society of America 2(2).
• Gama, J., Medas, P., Castillo, G. & Rodrigues, P. (2004). Learning with Drift Detection. SBIA.
• Guyatt, G. et al. (2008). GRADE: an emerging consensus on rating quality of evidence and strength of recommendations. BMJ 336.
• Hwang, C.-L. & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications. Springer.
• Keeney, R. L. & Raiffa, H. (1976). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Wiley.
• Li, L., Chu, W., Langford, J. & Schapire, R. E. (2010). A Contextual-Bandit Approach to Personalized News Article Recommendation. WWW.
• Samuelson, P. (1938). A Note on the Pure Theory of Consumer’s Behaviour. Economica 5(17).
• Saaty, T. L. (1980). The Analytic Hierarchy Process. McGraw-Hill.
• Salton, G., Wong, A. & Yang, C. S. (1975). A Vector Space Model for Automatic Indexing. CACM 18(11).
• Shafer, G. (1976). A Mathematical Theory of Evidence. Princeton University Press.
• Thompson, W. R. (1933). On the Likelihood that One Unknown Probability Exceeds Another in View of the Evidence of Two Samples. Biometrika 25(3/4).
• Pius XI (1931). Quadragesimo Anno.
• Treaty on European Union, Article 5(3) (subsidiarity).
• DPF internal: docs/architecture/trusted-ai-kernel.md; docs/design/golden-triangle-design.md; docs/superpowers/specs/2026-07-23-decision-tier-rebalance-and-vector-epistemology-design.md; docs/superpowers/specs/2026-07-24-job-specific-intelligence-fluid-weight-layer-design.md; docs/superpowers/plans/2026-07-24-weight-inference-from-rulings.md; docs/superpowers/specs/2026-06-09-wsid-coworker-professional-corpus-design.md.