Theories, theorems & principles

The foundations of a recursive epistemic system for books, evidence, interpretation and inquiry.

A plurality of mathematical theorems, scientific and psychological theories and principles, a provenance standard and philosophical frameworks, each named for what it is, with its formal model, a demonstration you can change, and what it means in Folium X. Back to the example.

1Foundationstheorems, theories, principles and frameworks, each named for what it is
2Formal modelvariables and assumptions, stated
3Demonstrationchange an assumption; watch the consequence
4Product correspondencewhat it means in the product, and its status

Five status labels keep kinds of claim apart: ESTABLISHED THEORY STRUCTURAL CORRESPONDENCE DESIGN PRINCIPLE EXPLORATORY HYPOTHESIS IMPLEMENTED MECHANISM. IMPLEMENTED MECHANISM is used nowhere on this page. Kind words (theorem, theory, principle, framework, standard) stay beside each title; our own formulas are marked Φ Framework notation.

Theories, theorems & principles

A plurality of foundations, connected.

Two mathematical theorems, scientific and psychological theories and principles, a provenance standard and philosophical frameworks each do a distinct job here. None is decoration, and none stands alone: select one to see its role, its kind, and how it connects to the others.

The foundations and their connections. Each link says what one does for another.

The connections as text

i · Graph theory and provenance

A library is a graph that remembers how it knows.

A record in isolation says little. Meaning lives in relationships and in the paths between them: this passage belongs to that edition, supports this reading, complicates that one, and was added on a particular evening by a particular hand.

Foundations

Graph theory (theory) ESTABLISHED THEORY: Vertices joined by typed edges. Give the edges types and times, and the structure holds which things are connected, how, and when each connection entered the record.
The W3C PROV data model (standard) ESTABLISHED THEORY: Entities are used and generated by activities and attributed to agents, and are linked by derivation, quotation and revision. The correspondence is close; conformance is not asserted. PROV-DM (opens in a new tab)

Formal model (Φ Framework notation) DESIGN PRINCIPLE

G_{t} = (V_{t}, E_{t})

e = (u, v, r, τ, π, κ, δ)

V_t: vertices recorded by time t: works, editions, individual copies, passages, concepts, people, places, memories, inquiries and readings. (V in G = (V, E) means the vertices; the V over the arrow in part vi means validation, not a set of vertices.)
u, v: the endpoints of an edge; r its relation type (edition-of, copy-of, located-at, contains, asks-about, supports, complicates, contradicts, compares, recalls, proposes, answers).
τ, π, κ, δ: the time recorded; provenance (who or what asserted it); the origin κ, a kind of claim (quoted, remembered, inferred, invented), not a confidence score; and the reader’s decision δ (undecided, accepted, rejected, revised), kept in a separate field so that accepting never rewrites an origin.

In words. In this model, the library at a moment is its things and the typed, dated, attributed connections among them. Provenance, origin and decision travel with every edge, so an accepted inference cannot quietly become a quotation. Only derivation relations (edition-of, contains, asks-about, answers, and supports where it records a reading’s own grounds) are traced as provenance; supports found later, complicates and contradicts are evidence, shown apart as E⁺ and E⁻; compares, located-at and recalls describe. At every moment, every edge’s two endpoints are already recorded.

Demonstration

Inspect the graph of this page’s inquiry. Choose a moment, select a claim, and trace its provenance back to sources or forward to what depends on it.

Moment in the library’s history

The graph as text

Product correspondence DESIGN PRINCIPLE

Works, editions and individual copies are distinct records. An interpretation stays connected to its passage, edition, author, the transformation that produced it and its revision history. The encounter already draws some of these relations: green threads join three passages that share an image (a comparison, not evidence of influence), and a copper thread marks a book that adds a limit.

The existing 3D encounter at its third step: passages from The Odyssey, Walden and The Prophet joined by green threads, and a copper thread to a slip from The Secret Garden labelled Adds a limit. — The existing encounter (unchanged), third step. The threads are typed relations; this page names them and makes them inspectable.

ii · Information theory and the data-processing inequality

Distinctions carry information.

“The text states this.” “I remember this.” “The system inferred this.” Two statements can read identically and still make different claims. Strip the label and the difference is gone, and no later processing of the words can bring it back.

Foundations

Information theory (Shannon) (theory) ESTABLISHED THEORY: Information is reduction of uncertainty, on average. A representation keeps information about a variable to the degree that knowing it reduces uncertainty about that variable.
The data-processing inequality (theorem) ESTABLISHED THEORY: For a Markov chain X → Y → Z, processing Y alone cannot increase its information about X. Adding another source changes the setup. Polyanskiy and Wu, strong data-processing inequalities (opens in a new tab)

Formal model (theorem) ESTABLISHED THEORY (exact illustrative model) STRUCTURAL CORRESPONDENCE

I (X; Z) \leq I (X; Y)

I (S; T) = 0, I (S; T, P) = 1 bit

S: whether a statement was quoted or inferred, equally likely (so H(S) = 1 bit).
T: the displayed text. In this deliberately specified model both kinds produce the same words, so T carries nothing about S.
P: a provenance label that records S exactly. With it, S is recovered with certainty: 1 bit.

In words. The words alone cannot reveal the source distinction; the label preserves it perfectly in this model. It is an actual information-theoretic example, not a measured, product-wide guarantee.

Demonstration

Two identical statements. Remove their provenance, then restore it.

Generalize: three kinds of claim, under a distribution you set

Illustrative distribution of claim status in a library

Product correspondence DESIGN PRINCIPLE

Quotation, testimony, inference and invention are designed to remain distinguishable states, and each answer to say which input it came from: full text the app can read, the reader’s notes, or catalogue records only. The prepared ordinary-shelf example keeps this visible: “A lead, not a finding.”

iii · Bayes’ theorem and Bayesian epistemology

An interpretation stays answerable to evidence.

A passage matters because of how differently competing interpretations would lead us to expect it, not merely because it sounds supportive.

Foundations

Bayes’ theorem (theorem) ESTABLISHED THEORY: Relates belief in a hypothesis before and after evidence, through how expected the evidence is under each hypothesis.
Bayesian epistemology (framework) ESTABLISHED THEORY: Treats degrees of belief as answerable to evidence. The evidence discipline proposed here resembles it more directly than it establishes a Bayesian computation. Seeing Theory: Bayesian inference (opens in a new tab)

Formal model (theorem) ESTABLISHED THEORY (illustrative numbers) STRUCTURAL CORRESPONDENCE

P (H ∣ E) = \frac{P (E ∣ H) P (H)}{P (E)}

\frac{P (H_{1} ∣ E)}{P (H_{0} ∣ E)} = \frac{P (H_{1})}{P (H_{0})} \frac{P (E ∣ H_{1})}{P (E ∣ H_{0})}

H₁, H₀: competing readings: he stays because he cannot leave and he stays because he wishes to.
Bayes factor: the last fraction, Λ_i for passage E_i. With several passages the factors multiply, assuming the passages are conditionally independent given each reading, and that exactly one of the two readings holds, so P(H₀) = 1 − P(H₁).
Assumptions: every prior and factor below is an assumption you can change, chosen for illustration. Displayed probabilities belong to this model, not to the truth of a literary meaning.

In words. Posterior odds equal prior odds times the Bayes factor. Folium X is not claimed to compute calibrated probabilities, and accepting a reading does not turn it into an observed fact.

Demonstration

Include or exclude a passage, change an assumed Bayes factor or the prior, and watch the posterior of this illustrative model.

Prior P(H₁)

Product correspondence DESIGN PRINCIPLE

Competing readings, counterevidence, corrected attributions and explicitly revisable interpretations. A reading keeps the passages that support it and those that complicate it, in plain view.

iv · Cybernetics, requisite variety and double-loop learning

You govern the loop, and the loop can change its own rules.

Observe, propose, evaluate, intervene, observe again. Different disturbances need different responses when no single response serves them all. And when the answers keep failing, the reader can revise the distinction that produced them.

Foundations

Cybernetics (theory) ESTABLISHED THEORY: Regulation through feedback: an output returns to change what produced it.
Ashby’s law of requisite variety (principle) ESTABLISHED THEORY: When no single response serves different disturbances, effective regulation needs responses as varied as the disturbances it meets. It does not prove that any particular number of AI agents is necessary. Ashby, An Introduction to Cybernetics (PDF, opens in a new tab)
Double-loop learning (Argyris and Schön) (theory) ESTABLISHED THEORY: Single-loop learning corrects an action within fixed assumptions; double-loop learning revises the assumptions, categories or methods themselves. Argyris, 1977 (opens in a new tab)

Formal model (Φ Framework notation) DESIGN PRINCIPLE

x_{t + 1} = F (x_{t}, d_{t}, u_{t})

single loop: y^{'} = f_{θ} (x^{'})

double loop: y^{'} = f_{θ^{'}} (x)

x_t, d_t, u_t: the inquiry’s state; newly encountered material; the reader’s authorized intervention.
f_θ: the method that turns material into an answer, governed by θ, its organizing distinction. Single loop keeps θ: a new input x′, or a corrected answer, gives y′. Double loop changes θ itself.

In words. The loop runs through the reader. Feedback alone does not prove stability or optimal control; the notation is our architectural interpretation, not a quotation or theorem from Argyris.

Demonstration

Requisite variety: choose a disturbance, and see the distinct response it needs, still under the reader’s control.

Double loop: Folium X proposes a reading. Respond, and watch a classification fail until the governing distinction is revised.

Governing question and distinction θ

Product correspondence DESIGN PRINCIPLE

The reader accepts, rejects, corrects, changes scope or suspends an inquiry; interventions alter the process without erasing its history. Ambiguity, contradiction, missing sources and denied access each get a distinct response. Proposals from Folium X are advisory, within the reader’s authority. The essay on this site makes the double-loop move in its own words: “The first question asks why he would leave. The poem first asks whether he can.”

v · Hermeneutics

A passage changes the book; the book changes the passage.

Understanding moves between part and whole: passage, work, collection, reader, world. Each move changes the others, and the route back to the evidence must stay open.

Foundations

The hermeneutic circle (Schleiermacher, Dilthey, Gadamer) (framework) ESTABLISHED THEORY: A part is understood through the whole, and the whole through its parts; reading is repeated movement between them, and every pass can revise both.

Formal model (conceptual structure) STRUCTURAL CORRESPONDENCE

Passage Work Collection Reader World

In words. A philosophical part–whole structure, represented as bidirectional relations between scales, not as a numerical theorem.

Demonstration

Hold one passage fixed and move between scales. Use the buttons or the arrow keys.

Product correspondence DESIGN PRINCIPLE

A reader can move from a passage to its work, to the collection and to their own kept readings without losing the route back to the exact lines and their source. “A reading you keep stands with the books that gave it.”

vi · Validated generativity

A discovery can become an instrument, after it has earned that role.

What you discover can change what you can discover — after it has earned that role. A discovery becomes an instrument for further discovery only after the reader validates it.

Foundations

Validated generativity (Φ Framework notation) DESIGN PRINCIPLE: A discovery may yield a method only through V: the reader’s explicit authorization and the discovery’s epistemic qualification (purpose, scope and assumptions stated; its basis recorded; checked against a resisting case, or the absence of one recorded; its origin kept). The method, applied to the library, yields proposals for review, and sometimes a further discovery. Distinction → instrument → model → creation → material for another inquiry.

Formal model (Φ Framework notation) DESIGN PRINCIPLE

D_{t} \overset{V}{⟶} f_{D_{t}}

D_t: a discovery (a distinction, a pattern or a question) with its origin and basis; f_{D_t} the instrument it may become: a function from the library and a scope to proposals.
V over the arrow: validation / reader authorization / epistemic qualification; not a set of vertices. (V in G = (V, E), in part i, means the vertices.)
Outcome: if V holds, the instrument joins the methods; otherwise the methods are unchanged and the refusal or request for revision is recorded. Applied to the library, the instrument yields proposals for review, never trusted state.

In words. The architecture accumulates ways of investigating alongside accumulated material, and only the ones the reader has validated. This is Φ Framework notation for a design principle, not a theorem.

Demonstration

Carry one concrete example through the whole sequence.

Product correspondence DESIGN PRINCIPLE

A distinction you draw is designed to become a reusable instrument, a question, a model or a way of reading, only after you validate it, and then to be applied to the rest of your library, with the path from its origin preserved. The encounter’s “Who holds the door?” is this move, prepared.

vii · Extended cognition and scaffolding

An environment that takes part in thought.

Appropriately coupled external resources can participate in cognitive processes, a claim more specific than “tools are useful”. Assistance can support work beyond what a reader would do unaided, and then step back.

Foundations

The extended mind thesis (Clark and Chalmers) (framework) ESTABLISHED THEORY: External resources, appropriately coupled, can be part of a cognitive process. The Extended Mind, 1998 (opens in a new tab)
Scaffolding (Wood, Bruner and Ross) (principle) ESTABLISHED THEORY: Assistance that supports activity beyond unaided performance. The original tutoring research is not direct evidence about AI-assisted adult inquiry. The Role of Tutoring in Problem Solving, 1976 (opens in a new tab)
The Zone of Proximal Development (Vygotsky) (theory) ESTABLISHED THEORY: What a learner can do with guidance, just beyond what they can do alone.

Distinct modes (Φ Framework notation) DESIGN PRINCIPLE

Demonstration, guided practice, challenge and independent investigation are distinct modes, not one dial of “more help”. What changes between them is who holds the method.

In words. The measure of good assistance here is the reader gaining control over the method, not receiving more output. No learning gain is claimed.

Demonstration

The same inquiry at four degrees of assistance. Watch who holds the method.

Product correspondence DESIGN PRINCIPLE

Persistent annotations, arguments, source trails and unfinished inquiries externalize structures the reader can revisit and manipulate. Assistance is offered in distinct modes, and the reader chooses how much.

Further connected investigations

These keep distinct roles in the argument. Some are established; others are fruitful but need more specified mappings before any stronger statement. Prioritized by correspondence, not deleted.

How understanding is built

Assimilation and accommodation (Piaget) (theory) ESTABLISHED THEORY: An encounter can be absorbed into an existing schema, or force the schema to change. Here: “the loveliest place is where one wants to stay” can absorb the island, or be restructured by “The unwilling by the fond”.
Constructivism (theory) ESTABLISHED THEORY: Knowledge is built by the learner. In Folium X, as a design principle: the reader is designed to construct, test, revise and retire interpretations.
Enactivism (framework) ESTABLISHED THEORY: Meaning arises in interaction among reader, material, world and system.

How interpretations change

Evolutionary epistemology (framework) ESTABLISHED THEORY: Variation, criticism, selection, retention and recombination, of interpretations and of methods.
Path dependence (theory) ESTABLISHED THEORY: Where an inquiry arrives depends on the route it took; the history of a reading is designed to be kept, not only its latest form (a design principle).
Dynamical and complex adaptive systems (analogy) EXPLORATORY HYPOTHESIS: Changing relationships, and larger patterns arising through interaction. A literal claim would need actual variables and dynamics; none is claimed.

Qualified analogues, needing specified mappings

Active inference and predictive processing (analogy) EXPLORATORY HYPOTHESIS: Minds as prediction-correcting systems; no implemented mechanism is claimed.
Hebbian association (analogy) EXPLORATORY HYPOTHESIS: Connections strengthened by co-occurrence. Useful for noticing; frequency is not truth.
Autopoiesis (analogy) EXPLORATORY HYPOTHESIS: A system that maintains the processes producing it: an image for a self-sustaining inquiry practice.
Simulated annealing (speculative) EXPLORATORY HYPOTHESIS: Broad exploration that narrows over time: a possible image for how an inquiry settles, not an algorithm the app runs.

Speculative research connections

Multiscale modelling and renormalization (speculative) EXPLORATORY HYPOTHESIS: Traceable movement across scales, from passage to library, without losing the finer grain.
Category theory (speculative) EXPLORATORY HYPOTHESIS: Structure-preserving transformations between representations: reading to model to creation. A research connection, not a claimed result.

Theories, Theorems & Principles

What corresponds to what.

Each foundation, what kind of thing it is, its formal model, the demonstration above, its product correspondence, and the status of that correspondence today.

Foundation	Kind	Formal model	Demonstration	Product correspondence	Status
Graph theory · W3C PROV	Theory · Standard	e = (u, v, r, τ, π, κ, δ) in G_t	Inspectable graph; provenance traced back and forward	Works, editions, copies; interpretations tied to passage, edition, transformation and revisions	Planned; illustrated with prepared and fictional records; no PROV conformance asserted
Data-processing inequality	Theorem	I(X;Z) ≤ I(X;Y); toy: 0 → 1 bit	Two identical statements, labels removed and restored	Quoted, remembered, inferred and invented stay distinguishable	Planned; exact toy model, not a product-wide measurement
Bayes’ theorem · Bayesian epistemology	Theorem · Framework	Posterior odds = prior odds × Bayes factor	Evidence and assumptions you can change	Competing readings, counterevidence, revisable interpretations	Planned; no calibrated probabilities claimed
Cybernetics · requisite variety	Theory · Principle	x_t+1 = F(x_t, d_t, u_t)	Distinct disturbances, distinct responses	Reader interventions change the process without erasing history	Planned; no stability or agent-count claim
Double-loop learning	Theory	f_θ → f_θ′	A classification fails until θ is revised	Questioning the category or method, not only the answer	Planned; our notation
Hermeneutic circle	Framework	Passage ⇄ work ⇄ collection ⇄ reader ⇄ world	Scale ladder with a route back	Moving across scales without losing the source	Planned; conceptual structure
Validated generativity	Φ Framework notation	D_t →^V f_{D_t}	A discovery becoming an instrument, only after validation	Keeping a distinction as a reusable way of reading	Planned
Extended mind · scaffolding · ZPD	Framework · Principle · Theory	Four distinct modes of assistance	Who holds the method, mode by mode	Persistent structures to revisit; assistance the reader sets	Planned; no learning gain claimed

Where things stand

In development. Folium X has not been released. First release planned for the United States, for adults 18 and over; membership enrollment is closed, and pricing and the release date have not been announced.