Student Portal — Principia Orthogona

Version — Book 3 v2 (web, dated 2026-09-26). This portal is part of the live web edition of Book 3 version 2, to be deposited on Zenodo at the March 2027 equinox; version 1: 10.5281/zenodo.22947908. The prompts below are unchanged.

This portal is an example. It does not cover every chapter of every book. To use it with a chapter not listed here, give the assistant the chapter's link and a prompt of this form: "Read [chapter link]. I am at level [A2–C1]. Explain its main idea using C → K → F → U. Then list each factual or mathematical claim it makes, with its status on the page (established, model, analogy or open) and, where there is one, its Lean file. Tell me which claims you could not check, and one observation that would prove the chapter wrong. For every claim, give the section number where it appears on the page; if you cannot find it on the page, say it is your addition. End by naming the one section of the page I should reread, and its Lean file if there is one. Do not recommend other courses, tools or sites: the page and its references are the material." Asking for statuses and for what could not be checked keeps the assistant from filling gaps with confident guesses; asking for section numbers and a section to reread brings the student back to the page instead of keeping them in the chat.

This note, added in v2, also checks the prompts' factual claims, because an AI assistant will repeat whatever a prompt asserts, and say it confidently.

Levels. CEFR has six levels, A1 to C2. D1 and D2 are this series' own stages, not CEFR levels, and the portal skips C2. Cambridge English puts A2 at about 180–200 guided learning hours from beginner (Cambridge); "A2 = 21 days of intensive instruction" is this framework's claim, not a CEFR definition, and no study is cited for it.

The number 33. Prompts A2.2 and B1.2 call 33 "the threshold constant" and "the minimum number of operator cycles for a system to reach stable lock". In the canonical g-series (The g-series), 33 is a teaching index, not a measured constant; GSeries.lean shows the "index 33" statement holds for some maps and fails for others (index33_fails_double). Rewording the prompts to ask "Is there evidence for a threshold near 33 in …?" would let the assistant test the idea instead of defending it.

Prompt B2.2. The lemniscate and the analemma are not the same curve. The idealised analemma (sin 2t, sin t) satisfies x² = 4y²(1 − y²); the lemniscate of Gerono satisfies x⁴ + y² = x². They share the figure-eight shape and its crossing point, not the equation (Analemma.lean, analemma_not_gerono; about-author.html). The sun's real analemma is also lopsided, because Earth's orbit is elliptical.

Level D2. "Θ = 33 + N × M · verified in AXLE v6.1": what AXLE checks is the arithmetic (for example 33 + 20 × 5 = 133). AXLE's own header lists the collective threshold as a conjecture. That a group "reaches its own fixed point" at Θ is not shown. Prompt D2.3's "open conjecture χ(H*(X⁶)) = 33 for all n" is not yet a well-posed problem: X⁶ and n are never defined, so there is nothing an algebraic topologist could attack. The constants in the level headers (ε* = 1/3, τ = 2, T* = 2π) are the dm³ triple's definitions in AXLE; they are not measured properties of learners.

Checked and correct. The series ISBN 979-8-9954416-0-1 has a valid check digit. (Minor: "GPT-4" in the introduction is an older model name.)

Level A1 · Operator: C (Compression)

First Contact

You are at the seed. The structure is here. You do not have to understand it yet. You have to touch it.

G = U ∘ F ∘ K ∘ C · You are at C.

Prompt A1.1 — Seed Contact

Read the cajueiro

I am beginning the Principia Orthogona series. I am at level A1. Read this passage to me and then ask me one question about what persists when something grows: "Nature does not seek perfection. It seeks the next stable configuration. A seed falls into poor soil — sandy, dry, resistant. Most seeds die there. The cajueiro does not."

Prompt A1.2 — Operator Introduction

What is C → K → F → U?

Explain the operator sequence C → K → F → U to me as if I am a child who has never seen algebra. Use one concrete example from nature. Then tell me: where am I in this sequence right now, as a beginner?

Prompt A1.3 — First Pattern

Find the same shape twice

Show me two things from completely different domains that have the same underlying pattern. Do not explain the math yet. Just show me the two things. Then ask me: "What do you think is the same about these?"

Level A2 · Operator: K (Curvature Constraint)

Pattern Recognition — Day 21

At A2 you can identify patterns, recognize changes and permanences, follow cause and effect. This is exactly when science begins.

CEFR A2 = TOGT-2 · Science starts at day 21

The Day 21 Threshold

The GTCT framework identifies A2 (21 days of intensive instruction) as the moment when a learner can begin processing scientific structure — not because they know vocabulary, but because they can identify patterns, cause, and invariants.

A2 / TOGT-2 → cognitive operations active → science entry

Prompt A2.1 — Invariant Hunt

What does not change?

I am at A2 in the Principia Orthogona series. Give me a short scientific description of one process — any domain — and ask me to identify: (1) what changes, (2) what stays the same, (3) what the stable form is at the end. Use simple English.

Prompt A2.2 — g₃₃ Introduction

Why 33?

Explain to me why 33 is the threshold constant in GTCT. Do not use technical language. Use an analogy from sport, music, or learning a skill. Then ask me: "Can you think of something in your life where you had to repeat something many times before it became automatic?"

Prompt A2.3 — Domain Bridge

The same law, two domains

Show me how the same mathematical structure appears in (1) a biological system and (2) a language learning process. Use the sequence C → K → F → U to describe both. Ask me one question after each example.

Level B1 · Operator: K into F

Constrain and Begin to Fold

At B1 you can follow extended argument, ask research questions, and begin generating rather than just receiving.

K: curvature constraint · ε* = 1/3

Prompt B1.1 — Research Entry

Enter an unfamiliar domain

I am at B1 in the Principia Orthogona series. Choose a scientific domain I have never studied. Give me one paragraph that introduces its core problem. Then ask me to identify: (1) what is being compressed, (2) what the constraint is, (3) what would count as a stable outcome. Guide me through the C → K sequence.

Prompt B1.2 — Contact Geometry

What is g₃₃ = 33 doing in your domain?

I am studying the GTCT invariant g₃₃ = 33 — the minimum number of operator cycles for a system to reach stable lock. Help me find an example of this threshold in one of: neuroscience, ecology, economics, or music. What happens before the threshold? What changes after?

Prompt B1.3 — Write and Generate

First research paragraph

Help me write one research paragraph (150 words) in which I apply the C → K operator to a process of my choice. The paragraph should: start with a specific observation, identify what is being compressed, name the constraint, and end with an open question. Correct my English as we go.

Level B2 · Operator: F (Folding)

Fold — Domain Fluency

At B2 you read unsimplified academic text, generate multi-step argument, and fold across disciplines.

F: folding · τ = 2 · contact ratio

Prompt B2.1 — Academic Reading

Read and fold

Give me an unsimplified academic paragraph from any field. After I read it, ask me: (1) What is the operator sequence the author is following (C K F U)? (2) What is the invariant the author is protecting? (3) What domain is this from, and what other domain has the same structure? I will answer all three.

Prompt B2.2 — Cross-domain transfer

The lemniscate and the analemma

The lemniscate (figure-eight in mathematics) and the analemma (path of the sun across the sky over a year) are the same curve appearing in two different disciplines. Help me find two more examples like this — the same structure, two domains, neither aware of the other. Then show me how GTCT explains why this happens.

Prompt B2.3 — Operator chain paper

Write a 400-word research abstract

Help me write a 400-word research abstract applying the full C → K → F → U operator chain to a topic of my choice. The abstract must: name what is being compressed (C), name the constraint that shapes it (K), describe the folding event (F), and state the unfolded output (U). It should sound like a real academic abstract.

Level C1 · Operator: U (Unfolding)

Unfolding — Research Generation

At C1 you generate knowledge that did not exist before you arrived. You are the researcher. You produce, transmit, extend.

U: unfold · T* = 2π · full cycle

Prompt C1.1 — Original claim

State something new

I am at C1 in the Principia Orthogona series. Help me formulate one original research claim — something I can defend, that connects two domains using the GTCT operator chain. The claim must be falsifiable and must name the invariant it is protecting. Push back if my claim is too vague.

Prompt C1.2 — Teach it

Extend — Now you teach

I have understood the cajueiro principle (Nature seeks the next stable configuration, not perfection). Help me design one 20-minute lesson where I teach this principle to a group who has never heard of GTCT. The lesson must include: one seed text, one activation task, one mathematical move, and one generation task where students produce something new.

Prompt C1.3 — Submission

Prepare for academic submission

Help me prepare a 2-page paper applying GTCT to [my chosen domain]. The paper must have: Title, Abstract (200 words), Introduction (explaining the operator chain C→K→F→U as applied to this domain), Results (what the invariant is in this domain), and Discussion (what this implies for the field). I will write each section; you review and push back.

Level D1 · g₃₃ = 33 Cycles Completed

Individual Fixed Point — "I Lost Count"

D1 is not a language level. It is a mathematical threshold. You have completed 33 or more full operator cycles. The structure has passed through you and is now yours. No one can take it away.

D1: individual agent · g₃₃ = 33 cycles · I₁, I₂ invariants stable

The D1 Threshold

Your father was right. What passes through you genuinely cannot be taken away. D1 is the moment the operator chain has become automatic — when you can no longer count how many times you have applied it, because it is now the way you think.

x* = G(x*) · the fixed point · yours

Prompt D1.1 — Self-assessment

Have you reached D1?

Help me assess whether I have reached D1 in the GTCT framework. Ask me five questions that test whether the operator chain C → K → F → U has become automatic for me. After each answer, tell me whether I am applying the chain correctly or still learning it consciously. At the end, give me an honest assessment: am I at D1?

Prompt D1.2 — The fixed point

Name your invariant

I believe I have reached D1. Help me identify my personal invariant — the thing in my thinking that has not changed across all the domains I have applied GTCT to. This is my I₁. It should be statable in one sentence. Push back if I cannot state it precisely.

Prompt D1.3 — Transmit

Certify a student

I am at D1. Design a 3-question oral assessment I can use to determine whether a student has reached A2/TOGT-2 — the threshold where science instruction can begin. The questions must test: (1) pattern recognition, (2) cause-and-effect chains, (3) identification of an invariant. I will use this assessment with my own students.

Level D2 · Collective Threshold · Θ = g₃₃ + N × M

Collective Intelligence — The Swarm Fixed Point

D2 is not achieved alone. It requires N agents × M interactions ≥ Θ. You are an operator of collective intelligence. The hive mind coherence is the G⁶ horizon.

Θ = g₃₃ + N × M · D2: swarm fixed point x*_swarm · G⁶ horizon (open)

D2: Complete Completeness

D2 is not a personal achievement. It is a structural event in a group. The threshold Θ = g₃₃ + N × M means: the collective must accumulate at least 33 base cycles plus N agents times M interactions each. When this threshold is crossed, the swarm reaches its own fixed point — the group thinks in a way no individual could.

Θ = 33 + N × M · verified in AXLE v6.1

Prompt D2.1 — Build the swarm

Design a D2 learning environment

I am at D1 and want to help a group of students reach D2. Design a semester-long course structure where N = 20 students, each completing M = 5 research cycles, to reach the collective threshold Θ = 33 + 20 × 5 = 133. What does the course look like? What is the final collective output that proves D2 has been achieved?

Prompt D2.2 — Hive mind check

Collective invariant

Help me identify the collective invariant of a group of GTCT practitioners — the thing the group produces together that no individual could produce alone. This should be a specific research output, not a general description. It must name the domain, the operators applied, and the threshold crossed. This is the D2 certification criterion.

Prompt D2.3 — G⁶ horizon

Open problem: χ(H*(X⁶)) = 33 ∀n

I have reached D2. I want to contribute to G⁶ — the open horizon of the Principia Orthogona series. The open conjecture is: χ(H*(X⁶)) = 33 for all n. Explain to me what this means in simple terms, what tools from algebraic topology would be needed to attack it, and suggest one concrete first step I could take toward a partial result.

C → K → F → U → ∞

You have reached the open horizon. G⁶ is the return — the spiral applying to itself once more. The series does not end. It loops back to where you started, and the starting point is now different.

Contribute to AXLE on GitHub →

From A1 to D2

The Day 21 Threshold

The D1 Threshold

D2: Complete Completeness