Paradigm Sage

World Ribbons

Mon, 06 Apr 2026 00:00:00 +0000

Chapter 14 - World Ribbons

Portent

The conventional metaphor for breaking out of a paradigm is thinking outside the box. The allusion is that the walls of the box block a view of what is outside the box. This is harder than it looks, it requires inspired guessing, and ruthless pruning. Most ideas are…junk, one can find oneself not just out of the box, but off the table and onto the floor. Usually, the current box does present some clues about how to do this, unsolved problems that focus our attention, threads to pull on that tug on some hidden assumption. There is a chance. Once in the new box, it too will eventually have unsolved problems, clues that lead to the next box, etc. - you are always in a box.

12 Light Seconds

Mon, 30 Mar 2026 00:00:00 +0000

Chapter 13 - 12 Light Seconds

Portent

Physicists love their invariants. They hunt for them like buried treasure. They lovingly name them, “Conservation of Energy”, “Conservation of Mass”, etc. An invariant is that which doesn’t change while all around it does. They are reliable guideposts in the search for the laws of physics.

Primary Docs

Paradigm Discourse:
Open the Red White & Blue discourse in the next tab

Technical Chapter:
Open the 12 Light Seconds chapter in the next tab

By Contradiction

Mon, 23 Mar 2026 00:00:00 +0000

Chapter 12 - By Contradiction

Portent

There is no shortage of proofs that quantum nonlocality cannot be used to send superluminal signals. Eberhard’s paper of 1989 is probably the definitive treatment, comprehensive, succinct and devastating. Randomness defeats any possible combination of measurements, no matter how clever.

But the problem with impossibility proofs is that they require a very strong model, and a strong model is a de facto paradigm. If you are going to go ahead and do the impossible anyway, you are going to have to subvert the paradigm.

Quantum Tic-Tac-Toe

Mon, 16 Mar 2026 00:00:00 +0000

Chapter 11 - Quantum Tic-Tac-Toe

Portent

This, then that. This event caused that event: cause -> effect. In relativity, events are spacetime events, tightly local in both location (space) and duration (time). What are we to make of happenings that are not events, that are not localized? Can we use games to dig under the paradigm of events (point-like causes) as the only possible type of cause?

Primary Docs

Paradigm Discourse:
Open the Out Smarted discourse in the next tab

Objections

Mon, 09 Mar 2026 00:00:00 +0000

Chapter 10 - Objections

Portent

Ambiguity. Most of us have a low tolerance for it. We like the cognitive clarity of black and white - it’s efficient. But on the eve of a paradigm shift, ambiguity is our friend. A deliberate strategy to let the web of ideas find a new local minimum.

Primary Docs

Paradigm Discourse:
Open the Journal discourse in the next tab

Technical Chapter:
Open the Objections chapter in the next tab

R&D

Mon, 02 Mar 2026 00:00:00 +0000

Chapter 9 - R&D

Portent

Risk. How do new theories come into existence? How does someone, anyone, let go of a successful past to chart a new and uncertain future? The lure of the quest; challenge, purpose, reward…and risk. Perhaps evolution has built this algorithm into us as well.

Primary Docs

Paradigm Discourse:
Open the Quest discourse in the next tab

Technical Chapter:
Open the R&D chapter in the next tab

Quantum Temporal Paradox (QTP)

Mon, 23 Feb 2026 00:00:00 +0000

Chapter 8 - Quantum Temporal Paradox (QTP)

Portent

Entanglement cannot be explained by timelike causality nor by common cause. But those are the only two types of causes we know about. Well, except for spacelike causality (and its equivalent common cause variation), but that can’t be allowed - can it? If it is, the specter of causal paradox cannot be ignored. A choice must be made: either assume it is insurmountable and thus a definitive barrier, or decide that it must be incorporated into physical theory.

Entanglement

Mon, 16 Feb 2026 00:00:00 +0000

Chapter 7 - Entanglement

Portent

Entanglement is the unexpected phenomenon that individual particles can lose their individuality. They no longer have independent states, but only a joint state. They have become entangled. Mathematically, their states are no longer separable. This disturbs the statistics of their possibilities from classical expectations. More jarringly, it violates our fundamental assumptions about identity, individuality, and distinguishability.

This was baked into the math of quantum mechanics from the very beginning, but was hidden by the expectation that common cause was sufficient; it was not. Entanglement cannot be reduced to common cause, although the difference is mathematically tiny and requires the non-traditional application of measurement bases. Not only was the theory hard, but so was experimentally proving it. Closing the remaining loopholes proved even more technically challenging.

Interference

Mon, 09 Feb 2026 00:00:00 +0000

Chapter 6 - Interference

Portent

Logical linear western thought. If a allows c, and b allows c, then a and b allows c. What could be more natural, more obvious, it’s simple logic - commonsense demands this conclusion. And it works - in classical systems. Not so much in quantum ones. The problem is not interference, that is a well understood wave phenomena. The problem is destructive interference. Two is not always better than one - sometimes its zero.

Uncertainty

Mon, 02 Feb 2026 00:00:00 +0000

Chapter 5 - Uncertainty

Portent

There is tension between observation and abstraction. Which one should dominant depends on where one is along the historical arc of a technical field. Sometimes more is less - and in perfect symmetry - sometimes less is more.

Primary Docs

Paradigm Discourse:
Open the Ignoring discourse in the next tab

Technical Chapter:
Open the Uncertainty chapter in the next tab

Significance

Too much information can create a mental box from which it is very hard to escape. An abundance of observations can obscure patterns. Too many facts can bury hidden assumptions even deeper.

Quantum Mechanics

Mon, 26 Jan 2026 00:00:00 +0000

Chapter 4 - Quantum Mechanics

Portent

The uncertainty principle is not about measurements disturbing a system, although it is often taught that way. The real situation is much deeper, and more mysterious. The uncertainty principle is a consequence of conjugate bases. Conjugate bases do not have to have the same units. When they do, we find them tolerable, even understandable. When they don’t, when their units are different, our commonsense balks. It is counterintuitive that different things are components of each other.

Relativity

Mon, 19 Jan 2026 00:00:00 +0000

Chapter 3 - Relativity

Portent

The speed of light is more fundamental than just a velocity, it is the thread that weaves space and time together. In relativity (Minkowski space), space and time are supposed to be treated on an equal footing. Yet, spacetime presents a unique symmetry/assymmetry signature: past and future break the isotropic nature of time, but left and right do not break the isotropic nature of space. From symmetry principles, timelike and spacelike causes (and their common cause variants) should be legitimate, but a priori, because of the spectar of causal paradox, they are declared verbotten. Is the obstacle contradiction, or is the obstacle that we don’t understand paradox?

Paradox

Mon, 12 Jan 2026 00:00:00 +0000

Chapter 2 - Paradox

Portent

Self-reference has been at the heart of the major limit theorems of the last century, from Russell’s paradox in set theory, to Gödel’s incompleteness theorems in math, to the halting problem in software engineering. In any sufficiently complex system, self-reference intrudes, unwanted, unwelcome, and unexpected, violating assumptions we were absolutely certain of. It is the ultimate revealer of paradigm barriers.

A first attack was achieved last century by G. Spencer-Browns in his seminal work, The Laws of Form where he solved the logical paradox problem by introducing imaginary truthvalues. Their applicability to quantum systems is one of the theses of this work.

Paradigms

Mon, 05 Jan 2026 00:00:00 +0000

Chapter 1 - Paradigms

Portent

Paradigm is a technical term, introduced by Thomas Kuhn in his seminal work The Structure of Scientific Revolutions. It is derived from the Greek word for pattern. The modern idiom is thinking out of the box. Many puzzles are paradigm puzzles, they are quite opaque until you question the blocking assumption. This is a skill which can be developed. The great revolutions in science were all paradigm shifts. Solving the quantum measurement problem likely requires a major paradigm shift.

Mon, 01 Jan 0001 00:00:00 +0000

Of all the paradigm shifts that science has muddled through, quantum mechanics has to rank as the most traumatic. It violates our common sense again and again, and then yet again in some completely new and unexpected way, constantly challenging rational, logical, fundamental assumptions about the nature of reality.

It challenges taxonomies — is it particle or is it wave?
It challenges observation — how can it possibly not have both position and momentum?
It challenges continuity — how can it go from here to there yet never be anywhere in between?
It challenges motion — what do you mean it took both paths?
It challenges cause and effect — how can uncaused causes cause random effects?
It challenges locality — how can something over there, spacelike separated, possibly affect something over here?
It challenges measurement — don’t tell me it ain’t there until I look at it; what’s wrong with you?

How is it that our common sense feels violated by quantum systems? Why is it not a trustworthy source of intuition?

2006 AJP Paper

Mon, 01 Jan 0001 00:00:00 +0000

Peer Reviewed Paper

This paper introduced Quantum Tic-Tac-Toe as a pedagogical game for illustrating superposition and collapse.

It is included here as prior work, not as a statement of the current QT3 program. The present site revisits the game’s deeper conceptual implications, which could not be explored in the original venue.

Published Paper:
Download the original 2006 AJP paper (PDF)

Clues

Mon, 01 Jan 0001 00:00:00 +0000

(Occasional Posts)

To achieve play that feels like chess requires:

a formal line of development, with geometrical and logical rigor, that respects the game
the penetration of two paradigm barriers

Clues

4/8/26 - If the 2D board is a square of squares (8x8), then the 3D board should be a…?

Dev

Mon, 01 Jan 0001 00:00:00 +0000

Dev (INWORK)

A playable version of chess scaled up to three dimensions.

(Game has not yet been introduced in the PoP narrative.)

Target release date is fall, 2026. For the terminally impatient, I will post clues from time to time.

Can you discover the rules with only minimal hints?

Log

3/15/26 - AI axioms, hello world, python port plan.

Week 1:

3/16/26 - Transfer DNS records from WordPress to Hugo.
3/17/26 - 3DC: directory architecture.
3/18/26 - 3DC: regression tests, port half of foundation/coords.py to coord.js.
3/19/26 - 3DC: resolve coord and anchor ambiguity, port rest of coords, improve regression framework.
3/20/26 - Port colors and port rays.
3/21/26 - Explore refactor options for quads and quandMaps.

Week 2:

3/22/26 - Establish template files (*.md, *.json, *.js, and *.test.js).
3/23/26 - Planes: specs, data/code/test slices partway through para 4.
3/24/26 - Finish porting over the planes module.
3/25/26 - Long negotations with AI about code purity.
3/26/26 - Finished quads module, data, code, & tests; specs still need work.
3/27/26 - Finished perims module.

Week 3:

3/29/26 - Start on advSqs module.
3/30/26 - Finish the overlapTiles module.
3/31/26 - AdvSq first stable point.
4/01/26 - AdvSq under good set of regression tests, more to do.
4/02/26 - POC on tiles, the 8-color board unitcell, and basic POVs.
4/03/26 - 8x8x8 tiles, plumb dragable 2D control canvases, with names.
4/04/26 - Nail down state for undo/redo.

Week 4:

4/6/26 - Condensed regression reporting, simpler 3dc page, start on model/state tests.
4/7/26 - Bishop rays for slip & slide, control/view callback arch, and decorator POC.
4/8/26 - Demo decorators, click toggle on src/dst, define basic decorators.
4/9/26 - Port render code from initThree.js to specialized modules.

Expositions

Mon, 01 Jan 0001 00:00:00 +0000

Expositions

A place for short articles on the principles, premises, promises, and perils of QT3.

Intrepid Reader

Ask a question. What is confusing, unclear, intriguing, or questionable. Or even “why didn’t you think of ’that’?” Send an email to

contact [at] paradigmsage [dot] com

Who knows, a member of Understanding’s team just might post a response here.

SamIam

Q: “I’m not buying entanglement in time - that’s pure science fiction.”

Instructions

Mon, 01 Jan 0001 00:00:00 +0000

Instructions

Rules

There are two players:

X moves first, his spooky marks are subscripted with odd integers.
O moves second, her spooky marks are subscripted with even integers.

There are two types of moves:

Placement moves: click in two squares to place a pair of spooky marks.
Collapse moves; click on a spooky mark on the loop (purple) to collapse that move to that square.
- The other player makes the collapse selection.
- All moves entangled with that one will also collapse to classical positions.

Restrictions

Limitations

Mon, 01 Jan 0001 00:00:00 +0000

Limitations

Critical discussions of where QT3 is limited as a toy universe.

Only One Basis

Quantum tic-tac-toe asks the players to place each move into a superposition of two squares. The nine squares form the basis states of the quantum system. It is the only basis, there are no conjugate bases. One could envision an alternative basis, say the ‘diamond’ basis, but its relation to the ‘square’ basis would be intricate and require a very specific relationship to qualify as a conjugate basis.

Play

Mon, 01 Jan 0001 00:00:00 +0000

Play (INWORK) A playable version of 3D Chess with planar moves and advancement squares. The 3D board is a cube of cubes (8x8x8). Each tile is the bottom of a cube. An 8 color board includes the 2 bishop colors (tile faces) and the 4 duke colors (tile edges). Board is not yet active, just a POC for the render engine.

(Game has not yet been introduced in the PoP narrative.)

Play

Mon, 01 Jan 0001 00:00:00 +0000

Play

An implementation of QT3. This is where the Intrepid Reader can interact with a playable version of QT3.

Ask your own questions of the game.

Demo

Copy this state string

X1+(1,2); O2+(2,3); X3+(3,1)[132]; O3@X1(1)!X1(1)!O2(2)!X3(3); O4+(4,5); X5+(4,6); O6+(4,6)[56|4]; X6@X5(4)!O4(5)!X5(4)!O6(6); X7+(7,8); O8+(8,7)[78]; X8@X7(7)!X7(7)!O8(8); {X=1,O=0.5}

and paste it into the state box (black edit box below). Then hit the load button and the rerun button. Then re-run the game by using the redo button. A placement move requires a pair of spooky marks. A cyclic entanglement forces a choice, a collapse move. Note how the quantum and classical listings differ. Note how the classical ensemble grows (x2), but also how it shrinks; pruned by contradiction (gray) and pruned by collapse (yellow.)

Rationale

Mon, 01 Jan 0001 00:00:00 +0000

Rationale

Problem:

Chess in three dimensions doesn’t feel like chess. The pieces lose too much power, a one pawn advantange no longer yields the game. Every contest is a draw. Boring, frustrating; in equal measure.

Conundrum:

Over the last hundred years lots of rule sets have been proposed, a few are interesting, a couple even have followings. But none feel like chess. Is chess in three dimensions impossible?

Quest:

Assume it is possible, but that one or more paradigm barriers block the way. The goal is not to invent another way to move chess pieces around a three dimensional manifold, but to discover the rules that lead to play which feels like chess. Rules which are compelling, even if…a little odd, even if unlike any other game, ever.

Superposition

Mon, 01 Jan 0001 00:00:00 +0000

Superposition

Quantum Tic-Tac-Toe was invented with three strategic objectives:

Demonstrate superposition in its simplest possible form.
Demonstrate that collapse can be forced by self-reference.
Demonstrate that an objective measurement process is conceivable.

Spooky Marks

The key to superposition is multiple possibilities. The smallest multiple possibilities is two. Therefore, just two spooky-marks, not three, not more, not variable. The simplest possible weighting is to make the weights equal. Superposition in its simplest possible form - pure scientific reductionism.

Variations

Mon, 01 Jan 0001 00:00:00 +0000

Variations

Present variations on QT3, such as

N-way QT3
Unitary evolution
Probabilistic collapse
Hybrid
Indistinguishable QT3

N-Way QT3

There are two possibilities here: fixed N (like 3) and variable N (N>1). The simple cyclic entanglements of classical quantum tic-tac-toe (gotta love language) are no longer the only kind of entanglement that requires collapse.

Unitary Evolution

In this variation, no player gets to make a placement move. Spooky-marks are placed on the board by chance, or an algorithm, but not by choice. The only choice is that when a cyclic entanglement occurs the appropriate player is required to make a collapse move.

Paradigm Sage

World Ribbons

Portent

12 Light Seconds

Portent

Primary Docs

By Contradiction

Portent

Quantum Tic-Tac-Toe

Portent

Primary Docs

Objections

Portent

Primary Docs

R&D

Portent

Primary Docs

Quantum Temporal Paradox (QTP)

Portent

Entanglement

Portent

Interference

Portent

Uncertainty

Portent

Primary Docs

Significance

Quantum Mechanics

Portent

Relativity

Portent

Paradox

Portent

Paradigms

Portent

2006 AJP Paper

Clues

Clues

Dev

Log

Week 1:

Week 2:

Week 3:

Week 4:

Expositions

Intrepid Reader

SamIam

Instructions

Rules

Limitations

TOC

Only One Basis

Play

Play

Rationale

Problem:

Conundrum:

Quest:

Superposition

Spooky Marks

Variations

N-Way QT3

Unitary Evolution