Our Science / Forced elements
Why do nature's rules take the form they do? We look for conditions that leave only one answer. Five examples show how that reasoning works, from counting marks to comparing lengths and linking loops.
Published research
Two published papers establish the comparison formula and the golden scaling described below. Each identifies the conditions that exclude a different answer.
This paper starts with a rule for combining comparison costs and a scale for small mismatches. It proves that these conditions allow only J(x) = ½(x + 1/x) − 1, where x is the ratio being compared.
Uniqueness of the Canonical Reciprocal Cost
See the comparison example ↓If each level grows by the same factor, and the third level equals the first two added together, the factor must be the golden ratio. The paper also covers sequences that approach these two rules as they continue.
See the golden-ratio example ↓Take three switches. Each can be on or off, independently of the others. There are exactly eight possible settings. You can check by listing them. Getting nine would require changing the setup.
The setup leaves no ninth setting. We use the same reasoning to ask a deeper question: which features of reality are necessary because their alternatives cannot satisfy the conditions?
A proof shows what follows from its starting conditions. Connecting the proof to nature means establishing that those conditions describe the physical process.
Once the conditions hold, the answer cannot change. Three independent on/off switches have eight settings.
The activity itself uses the fact in question. Comparing two readings requires information about both.
The second idea is what we mean by “for all practical purposes, forced.” It can also be expressed precisely in mathematics.
01 / Counting
Imagine recording each flash of a lamp with a mark on paper. Begin with no marks. Add one for each flash:
0 → 1 → 2 → 3 → …
Put a group of two marks beside a group of three, and you have five: addition. Make three groups of two, and you have six: multiplication. Comparing counts gives differences; comparing their sizes gives ratios, such as one-half.
RS's δ-calculus is a mathematical way to build these operations from records and a rule for adding the next mark. The symbol δ names that step.
Change the marks from strokes to dots and two plus three still gives five. Under the stated rules, every such counting system has the same arithmetic.
Read the construction. The δ-calculus, Theorems 1.6–1.7 and Sections 2 and 5. The accepted manuscript gives the counting proof and the extension to signed counts and fractions. [1]
Start with an empty record. Adding a mark never gives the empty record, and two different counts must remain different after the same step. Every record in the construction must be reachable by finitely many such steps. With the stated induction rules, these conditions determine the counting structure.
The proof uses background mathematics, including equality, induction, recursion, and quotients. It does not claim that two different objects alone supply all these rules, or that a physical notebook has unlimited space. Signed counts and fractions require the further constructions given in the paper.
Read The δ-calculus: from distinction to arithmetic, or explore the construction.
02 / Comparing two quantities
Compare a length of 10 cm with a length of 5 cm. Their ratio is 10 ÷ 5 = 2. Reverse the comparison and the ratio is ½. Compare equal lengths and it is 1.
RS assigns a number to the mismatch. We call it the comparison cost. Here, “cost” is a number with no units; it isn't automatically an energy measured in joules.
The RS proof specifies how costs combine when comparisons are made together, and fixes the scale of the cost near a match. Those conditions allow just one formula. If x is the ratio:
J(x) = ½(x + 1/x) − 1.
To use it, add the ratio to its reciprocal, divide by two, then subtract one. Equal lengths give J(1) = 0. A ratio of 2 gives J(2) = ¼. Reversing that comparison gives the same result: J(½) = ¼.
Keep the combination rule and the scale near a match, and the formula has no alternative. That is the necessity the proof establishes.
Read the published proofs. The uniqueness theorem selects J from the composition rule and calibration. [2] The d’Alembert paper, Theorem 3, classifies the symmetric quadratic rules for combining comparisons. [3]
Let F stand for a possible cost function. For positive ratios x and y, the composition rule is:
F(xy) + F(x/y) =
2F(x)F(y) + 2F(x) + 2F(y).
“Quadratic calibration” fixes the scale close to a match: 2F(eᵗ)/t² tends to 1 as t tends to 0. Together, these conditions select J. Reversing comparisons without changing the cost is not enough on its own to select this formula.
A further RS derivation connects J to the specified comparison apparatus and reference unit. Using it to calculate a physical energy, rate, or material property requires the corresponding physical relationship.
03 / A fixed ratio between levels
Picture a sequence of positive sizes. Each step multiplies the size by the same number, r. Start at 1, and the first three sizes are 1, r, and r².
Now require the third size to equal the first two added together. That gives:
r² = 1 + r.
Solving this equation gives only one positive answer: r = (1 + √5) / 2, about 1.618. This number is called the golden ratio, written φ.
In RS, the work is explaining why the levels must obey these two requirements. The hierarchy proof gets a fixed multiplier from preserving comparisons between levels. Its first join of two different levels supplies the addition rule. Together, they force φ.
Read the published proof. The coherence theorem, Section 4, Theorems 5–6, gives the exact result and its extension to sequences that approach the scaling rules. [4]
The proof uses an additive reading: joining two records adds their assigned values. It also requires the first join to use earlier levels and to have nonzero comparison cost.
Joining a level to itself would have zero cost. That leaves the first and second levels as the pair forming the third. The proof also derives uniform scaling from an update that preserves positive comparisons.
These conditions determine the golden scale for this hierarchy. Applying the result to a physical system means showing that it has these properties. The argument does not say that every pattern in nature has golden proportions.
The published scaling proof is in the coherence paper, Theorems 5–6. [4]
04 / Linked loops and space
Think of two linked rings in a chain. Bend or move them without cutting them or passing one through the other, and they stay linked. Their shapes can change while the link survives.
For closed loops in the kind of space used by the RS construction, three dimensions have a special role. On a flat sheet, separate loops cannot thread through each other this way. In four or more spatial dimensions, an extra direction allows a link of this type to come apart.
The RS result requires a relation between two loop-shaped traces to survive the allowed changes in shape. With the stated connection between that relation and loop linking, the proof selects three spatial dimensions.
This gives the spatial argument its physical purpose: a relation can preserve a record while the shapes carrying it change.
Our paper, From One Generator to Loop Order on the Three-Dimensional Cube, develops a related question: what must a record retain when the order of loops changes? Two routes around the cube can cross every directed edge the same number of times yet have different reduced loop words. The paper proves that adding up crossings loses this order information. Explore the result in our interactive guide.
The theorem uses a specified spatial realization: a mathematical account of the loops, their allowed movements, and how their relationship is recorded. It requires a nonzero integer pairing, meaning a whole-number measure of the relation that survives those movements.
The proof connects this measure to linking through homology, a branch of mathematics that studies features such as holes. It requires the relation to persist under deformations that add no new record. Within these conditions, the dimension is forced to be three.
This is a result about that loop-based description of recognition. It does not rule out every other kind of space or every mathematical use of extra dimensions.
05 / Counting the possible states
Return to the three switches. Each has two possible settings, so together they have 2 × 2 × 2 = 8. The same count applies to three independent binary positions in the RS construction. Binary simply means each position has two possible values.
Write 0 for off and 1 for on. Here are all eight settings in an order that changes only one switch at a time:
Read from left to right, then continue on the next row. Returning from 100 to 000 also changes just one switch.
Seeing every setting requires at least eight state visits, counting the starting setting. The sequence shows that eight are enough. That count is exact; how long a real system takes depends on its update rules.
Read the published proof. Coherent Comparison as Information Cost, Section 3.7, Theorems 7–8, proves the coverage bound and gives this one-switch-at-a-time cycle. Table 3 counts the states; Figure 2 shows the same eight-step sequence. [7]
A process might skip states, pause, or revisit a state. Three binary positions still give eight possible states, but the timing can differ. To establish an eight-tick repeating cycle, we must also establish how the process moves and that it completes the required tour.
What an investigation already requires
An investigation already relies on some facts before its first calculation. A question distinguishes possible answers. A comparison uses information. Reading a record is another event.
That is what “for all practical purposes, forced” means here. It doesn't mean “probably true” or “almost proved.” It means the activity requires the fact we are describing.
“The light is on” and “the light is off” say different things. Asking which is true already uses that difference. This establishes that some distinctions exist here, without claiming that we can tell every possible pair of states apart.
To compare today's temperature with yesterday's, you need information about yesterday's reading. A notebook, a memory, or an instrument can hold it. Something must preserve that information long enough for the comparison to happen.
A log of everything that happened through noon doesn't include the later event of your reading it at one. The log could predict that you will read it. But writing the prediction and carrying it out are two different events.
RS represents questions, comparisons, and records mathematically, then proves the corresponding statements. Their practical force comes from identifying those features in the activity itself.
They provide a starting point. Reaching the comparison formula, golden scale, or spatial result still takes the further arguments described above.
Suppose a device stores both “red” and “blue” as the same symbol, 0, and retains nothing else. Later, seeing that 0 cannot tell you which color came in. Processing the same stored symbol again won't restore the lost difference. You would need another source of information.
The missing distinction explains why recovery fails. We seek that kind of reason for physical laws: a chain in which each result follows from the preceding conditions and each physical connection can be examined.
Read the published mathematics. Recognition Geometry, Section 2.7, Theorem 1 and Proposition 1, identifies the observable states with the distinctions a recognizer retains. That is the mathematical basis of the red-and-blue example. [8]
Follow a reference to the full paper. Each entry identifies the result used here and whether the source is a published journal article or an arXiv preprint.
Constructs the counting structure, proves its unique correspondence with systems satisfying the stated counting rules, and extends it to signed counts and fractions.
Back to the example ↑Proves that the composition law and local quadratic calibration uniquely determine the comparison cost J.
Back to the example ↑Classifies symmetric composition rules of degree at most two, giving the d’Alembert form used by the comparison law.
Back to the example ↑Uniform scaling and the addition rule select the golden ratio, exactly or under the paper’s corresponding limiting conditions.
Back to the example ↑Constructs golden-ratio and related metallic structures from reciprocal-cost geometry. This is a related geometric result.
Back to the example ↑Applies a golden-ratio ladder to water’s timescales and terahertz bands with an experimental time anchor and measured participation law; Section 3.2 compares alternative spacings.
Back to the example ↑Counts binary states and proves the coverage bound and cyclic one-bit traversal for the specified cube scheduler.
Back to the example ↑Identifies observable states with the distinctions retained by a recognizer, grounding the example of two inputs stored as one symbol.
Back to the example ↑