Where the derivation starts
The discovered architecture of reality.
Reality has one forced shape. A recognition is one thing told from another: a comparison actually made. Recognition Science starts from three facts that every question already uses, adds the stated conditions of each part of that shape, and proves that under them only one answer is possible. We follow that shape into gravity, matter, morality and meaning.
- Start with the overview
- Read the foundation paper
The Forced Core of Physical Reality. Manuscript (PDF), not yet peer reviewed.
The three facts
- Some things can be told apart.
- Comparing two readings needs both.
- Reading an old record is a new event.
The forced shape in one view
Five answers, each the only one its conditions allow.
- 01 · Counting0 → 1 → 2 → 3 → …
- 02 · The costJ(x) = ½(x + 1/x) − 1
- 03 · The golden ratioφ = (1 + √5)/2
- 04 · SpaceThree dimensions
- 05 · Eight states2 × 2 × 2 = 8
Five working drawings follow, one for each answer. In each, move what can be moved and see what stays fixed. Each answer links to its worked example on the overview, and each example links to its proof.
Colours in the drawings
Live · computed in your browserWhat you moveCost or gapExact resultThe golden ratio
01 · Counting
Counting has one arithmetic.
0 → 1 → 2 → 3 → …
Start empty, always add one more mark, reach every count in finitely many steps and never repeat one: every such system has the same arithmetic (Dedekind, 1888).
A + B in binary
101
Tally: 2 + 3 = 5 marks. Binary: 10 + 11 = 101. Tally and binary agree.
What you are looking atTwo records kept two ways: as tally marks on the left, as binary digits on the right. Under the line the marks of A and B are joined into one row, and the two binary numbers are added column by column. The last line counts the joined marks one at a time.
Join two tally records, add the same two in binary, and the totals match.
More numbers
tally marks: 2 + 3 = 5
marks counted in binary: 101
02 · The cost
A comparison has one cost.
J(x) = ½(x + 1/x) − 1
J is the only cost of a positive ratio x that obeys both the rule for combining comparisons, J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y), and the calibration at a match: near x = 1 the cost is ½(ln x)² to leading order. It follows that twice and half cost the same, the cost is never negative, and only a match costs nothing.
Cost J(x)
0.250
x = 2.000 and 1/x = 1/2.000: x and 1/x cost the same.
What you are looking atThe cost J over every ratio from one quarter to four. The filled dot is the ratio x, and the open dot is its mirror, 1/x. The bar under each dot is its cost. The two dots always sit at the same height, and the curve touches zero in one place: the match, x = 1.
More numbers
J(2.000) = J(1/2.000) = 0.250
J(x²) = 2J(x)² + 4J(x) = 1.125
03 · The golden ratio
One ratio lets the levels join.
φ = (1 + √5)/2
Positive levels that grow by one fixed ratio, with level 2 the sum of levels 0 and 1: the ratio solves λ² = λ + 1, whose one positive root is φ = 1.618…
Gap at level 2
0
At φ the gap closes: the join of levels 0 and 1 is exactly level 2.
What you are looking atThree levels of a ladder, as bars. Each level is the one before it, stepped up by the ratio λ. The bottom row lays levels 0 and 1 end to end. Off φ, a shaded piece shows the gap between level 2 and that sum. Move the step and try to close it anywhere else.
A level joined to itself costs nothing, so the costly join that makes level 2 joins levels 0 and 1.
GivenLevels are positive sizes that add. Each step respects the adding: stepping a sum gives the same size as stepping each part and then adding. If adding and subtracting whole-number multiples of levels gives a positive size, the step keeps it positive. Level 2 is one join of two earlier levels, and comparing its two parts costs something.
More numbers
λ = φ ≈ 1.6180339887
λ² = λ + 1, join 0 + 1: J(φ) ≈ 0.118
04 · Space
Two loops stay linked only in three dimensions.
Three dimensions
In a closed space without one-dimensional holes, a lasting pair of loops can wind through each other a nonzero number of times, a count no bending changes while the loops stay apart, only in three dimensions.
Linking number
1
2 crossings between the loops: +1 +1. Counted from this view: linking number 1.
What you are looking atTwo closed loops in three dimensions, seen from one side. Where one loop passes in front of the other, the line behind is broken. Each such crossing counts +1 or −1 by its handedness, and half the sum is the linking number.
Turn the loops or reshape one: the crossings move, the linking number stays 1.
More numbers
half the sum of +1 +1
2 crossings between the loops
05 · Eight states
Three switches, eight states, eight ticks.
2 × 2 × 2 = 8
Give each of the three dimensions one on/off switch: 2 × 2 × 2 = 8 settings. Each tick flips one switch, so the shortest pass through all eight settings and home takes eight ticks.
Corners visited
8 of 8
Home after 8 ticks, every corner once: a Gray cycle.
What you are looking atA cube. Each corner is one setting of the three switches, written as three digits. Each edge joins two settings that differ in one switch. The large dot is the setting now, and filled corners have been visited. The path shown visits all eight and comes home in eight ticks. Move the tick to follow it one flip at a time.
Eight is the fewest: seven ticks to reach the seven other corners, one more to come home.
More numbers
tick 8, at 000, home
one switch flips per tick