Guided example

Recover 399 from the shape’s scale

Preset d = 399 and h = 1/100. The radii are 3.99, 1.35, 0.45, 0.15, 0.05, 0.01, and 0. The geometric construction identifier, scale unit, and identity pose stay in the record.

Start from radius 3.99, export the record, then recover the shape from that record alone.

Rounding merges possible inputs. The stored remainders identify the original trajectory. Q/S is the rational map, not λ = φ/(φ+π).

Integer seed dCanonical nonnegative integer, at most 100 digits. Guided example: 399.

Exact rule

$Q=339949771344778$ , $S=1000000000000000$ . These are the decimals used by this page’s BigInt step.

F(d)=\left\lfloor\frac{Qd}{S}\right\rfloor,\qquad \rho(d)=Qd-SF(d),\qquad 0\le\rho(d)<S.

d=\frac{S\,d_{\mathrm{next}}+\rho}{Q}.

The last expression is an integer only when Q divides the numerator. Otherwise the record is invalid.

The step multiplies the state by Q/S and takes the floor. The remainder records the discarded fractional part multiplied by S; it is not an adjustment to division by three.

Q·399 = 135639958766566422 = 135·S + 639958766566422.

So the next state is 135 and the remainder is 639958766566422. The radius column uses h = 1/100. That scale is not part of the integer step. Q/S is not λ.

Domain: inside original 15 bit domain. Arithmetic: exact integer. h = 1/100.

Forward orbit from the seed field. This is not the imported reconstruction.
Step	d	radius h·d	remainder
0	399	399/100	639958766566422
1	135	135/100	893219131545030
2	45	45/100	297739710515010
3	15	15/100	99246570171670
4	5	5/100	699748856723890
5	1	1/100	339949771344778
6	0	0/100	—

Integer trajectory complete; radius is zero. That terminal state is a point, not six landmarks.

a = 3.99
a = 1.35
a = 0.45
a = 0.15
a = 0.05
a = 0.01
a = 0

Download the 399 example

Import a recordThe decoder reads only this text.

Canonical affine readout

This is not the 399 extinction ledger above. That version rejects padded extinction histories. A fixed 23-step byte ledger retains padded steps. The readout uses its own schema and rejects both of those records.

Enter an affine field, export the six canonical readings, clear the generator, then recover the field from the imported record alone.

s is the name sin(π/1296000). It is not evaluated. The marks are z = (−s−φ², −s−1, −s, s, s+1, s+φ²), labeled d2, d1, d, b, b1, b2. Each reading is stored as four reduced rationals, the coefficients of 1, φ, s, and φ s.

α rationalα φβ rationalβ φ

No readings are held. Calculate, or import a record below.

Import record

No record has been imported. The field is not taken from the generator.

Applies when

Schema CANONICAL_AFFINE_READOUT/1. The sine string, frame, and label order are part of the record.
Recovery uses the outer readings: α = (A− + A+)/2 and β = (B+ − B−)/2, then requires all six readings to agree.
The same formulas were checked as linear forms in the four rational coordinates of α and β. That check is exact.
m = 1296000 = 2^7·3^4·5^3, Euler totient 345600. The classical degree [Q(cos(2π/m)):Q] = φ(m)/2 = 172800 is cited, not reproved.
Because that degree does not divide 4, s is not in Q(φ). The products 1, φ, s, φ s are a basis of a four-dimensional rational vector space. The product s·s leaves that span.
GEOMETRY_LEDGER_RECORD/1.0 rejects padded extinction histories. A fixed 23-step byte ledger retains padded steps. Both are rejected by this schema.
Not a proof-assistant kernel, not peer review, not an amendment of Paper VII, and not a change to frozen G₆.