← scholia citrinitas v1.3 5 min · 1,158 words
Substrate Regime Bifurcations
One kernel produces three different pictures when its numerical observer changes. Accurate trigonometry preserves the bounded disk; the ancestral MSVC fast-sincos branch produces AXIS_MUNDI_FREEFORM; binary64 overflow plus finite-value filtering produces a separate escape ecology.
Two floors
Picture the engine standing on two floors at once. The upper floor is the mathematics: one bounded trigonometric kernel. The lower floor is the execution contract: number format, argument reduction, compiler branch, finite-value policy, seed stream, and raster window.
The historical C renderer adds an operation to the mathematics: MSVC
fast fused-sincos evaluation at enormous arguments. We recovered the
original source card and reproduced that branch headlessly. The exact
map does not change regime at 2⁵³ or DBL_MAX, but the ancestral execution
does cross a reproducible observer threshold when |alpha * state|
reaches 2⁵⁴. Literal DBL_MAX adds a different overflow aperture.
Read downward while holding the kernel fixed. Adequate precision preserves the disk; the ancestral fast-sincos branch reorganizes the same recurrence into the Axis family; overflow filtering opens the map and leaves a sparse husk. The diagram separates the three contracts before the details begin.
The three-layer read
| layer | numerical contract | visible result | what may be claimed |
|---|---|---|---|
| Exact map | accurate trigonometric reduction; adequate exponent range | bounded two-phasor disk | substrate-independent dynamics |
| Ancestral observer | MSVC /fp:fast; fused sin/cos behavior above the measured argument threshold | AXIS_MUNDI_FREEFORM | reproducible historical observer family |
| Overflow ecology | binary64 near DBL_MAX; non-finite trajectories discarded | POLAR / HUSK | open-system artifact of encoding and filtering |
These are not three phases of one exact attractor. They are three readings of one formula under three execution contracts. That distinction is the scholion.
Name binding
Here the historical phrase AXIS MUNDI LATTICE is rebound as
AXIS_MUNDI_FREEFORM: the reproducible C-v1 observer family rooted at
alpha=1e307, mu=+1, eps=0, with a reflected mu=-1 sibling. It is not
a substrate-independent attractor class, but it is more than styling: its
piecewise recurrence is fixed by the ancestral compiler and renderer. It is distinct
from the older AXIS_MUNDI glyph class—a vertical spine through
three horizontal realms.
The exact fossil is FREEFORM_20260422_163434_CARD.png, exported at
2026-04-22 16:34:34. Its archived tuple is 1e307, +1, 0. The adjacent
1e308, +1, 0 card is a reduced near-ceiling sibling, not the same form;
no literal-DBL_MAX card was found among the 229 preserved C-v1 cards.
The observer contracts in detail
I · Accurate reduction
For finite parameters, the real-valued kernel remains bounded by its trigonometric amplitudes. With adequate argument reduction and exponent range, the enormous-α sweep retains the underlying two-phasor disk. There is no exact-map transition at 2⁵³ or DBL_MAX.
For context, the tested large-α settings showed broadly filled, chaotic-looking orbits (sampled α = 5…10¹⁵: coordinate standard deviation ≈ 1.0, grid-fill ≈ 58%). Those finite measurements do not establish ergodicity throughout |α| < 2⁵³. The filled-disk + small-astroid form appears only at small α (e.g. α ≈ 1.68), nowhere near the 2⁵³ wall; the region is not a distinct mathematical-dominant regime bounded at 2⁵³.
The 2⁵³ landmark · OMPHALOS
The mantissa-integer cliff: beyond 2⁵³, binary64 can no longer represent every consecutive integer. That arithmetic fact does not by itself force a transition in the real-valued product α·y. The recorded comparison found no change in concentration statistics across 2⁵³. The geometric impression depends in part on Egregore’s rendering pipeline. The name OMPHALOS belongs to a historical geometric-organization impression in those renders: a squircle envelope, an axial 4-cusp astroid (axes aligned to x/y, not 45°), a 45° diagonal transit line (the swap-symmetry eigenaxis), and ambient dust.
Named for the Delphic ὀμφαλός (navel-stone), the world’s center in the Delphic tradition. Here the name is a reading of a centered, patterned form; it does not claim that the historical stone has a squircle outline.
II · The ancestral observer · AXIS_MUNDI_FREEFORM
The historical source calls sin and cos on the same huge arguments and
was built with /fp:fast. On the reproduced MSVC path those calls fuse;
above |t| = 2⁵⁴, the cosine lane empirically behaves as sin(|t|).
At mu=1, eps=0, writing s_x=sin(alpha*x) and s_y=sin(alpha*y), the
executed recurrence becomes
X = s_y + sign(x)s_x, Y = s_x - sign(y)s_y.
Same-sign quadrants fill the diamond. Opposite-sign quadrants collapse
exactly onto the two diagonals. Their returns generate the cardinal axes,
parallel rays, central lattice cycle, and faint caustic arcs. Historical
seeds span [-3,3], so partial onset begins near 2⁵⁴/3; this explains
why several nearby tuples reveal subtly different Axis forms.
III · The overflow observer · POLAR / HUSK
This is not the true Axis. At literal DBL_MAX, products outside the finite
square |state| <= DBL_MAX/alpha overflow. Discarding those trajectories
turns the renderer into an open dynamical system with a square escape
aperture. The later density ladder retained a sparse husk after that escape;
adequate arbitrary precision retained the filled disk. POLAR SWAP therefore
names a representational-ceiling ecology, not an exact-map bifurcation.
Inputs with |α| > DBL_MAX are not representable as finite binary64 α and therefore belong outside this observer’s domain.
Reading
Three layers must now remain separate: accurate trigonometry gives the two-phasor disk; ancestral MSVC fast-sincos gives AXIS_MUNDI_FREEFORM; literal DBL_MAX plus finite-value filtering gives the POLAR/HUSK escape ecology. None is a Penrose compactification, and only the first describes the exact map independently of its execution substrate.
Under adequate arbitrary precision, the tested orbit statistics remain uniform through both historical thresholds. That result is preserved. The recovered Axis adds the missing observer contract: compiler flags, math-library branch, stream seeding, short deposition window, and raster.
The observer and numeric encoding are part of a rendered execution, but they are not the exact map itself.
:: ARKHEIN ::
Lineage
Those who arrived at the same question before. This is not the first arrival.
- 1985
IEEE 754
The standard codifying double-precision floating-point. The 2⁵³ consecutive-integer cliff and DBL_MAX exponent ceiling are properties of binary64. Historical Egregore sessions named cards observed around those execution limits.
- 700 BC
Delphic Omphalos
The navel-stone at the Pythian sanctuary, marking the world’s center. The first historical threshold card was named OMPHALOS after this stone; its rendered squircle supplied the visual correspondence.
Orientation under μ → −μ
It changes orientation, not membership in the family. The true fossil is
mu=+1; the preserved mu=-1 card is its reflected sibling. Accurate
trigonometry gives conjugate disk measures at both signs, while the
ancestral fast-sincos branch gives conjugate Axis observers. This agrees
with the existing mu-flip result without promoting either image to an
exact-map attractor phase. The Penrose null remains intact at both signs.
:: APPARATUS ::
Sources for this page
-
Cosmopoiesis · living edition
The public kernel definition, parameter meanings, iteration protocol, and dynamical vocabulary used throughout the scholia.
Read the live canon -
Mathematica · living edition
The canonical constants, algebraic identities, and computed closures used by the engine and its derived findings.
Read the live canon