C - Intercalation & Expansion-Joint Tables

How Solarian Keeps Civil Calendars Aligned to Real Orbits

C.1 Purpose

Appendix C defines the canonical Solarian calendar profiles. This appendix explains how those calendars remain aligned to their physical orbits.

There are two separate questions:

1. Which civil year form applies — Common or Leap?

2. Where does the corresponding week difference go inside that year?

Solarian treats them separately.

The first is determined from the underlying astronomy. The second is fixed by the implementation profile.

In short:

Astronomy selects the year form.
The profile selects the internal structure.

That distinction makes the calendars astronomically anchored without requiring users to think about astronomy every time a year changes.

C.2 Common and Leap forms

Every standard Solarian astronomical profile uses two adjacent whole-week year lengths.

One contains an even number of weeks and is called the: Common Year

The other contains an odd number and is called the: Leap Year

Leap therefore describes parity, not greater length.

Mars, for example, uses: 96-week Common Years

And: 95-week Leap Years

Uranus similarly has a Leap form that is one week shorter than its Common form.

The two forms are sufficient because a physical orbital year that falls between two whole-week lengths can be represented over time by an appropriate mixture of those adjacent values.

No third civil year form is required.

C.3 The deterministic year-selection rule

Solarian does not define arbitrary rules such as: every tenth Venusian year is Leap.

Those ratios are useful summaries, but they are not the underlying rule.

The physical Solarian New Year occurs when the world crosses the common heliocentric reference direction: λ₀ = 100°

That crossing will almost never occur exactly on a permitted civil week boundary.

For each physical crossing, the implementation therefore selects: the nearest permitted local week boundary

For a standard-day world, these are the Monday boundaries of the Solarian Week Index.

For Mars, they are boundaries of its continuous seven-sol local week.

If a crossing were exactly halfway between two permitted boundaries, the later boundary would be selected.

The interval between two consecutive selected boundaries determines the civil year length.

That length will be one of the profile's two approved forms and therefore identifies the year as Common or Leap.

For example:

Earth → 52 or 53 weeks
Mars → 95 or 96 Martian weeks
Jupiter → 618 or 619 weeks
Saturn → 1,536 or 1,537 weeks

The exact future sequence is therefore generated from the orbit rather than imposed by a separate mnemonic cycle.

C.4 Bounded quantisation

Rounding an astronomical crossing to its nearest week boundary introduces deliberate civil approximation.

The maximum New Year displacement is: half one local week

For standard-day calendars, that is approximately: 3.5 days

For Mars: 3.5 sols

This is the price of preserving complete weeks. The alternative would be to break the week in order to follow the astronomical event more precisely.

Solarian deliberately chooses not to do that.

C.5 Expansion joints

Once the year form is known, its internal structure is predetermined.

In a synchronized calendar, the normal minth contains four weeks.

A designated expansion-joint minth may instead contain: 3 weeks

Or: 5 weeks

The primary expansion joint is: Minth L — the final minth of a mear

A contraction is written: L− and changes L from four weeks to three.

An expansion is written: L+ and changes L from four weeks to five.

Where the correction exceeds the available Minth-L capacity, secondary joints may be used according to the distribution rules in the Solarian Protocol.

Earth is the principal canonical example.

C.6 Canonical synchronized patterns

The following table contains the fixed Common and Leap structures for the canonical synchronized profiles.

M3 L+, for example, means that Minth L of Mear 3 expands from four weeks to five.

The general pattern is worth noticing. Common and Leap forms are normally almost identical. The difference between them is one week and therefore usually one adjusted minth.

C.7 The simple profiles

Several canonical calendars need almost no machinery at all.

Mars

The R2 nominal structure is exactly: 96 weeks So a Common Year contains: 48 + 48 weeks across its two mears. A Leap Year contracts only the final Minth L: 48 + 47 The final Martian orbital month therefore contains seven weeks rather than eight.

Ceres

The nominal R5 structure is: 240 weeks So the Common Year requires no adjustment. The Leap Year expands only the final Minth L of Mear 5, producing: 241 weeks

Saturn

The R32 nominal structure is: 1,536 weeks Again, the Common Year needs no correction. The Leap Year simply expands the final Minth L of Mear 32.

These profiles illustrate the mechanism in its cleanest form: normal structure most of the time; one controlled week where needed.

C.8 Earth and secondary joints

Earth is structurally R1, but its approximately 52-week year sits several weeks above the nominal 48-week mear.

A Common Year therefore uses: C+ · F+ · I+ · L+

Giving:
4 · 4 · 5 | 4 · 4 · 5 | 4 · 4 · 5 | 4 · 4 · 5

This produces four thirteen-week quarters.

The Leap Year adds one further expansion at Minth K:
4 · 4 · 5 | 4 · 4 · 5 | 4 · 4 · 5 | 4 · 5 · 5

Earth therefore demonstrates why secondary expansion joints exist.

The underlying R1 structure survives even though Minth L alone cannot absorb the full difference.

C.9 Distributed outer-world corrections

Larger Repeat Factors provide many possible Minth-L joints through which small corrections can be distributed.

Jupiter

R13 has a nominal length of: 624 weeks

The actual year requires contraction.

The Common form contracts Minth L in Mears: 2, 4, 6, 8, 10 and 12 producing 618 weeks.

The Leap form omits the final contraction and therefore contains: 619 weeks

Uranus

The R91 nominal structure is: 4,368 weeks

The 4,384-week Common form requires sixteen expansions, distributed through the year.

The 4,383-week Leap form uses fifteen.

The Common form's expanded mears are:

5, 10, 16, 21, 26, 32, 38, 43, 49, 55, 61, 67, 73, 79, 85, 91

The Leap form uses the same sequence except for Mear 91.

Neptune

R179 begins from: 8,592 weeks

The 8,598-week Common Year expands Minth L in Mears:

14, 44, 74, 104, 134 and 164

The 8,599-week Leap Year adds: Mear 179

Only six or seven minths out of 2,148 are therefore different from their normal four-week length.

Large orbital years do not require more kinds of calendar machinery.

They simply give the same machinery more places to operate.

C.10 Independent calendars

Venus and Mercury do not use Repeat Factors.

Their human operating calendar remains: 12 four-week minths = 48-week mear without astronomical adjustment.

Only the separate orbital calendar intercalates.

Venus

Common astronomical year: 32 weeks

Month pattern:

3 · 3 · 2 | 3 · 3 · 2 | 3 · 3 · 2 | 3 · 3 · 2

Leap astronomical year: 33 weeks with the distinguishing week placed in Month 12:

3 · 3 · 2 | 3 · 3 · 2 | 3 · 3 · 2 | 3 · 3 · 3

Mercury

Common astronomical year: 12 weeks with twelve one-week orbital months.

Leap astronomical year: 13 weeks with Month 12 expanded to two weeks.

In both cases, the human mear continues unaffected across astronomical New Year.

That is the defining feature of the independent regime.

C.11 Pluto receives no standard intercalation profile

Pluto's candidate R270 arithmetic could support: 12,938-week Common Years

And: 12,937-week Leap Years through twenty-two or twenty-three Minth-L contractions.

But the arithmetic is not the problem. The standard twelve-month astronomical layer fails the required orbital-phase test.

Solarian Protocol v1.0 therefore does not issue Pluto with a standard intercalation profile.

Its human operating calendar remains available. Its astronomical representation requires specialised treatment.

C.12 Long-run ratios are descriptions, not rules

The mean orbital periods produce useful approximations such as:

Earth — roughly one 53-week year every 5–6 years
Mars — roughly 18 Common : 17 Leap
Ceres — roughly 4 Common : 1 Leap
Jupiter — roughly 1 Common : 16 Leap
Saturn — approximately 47% Common : 53% Leap
Venus — roughly 9 Common : 1 Leap
Mercury — roughly 3 Common : 4 Leap

These are helpful for understanding the calendars. They are not the authoritative leap algorithms.

The actual sequence follows the physical λ₀ crossings and permitted whole-week boundaries.

That avoids inventing a different arbitrary arithmetic leap cycle for every world.

C.13 Published schedules

Ordinary software should not need to calculate planetary ephemerides in order to display next year's calendar.

Canonical implementation profiles should therefore publish precomputed Common/Leap schedules for practical use.

Those schedules are generated data.

They are not the defining rule.

The full reference data, machine-readable profiles, generated schedules and implementation test vectors belong in the accompanying Solarian Technical Record.

That material can be updated and versioned independently of this book while preserving the normative protocol and historical calendar records.

C.14 The mechanism in four steps

A Solarian astronomical calendar therefore runs as follows.

1. Measure the orbit. Identify the physical λ₀ crossing.

2. Round at the week. Assign the crossing to the nearest permitted civil week boundary.

3. Select the year form. The interval between successive civil New Years determines whether the year is Common or Leap.

4. Apply the fixed profile. Use the published expansion-joint or orbital-month pattern for that year form.

There is no stray day. There is no weekday reset. There is no universal planetary leap formula pretending all orbits are the same.

The astronomy chooses between two permitted whole-week forms. The profile decides where the one-week difference goes.

Measure the orbit.
Round at the week.
Distribute the difference.
Preserve the structure.