Reading the notation

The symbol

Every PROTIUM unit symbol has three parts: the H prefix (the PROTIUM marker), an axis letter that encodes both which axis and which tier, and the ☉ suffix (the protium mark). The letter’s case is load-bearing — it is the 10¹⁰, not a style choice.

The four axis letters are t (time), l (distance), e (energy), and m (mass). Lowercase is base tier; uppercase is named tier. The H…☉ frame disambiguates what would otherwise collide: He☉ is energy, not helium; HM☉ is mass, not a variable M. A receiver who sees the frame knows it is a PROTIUM quantity without external context.

The exponent

All PROTIUM exponents are log₁₀ of the base-unit count. The named tier sits at exponent 10 on every axis, so tier conversion is ±10 on the exponent. For any light-travel relationship, the time and distance exponents are equal — a direct expression of c = 1.

Calculating: lower to base first

The named tier is for reading and reporting; the base tier is for calculating. Because the two tiers are the same structure at different scales — the c = 1 survives the tier change, proven on the previous page — you can always lower a named-tier expression to base, compute there, and raise the result back for presentation.

The default method is three steps:

Lower. Replace each named symbol with its base form: HT☉ → 10¹⁰ Ht☉, and likewise for the other axes. The expression is now entirely in base units.

Compute. Do the physics in base units, where every standard equation holds as written — c = 1, h = 1 He☉·Ht☉, no stray tier factors to track.

Raise. If a human-readable result is wanted, convert back to named units — a single ±10 coefficient shift, applied once, at the end.

Working this way, the awkward question of what an exponent “means” on a named symbol never arises during calculation: by step 1, every named symbol has already become a base symbol with an unambiguous exponent. And it needn’t arise when reading or reporting either — because exponents belong to the axis, not to the units. The log-scale coordinate carries exponents (Hl☉^11.85); named units carry plain multipliers (153 CLIP). A named symbol never takes an exponent in either mode: you lower it away to calculate, and you never write one to report. The base tier is where the physics lives; the named tier is how you read it.

Reading expressions

At the base tier, Planck’s constant is unity:

At the named tier, two named units multiply — each carries 10¹⁰, so the product carries 10²⁰:

This is the count rule: count named-tier symbols top and bottom of an expression. If the net count is zero (as in c = HL☉/HT☉), the quantity is tier-invariant — it reads the same at either tier. If non-zero, the leftover 10¹⁰ factors appear as a coefficient. The count rule lets you predict that coefficient — or check a conversion — without lowering the whole expression, which is why c = 1 and the unity of h at base tier both fall straight out of it.