GENSOKYO’S SANZU RIVER PROBLEM

According to Bohemian Archive in Japanese Red/Ran, Ran Yakumo stated that:

"The Sanzu River is not a normal river, but one that carries away the past. This is why the actual length is different from the observed length. It is necessary to know the histories of the deceased to calculate that distance. The breadth of the river, depending on the person, can take either a mere instant to cross or a practically interminable amount of time. Humans may not know what factor determines this, but it has been said to be related to the ferryman's fee. The more you pay the ferryman, the shorter the distance. Plotting the curve of this inversely proportional function, we see that the closer to zero the ferryman receives, the closer the distance approaches infinity. Conversely, the more the ferryman is paid, the closer the distance approaches zero."

To interpret her statement:

• -The river’s actual width differs from the observed length and depends on the deceased’s history.
• -The width (D) varies inversely within the ferryman’s fee. The more you pay, the shorter the width. In other words, as payment nears zero, the width nears infinity.
• -The fee itself is an emergent quality determined by many factors, which is metaphysical.

To compute the mapping from history of the deceased to D requires many specialized rules, as stated by Aya Shameimaru in Bohemian Archive in Japanese Red/Ran:

“Humans cannot bring their fortune from this world to the next when they die. That is why it seems the amount paid is determined by other factors. From here on, the calculations take into account factors such as the histories of the deceased, where several thousand formulae are needed to solve the equation.”

VARIABLES

These are the variables that would be used and what they mean.

D -- Actual Sanzu width (the time or effective distance a soul experiences while crossing). --Units: Kilometers.

𝑊 -- Effective ferryman fee (the payment that shortens crossing). --Units: Soul Currency Units (SCU) conceptual; not earthly money.

H -- History factor (a scalar summarizing how much past the soul carries). --Units: Dimensionless, 𝐻 ≥ 0.
--H ∝ W

𝛼-- scale constant (converts non-dimensional ratios into CTU). --a > 0

β>0 -- sensitivity exponent (how strongly W reduces D). --𝛽 = 1 = perfect simple inverse proportionality; 𝛽 ≠ 1 = power-law scaling.

W0 ≥ 0 -- baseline fee / regularizer (small constant to avoid division-by-zero singularities.).

η -- random residual or noise (represents the atmosphere of the river or other minor, passive factors. Could be zero for deterministic analysis.)

Variables used to define H and W:

A: attachment score (dimensionless)

R: regret/unfinished business score

M: memorial/rites performed by the living (SCU-equivalent)

S: social memory / reputation score

G: guilt / moral debt score

𝑆 = ( 𝑆1 , 𝑆2... 𝑆n ) vector of life aka history features.

Canonical Formula.

The simplest canonical formula that exactly matches Ran’s qualitative rules is:

𝐷 = 𝛼 ⋅ 𝐻 / (𝑊 + 𝑊0 )𝛽 + 𝜂

with 𝛼 > 0, 𝛽 > 0, 𝐻 ≥ 0, 𝑊 + 𝑊0 > 0.

Recommended to use 𝜂 for stochasticity (having a random probability distribution or pattern that may be analysed statistically but may not be predicted precisely). Set 𝜂 = 0 for deterministic proofs.

Behaviour: As 𝑊 increases, denominator increases → D decreases.

As W→0 and 𝑊0 = 0, denominator → 0 and 𝐷 → ∞ (which matches Ran’s statement of the river’s width is capable to reach till infinity.)

H acts multiplicatively to scale difficulty by life-history.

Proofs for Ran’s claims

For clarity, set 𝜂 = 0 for during proofs to preserve accuracy.

Claim A -- Inverse behavior: D decreases with 𝑊

Proposition A (Monotonicity). For fixed 𝐻 > 0 , 𝛼 > 0 , 𝑊0 ≥ 0 , 𝛽 > 0, the function

𝐷 ( 𝑊 ) = 𝛼 𝐻 / ( 𝑊 + 𝑊0 )𝛽

is strictly decreasing in W over 𝑊 > − 𝑊0.

Compute derivative with respect to 𝑊:

𝑑𝐷 / 𝑑𝑊 = 𝛼𝐻 ⋅ d/𝑑𝑊 ( ( 𝑊 + 𝑊0 )−𝛽 ) = 𝛼𝐻 ⋅ (−𝛽) (𝑊 + 𝑊0)−𝛽−1

Since 𝛼 > 0 , 𝐻 ≥ 0 , 𝛽 > 0 , ( 𝑊 + 𝑊0 )− 𝛽 − 1 > 0, we have

𝑑𝐷 / 𝑑𝑊 = − 𝛼𝛽𝐻 ( 𝑊 + 𝑊0 )− 𝛽 − 1 ≤ 0,

and strictly negative when 𝐻 > 0. Therefore D is strictly decreasing in 𝑊 W.

This proves the textual claim “more you pay, shorter the distance.”

Claim B -- Limits: 𝑊 → 0 gives 𝐷 → ∞; 𝑊 → ∞ gives 𝐷 → 0

Proposition B.1 (Limits). For 𝐻 > 0 , 𝛼 > 0 , 𝛽 > 0, with 𝑊0 = 0:

lim 𝑊→0+ 𝐷( 𝑊 ) = ∞ and lim 𝑊→∞ 𝐷( 𝑊 ) = 0

If 𝑊 0 = 0,

As 𝑊 → 0+, 𝑊−𝛽 → ∞ (since 𝛽 > 0 ), so 𝐷 → ∞.

As W→∞, 𝑊 −𝛽 → 0, so 𝐷 → 0

If 𝑊0 > 0 then lim 𝑊 → 0+ 𝐷 = 𝛼𝐻 / 𝑊0𝛽 (finite), but Ran’s statement implies singular behavior, thus 𝑊0 is canonically 0 for exact canon behavior.

This proves Ran’s statement that distance → ∞ as fee → 0 and distance → 0 as fee grows very large.

Claim C -- Invertibility: recover W from 𝐷

If 𝛽 > 0 and H>0, D(W) is strictly decreasing and continuous on

𝑊 > − 𝑊0, so it is invertible onto (0,∞) or (αH/W0β ,∞) if 𝑊0 > 0).

Inverse formula (explicit): For 𝜂 = 0,

𝐷 = 𝛼 𝐻 / ( 𝑊 + W0)𝛽 ⟹ W = ( 𝛼𝐻 / 𝐷 )1 / 𝛽 − 𝑊0.

(This formula is valid for 𝐷 > 0.)

To rearrange algebraically: ( 𝑊 + 𝑊0 )𝛽 = 𝛼𝐻 / 𝐷 ⇒ 𝑊 + 𝑊0 = (𝛼𝐻/𝐷 )1 / 𝛽

This can be interpreted as one can compute the required payment W to get a desired crossing D.

Claim D -- Sensitivity / elasticity

We want to quantify how sensitive D is to small fractional changes in W. Define elasticity 𝜀𝐷,𝑊 = ∂𝐷 / ∂𝑊 ⋅ 𝑊/𝐷.

Compute:

∂W / ∂D = −αβH(W+ 0 )−β−1

and

D = αH(W+W0 )−β .

Thus

εD,W = (−αβH(W+W0) −β−1 ) ⋅ W / αH(W+W0)−β = −β⋅ W / W + W0

Key observations:

If 𝑊0 = 0, εD,W = −β (constant). For β=1, a 1% increase in 𝑊 reduces D by 1%.

If 𝑊0 > 0, elasticity magnitude is smaller in the small-W regime because 𝑊 / ( 𝑊 + 𝑊0 ) < 1

This gives a clear quantitative measure of the question “how much does paying more help?”.

Constructing H and W from life-history: forms and examples

According to Aya Shameimaru’s statement, one need thousands of formulae. That’s because H and W are compressions of a very high-dimensional life narrative.

Below are the mathematically plausible constructions.

Linear aggregator (simple, interpretable) Let life features 𝑆 = ( 𝑆1 ,...., 𝑆n ) Choose non-negative weights 𝜃𝑖.

𝐻 = 𝐻0 + n∑ 𝑖 = 1 𝜃𝑖𝑆𝑖

Typical features: S1: unresolved attachments S2: number of harmed people S3: depth of grief Sn: all life features and events.

Similarly for W:

𝑊 = 𝑊0′ + m∑ j=1 𝛾𝑗M𝑗,

where Mj are memorials, rituals, atonements, social memory measures. But linearity is often not enough because life events interact nonlinearly.

Nonlinear / interaction model

One may include terms like: 𝐻 = 𝐻0 + ∑𝑖 𝜃𝑖𝑆𝑖 + ∑ 𝑖<𝑗 𝜃𝑖𝑗 𝑆𝑖𝑆𝑗 + ...

Interaction terms quickly blow up combinatorially. If 𝑛 = 20 n=20 features, the number of possible interaction subsets is 220 − 1 = 1,048,575, which is over a million so the statement ‘thousands of formulae’ is kinda mild.

Piecewise rules (cultural/legalistic) Certain cultural rules might say: if you are a priest and you performed ritual x, then your W gets a multiplier; if you committed crime y, H incurs a non-linear penalty. These are naturally encoded as case-by-case formulae.

Worked numeric examples (step-by-step arithmetic)

I will use the simple canonical case: α=1, β=1, W0=0, η=1, so D=H/W Example 1: moderate burden, moderate payment

H=3.0

W=0.5

Compute D=H/W =3.0/0.5

3.0 ÷ 0.5 6.0 D = 6.0 km

Interpretation: Because W is small, crossing is long.

Example 2: same burden, larger payment

H=3.0H

W=10.0

Compute D=3.0/10.0

So, D=0.3

0.3D = 0.3 km

PS: Large payment shortens crossing drastically.

Summary

A mathematically faithful and provable representation of Ran Yakumo’s description is:

𝐷 = 𝛼 ⋅ 𝐻 / (𝑊 + 𝑊0 )𝛽 + 𝜂

Properties proven above:

D is strictly decreasing in W (proof via derivative).

If W0 = 0, D→∞ as W→0 (limit proof).

D→0 as W→∞

Elasticity is εD,W = −β⋅W/W+W0 (so -𝛽 when W0 = 0)

D is convex in W (second derivative positive).

The complexity Ran mentions arises because H and W must be computed from high-dimensional life histories; modeling their interactions generally requires many rules or a highly-parametric function — hence “several thousand formulae” is mathematical common sense.

all of this calculating is such a pain in the ass