[Sad Catapella N1][Scientific precision] 'Exact' int64/uint64 histogram becomes inexact when representatives are stored as float64 #28

Scope / classification

Normalization Batch 1 at f784c95e3049483f87466150ffc5764375fa631d. Confirmed numerical precision limitation conflicting with the 'exact' language, downstream fit policy requires decision. No full-repository tests executed.

Evidence

• PlaneHistogram.representatives, lines 159–169 makes an np.float64 array and computes integer representatives using self.binning_min + (a+b)/2. The true singleton value cannot be represented for every int64/uint64 integer larger than 2**53.
• PlaneHistogram.is_exact, lines 174–207 reports exactness based only on integer bin widths, ignoring actual representability of values in the float64 downstream model; max representative error is reported as 0.
• PooledHistogram.__init__, lines 107–136 takes those rounded representatives and percentile(), lines 166–170 returns the rounded float64, causing an objectively incorrect 'exact' percentile for 64-bit input.
• Tests only check that full-range int64/uint64 counts sum correctly and representatives are finite, not that large integer quantiles are exactly preserved.

Counterexample

For np.array([2**53+1], dtype=np.uint64), the histogram has one occupied singleton bin and reports is_exact=True, but its representative becomes 9007199254740992.0, not 9007199254740993. An isolated NumPy float-conversion probe confirmed this loss. The counts and integer bin boundaries themselves can remain correct; the numerical downstream summary is inexact.

Impact / policy

Incorrect quantile value and false zero-error guarantee for high-bit-depth integers; grouping nearby representable and unrepresentable values may coalesce. Exact float output for arbitrary uint64 values is impossible, so either retain exact integer order-statistic values in a dtype-aware representation or explicitly qualify 'exact' as counts/partition only with accurate summary error bounds/refusal for nonrepresentable outputs.

Fix / acceptance

Choose a versioned contract policy, preserve exact 64-bit bin identities and quantile ordering where feasible, document any intentional float64 approximation for normalization parameters, and ensure error reporting never claims 0 if a represented singleton differs from the pixel. Test 2**53 ±1, near signed/unsigned extrema, adjacent integers collapsing to one float, single-bin and pooled quantiles, CSV round-trip, and exactness flags. No Numba benefit until numerical semantics are defined.