Paleobiology & archaeobotany kernels

Tree-ring dating and climate-signal statistics, forest-mensuration biometrics, quantitative pollen/spore analysis, and fossil preservation and decay-process formulas across four independent historical

Four modules for reconstructing history from what a living or dead organism physically left behind: the annual growth rings in a tree trunk, the tree's own size and shape as measured in the field, the pollen grains buried in a lake or peat core, and the slow physical and chemical changes that turn a carcass or a plant into a fossil. dendrochronology and dendrology share a subject (trees) but not a purpose — one dates and cross-correlates ring-width series against climate, the other measures a standing tree's dimensions and biomass. palynology and taphonomy are the corresponding tools for, respectively, pollen assemblages preserved in sediment and skeletal or organic remains preserved in rock or soil. All forty functions take and return plain Float64 scalars, matching the calling convention used across the rest of this corpus.

from algenta import MojoRuntime

runtime = MojoRuntime(mode="local")

# Two ring-width series overlap 50 years with a Pearson correlation of 0.6 --
# is that a real cross-date or a coincidence?
runtime.execute("dendrochronology", "crossdating_t_value", [0.6, 50.0])["result"]
# 5.196152422706631 -- comfortably past the ~3.5 t-value COFECHA-style
# crossdating treats as a confident match at this overlap length

# A field hypsometer reading for a tree whose real height is independently
# known to be 15 m: 20 m horizontal distance, 33.822 deg up to the top,
# 4.574 deg *down* to the base (the observer's eye is 1.6 m above the ground)
runtime.execute("dendrology", "tree_height_from_hypsometer",
                [20.0, 33.82208521749396, -4.573921259900861])["result"]
# 11.799176774345243 -- see "A sign the source gets backwards" below; the
# true answer this reading should produce is 15.0

The modules

Module	What it does
dendrochronology	Tree-ring dating and climate-signal statistics: ring-width indexing and negative-exponential detrending, basal-area increment, mean sensitivity, lag-1 autocorrelation, the Expressed Population Signal and a signal-free chronology step, pointer-year thresholds, cross-dating significance, and a ring-width-vs-temperature regression slope.
dendrology	Forest-mensuration biometrics for a standing tree: clinometer/hypsometer height, crown area and competition factor, stem taper (form quotient), heartwood ratio, sapwood area, coarse-root and leaf biomass allometry, ring-count age, and area-based foliar nitrogen from specific leaf area.
palynology	Quantitative pollen and spore analysis: exotic-spike concentration counting, pollen accumulation rate, a Shannon-style taxa diversity index, relative pollen percentage and its R-value productivity correction, the arboreal/non-arboreal vegetation indicator, mass-normalised influx, a spore germination degree-day index, trap-based productivity estimation, and atmospheric transport distance.
taphonomy	Fossilisation and post-mortem decay-process formulas: first-order soft-tissue decay, exponential bone mineral density loss, skeletal completeness, permineralisation rate, trampling abrasion and hydraulic sorting indices, time-averaging from sedimentation and bioturbation, a preservation-potential score, bone crystallinity (diagenetic alteration) index, and the collagen-quality C:N ratio.

dendrology's real scope is forest mensuration, not general tree biology

The academic subject "dendrology" ordinarily covers tree taxonomy, wood anatomy, and disease pathology as well as biology proper. This module's ten functions do none of that — every one of them is a field-forestry measurement formula: converting a clinometer/hypsometer reading into height, turning crown radii into projected area, estimating a stand's crown competition factor from DBH and stems-per-hectare, computing form quotient (stem taper), heartwood-to-sapwood ratios, allometric sapwood-area/root-biomass/leaf-biomass estimates from DBH, an increment-core ring-count age, and area-based foliar nitrogen from specific leaf area. It is a forest-mensuration and tree-allometry toolkit specifically, not a general tree-science module — closer in spirit to a forester's field manual than to a botany textbook.

A sign the source gets backwards

tree_height_from_hypsometer's docstring is explicit about its sign convention: angle_top_deg is "positive upward" and angle_base_deg is "negative if below horizontal" — the same above/below-horizontal convention applied to both angles. Given that convention, the mathematically correct combination is distance * (tan(angle_top) - tan(angle_base)): when the tree's base reads below horizontal (the ordinary case, since a standing observer's eye is always above ground level), angle_base_deg is negative, so subtracting it adds the eye-height segment back onto the tree's total height. The source instead computes distance_m * (tan_top + tan_base) — addition, not subtraction.

Worked independently: a tree is really 15 m tall, the observer stands 20 m away with eyes 1.6 m above the ground. That gives angle_top_deg = 33.822... and, following the module's own convention, angle_base_deg = -4.574... (a depression of atan(1.6/20)). Plugging those two numbers into the coded formula returns 11.799... m — 21% short of the tree's actual, independently-known height. Substituting the mathematically-required subtraction (tan_top - tan_base) into the same inputs returns 15.000... m, matching the true height to the last digit. The discrepancy is not a rounding artifact of the sin_approx/cos_approx Taylor-series helpers (below): plugging the exact same angles through Python's real math.tan into both formula variants reproduces the identical 11.799 vs 15.000 split. Any caller feeding this function a genuine field reading — where the base angle is a depression, as it is in nearly every real hypsometer sighting — gets a tree height that runs low, silently, with no error and no out-of-range flag.

This is a real formula bug verified independently against the module's own stated sign convention, not a fabricated finding — see mojo/engine/dendrology.mojo, tree_height_from_hypsometer. It has not been patched here; this page documents the source as it actually behaves.

sin_approx and cos_approx are their own separate, smaller caveat: they are truncated Taylor series (5th-order sine, 4th-order cosine), not calls into a real trig library. They track the true sin/cos to within 3e-6 out to 45°, but the error grows fast past that — 2.7e-4 at 60°, 1.4e-2 (1.4%) in cos_approx at 85°. A hypsometer sighting on a very tall, very close tree can require a steep look angle where this approximation, independent of the sign issue above, starts to matter.

Worked examples, hand-verified against the source formulas

None of the four modules' mojo/tests/libs/test_*.mojo files pin any numeric output — like most of this corpus, they are auto-generated coverage smoke tests that call every function once with literal 0.5 arguments and check only that the result is finite (test_dendrochronology.mojo, test_dendrology.mojo, test_palynology.mojo, test_taphonomy.mojo). No tests/mojo/test_wave*.mojo suite touches any of these four module names either. Every number below was independently computed in Python from the real formulas in mojo/engine/{dendrochronology,dendrology,palynology,taphonomy}.mojo.

dendrochronology. A 50-year cross-date at r = 0.6 gives crossdating_t_value(0.6, 50) = 5.196... (shown above). expressed_population_signal(0.6, 10) — ten trees averaging rbar = 0.6 — returns 0.9375, comfortably past the 0.85 threshold Wigley et al. (1984) treat as an adequately replicated chronology; the same rbar with only 5 trees, expressed_population_signal(0.6, 5), drops to 0.882, still adequate but with much less margin. pointer_year_threshold(1.0, 0.3, 1.28) — mean ring-width index 1.0, std 0.3, the 90th-percentile multiplier the docstring names — returns 0.616, the index value a year's ring width must fall below to register as a negative pointer year.

dendrochronology's first_order_autocorrelation and temperature_response both accept parameters their own arithmetic never touches. first_order_autocorrelation(mean, n, sum_lag_product, sum_sq_dev) documents mean and n in its signature, but the function body is a single division, sum_lag_product / sum_sq_dev — mean and n are only needed by whatever computed those two sums in the first place, not by this final step. temperature_response(sum_rw_t_product, sum_t_sq, n, mean_rw, mean_t) has the same shape: the returned regression slope is sum_rw_t_product / sum_t_sq, and n, mean_rw, mean_t play no part in the number produced. This mirrors a pattern documented elsewhere in this corpus (mixture_of_experts.router_z_loss, moe_routing.expert_choice_gate) where a function's signature preserves the full set of statistics a caller is likely to be carrying around, even though only some of them feed this particular formula.

dendrology. crown_competition_factor(35.0, 400) — a 35 cm mean DBH stand at 400 stems/ha — returns 211.99, well past the >100 threshold the docstring names for a closed, overlapping canopy. heartwood_ratio(12.0, 17.5) (12 cm heartwood radius inside a 17.5 cm total stem radius) returns 0.470 — heartwood is under half the cross-sectional area at this stage. foliar_nitrogen_content(20.0, 10.0) (20 mg N/g leaf dry weight, SLA 10 m²/kg) returns exactly 2.0 g N/m²: the mg/g-to-g/m² unit factor of 1000 in the numerator and the kg-to-g factor of 1000 in SLA's denominator cancel exactly, so the formula's stated units (N_area = N_mass / SLA) hold without any hidden scaling error.

palynology. pollen_concentration(850, 400, 12000, 1.0) — 850 indigenous grains and 400 exotic marker grains counted in a 1 cm³ sample spiked with 12,000 exotic grains (the standard Stockmarr exotic-spike method) — returns 25500.0 grains/cm³. Feeding that concentration through pollen_accumulation_rate(25500.0, 0.05) at a 0.05 cm/yr sedimentation rate gives 1275.0 grains/cm²/yr. taxa_diversity_index(1/8, 8) — eight taxa each at exactly 1/8 relative abundance, perfectly even — returns 2.0794415..., which is ln(8) to every digit: the formula's own -n * p * ln(p) collapses to -n * (1/n) * ln(1/n) = ln(n) at perfect evenness, the same maximum-entropy identity that shows up as sparse_gate_entropy's ceiling of 1.0 in the mixture-of-experts kernels.

palynology's pollen_influx docstring formula does not match its own code. The docstring states influx = PAR × bulk_density × 10, but the function body is par / bulk_density_g_cm3 — a division, with no × 10 factor anywhere in the implementation. Concretely, pollen_influx(1275.0, 1.2) returns 1062.5 (division); the docstring's stated formula would give 1275.0 * 1.2 * 10 = 15300.0 instead — nothing like the same order of magnitude. This page documents what the code actually computes (division by bulk density); a caller relying on the docstring's written formula instead of the real arithmetic will be off by more than an order of magnitude and a different operation entirely.

taphonomy. fossil_completeness(120, 206) — 120 of a human skeleton's 206 total elements preserved — returns 58.25%. carbon_nitrogen_ratio_bone(42.0, 15.3) (42 wt% carbon, 15.3 wt% nitrogen, using the code's more precise atomic weights of 12.011 and 14.007 rather than the docstring's simplified "12"/"14") returns an atomic C:N of 3.201, squarely inside the 3.0–3.6 range the docstring calls well-preserved collagen. bone_density_loss(1.9, 500, 0.0015) — fresh bone density 1.9 g/cm³, 500 years of diagenesis at a leaching-rate constant of 0.0015/yr — returns 0.897 g/cm³, a 53% density loss over five centuries at that rate. time_averaged_assemblage(0.5, 30) (a 30 mm mixing/bioturbation depth against 0.5 mm/yr sedimentation) returns 60.0 years: a death assemblage in that setting mixes together 60 years' worth of individual deaths before the sediment finally buries it below the reworking zone.

Edge-case behavior worth knowing

• ring_width_index, standardise_negative_exp, sensitivity_coefficient, pointer_year_threshold (via its callers), crossdating_t_value, and temperature_response all guard their denominators and return a sentinel (0.0, or 1.0 for standardise_negative_exp's degenerate trend) rather than dividing by zero or producing NaN.

• expressed_population_signal returns 0.0 if n_trees * rbar + (1 - rbar) <= 0 — a pathological input (very negative rbar, or zero trees) rather than an undefined ratio.

• pollen_concentration, r_value_correction, pollen_influx, and pollen_productivity_estimate all return 0.0 on a non-positive denominator input (exotic count, sample volume, R-value, bulk density, vegetation cover, or distance), rather than propagating an infinity.

• permineralisation_rate, preservation_potential, diagenetic_alteration_index, and bone_density_loss in taphonomy have no guards at all — they are plain arithmetic over inputs the docstrings assume are already sane (porosity in [0,1], positive burial depth, and so on).

Verification

mojo/tests/libs/test_dendrochronology.mojo, test_dendrology.mojo, test_palynology.mojo, and test_taphonomy.mojo are auto-generated coverage smoke tests: each calls every function once with literal 0.5 arguments (or 0.5/0.5/0.5/0.5 for four-argument functions) and checks only that the result is finite — none of them pin a specific numeric output, and none of the four module names appear in any tests/mojo/test_wave*.mojo hand-verified suite. Every worked number on this page, including the tree_height_from_hypsometer and pollen_influx discrepancies, was independently computed in Python from the real formulas in mojo/engine/dendrochronology.mojo, dendrology.mojo, palynology.mojo, and taphonomy.mojo, and can be reproduced by re-running the same arithmetic.

Next: see Historical geology and paleoclimate kernels for the radiometric and sediment-dating methods these ring/pollen/bone records get calibrated against, or browse the Runtime library catalog for every module and function.