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The sperm — physics dictates form at Re ≪ 1

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An engraved plate in six panels. THE REGIME (I) is computed on the plate rather than asserted: Re = ρuL/µ with ρ = 1.0×10³ kg/m³ (water), u = 6.2×10⁻⁵ m/s (CN06-01), L = 5.5×10⁻⁵ m (CN06-06, itself NOT-MEASURED) and µ = 7.0×10⁻⁴ Pa·s (CN06-08) gives 4.9×10⁻³ (CN06-10, MODELED); every input and product was re-evaluated for this plate and reproduces. The Stokes coast is 0.45 Å (CN06-12, MODELED), drawn to scale against the 1.54 Å carbon–carbon bond: inertia is not small here, it is absent. What is load-bearing is the EXPONENT, not the digits — Re ~ 10⁻³ and d_coast ~ 10⁻¹¹ m survive any plausible revision of the inputs. THE AXONEME (II) is drawn with NO DIMENSION AT ALL, because CN-06 carries none: the 9+2 arrangement is sourced, the metric is NOT-SOURCED (a gap in our homework, not in the literature), and the cross-section is not to scale. Two of the nine doublets are shaded as the HYPOTHESIZED active fraction (67,500 × 2/9 = 15,000 — CN06-14), an arithmetic consequence of the two-groups-of-doublets hypothesis and not an observation of which doublets are active. The Summers & Gibbons (1971) trypsin ablation is recorded with its extension factor NOT-SOURCED (CN06-16) and with an attribution correction: S&G did NOT identify what trypsin removed — that the targets are the radial-spoke heads and nexin links is later work (Witman et al.), and the ablation licenses the conclusion without licensing the mechanism being read back into the 1971 paper. THE SCALLOP THEOREM (III) is drawn as the REFUTED alternative: a reciprocal paddle whose net displacement is EXACTLY ZERO — not inefficient, zero — with the reason drawn in configuration space (one hinge = one degree of freedom = a cycle that must retrace its own path and enclose no area; Purcell's own escape adds a second hinge and the cycle encloses area). That distinction is asserted QUALITATIVELY, as CN-06 states it; no proportionality between enclosed area and displacement is printed, because the corpus carries none. The drag anisotropy all flagellar thrust is bought from (CN06-13) is printed verbatim from the ledger, with a plate note recording that the row's own formula and its own printed sub-range do not reproduce one another. The panel carries its fence: cervical mucus is VISCOELASTIC, so the theorem's Newtonian premise is violated in vivo and viscoelasticity is a named escape in Lauga (2011) — inside the theory, not a refutation of it. THE TRAVELLING WAVE (IV) is drawn to scale in λ and L only, at 13.5 px per µm, on the only MEASURED flagellar length (41 µm, CN06-07); amplitude and envelope are NOT-SOURCED and schematic. The frequency halves from 23 to 11 Hz and the wavespeed drops 4.4-fold while the swimming speed does not change — the same cells, the same 37 °C, only the fluid differs. THE HELIX (V) is drawn to scale from CN06-32 as the sampling instrument that converts a static spatial gradient into a periodic temporal signal, and the detour is priced at 33 % of path length (MODELED, computed on the plate). Its four parameters are the UNSTIMULATED BASELINE — no gradient present — and the gradient drawn behind the helix is the argument, not the condition the geometry was measured under. Species is a fence, not a detail: I and IV are HUMAN, II and V are SEA URCHIN, and CN-06 records the crossing as a first-class conflation trap. Every number carries units, a ledger row and an evidence class. THE PLATE CONTAINS ZERO UNI CLAIMS: a nature citation is never a UNI gate, and the honest program position is unchanged by it — ~2 of 11+ developmental rungs earned.

The sperm — physics dictates form at Re ≪ 1

Claims

ref symbol value units scope
u 62 (low visc) / 65 (high visc) µm/s migrating human sperm, 37 °C, n = 16 / 19
f_beat 23 (low visc) / 11 (high visc) Hz same cells, 37 °C; low-visc buffer vs ~0.14 Pa·s analogue
λ 39 (low visc) / 18 (high visc) µm human sperm flagellar wavelength, 37 °C
c_wave 890 (low visc) / 200 (high visc) µm/s wavespeed = f × λ
L_cell ~50–60 (head 4–5, midpiece ~7–8, tail ≥45) µm derived from WHO 2021 normative morphometry criteria — what counts as a normal-form spermatozoon — not a measured distribution over a population; used as an input to Re regardless
L_flag ~41 µm human flagellum, tethered-cell imaging
µ_buffer 0.7 (0.73 ± 0.01 for HTF) mPa·s aqueous buffer, 37 °C
µ_mucus ~0.14 (analogue) / ~0.2 (midcycle, Day 0) / ~0.68 (Day 5) Pa·s Maxwell fit to measured G′/G″ at ~5 Hz, 37 °C
Re 4.9 × 10⁻³ (bracket 2–6 × 10⁻³) dimensionless human sperm, ρuL/µ; ρ=10³, u=6.2e−5, L=5.5e−5, µ=7.0e−4 (SI)
Re_mucus ~2.6 × 10⁻⁵ dimensionless ρuL/µ; ρ=10³, u = 6.5e−5 (65 µm/s — the high-viscosity migrant velocity; NOT the 6.2e−5 used for Re), L=5.5e−5, µ = 0.14 Pa·s (SI)
d_coast ~0.45 Å (4.5 × 10⁻¹¹ m), τ ≈ 0.7 µs m Stokes coasting, τ=m/(6πµa); V≈17 µm³, ρ_cell≈1.1×10³, a≈2 µm
ζ⊥/ζ∥ ≈2 asymptotic only; ~1.5–1.8 at realistic aspect ratios. ζ⊥/ζ∥ = 2(ln(2λ/a) − 0.5)/(ln(2λ/a) + 0.5) — strictly < 2 for any real filament dimensionless resistive-force-theory coefficient, slender filament, far from a boundary — a derived coefficient, not a measurement
n_dynein ~67,500 total; ~15,000 active/beat motors sea-urchin sperm flagellum; active count is a hypothesis (67,500 × 2/9)
d_dynein ~8 nm/power stroke axonemal dynein, single-molecule
axoneme extension on trypsin + ATP — (the 'five or more times original length' previously printed here is withdrawn as unsourced; secondary accounts conflict: several-fold / ~7× / nine-fold) fold of original length demembranated sea-urchin axoneme, brief trypsin digestion, + ATP
n_ATP/beat (2.3 ± 0.2) × 10⁵ (low visc) → (3.2 ± 0.5) × 10⁵ (0.5 % MC) ATP/beat demembranated sea-urchin sperm axoneme, single-cell, [ATP] = 20 µM, viscosity as stated — this quantity is not viscosity-independent
r_ATP (2.4 ± 0.3) × 10⁶ active; (9.2 ± 0.2) × 10⁵ inactive ATP/s demembranated Lytechinus sperm axoneme
ε_hydro 0.004 (low visc) → 0.013 (high visc) dimensionless demembranated sea-urchin axoneme, 20 µM ATP, buffer vs 0.5 % MC
ε_chemo 0.34 → 0.6 dimensionless as above
ε_swim 0.001 → 0.008 dimensionless as above; ε_swim = ε_chemo · ε_hydro
ε_hydro,human — dimensionless intact human spermatozoon
resact sensitivity 1 bound molecule evokes a Ca²⁺ response; 50–100 saturate molecules Arbacia punctulata sperm
min. gradient 0.8 fM/µm A. punctulata, resact
T_sample 0.2–0.6 s A. punctulata, measured Ca²⁺-response latency, which the authors read as a sampling window. Equating it with Berg & Purcell's integration time T is a MODELED identification, not an observation
helix r = 8.4 ± 3.1 µm; period 0.38 ± 0.07 s; pitch 47.6 ± 9.1 µm; u = 200 ± 57 µm/s — A. punctulata, free 3-D swimming far from boundaries with NO gradient present (the unstimulated baseline), holographic tracking, n = 20, 1 s tracks
human chemotaxis progesterone → CatSper → Ca²⁺ influx replicated; progesterone as the chemoattractant disputed — human sperm
δc/c scaling ∝ (D·a·c·T)^(−1/2) dimensionless diffusion-limited chemoreception — see NA-08 for the contested prefactor
detour cost 33 % of path length (path per turn = √(pitch² + (2πr)²) = 71.1 µm buys 47.6 µm of axial progress; 47.6/71.1 = 67.0 %) % of path length Arbacia punctulata helical swim, computed on this plate from CN06-32’s four measured parameters
path speed cross-check 71.1 µm / 0.38 s = 187 µm/s, against a measured u = 200 ± 57 µm/s — inside the band µm/s internal consistency of CN06-32: geometry (r, pitch) and kinematics (period, u) were measured separately and close
ζ⊥/ζ∥ — formula vs printed range The row prints “~1.5–1.8 at realistic aspect ratios” AND the formula ζ⊥/ζ∥ = 2(ln(2λ/a) − 0.5)/(ln(2λ/a) + 0.5) at the row’s own stated ln(2λ/a) ≈ 3–5. Evaluating that formula over that range gives ≈1.43–1.64, not 1.5–1.8; reaching 1.8 needs ln(2λ/a) ≈ 9.5. dimensionless SURFACED, NOT PATCHED. Both readings sit inside the row’s OWN falsifier bracket (1.4–2.0), so the row is NOT refuted here; but the printed sub-range and the printed formula do not reproduce one another. The plate prints the ledger value verbatim and carries this note beside it.
axoneme cross-section geometry — no value carried nm (would be) CN-06 states the 9+2 ARRANGEMENT (nine doublet microtubules around a central pair) and carries NO diameter, NO doublet spacing and NO doublet size. The plate therefore draws the arrangement and asserts no metric; the cross-section is NOT TO SCALE, and doublet numbering/orientation are drawn for reference only.
flagellar beat amplitude / envelope — no value carried µm (would be) CN-06 carries no beat amplitude. The wave at IV is drawn to scale in WAVELENGTH and LENGTH only; the amplitude and the envelope shape are schematic and assert nothing.

What this plate does NOT claim

  • Not claimed: ANY axoneme dimension. CN-06 carries none. The 9+2 arrangement is drawn; the metric is NOT-SOURCED and the cross-section is not to scale. Nothing on panel II may be measured with a ruler. Doublet numbering and orientation are drawn for reference only — CN-06 carries no numbering convention.
  • Not claimed: that the scallop theorem governs the female reproductive tract. Cervical mucus is viscoelastic, the theorem's Newtonian premise is violated there, and viscoelasticity is a NAMED ESCAPE in Lauga (2011) — inside the theory, not a refutation of it. The theorem explains why the flagellum is a travelling wave; it forbids nothing in vivo, and the plate draws it as a constraint on DESIGN, not as a law of the oviduct.
  • Not claimed: any proportionality between the area a configuration-space cycle encloses and the net displacement it produces. The plate draws the zero-area/enclosed-area distinction qualitatively, exactly as CN-06 states it. The quantitative relation is not in the corpus and is not printed.
  • Not claimed: that Summers & Gibbons (1971) identified what trypsin removed. They did not. That the targets are the radial-spoke heads and nexin links, with dynein arms intact, is LATER work (Witman et al.). The ablation licenses the conclusion — sliding is not bending; constraint is the transmission — and does not license reading the mechanism back into the 1971 paper. SIGNUM SIGNUM MANET: the signal and the commentary about it travel separately. The extension factor is NOT-SOURCED (CN06-16) and is drawn without one.
  • Not claimed: that the ~15,000 active dyneins per beat is an observation. It is 67,500 × 2/9 — an arithmetic consequence of the two-groups-of-doublets HYPOTHESIS (CN06-14). The 67,500 total is OBSERVED-REPLICATED by cryo-ET; the active fraction is not. The two shaded doublets on panel II mark the hypothesis, not a measurement of which doublets are active.
  • Not claimed: that a sperm decides, wants, seeks, knows, or navigates in any sense involving experience. "EPISTEMIC ACTION" HERE IS A STATISTICAL DESCRIPTION — a policy whose selection weight includes an uncertainty-resolving term. The helix is a sampling geometry. Nothing on this plate bears on whether anything is like anything.
  • Not claimed: that human sperm perform chemotaxis. The progesterone → CatSper → Ca²⁺ physiology is replicated; whether progesterone is THE chemoattractant is disputed (CN06-33, OBSERVED-CONTESTED) and is carried as contested. Panel V is SEA URCHIN and its numbers do not transfer.
  • Not claimed: that sperm sample temporally BECAUSE they are too small to read a gradient across the body. That is the folk version — HYPOTHESIZED — and the measurements point elsewhere: Kashikar et al. report electrotonic blurring (CNGK distributed along the flagellum; hyperpolarisation spreads in milliseconds), and Tan & Chiam (2018) conclude the choice turns on speed-to-size, not size — sperm are relatively BIG and sample temporally because they are FAST. Three claims, three evidence classes, never merged.
  • Not claimed: that the 0.2–0.6 s figure is Berg & Purcell’s integration time T. It is a MEASURED Ca²⁺-response LATENCY which the authors read as a sampling window; equating it with T is a MODELED identification that assumes the latency is set by counting statistics rather than transduction kinetics (CN06-30, a compound row). The epistemic-action reading survives either way.
  • Not claimed: that the sea-urchin efficiency and ATP numbers transfer to humans. They are demembranated sea-urchin axonemes at [ATP] = 20 µM. Quoting them bare drops the species, the membrane, the [ATP] AND the load. ε_hydro for an intact human spermatozoon is NOT-MEASURED (CN06-22).
  • Not claimed: that dynein’s ~8 nm power stroke (CN06-15) is set by the tubulin-dimer repeat. NA06-13 records the 8 nm KINESIN step as the tubulin-dimer repeat (Svoboda et al. 1993). Those are two separate rows and the corpus does not join them. A reader will notice the coincidence; the coincidence is not an argument, and this plate does not make it one.
  • Not claimed: that this vehicle is OPTIMAL, or that "nature does it this way" is an argument. Per Gould & Lewontin (1979), drift, phylogenetic inertia, developmental constraint, pleiotropy and frozen accidents produce features that solve nothing. Nature’s authority here is precise and limited: it already ran a very long parallel search under real physical constraints in which the failures were deleted. That makes convergence evidence of a CONSTRAINT-optimum and makes every number on this plate a HYPOTHESIS GENERATOR, never a proof. ε_hydro ≈ 0.4–1.3 % sits in CN-06 as a standing counterexample to reading this cell as an optimised propeller, and it rises in high-viscosity media sea-urchin sperm never encounter — which the authors themselves call counterintuitive. Per repo rule M7, a biomimetic design taken from this plate must beat a TUNED CONVENTIONAL BASELINE on a PRE-REGISTERED metric, with a discriminator that collapses the gain, or it is recorded NEGATIVE.
  • Not claimed: that the printed Re is the last word on the regime. Its L input is NOT-MEASURED (normative WHO criteria, not a morphometric distribution), and CN-06 carries a standing three-way disagreement on flagellar length (41 µm measured / ≥45 µm normative / 50 µm in the coasting volume). If the measured 41 µm is right, the printed Re is ~20 % high. The exponent survives either way, and the exponent is the load-bearing part.
  • Not claimed: that anything here raises any UNI rung. A NATURE CITATION IS NEVER A UNI GATE. Reading Purcell 1977 does not make any UNI claim proven, designed, or built. The NATURA classes and the UNI ledger’s four values (proven / designed / hypothesized / not-yet-built) describe different kinds of claim and never merge. This plate contains ZERO UNI claims, and the honest program position is unchanged by it: ~2 of 11+ developmental rungs earned; a developmental active-inference SIMULATION; a toy world, never a person. "Full human" and "beyond human" appear nowhere as targets — they are permanent open questions.

Design fence — {"static":"No script elements, no on-event handlers, no external references (no xlink to a URL, no @import, no webfont, no , no ). Pure static SVG; every url() resolves to an internal fragment id. Audited by regex over the emitted file: 0 hits on every prohibited construct.","responsive":"viewBox-based (0 0 1440 3865), width=100%, no fixed height. Renders standalone AND inline — both were loaded in a browser and measured.","theme":"CSS custom properties with literal fallbacks in every var(); flips on prefers-color-scheme and on :root[data-theme] in both directions. MEASURED, not assumed: standalone under prefers-color-scheme:dark resolves ink rgb(233,229,219) on bg rgb(20,22,26); inlined in an HTML document with a dark OS preference, data-theme="light" forces the light palette back (bg rgb(246,242,233)) and data-theme="dark" holds dark — the override wins in both directions. Note that in a STANDALONE svg document :root IS the element, so the data-theme rules cannot match there by construction; prefers-color-scheme carries that case.","greyscale":"Every distinction colour carries is also carried by a glyph mark (— † ‡ △ ○ ◇ ∅ ✗), dash pattern, hatch density and line weight. Species is carried by a word (HUMAN / SEA URCHIN) in every panel header, never by colour.","verification":"Geometry computed from the sourced constants, not eyeballed: the wave at 13.5 px/µm from λ = 39/18 µm and L = 41 µm; the helix at 6.5 px/µm from r = 8.4 µm and pitch = 47.6 µm; the coast bar at 0.45 Å against a 1.54 Å C–C bond. Layout MEASURED by browser bbox readback with transforms applied: 0 canvas overflows, 0 panel overflows, 0 text–text collisions across 362 text nodes. THE COLLISION DETECTOR WAS POSITIVE-CONTROLLED by injecting a known overlap and confirming it fired, then confirming it returned to 0 on removal — the negative result is therefore an observation, not a silent instrument. An earlier run of the same detector found 16 real collisions, which were fixed; the clean result is the second reading, not the first. Every numeric token rendered on the plate (146 distinct) was diffed against CN-06 + its ledger rows: 11 did not appear verbatim and each was accounted for — 2 superscript-normalisation artifacts (47.6², 52.8²), 8 values computed ON the plate and marked MODELED with their assumptions named (67.0, 71.1, 187, 5.9, 897, 1.43, 1.64, 9.5), and 1 git commit hash (f1be794). ZERO unsourced numbers.","arithmetic_reproduced":"Every headline computation in CN-06 was independently re-evaluated for this plate and reproduces: Re = 4.87×10⁻³ ≈ 4.9×10⁻³; Re_mucus = 2.55×10⁻⁵ ≈ 2.6×10⁻⁵ at u = 65 µm/s and 2.44×10⁻⁵ ≈ 2.4×10⁻⁵ at u = 62 µm/s; 6πµa = 2.64×10⁻⁸ kg/s; τ = 7.2×10⁻⁷ s; d_coast = 4.46×10⁻¹¹ m ≈ 0.45 Å (29 % of a 1.54 Å C–C bond); head ellipsoid 7.07 µm³ + flagellar cylinder 9.82 µm³ ≈ 17 µm³; c = fλ at 897 ≈ 890 and 198 ≈ 200; ε_swim = ε_chemo·ε_hydro at 0.00136 ≈ 0.001 and 0.0078 ≈ 0.008; 67,500 × 2/9 = 15,000 exactly; Re at L = 46 µm is 4.07×10⁻³, i.e. the printed Re is ~20 % high if Saggiorato's measured length is right — as CN-06 itself states. TWO DISCREPANCIES FOUND AND CARRIED, NOT SMOOTHED: (1) CN06-13's printed formula does not reproduce its printed sub-range (see the plate note); (2) progression per beat 5.8 µm vs u/f = 65/11 = 5.9 µm — rounding in the source's inputs, printed as CN-06's table gives it and explicitly NOT manufactured into a dispute."}

Source chapters

  • cookbook/recipes-natura/CN-06-sperm.md
  • encyclopedia/NATURE-LEDGER.md (rows CN06-01…CN06-39 — the authority for every value; where it and the chapter disagree, the ledger wins)
  • encyclopedia/wing-NATURA/NA-00-how-to-read-this-wing.md (evidence vocabulary; amended 2026-07-15-A)

sha256 54c61897792f40fd — of the original file, so what was ingested stays checkable.