E5 The frequency ladder, quantified — a ~13-order collapse
ν_light ≈ 10¹⁴ Hz drops ~13 orders to the ~10–100 Hz neural band by event-detection + low-pass, NOT mixing: the output is carrier-frequency-invariant and the surviving band scales as 1/τ.
What this rung establishes
The whole collapse, quantified. The carrier is ν = c/λ ≈ 10¹⁴ Hz — red 632.99 nm gives 4.736e+14 Hz (energy 1.959 eV), green 532 nm gives 5.635e+14 Hz (2.331 eV) — yet the ganglion band is ~10–100 Hz, a drop of about 12.7–13.7 orders of magnitude. The receptor cannot oscillate at 10¹⁴ Hz; it absorbs E = hν as one quantum of drive. The ratio is forced; the absolute neural Hz is a named [O]. [L][F]
Mechanism = event-detection + low-pass, not mixing
- Intrinsic rhythm (no carrier)
- f0 = 0.01050 /unit-time
- Carrier-invariance (no-mixing test)
- output moves only 11.3% across a 20× carrier span
- The cubic is the all-or-none event
- delete −s³ ⇒ flip +2.151 becomes -50.000 (no threshold)
- Band ∝ 1/τ (τ_s = 20)
- out_dom = 0.01875
- Band ∝ 1/τ (τ_s = 160)
- out_dom = 0.00300
On the frozen Neuron a fast carrier (24×–476× the intrinsic rhythm) is averaged away (out/f_c → 0); the output is carrier-frequency-invariant (a mixer would track it); deleting the cubic destroys the threshold; and the surviving band is set by the recovery τ, scaling as 1/τ. The carrier frequency is discarded — colour survives only because it rides the angle (geometry), not a frequency the cell could follow. [F][V]
Grades (VP-SPEC C3 — honest)
| [F] | Down-conversion is event-detection + low-pass, not mixing: the output is carrier-frequency-invariant, the cubic is the all-or-none event, and the surviving band scales as 1/τ. |
| [V] | On the frozen Neuron the fast carrier is averaged away (out/f_c→0); deleting −s³ removes the threshold; the band moves with τ. |
| [L] | The carrier ν=c/λ and quantum E=hν at the committed wavelengths; the cited ganglion range ~10–100 Hz. |
| [O] | The absolute biological Hz at each rung (carrier → event → cascade τ → spike rate); only the collapse RATIO and the mechanism are forced. Each obstacle named. |
Reproducibility
Every number on this page is the code’s own output. The transcript below is the verbatim, hash-pinned stdout of the listed module(s); tools/gate_volume.py re-runs them and asserts HTML↔code drift 0.
research/E5-frequency-ladder/run.pysha256 a3e625a8a830d561d39b6093…
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E5 — THE FREQUENCY LADDER, QUANTIFIED (10¹⁴ Hz → ~10–100 Hz)
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inherited rung-1 canon: invariant D = 4.852620 pm; SEED=19
(0) RUNG 1 — the carrier (ν=c/λ) and the quantum of drive (E=hc/λ):
red 632.99nm → ν = 4.736e+14 Hz E = 1.959 eV (angle χ=89.9378°)
green 532.00nm → ν = 5.635e+14 Hz E = 2.331 eV (angle χ=89.8248°)
the receptor cannot oscillate at 10¹⁴ Hz; it absorbs E=hν as ONE quantum of drive. [L]/[F]
(1) THE COLLAPSE — carrier vs neural band (cited ganglion range ~10–100 Hz):
ν_light/neural ≈ 4.74e+14/10–100Hz = 4.7e+12–4.7e+13 ≈ 12.7–13.7 orders of magnitude
a ~13-order frequency collapse: the RATIO is forced; the absolute neural Hz is [O].
(2) MECHANISM = EVENT-DETECTION + LOW-PASS, not mixing (frozen Neuron, substrate-time):
intrinsic rhythm (no carrier): f0 = 0.01050 /unit-time
(2a) LOW-PASS — feed a fast carrier f_c, watch the output:
f_c= 0.25 ( 24× f0) out_dom=0.01175 out_rate=0.01200 out/f_c=0.0470
f_c= 0.50 ( 48× f0) out_dom=0.01075 out_rate=0.01100 out/f_c=0.0215
f_c= 1.00 ( 95× f0) out_dom=0.01075 out_rate=0.01075 out/f_c=0.0108
f_c= 2.00 ( 190× f0) out_dom=0.01050 out_rate=0.01075 out/f_c=0.0053
f_c= 5.00 ( 476× f0) out_dom=0.01050 out_rate=0.01075 out/f_c=0.0021
→ output stays at the intrinsic LOW band; out/f_c → 0. The cell cannot follow it. [V]
(2b) CARRIER-INVARIANCE (the no-mixing test):
carrier spans 20× ; output rate moves only 11.3% (min=0.01075, max=0.01200)
→ output is orthogonal to carrier frequency. A mixer would TRACK f_c; this does not. [F]
(2c) THE CUBIC IS THE ALL-OR-NONE EVENT (delete −s³ → no threshold):
CUBIC : s(0.95h*)=-0.794 → s(1.05h*)=+1.357 (discontinuous flip Δ=+2.151)
LINEAR: s(0.95h*)=-50.000 → s(1.05h*)=-50.000 (no two-basin threshold)
→ the carrier collapses onto ONE discrete flip-event; the cubic is what makes it all-or-none. [F]
(3) THE BAND IS SET BY THE RECOVERY τ, not the carrier (carrier fixed at f_c=2.0):
τ_s= 20 out_dom=0.01875
τ_s= 40 out_dom=0.01050
τ_s= 80 out_dom=0.00575
τ_s= 160 out_dom=0.00300
→ output band ∝ 1/τ: manufactured by the cascade/recovery, NOT inherited from the light. [F]
(4) THE LADDER, end to end:
ν_light ~10¹⁴ Hz ──(E=hν → ONE isomerisation; R19 event-detector, §E2)──▶ discrete flip
discrete flip ──(cascade LOW-PASS, recovery τ; carrier averaged away)──▶ graded ~Hz
graded ~Hz ──(spike-rate re-quantisation, §E8)──────────────────────▶ ~10–100 Hz
collapse ≈ 13 orders. RATIO forced [F]; absolute Hz at each biological rung [O].
LEARNED: the eye does NOT mix the carrier down — it DETECTS the photon as a single energy
event (the R19 all-or-none flip) and LOW-PASSES the event stream into a band set by
its own recovery time-constant. The carrier frequency is discarded; colour survives
only because it rides the ANGLE (geometry), not a frequency the cell could follow.
sha256: c2bfd1598f9d7d9dfaf95d9fc89136d0e3b11e43242b5edea2c7c6f38082a192