E7 The cascade as the band-setting low-pass — f_c = β/(2πτ)
The frozen recovery law w ← w + dt·(s−βw)/τ_s IS a single-pole low-pass; the surviving band is the cutoff f_c = β/(2πτ_s), invariant to both the carrier and γ — it is τ that fixes the band.
What this rung establishes
The band the eye speaks in is the cutoff of the transduction low-pass. The frozen recovery law w ← w + dt·(s − βw)/τ_s (quoted verbatim) is a first-order leaky integrator — a single-pole low-pass H(f) = 1/(β + i·2πf·τ_s). Its one claim, the band is set by the recovery time-constant f_c = β/(2πτ_s), is read off the substrate four ways that agree. [F] [V]
Four agreeing reads, and band ∝ 1/τ
- DC gain = 1/β
- closed form 2.0000, measured 1.99902
- Measured cutoff vs β/(2πτ)
- 0.001993 vs 0.001989
- Phase lag at cutoff
- 44.99° (one pole ⇒ 45°)
- High-frequency roll-off
- -19.96 dB/decade (one pole ⇒ −20)
- f_c·τ invariant (= β/2π)
- 0.079577, spread 0.27%
- Cutoff halves with τ (20→80)
- 0.003985 → 0.000999
Sweeping τ gives f_c ∝ 1/τ exactly, and the recovery law carries no γ term and no carrier term — so the band is purely τ: orthogonal to the carrier (E5) and to γ. The full frozen neuron inherits this (its rhythm falls monotonically in τ); γ is a secondary mover via dwell ∝ γ^1.5. The single-pole idealisation vs. the real multi-stage cascade and the absolute τ→Hz are named [O]. [F]
Grades (VP-SPEC C3 — honest)
| [F] | The surviving band is f_c = β/(2πτ_s), set by the recovery time-constant; it is invariant to both the carrier and γ — τ, not the carrier and not γ, fixes the band. |
| [V] | DC gain 1/β; the measured cutoff matches β/(2πτ); phase lag 44.99°; roll-off −19.96 dB/decade; f_c·τ constant ⇒ f_c ∝ 1/τ. |
| [L] | The frozen Neuron constants β=0.5, τ_f=1.0, τ_s=40.0; γ(GUCY2D)=1.365 read byte-equal from the atlas. |
| [O] | The single-pole idealisation vs. the real multi-stage cascade (rhodopsin→transducin→PDE→cGMP→CNG); the absolute τ→Hz calibration; the full-neuron rhythm is only approximately ∝1/τ. 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/E7-cascade-lowpass/run.pysha256 2c1e4c8ae0076c86247f550b…
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E7 — THE CASCADE AS THE BAND-SETTING LOW-PASS (cutoff f_c = β/(2π·τ))
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(0) THE FILTER IS THE SUBSTRATE'S RECOVERY (verbatim): w ← w + dt·(s − β·w)/τ_s
⇒ a single-pole low-pass H(f) = 1/(β + i·2πf·τ_s), β=0.5 (frozen Neuron β); SEED=19
forced (closed form): DC gain = 1/β = 2.0000 ; cutoff f_c = β/(2π·τ_s) ; f_c·τ_s = β/2π = 0.079577
phase(f_c) = −45° ; roll-off = −20 dB/decade (one pole)
(1) MEASURED transfer function on the FROZEN recovery law (τ_s=40 = Neuron default):
(1a) passband DC gain = 1.99902 (closed form 1/β = 2.00000) [V]
(1b) cutoff f_c(meas) = 0.001993 vs β/(2π·τ) = 0.001989 ratio = 1.0016 [V]
(1c) phase lag at f_c = 44.99° (single-pole ⇒ 45°) [V]
(1d) roll-off high-f slope = -19.96 dB/decade (one pole ⇒ −20) [V]
→ the cascade IS a first-order low-pass; its band edge is the explicit number β/(2π·τ). [F]/[V]
(2) THE BAND ∝ 1/τ — sweep the recovery τ_s, the cutoff halves with it (carrier never enters):
τ_s= 20 f_c = 0.003985 β/(2π·τ) = 0.003979 f_c·τ_s = 0.079695
τ_s= 40 f_c = 0.001993 β/(2π·τ) = 0.001989 f_c·τ_s = 0.079702 (↓ with τ)
τ_s= 80 f_c = 0.000999 β/(2π·τ) = 0.000995 f_c·τ_s = 0.079913 (↓ with τ)
f_c·τ_s constant = 0.079770 (β/2π = 0.079577); spread 0.27% ⇒ f_c ∝ 1/τ EXACTLY
the recovery law carries NO γ and NO carrier term ⇒ the band is PURELY τ: ⟂ carrier (E5) and ⟂ γ. [F]
(3) THE LOW-PASS *IS* E5's CARRIER-REJECTION (closed-form |H| at E5's fast carriers, τ_s=40):
carrier f= 1.0 (= 503× f_c) → |H|/DC = 0.00199 (rejected ∝ 1/f)
carrier f= 5.0 (= 2513× f_c) → |H|/DC = 0.00040 (rejected ∝ 1/f)
E5 said the carrier is 'averaged away (out/f_c→0)'; here it IS the −20 dB/dec roll-off. [F]
the REAL ~10¹⁴ Hz carrier sits ~17 orders above any plausible f_c ⇒ rejection is total
(the RATIO is forced; the absolute f_c in Hz is the ladder's standing [O]).
(4) THE FULL FROZEN NEURON inherits the band — emergent rhythm vs τ_s (constant drive d0=0.4):
τ_s= 20 out_dom = 0.018750 out_dom·τ_s = 0.3750
τ_s= 40 out_dom = 0.010500 out_dom·τ_s = 0.4200
τ_s= 80 out_dom = 0.005750 out_dom·τ_s = 0.4600
τ_s= 160 out_dom = 0.003000 out_dom·τ_s = 0.4800
out_dom ↓ monotonically with τ; out_dom·τ DRIFTS upward to an asymptote ⇒ only APPROX ∝ 1/τ:
a relaxation period = (slow recovery ∝ τ) + (τ-independent fast transit). The CLEAN 1/τ is the
explicit filter's; the neuron inherits it. [V]
γ is the SECONDARY mover (the switch nonlinearity; dwell ∝ γ^1.5), not the band-setter — at
fixed τ_s=40 the eye γ's shift the rhythm only mildly while τ sets the decade:
CNGB3 γ=1.2425 dwell∝γ^1.5=1.2591 out_dom=0.009000
RPE65 γ=1.2842 dwell∝γ^1.5=1.3230 out_dom=0.008750
GUCY2D γ=1.3650 dwell∝γ^1.5=1.4498 out_dom=0.008250
RHO γ=1.4719 dwell∝γ^1.5=1.6234 out_dom=0.007500
(γ READ-ONLY from the frozen atlas; band channel = τ, size/settle channel = γ. [F]/[V])
LADDER PLACEMENT (E7 fixes the low-pass rung):
ν_light ~10¹⁴ Hz ──(E=hν → ONE flip; R19 event-detector, §E2)──▶ discrete flip-events
discrete flips ──(THIS low-pass, cutoff f_c=β/(2π·τ); carrier ∝1/f rejected)──▶ graded band
graded band ──(spike-rate re-quantisation, §E8)─────────────────────────────▶ ~10–100 Hz
the output band is the FILTER's (set by τ), never the light's. RATIO forced [F]; absolute Hz [O].
LEARNED: the band the eye speaks in is not chosen by the photon — it is the cutoff of the
transduction low-pass, f_c = β/(2π·τ). Raise the recovery τ and the band drops with it;
the carrier (and γ) leave the cutoff untouched. The carrier is detected as one event and
then filtered out; what survives is the recovery's own band. ONE pole vs the real cascade
(multi-stage) and the absolute τ→Hz are named [O].
sha256: 094fc06ab8d727bc46a66e830069ea159d98a1274ef647a4c92c8be49a14f2e8