E10 Light/dark adaptation — gain control as the switch on its saturating branch
Adaptation needs no new machinery: it is the same R19 switch on its steady-state ON branch, where the cubic makes the response saturate (s* → h^(1/3)) — so a huge background range is log-compressed (by the cubic order n=3), the incremental gain falls automatically as the surround brightens, and the FRACTIONAL (contrast) gain is constant at exactly 1/n = 1/3, a Weber-Fechner-like law forced by the cube and independent of γ.
What this rung establishes
Adaptation needs no new machinery. The eye works from starlight to noon — a range of many orders of magnitude — and the reason is already in the E2 switch. The steady state of the frozen field γ·s−s³+h, read on its upper (light-adapted) branch, is the real root of s³−γ·s−h = 0; for a large background drive the cubic dominates, so s*(h) → h^(1/3) — a compressive response. On the rod pigment RHO (γ = 1.4719, fold h* = 0.68733) the ON-branch steady state saturates onto that cube root: s*/h^(1/3) runs 1.467434 → 1.105351 → 1.022769 → 1.004906 → 1.001057 → 1.000228 as the background climbs. Every one of those is a true zero of the frozen field (residual ≈ 0). This is normal physiology — no clinical magnitude is produced. [F][V]
Key results
- Anchor switch (rod pigment RHO)
- γ = 1.4719, fold h* = spinodal(γ) = 0.68733
- ON-branch steady state saturates
- s*/h^(1/3) → 1.000228 (onto the cube root)
- Dynamic range (operating sweep)
- 5.000000 background decades → 1.500207 response decades
- Deep-saturation log-compression
- 6.000000 → 1.990223 decades = 3.014737 → n=3
- Incremental gain falls (light adapt.)
- 2.004737e-01 (dim) → 1.546844e-04 (bright), 1296.0×
- Contrast gain → 1/n (Weber)
- 0.136615 → … → 0.333182; pure-cube limit 0.333333
A ~5-decade background range is squeezed into a ~1.5-decade response, and deep in the saturated branch the log-compression is exactly the cubic order n = 3 (3.014737). That is how one switch spans the eye’s enormous intensity range without pinning at either end. [F]
Gain control, and the Weber law it forces
The incremental gain is ds*/dh = 1/(3s*²−γ) (the frozen field’s own slope). It falls monotonically as the background rises — 2.004737e-01 in dim light down to 1.546844e-04 in bright, about 1296.0× — so a bright surround automatically turns the switch down. No separate machinery is invoked; gain control is a property of the cube’s steady-state curve. The deeper consequence is the contrast gain (the response to a fractional change), (h/s*)·ds*/dh: on the saturated branch s*→h^(1/3), so it tends to (h^(2/3))/(3h^(2/3)) = 1/n = 1/3 exactly. In the pure-cube limit it is 0.333333 for any background; the real γ-carrying field converges to it (0.136615 → 0.270194 → 0.318497 → 0.330078 → 0.332629 → 0.333182). Equal fractional steps feel equal — a Weber-Fechner-like law, forced by the cube and independent of γ: a different gene’s threshold gives the same limit (0.333301 for RHO, 0.333306 for CNGB3 at a large background). [F][V]
Honest scope: this is the saturated branch. Near the fold (small background) the γ·s term still matters and the response is steeper than a clean power law — the Weber regime is the cube-root tail, stated, not hidden.
Two regimes; where γ and τ sit
Dark (operating point near the fold h* = 0.68733): the incremental gain is highest (2.004737e-01) and the single-photon flip (E2) gives the detection threshold. Bright (driven deep into the cube root): low gain (1.546844e-04), the response compressed against saturation. Adaptation is the operating point sliding from the fold into the cube-root tail. γ is read read-only: γ(RHO) = 1.4719 sets where the dark fold sits (a sensitivity offset), not the compression law — which is γ-independent, exactly as the band was τ-not-γ in E7 and γ was an offset in E8. The time course of dark re-sensitisation runs on the frozen Neuron’s recovery τ_s = 40.0 (β = 0.5), the same constant that set the band (E7) and the rate code (E8); its structure is forced, the absolute seconds are the inherited [O]. The felt brightness percept is the mind volume’s. [F]
Grades (VP-SPEC C3 — honest)
| [F] | Light/dark adaptation is the same R19 switch on its steady-state ON branch, where the cubic makes the response saturate (s*→h^(1/3)); the incremental gain ds*/dh = 1/(3s*²−γ) falls with the background (gain control); the CONTRAST gain → 1/n = 1/3 exactly (a Weber-like law), independent of γ; deep-saturation log-compression → the cubic order n=3; γ is an offset, not the law. |
| [V] | Every s* is a true zero of the frozen field (residual ≈ 0); s*/h^(1/3) → 1 monotonically; the gain falls monotonically dim→bright; the contrast gain rises to 1/3 and the pure-cube limit is exactly 0.333333; the two-gene asymptote agrees; τ_s/β and γ are byte-equal to the frozen substrate/atlas. |
| [L] | γ(RHO), γ(CNGB3) from NCBI promoters (cached, read-only); Weber’s law and the brightness power-law exponent (~1/3) are the cited psychophysics this matches. |
| [O] | The absolute background→drive→firing scale (inherited E2/E5) — so no absolute threshold, Weber fraction, or luminance unit is produced; the absolute τ→seconds of dark adaptation (inherited E7/E8); the molecular adaptation machinery (Ca²⁺ feedback, pigment bleaching/regeneration) — the substrate captures the compression, not that feedback loop; the felt brightness percept (→ mind volume). 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/E10-light-dark-adaptation/run.pysha256 7e3685b15c96d7148209d153…
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E10 — LIGHT/DARK ADAPTATION (gain control = the R19 switch on its saturating ON branch)
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SCOPE: theoretical, NON-CLINICAL — a NORMAL mechanism (not a disease chapter).
consumes (frozen): vp_substrate.sdot/spinodal (the R19 field) · Neuron τ_s,β · organ_gamma.json γ
re-derives: nothing; adds no γ. γ measured, never fitted; γ = promoter STRUCTURE only (firewall).
anchor gene RHO (rod pigment): γ = 1.4719 → fold h*(γ) = spinodal(γ) = 0.68733
the steady state of γ·s − s³ + h = 0 (E2's switch, ON branch) is the real root of s³ − γ·s − h = 0;
for a large background drive h the cubic dominates ⇒ s*(h) → h^(1/3): a COMPRESSIVE response.
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PART A — the ON-branch steady state s*(h) SATURATES onto the cube root (s*/h^(1/3) → 1)
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h (bg) s* h^(1/3) s*/h^1/3 gain=1/(3s^2-g) contrast
1 1.467434 1.000000 1.467434 2.004737e-01 0.136615
10 2.381407 2.154435 1.105351 6.434429e-02 0.270194
100 4.747275 4.641589 1.022769 1.511991e-02 0.318497
1000 10.049063 10.000000 1.004906 3.316980e-03 0.330078
10000 21.567120 21.544347 1.001057 7.173858e-04 0.332629
100000 46.426459 46.415888 1.000228 1.546844e-04 0.333182
s*/h^(1/3) decreases monotonically toward 1 (saturation onto the cube root): True
every s* is a true zero of the FROZEN field (max |γ·s−s³+h| = 2.3e-13). [V]
operating sweep 1 → 100000: background = 5.000000 decades, response = 1.500207 decades
→ compression = 3.332873 (a 5-decade background fits a ~1.5-decade response)
saturated sweep 100 → 100000000: background = 6.000000 decades, response = 1.990223 decades
→ compression = 3.014737 → the cubic order n=3 (pure cube-root log-compression). [F]
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PART B — gain control: the incremental gain ds*/dh = 1/(3s*²−γ) FALLS with the background
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gain at dim background (h=1): 2.004737e-01
gain at bright background (h=100000): 1.546844e-04
gain falls monotonically dim → bright: True
dim/bright gain ratio = 1296.0 (the bright surround turns the switch DOWN)
gain·h^(2/3) (≈ const ⇒ gain ∝ h^(−2/3)): 0.325749 (h=100) … 0.333257 (h=100000) → 1/3
→ a saturating response gives AUTOMATIC gain control: no separate machinery is invoked. [F]
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PART C — contrast gain (response to a FRACTIONAL change) → 1/n = 1/3 EXACTLY (Weber-like)
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contrast gain (h/s*)·ds*/dh along the sweep:
0.136615 → 0.270194 → 0.318497 → 0.330078 → 0.332629 → 0.333182
rises monotonically toward a constant: True
pure-cube limit (γ-term off ⇒ s*=h^(1/3) exactly): contrast = (h/s*)/(3s*²) for any h —
h=8 s*=2.000000 contrast = 0.333333
h=1000 s*=10.000000 contrast = 0.333333
h=1e+09 s*=1000.000000 contrast = 0.333333
→ the cube-root response makes the contrast gain EXACTLY 1/n = 0.333333 (n=3, the cubic). [F]
γ-INDEPENDENCE (the compression is the cubic ORDER, not the threshold γ):
RHO (γ=1.4719): contrast gain at h=1000000 = 0.333301 → 1/3
CNGB3 (γ=1.2425): contrast gain at h=1000000 = 0.333306 → 1/3
→ equal fractional steps feel equal regardless of γ: a Weber-Fechner-like law, FORCED not fitted.
[honest scope] this is the SATURATED branch; near the fold (small h) the γ·s term matters and the
response is steeper than a clean power law — the Weber regime is the cube-root tail, stated not hidden.
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PART D — two regimes (dark/fold vs bright/saturated); γ sets the offset, τ sets the recovery
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DARK (operating point near the fold h*): high incremental gain + the single-photon FLIP
(E2 detection threshold at h*=0.68733); the ON-branch gain here is the
largest in the sweep (2.004737e-01 at h=1).
BRIGHT (driven deep into the cube root): low gain (1.546844e-04), the response
compressed against saturation. Adaptation = the operating point SLIDING fold → cube-root.
γ READ-ONLY: γ(RHO)=1.4719 sets the fold/offset (where dark sensitivity sits); the
compression law (contrast gain → 1/3) is γ-INDEPENDENT (PART C) — γ is an OFFSET, as in E7/E8.
τ READ-ONLY: dark re-sensitisation runs on the FROZEN Neuron's recovery τ_s = 40.0 (β = 0.5),
the same time-constant that set the band (E7) and the rate code (E8). Structure [F]; absolute
seconds are the inherited [O]. The felt brightness percept is OUT OF SCOPE (→ mind volume).
[no-drift] anchor γ byte-equal to frozen atlas (no fitting): True
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E10 GRADES (VP-SPEC C3) — honest
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[F] forced : the ON-branch steady state saturates onto the cube root s*→h^(1/3); the
incremental gain ds*/dh = 1/(3s*²−γ) falls with the background (gain control);
the CONTRAST gain → exactly 1/n = 1/3 (a Weber-like law) and is γ-independent;
deep-saturation log-compression → the cubic order n=3; γ is an offset, not the law.
[V] verified : every s* is a true zero of the FROZEN field (residual ≈ 0); s*/h^(1/3) → 1
monotonically; the gain falls monotonically dim→bright; the contrast gain rises to
1/3 and the pure-cube limit is exactly 0.333333; τ_s/β byte-equal; γ byte-equal.
[L] measured : γ(RHO), γ(CNGB3) (NCBI promoters, cached, read-only); Weber's law and the
brightness power-law exponent (~1/3) are the cited psychophysics this matches.
[O] open : the absolute background→drive→firing scale (inherited E2/E5) — so no absolute
threshold, Weber fraction, or luminance unit is produced; the absolute τ→seconds
of dark adaptation (inherited E7/E8); the molecular adaptation machinery (Ca²⁺
feedback on guanylate cyclase, pigment bleaching/regeneration) — the substrate
captures the COMPRESSION, not that feedback loop; the felt brightness percept
(→ mind volume). Each obstacle named, not invented.
LEARNED: light/dark adaptation needs no new machinery — it is the SAME R19 switch read on its
steady-state ON branch, where the cube makes the response saturate (s*→h^(1/3)). That one
fact gives all three signatures: a huge background range is log-compressed by the cubic
order n=3 (dynamic range), the incremental gain falls automatically as the surround
brightens (gain control), and — the headline — the FRACTIONAL (contrast) gain is constant
at exactly 1/n = 1/3, a Weber-Fechner-like law FORCED by the cube and INDEPENDENT of γ.
γ only sets where the dark fold sits (an offset, as in E7/E8); τ sets the dark-recovery
time course (E7/E8); the absolute scales and the felt percept stay honestly open.
sha256: 97f9611cee3346554c5dcb266317758eb72091c7f7ff7b1c8dc13bdfd97c8c1d