E6 Why the band is visible — geometry places it, photochemistry pins it
Two unrelated constraints select ~380–750 nm: lattice geometry sandwiches visible in m between x-ray and infrared, but the angle never gates a band — the reversible 11-cis→all-trans energy window (~1.8–3.3 eV) pins the edges.
What this rung establishes
‘Visible’ is neither chosen nor geometric. Two unrelated constraints select the same window. The wavelength is the anchor, mapped exactly to ν = c/λ, period T = 1/ν, and E = hν with SI-exact constants — the HeNe red anchor 632.99 nm is 1.95870706 eV, the green channel 532 nm is 2.33053005 eV, both inside the window. [L][F]
Geometry places it; photochemistry pins it
- Gate (i) — placement in m
- visible (m≈10⁵) sandwiched between x-ray and infrared, near-transverse
- Gate (ii) — energy window
- reversible 11-cis→all-trans, 1.8–3.3 eV ⇒ λ = 375.70969222–688.80110241 nm
- Sawtooth, not a kink
- 633 nm deficit 0.0621945346° < 750 nm deficit 0.1187372397°
- IR excluded by energy, not geometry
- 10µm deficit 0.0491568932° is smaller than visible's, yet excluded
- Soft red edge
- 750 nm = 1.65312265 eV sits 0.14687735 eV below the 1.8 eV floor
The lattice geometry only places the band: the angle is a smooth sawtooth with no kink at the band edges, so it never gates a band — an infrared probe is more transverse than visible yet is excluded by the energy floor, not by any angle. What pins ~380–750 nm is the chromophore's reversible energy window: a floor to drive the flip, a ceiling before damage. The deep-red edge is soft and we say so. The molecular tuning of that window is coding/photochemistry — the promoter-γ this package reads has no lever on the retinal pocket, a named [O]. E6 invents no chromophore. [F][L]
Grades (VP-SPEC C3 — honest)
| [F] | Geometry PLACES the band (sandwiched in m, near-transverse) but the angle is a smooth sawtooth that never gates a band; the closure m·sinχ·D/λ=1 holds to 1e-15; the exact ν/T/E map from the wavelength anchor. |
| [L] | The cited reversible isomerisation window 1.8–3.3 eV; the already-measured visible anchors (633/532; opsin λmax 420/530/560). |
| [O] | The retinal chromophore energy that sets the floor (coding/photochemistry, outside the promoter-γ) — the soft red edge is a consequence; obstacle named, no chromophore invented. |
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/E6-why-visible/run.pysha256 5ba9f6c9ac00f81487faa4ff…
==============================================================================
E6 — WHY THE BAND IS VISIBLE (geometry PLACES it; photochemistry PINS it)
==============================================================================
SI-exact constants: c = 299792458 m/s, h = 6.62607015e-34 J·s; inherited D = 4.852620477366 pm; SEED=19
cited inputs: isomerisation window 1.8–3.3 eV [L]; conventional visible 380–750 nm [L]
(0) THE EXACT CARRIER MAP — already-measured λ mapped exactly to ν, period T, energy E:
(ν=c/λ, T=1/ν=λ/c, E=hν; constants SI-exact — no rounding)
HeNe red anchor λ= 632.990 nm ν=4.7361326087e+14 Hz T=2.1114273662e-15 s E=1.95870706 eV
green 2nd channel λ= 532.000 nm ν=5.6351965789e+14 Hz T=1.7745609865e-15 s E=2.33053005 eV
opsin S λmax λ= 420.000 nm ν=7.1379156667e+14 Hz T=1.4009691998e-15 s E=2.95200472 eV
opsin M λmax λ= 530.000 nm ν=5.6564614717e+14 Hz T=1.7678897046e-15 s E=2.33932450 eV
opsin L λmax λ= 560.000 nm ν=5.3534367500e+14 Hz T=1.8679589331e-15 s E=2.21400354 eV
→ carrier ≈ 4.7–7.1×10¹⁴ Hz, period ≈ 1.4–2.1 fs; T=1/ν and E=hν hold to 1e-15. [L]/[F]
(1) GATE (i) — PLACEMENT IN m (m=⌈λ/D⌉, monotone; inherited §10.9 canon, read-only):
band m χ (deg, full precision) (90°−χ) deficit (deg) closure
gamma 1pm 1 χ=11.8923903342° 78.1076096658° 0.999999999999999
x-ray 10nm 2061 χ=89.0938301623° 0.9061698377° 1.000000000000000
VISIBLE 380nm 78309 χ=89.7418532397° 0.2581467603° 1.000000000000000
VISIBLE 750nm 154556 χ=89.8812627603° 0.1187372397° 1.000000000000000
infrared 10µm 2060743 χ=89.9508431068° 0.0491568932° 1.000000000000000
radio 1m 206074224157 χ=89.9998972701° 0.0001027299° 1.000000000000000
→ VISIBLE is SANDWICHED in m between x-ray (smaller m) and infrared (larger m), in the
near-transverse manifold; gamma (m=1) is the only quasi-longitudinal band. [F]
(2) GATE (ii) — THE ENERGY WINDOW (reversible 11-cis→all-trans isomerisation, cited):
cited window 1.8–3.3 eV ⇒ λ = hc/E = 375.70969222–688.80110241 nm
every ALREADY-MEASURED visible anchor lands inside this window:
HeNe red anchor λ= 632.990 nm E=1.95870706 eV INSIDE
green 2nd channel λ= 532.000 nm E=2.33053005 eV INSIDE
opsin S λmax λ= 420.000 nm E=2.95200472 eV INSIDE
opsin M λmax λ= 530.000 nm E=2.33932450 eV INSIDE
opsin L λmax λ= 560.000 nm E=2.21400354 eV INSIDE
floor 1.8 eV: below it the photon cannot drive the flip (→ no transduction);
ceiling 3.3 eV: above it the photon ionises / photodamages (not reversible). [L]/[F]
(3) WHO PINS THE EDGES? geometry PLACES, energy PINS (the brutally-honest part):
(3a) deficits (90°−χ): 380nm=0.2581467603° 633nm=0.0621945346° 750nm=0.1187372397°
633<750 in deficit though 633<750 in λ ⇒ SAWTOOTH (m=⌈⌉ quantisation), not monotone
band-internal deficit span = 0.1959522257° vs gamma deficit 78.107610° (>78°)
(3b) violet edge 380nm → 3.26274206 eV (just under the 3.3 eV ceiling);
red edge 750nm → 1.65312265 eV (just under the 1.8 eV floor)
IR 10µm: E=0.12398420 eV ≪ floor (ENERGY excludes it) — yet its deficit 0.04915689° is
SMALLER than visible's (more transverse, larger m=2060743>154556); geometry would admit it
→ the edges are PHOTOCHEMICAL, not geometric. The angle (even to 10 dp) never gates a
band; it only labels colour. The chromophore's energy window is WHERE the band is. [F]/[L]
(4) THE SOFT RED EDGE + the named [O] (honesty, not a clean coincidence):
deep-red edge: 750nm = 1.65312265 eV sits 0.14687735 eV BELOW the
clean floor 1.8 eV ⇒ the conventional band overhangs the clean window on the red side
(deep red is low-efficiency, perceived only bright) — the red edge is SOFT and we say so.
[O] the chromophore energy that sets this floor is coding/photochemistry: the promoter-γ
this package reads has NO lever on the retinal pocket. E6 invents no chromophore.
LEARNED: 'visible' is not chosen and not geometric. The lattice geometry only PLACES the band
— it fixes the angle↔colour map and sandwiches visible in m between x-ray and infrared,
but the angle is a smooth sawtooth with no kink (the angle never gates a band). What
pins ~380–750 nm is the chromophore's REVERSIBLE energy window: a floor to flip the
switch, a ceiling before damage. The WAVELENGTH is the anchor — mapped exactly to a
ν≈10¹⁴ Hz / ~2 fs carrier — and the molecular tuning of that window stays a named [O].
sha256: d2fd0842b92f833453b2f69cde8b1b9aff94dafb53e618893a6091cec5fcc1d7