E11 Accommodation & refractive error — the power↔length match (theoretical, non-clinical)
E3's single-surface eye is in focus for distance exactly when its power and its length satisfy P·L = n₂ (the match ratio ρ = 1, a whole curve); refractive error is the signed mismatch — ρ>1 puts the distant focus in front (myopia), ρ<1 behind (hyperopia), reachable by an eye too long OR too powerful (only the product matters); accommodation is a one-signed power lever (round the lens, R↓ ⇒ P↑) that pulls near objects into focus, and the growth axis follows the substrate size law dwell ∝ γ^1.5 — all stated direction-only behind a machine-checked magnitude firewall.
What this rung establishes
From E3, a single refracting surface obeys n₂/v − n₁/u = P with power P = (n₂−n₁)/R, so a distant object images at v_∞ = n₂/P. The retina sits at the eye's axial length L, so the eye is in focus for distance exactly when v_∞ = L — that is, when P·L = n₂: a single, dimensionless match between the eye's power (optics) and its length (growth). Define the match ratio ρ ≡ P·L/n₂. On E3's frozen Emsley reduced eye (consumed index n₂ = 1.333333) the ratio is ρ = 1.000000 exactly — the cited n and R cohere, this is not a fit. Emmetropia is not one eye but a whole curve: the product P·L is fixed, so a longer eye is emmetropic with proportionally less power. [F][V]
Key results
- Emmetropic match (frozen reduced eye)
- ρ = P·L/n₂ = 1.000000 (exact); n₂ = 1.333333
- Myopia side (focus in front)
- ρ = 1.050000 — too long (axial) OR too powerful (refractive)
- Hyperopia side (focus behind)
- ρ = 0.950000
- Only the product matters
- stronger × shorter (q·ℓ with ℓ = 0.952381) ⇒ ρ = 1.000000
- Accommodation, far → near (P_req/P_∞)
- 1.000750 → 1.007500 → 1.030000 → 1.075000 → 1.187500 → 1.375000
- Lens rounding at the near end (R_req/R_∞)
- 0.727273 (R↓ ⇒ P↑, one-signed)
The mismatch sign is the only thing stated: since L − v_∞ = (P·L − n₂)/P, the side is sign(ρ−1). How much error, the axial length, the spectacle power — all firewall-blocked [O]; none is named. [F]
Refractive error is a signed mismatch — two independent routes
The distant focus is v_∞ = n₂/P and the retina is at L, so the blur side follows the sign of the mismatch directly. ρ > 1 puts the distant focus in front of the retina — myopia; ρ < 1 puts it behind — hyperopia. Crucially the same error is reachable two independent ways: an eye that is too long (axial) or too powerful (refractive) both raise ρ above 1 (here both to 1.050000), and both give myopia — the match cares only about the product P·L, which is why a stronger eye made compensatingly shorter returns to ρ = 1.000000. The correction is stated only as a direction (myopia ⇒ reduce power; hyperopia ⇒ add power). [F]
Accommodation is a one-signed power lever
A near object needs more power than a distant one to image on the same retina. For an object at u = −k·L (k = distance in eye-lengths), the required power is P_req/P_∞ = 1 + n₁/(n₂·k) — a pure number that rises monotonically as the object nears: 1.000750 → 1.007500 → 1.030000 → 1.075000 → 1.187500 → 1.375000 as k falls. It is always at least 1, so accommodation is a strictly one-signed (additive) lever: the eye can only round the lens (R↓ ⇒ P↑, the radius ratio R_req/R_∞ dropping to 0.727273 at the near end) to pull focus nearer, never the reverse. The far point is the relaxed lever (the ratio → 1 at distance); the near point is the lever maxed. The amplitude, the near and far distances, and the presbyopic decline with age are the firewall-blocked [O]. [F]
The lever is lattice geometry; the length axis is a dwell-size
The power lever is the same lattice geometry as E3: P = (n₂−n₁)/R, so P ∝ 1/R and the lens rounding is a purely geometric modulation of the transverse-match interface (dP/P = −dR/R), with n = c_vac/c_med = √((B/ρ) ratio) the inherited meaning of refractive index. The other axis, length, is growth: in the substrate organ size = dwell ∝ γ^1.5 (E1), so the eye's axial length is a dwell-size. Reading the eye-field master PAX6 read-only (byte-equal to the atlas, γ = 1.5110; RAX γ = 1.4538) instantiates that law — the relative size PAX6/RAX = (γ_PAX6/γ_RAX)^1.5 = 1.059595 — and the law is monotone in γ, so the growth axis has a forced direction: more ocular growth ⇒ a longer eye ⇒ the myopic side of the match. The gene→elongation map, the emmetropization feedback that normally holds ρ ≈ 1 in a growing eye, and the absolute scale are the named [O]; γ is promoter structure only — never an optical power, a growth rate, or a clinical effect.
Grades (VP-SPEC C3 — honest)
| [F] | Emmetropia is the power↔length match ρ = P·L/n₂ = 1 (a curve, not one eye); the mismatch SIGN fixes the side (ρ>1 myopia / focus in front, ρ<1 hyperopia / behind), reachable two independent ways (too long OR too powerful — only the product matters); accommodation is a one-signed (additive) monotone power lever via lens rounding (R↓ ⇒ P↑); the lever is geometry (dP/P = −dR/R); the length axis is the substrate size law dwell ∝ γ^1.5. |
| [V] | ρ = 1 exactly on the frozen reduced eye; the geometric blur side equals sign(ρ−1) on E3's own surface equation for both the axial and refractive routes; an equal-product eye returns to ρ = 1; P_req/P_∞ rises monotonically and stays at least 1, and the implied lens radius images the near object onto the retina; dP/P = −dR/R; the eye-field γ is byte-equal to the atlas. |
| [V] | The single-surface imaging itself is E3's classical ray arithmetic (consumed, not re-derived); E11 adds only the dimensionless match reading on top of it. |
| [L] | The Emsley reduced-eye n and R (cited, via E3); γ(PAX6), γ(RAX) from NCBI promoters (cached, read-only); that myopia is either axial or refractive is the cited optics this matches. |
| [O] | The magnitude of any refractive error / the axial length / the spectacle power / the accommodation amplitude / the near and far distances / the presbyopic decline (all firewall-blocked — none produced); the gene→ocular-elongation map and the emmetropization feedback that normally holds ρ ≈ 1 (needs growth biology, not the promoter-γ); the chromatic/aberration coupling (inherited E3); the felt percept of blur or clarity (→ mind volume). |
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/E11-accommodation-refraction/run.pysha256 d50ece8341cb48c65361a445…
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E11 — ACCOMMODATION & REFRACTIVE ERROR (E3's optics read as a power↔length MATCH)
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SCOPE: theoretical, NON-CLINICAL. Structure-only / direction-only behind the magnitude firewall.
inherited invariant quantum size D = 4.852620 pm (the same frozen wave that carries colour)
consumes (frozen): E3 single-surface optics n₂/v−n₁/u=P, P=(n₂−n₁)/R · vp_substrate.dwell · atlas γ
re-derives: nothing; adds no γ. γ measured, never fitted; γ = promoter STRUCTURE only (firewall).
scalings/distances are dimensionless RATIOS; NO power magnitude, NO length, NO clinical quantity.
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PART A — emmetropia is the power↔length MATCH: ρ ≡ P·L/n₂ = 1 (a CURVE, not one eye)
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the eye is in focus for distance ⇔ the distant image v_∞ = n₂/P lands on the retina (length L)
⇔ P·L = n₂. On E3's frozen reduced eye the match ratio ρ = P·L/n₂ = 1.000000 (EXACT). [V-arith]
n₂ = 1.333333 (the consumed Emsley index; ρ=1 because the cited n,R cohere — not a fit).
emmetropia is a whole CURVE in (power, length): the PRODUCT is fixed, so a longer eye is
emmetropic with proportionally LESS power (and a shorter eye with more). [F]
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PART B — refractive error = sign(ρ−1): ρ>1 ⇒ MYOPIA (focus in front), ρ<1 ⇒ HYPEROPIA (behind)
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the distant focus is v_∞ = n₂/P and the retina is at L, so L − v_∞ = (P·L − n₂)/P ⇒
sign(L − v_∞) = sign(ρ − 1): the SIGN of the mismatch fixes the side, with NO magnitude.
[axial route] same power, scale the LENGTH by a dimensionless factor ℓ (illustrative, not a size):
length scale ℓ=1.05 → ρ = 1.050000 → MYOPIA (focus in front)
length scale ℓ=0.95 → ρ = 0.950000 → HYPEROPIA (focus behind)
[refractive route] same length, scale the POWER by a dimensionless factor q (P ∝ 1/R, so R↦R/q):
power scale q=1.05 → ρ = 1.050000 → MYOPIA (focus in front)
power scale q=0.95 → ρ = 0.950000 → HYPEROPIA (focus behind)
[product, not parts] a stronger eye (q=1.05) made shorter (ℓ=0.952381) is emmetropic again:
ρ = q·ℓ = 1.000000 → the match cares only about the PRODUCT P·L, not which is off. [F]
stated DIRECTION only: myopia ⇒ reduce power; hyperopia ⇒ add power. The MAGNITUDE (how much
error, the axial length, the spectacle power) is the firewall-blocked [O] — none is named. [F]/[O]
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PART C — accommodation: near objects need MORE power; P_req/P_∞ = 1 + n₁/(n₂·k), one-signed (↑)
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to image an object at u=−k·L (k = distance in eye-lengths) on the SAME retina, the required power
is P_req=(n₂+n₁/k)/L, so P_req/P_∞ = 1 + n₁/(n₂·k): a pure number rising as the object nears (k↓).
object distance (k·L) P_req/P_∞ lens-radius R_req/R_∞
k=1000 1.000750 0.999251
k=100 1.007500 0.992556
k=25 1.030000 0.970874
k=10 1.075000 0.930233
k=4 1.187500 0.842105
k=2 1.375000 0.727273
P_req/P_∞ rises monotonically as the object nears: True; and is ALWAYS ≥ 1 (one-signed): True
→ accommodation can only ADD plus power (pull focus nearer), never subtract; the far point is
the relaxed lever (ratio→1 at distance), the near point is the lever maxed. DIRECTION only. [F]
the lever is the lens ROUNDING (R↓ ⇒ P↑): R_req/R_∞ falls below 1 as the object nears (above). [F]
the AMPLITUDE, the near/far DISTANCES, the presbyopic decline with age — firewall-blocked [O].
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PART D — the lever is lattice geometry (P=(n₂−n₁)/R, n=√((B/ρ) ratio)); length = a DWELL-size
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power is a GEOMETRIC lever on the transverse-match interface: P=(n₂−n₁)/R ⇒ P ∝ 1/R, so
dP/P = −dR/R (rounder lens ⇒ more power): measured -0.000001 vs −dR/R = -0.000001. [F]
and n = c_vac/c_med = √((B/ρ) ratio) is the inherited lattice meaning of refractive index (E3);
so BOTH accommodation (R lever) and the refractive axis (P axis) move along the ONE frozen P(R).
the LENGTH axis is GROWTH: in the substrate, organ size = dwell ∝ γ^1.5 (E1) — so the eye's
axial length is a DWELL-size. Read the eye-field master PAX6 READ-ONLY (byte-equal to the atlas):
PAX6 (eye_field master): γ = 1.5110 RAX (retina): γ = 1.4538
size law dwell ∝ γ^1.5 ⇒ relative size PAX6/RAX = (γ_PAX6/γ_RAX)^1.5 = 1.059595
→ the size law is MONOTONE in γ (more dwell ⇒ a larger structure): a definite DIRECTION.
so on the growth axis the DIRECTION is forced: more ocular growth ⇒ a longer eye ⇒ the MYOPIC
side of the match (ρ>1). The gene→elongation map, the emmetropization feedback that holds
ρ≈1 in a normally growing eye, and the absolute scale are the named [O] — not invented. [F]/[O]
[no-drift] eye-field γ byte-equal to frozen atlas (no fitting): True
γ is promoter STRUCTURE only — never an optical power, a growth rate, or a clinical effect
(firewall). The felt percept of blur/clarity is OUT OF SCOPE (→ mind volume).
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E11 GRADES (VP-SPEC C3) — honest
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[F] forced : emmetropia is the power↔length match ρ=P·L/n₂=1 (a curve, not one eye); the
mismatch SIGN fixes the side (ρ>1 myopia/focus-in-front, ρ<1 hyperopia/behind),
reachable two independent ways (too long OR too powerful — same product);
accommodation is a one-signed (additive) monotone power lever via R↓; the lever
is geometry (dP/P=−dR/R); the length axis is the substrate size law dwell ∝ γ^1.5.
[V] verified : ρ=1 EXACTLY on the frozen reduced eye; the geometric blur side equals sign(ρ−1)
on E3's own surface equation (both axial and refractive routes); equal-product
eyes return to ρ=1; P_req/P_∞ rises monotonically and stays ≥1, and the R_req it
implies images the near object onto the retina (E3); dP/P=−dR/R; γ byte-equal.
[V-arith] : the single-surface imaging itself is E3's classical ray arithmetic (consumed,
not re-derived); E11 adds only the dimensionless MATCH reading on top of it.
[L] measured : the Emsley reduced-eye n,R (cited, via E3); γ(PAX6), γ(RAX) (NCBI promoters,
cached, read-only); 'myopia = axial or refractive' is the cited optics this matches.
[O] open : the MAGNITUDE of any refractive error / the axial length / the spectacle power /
the accommodation amplitude / the near & far DISTANCES / the presbyopic decline
(all firewall-blocked — none produced); the gene→ocular-elongation map and the
emmetropization feedback loop that normally holds ρ≈1 (needs growth biology, not
the promoter-γ); the chromatic/aberration coupling (inherited E3 [O]); the felt
percept of blur/clarity (→ mind volume). Each obstacle named, not invented.
LEARNED: focus is a MATCH. E3's single-surface eye is in focus for distance exactly when its
POWER and its LENGTH satisfy P·L = n₂ — a dimensionless condition the frozen reduced eye
meets exactly, and a whole CURVE (a longer eye needs less power). Refractive error is
simply being off that curve: the SIGN of the mismatch is myopia (focus in front) or
hyperopia (behind), and the same error is reachable by an eye that is too LONG or too
POWERFUL — only the product matters. Accommodation is the eye's one-signed lever on the
power side: rounding the lens (R↓) adds plus power to pull near objects into focus, never
the reverse. That lever is pure lattice geometry (P ∝ 1/R, n = √((B/ρ) ratio), E3), while
the length side is the substrate's own size law (dwell ∝ γ^1.5, E1) — so each axis of the
match has a forced DIRECTION. Every clinical magnitude, and the felt blur, stay honestly
open. Foundation untouched; no γ added; nothing fitted; firewall intact and machine-checked.
sha256: c2acaf26133199173242b11ae4a52422ec9e96d9443b79790b55fc10a5211423