Dual-Kernel Convolution

A candidate Lindgren-DKC route with two normalized memory kernels, a bounded Hill response and an explicit open bridge between geometry and observables.

The candidate DKC FieldState calibration pipeline is implemented and produces values, and forecasts F1–F9 are content-addressed and locked. The evaluator also accepts explicitly uncalibrated sensitivity inputs. National mobile, broadband and urban series remain technology-timing proxies—not measured doses—and the Lindgren L2 geometry-to-observable operator remains open.

Publication gate: release not authorized

This is a candidate route that is not release-authorized. 0/10 critical points V1–V10 pass and the evidence-protocol audit has not passed. F1–F9 are locked candidate forecasts, not results of an authorized release artifact. The site builds with this state shown; the gate closes only with an evaluation bundle.

V1 FAIL · MISSING_EVALUATION
V2 FAIL · MISSING_EVALUATION
V3 FAIL · MISSING_EVALUATION
V4 FAIL · MISSING_EVALUATION
V5 FAIL · MISSING_EVALUATION
V6 FAIL · MISSING_EVALUATION
V7 FAIL · MISSING_EVALUATION
V8 FAIL · MISSING_EVALUATION
V9 FAIL · MISSING_EVALUATION
V10 FAIL · MISSING_EVALUATION

Rule: all(V1..V10 == PASS) AND protocolAudit.passed

1. Derivation boundary

L0 · LINDGREN 2025

PREMISE

g_mu_nu = eta_mu_nu + kappa A_mu A_nu

L1 · DERIVED

DERIVED

delta_g = kappa (A_bio⊗a_ext + a_ext⊗A_bio + a_ext⊗a_ext)

L1 · GEODESIC DEVIATION

DERIVED

D_u sqrt(-det g) = kappa (A·u) / sqrt(1 + kappa A^2)

L2 · LORENTZ → EUCLIDEAN/SCALAR

OPEN CHOICE

q = |A_bar| := sqrt(kappa)s [dimensionless spatial projection]

L1 · CHI ALGEBRA

DERIVED

chi_geo(q) = q / sqrt(1 + q^2); directional derivative evaluates chi_geo(|A_bar|) after the L2 reduction

L2 · OBSERVABLE BRIDGE

OPEN

O_r = integral K_r^(mu nu) delta_g_mu_nu dV

The algebraic χ_geo(q) shape retains its L1 status. Choosing q=|Ā| through a positive spatial Lorentz-to-Euclidean/scalar projection is an L2 reduction. Any measured quantity or proxy z instead needs an explicit dimensionless normalization q=N(z); choosing and calibrating N is the open L0→L2 step. Frequency weights, biological mechanisms, memory kernels and endpoint mappings keep their own provenance.

2. DKC calculation

T4 fast: k_B(s) = s^(n_B-1) exp(-s/tau_B) / (tau_B^n_B Gamma(n_B)), n_B in 2..6

annual T4 bins: K_B[l] proportional to CDF_Erlang(l+1) - CDF_Erlang(l)

slow: k_R(s) = exp(-s/tau_R) / tau_R

T3: v(a) = 5 (a<0), 4 (a<1), 3 (a<6), 2 (a<18), else 1

BL(t) = alpha integral k_B(s)x(t-s)ds + beta integral k_R(s)v(a)x(t-s)ds; beta = 1-alpha

sigma(BL) = BL^n / (x_half^n + BL^n)

Delta endpoint(t) = -gamma sigma(BL(t))

T4 uses a normalized Erlang fast kernel (integer n_B = 2–6) and a normalized exponential slow kernel. Beta is not free: beta = 1 − alpha, with tau_R > tau_B. T3 is the explicit cohort schedule v(a): 5 for a<0, 4 for 0≤a<1, 3 for 1≤a<6, 2 for 6≤a<18, otherwise 1.

Dual-kernel candidate explorer

Vary the candidate time scales and response shape against a synthetic annual technology-adoption curve. The chart is a sensitivity view, not an empirical country fit.

Model status

candidate calculation / national technology timing proxy / open L2 / locked F1–F9 falsification register

The registered DKC FieldState calibration pipeline is implemented and produces values; the route also publishes the locked F1–F9 register. This Explorer run deliberately uses a synthetic timing proxy, not a local physical measurement or calibrated dose.

Scenario parameters

Fast scale τB0.50 years

Slow scale τR12 years

Fast share α0.60

Derived slow share β

0.40

Hill coefficient n2.50

Erlang shape nB3

T4 uses the selected Erlang nB = 2–6 for the fast arm and a normalized exponential for the slow arm. β is constrained to 1 − α.

T3 cohort vulnerability (shown explicitly; not applied to this period-only curve): v(a)=5 for a<0, 4 for 0≤a<1, 3 for 1≤a<6, 2 for 6≤a<18, otherwise 1. Supply a cohort birth year to the evaluator to apply it to the slow input.

Candidate load and Hill saturation, 1990–2030

Four annual curves show the weighted fast arm, weighted slow arm, their total, and the normalized Hill saturation.

Fast armSlow armTotal loadHill saturation

Example construction: a midpoint-2013 sigmoid with 0.5/year steepness, multiplied by the continuous 0.33–1.00 duty-cycle scenario. A longer prehistory is calculated before the displayed window to avoid an artificial start-edge effect.

2030 scenario snapshot

Synthetic proxy: 1.000
Duty cycle: 1.000
Fast arm: 0.600
Slow arm: 0.292
Total load: 0.892
Hill saturation: 0.809
Seasonal CV: 0.000

3. T1–T12 refinement register

T1L3_IMPORTED

country-specific continuous duty cycle

d(t,c)=0.33+0.67/(1+exp(-k_s(t-t_mid(c))))

Stage 2 · free parameters: 0

T2L3_PHENOMENOLOGICAL

Hill response exponent

sigma(x)=x^n/(x_half^n+x^n)

Stage 3 · free parameters: 1

T3L3_IMPORTED

birth-cohort vulnerability

v(a)={5:a<0,4:0<=a<1,3:1<=a<6,2:6<=a<18,1:a>=18}

Stage 2 · free parameters: 0

T4L3_PHENOMENOLOGICAL

Erlang behavioural kernel

k_B(s)=s^(n_B-1) exp(-s/tau_B)/(tau_B^n_B Gamma(n_B))

Stage 3 · free parameters: 1

T5L3_IMPORTED

endpoint-specific testicular SAR multiplier

w_fertility(f)=SAR_testicular(f) coupling(f) modulation(f)

Stage 2 · free parameters: 0

T6L3_PHENOMENOLOGICAL

separate indoor Wi-Fi proxy

wifi(t)=w_wifi BB(t) D_wifi(t)

Stage 4 · free parameters: 0-1

T7L3_PHENOMENOLOGICAL

handset power-control correction

personal_eff=P_tx_max(1-eta ambient_norm)

Stage 4 · free parameters: 1

T8L3_PHENOMENOLOGICAL

network-density saturation

P_ambient=P_max N/(N_half+N)

Stage 4 · free parameters: 1

T9L3_PHENOMENOLOGICAL

spectral-complexity multiplier

H=-sum(p_f log2 p_f); multiplier=1+lambda_H H

Stage 4 · free parameters: 1

T10L3_PHENOMENOLOGICAL

seasonal-variation prediction

CV_seasonal=CV_0(1-duty_cycle)

Stage 2 · free parameters: 0

T11L3_PHENOMENOLOGICAL

melatonin-redox synergy

BL_syn=BL(1+epsilon_syn d_night chi_pineal)

Stage 2 · free parameters: 0

T12L3_PHENOMENOLOGICAL

explicit parental epigenetic carry-over

BL_final=BL_syn+epsilon_epi BL_parent_at_conception

Stage 4 · free parameters: 1

4. What is available now

Layer	Current state	Allowed interpretation
Geometry	2025 Weyl ansatz + exact tensor perturbation	Premise / derived algebra
L2	OPEN	No geometry-to-observable claim
National series	World Bank mobile, broadband, urban	TECHNOLOGY_TIMING_PROXY
FieldState	calibration pipeline implemented and producing values; uncalibrated sensitivity inputs also supported	Calibration output, proxy class and execution capability kept separate
DKC parameters	tau_B, tau_R, alpha and Hill parameters carry L3 provenance	They do not relabel L1-derived components
Predictions	F1–F9 content-addressed and locked	LOCKED_FALSIFIABLE_FORECAST

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Frequency weights Model comparison Falsification registry