Operational Information Geometry VIII — The Uniform Three-Parameter Atlas

A single relative error formula over all (h,ε,t) is false. Resolved smooth preparation, finite-cell lattice phase, and atomic diffusion tail are different operational charts. Zeros require absolute squared-profile estimates rather than division by a vanishing phase.

Sources: [1]

Let h=1/n be the mesh spacing, ε the preparation width, t the observation time, c=ε/h the width-to-mesh ratio, q=t/ε² the width-scaled time, and τ=t/h²=c²q the lattice-scaled time. The canonical cell-centred Neumann path uses A_h=h⁻²L_n, modulation V(x)=1+gx, a fixed cosine source φ_k, and the complete density-normalized target response N_h(t).

Sources: [1]

For exact cell-integrated deposition from an even compactly supported smooth kernel, with q in a compact subset of (0,∞), conservative Fourier consistency and the reduced modal response give the balanced estimate below. The ε² term is source diffusion; (h/ε)² combines target dispersion, cell integration, and source quadrature.

Sources: [1] [2]

For the canonical ramp/cosine response, the early chart adds q to the relative error. Odd and even ports differ by the first nonzero multiplication-moment order. Reflection makes the discrete first moment exactly zero for even k; a general continuum cancellation with m₁=0 is not enough unless the discrete moment obeys m₁,h=o(q).

Sources: [1] [2] [4]

When ε=O(h) and t=τh², a local probability profile Q retains its lattice characteristic function and produces a phase Ψ_Q(τ). Different placement or deposition rules need not agree at finite microscopic time. The safe uniform estimate is first order in h plus the ℓ¹ profile error, and one bound continues into the atomic tail.

Sources: [1] [3]

For an arbitrary target probability vector, let σ_h² be its physical preparation variance and d_h its barycentre distance from the reflecting boundary. The full finite response has the displayed error budget. Consequently, the four simultaneous hypotheses—not t/h²→∞ alone—give the direct joint interior atomic limit. If C_F>0, and only then, the norm scales as C_F t⁻¹/⁴.

Sources: [1] [3]

If d_h/√t→κ is finite, the interior constant is replaced by an image-profile boundary constant. At the reflecting endpoint κ=0, the squared constant doubles; the endpoint norm constant is √2 times the interior constant. Interior atomic universality therefore fails within O(√t) of a reflector.

Sources: [1] [3]

At microscopic time t=τh², if the first nonzero multiplication moment has order r, the critical width exponent is α_c(r)=4r/(4r+1). Odd ports use ε=h^(4/5) and converge at relative order O(h^(2/5)); even ports use ε=h^(8/9) and converge at O(h^(2/9)). For g=.8 the exact first corrections are γ₁=3.5 and γ₂=6.3. Because n=c^(4r+1), an asymptotically correct exponent can remain practically invisible on ordinary grids.

Sources: [1] [2]

The uniform bounds combine a weak Duhamel matrix-element estimate, conservative Fourier multipliers from exact cell integration, summable rescaled modal envelopes, and a barycentric characteristic-function bound |χ_h(k)−1|≤k²σ_h²/2. The overlap estimates prevent ambient dimension n from being mistaken for an error multiplier and join the finite-cell phase to the continuum atomic tail.

Sources: [2] [3]

The laboratory uses exact target-mode separation followed by symmetric tridiagonal source eigendecomposition, with a small dense n²-state exponential as an independent reduction check. It uses ordinary floating point and adaptive quadrature, so the results are regression and falsification evidence rather than interval certificates. The focused Stage VIII suites pass 20/20 tests; the complete repository passes 231/231 at the theorem commit and 287/287 after the current main integration.

Sources: [1] [5] [4]

The adversarial controls are part of the result's meaning. An atom and a half/half cell split remain distinguishable at finite τ; the continuum phase fails below mesh resolution; t/h²→∞ alone is insufficient; F=0 does not permit division by C_F; continuum moment cancellation does not guarantee discrete cancellation; and the multiplication-moment order is not a complete causal hierarchy.

• The proof is for fixed declared source ports; a growing-band response Gramian and uniform smallest-singular-value theorem are the next target.
• The model is a declared one-way parabolic Markov model, not a theory of spacetime, fields, matter, radio propagation, or the universe.
• Numerical fits are not interval certificates, independent specialist peer review has not occurred, and novelty remains provisional.

Sources: [4] [1]

The immutable repository commit contains the canonical synthesis, Stage VI and VII supporting manuscripts, resolved and lattice proof memos, executable laboratory, numerical report, focused tests, adversarial control, minimal dependency file, and reproducibility manifest. The full laboratory takes approximately 13 seconds on the reference environment.

• python -m pip install -r oig_viii_three_parameter_requirements.txt
• python -m py_compile oig_viii_three_parameter.py test_oig_viii_three_parameter.py test_oig_viii_adversarial_controls.py
• python -m unittest -v test_oig_viii_three_parameter.py
• python -m unittest -v test_oig_viii_adversarial_controls.py
• python oig_viii_three_parameter.py --fast
• python oig_viii_three_parameter.py
• python -m unittest discover -v

Sources: [1] [6]