Operational Information Geometry V — Scale Flow, Continuum Response, and Protocol Boundaries

The manuscript keeps proved analytic statements, independently tested computations, hypotheses supported by numerical evidence, and non-claims visibly separate. This is a research manuscript with computer-assisted evidence; it is not peer reviewed.

• Proved analytically: the path spectrum, scale limits, modal response reduction, smooth parity rule, smooth-versus-atomic scaling, rank bound, leakage envelopes, and protocol boundaries.
• Computed with independent tests: response-Gramian convergence, leakage radii, reflection-symmetric rank control, endpoint silence versus path activity, and the interval certificate.
• Hypothesized: uniform positive-time fixed-mode operator convergence and positive-time convergence for appropriately smoothed atomic targets.

Sources: [1]

Fixed-domain refinement, infinite volume, and joint continuum/infinite-volume limits are distinct experiments. The third is not a formal consequence of the first two and requires uniform localization and modulation assumptions.

Sources: [1]

For the cell-centred path, the discrete cosine vectors form an orthonormal eigenbasis. For 0≤k<n, μ_(k,n)=4n² sin²(πk/(2n)), and every fixed nonzero mode obeys 0≤(πk)²−μ_(k,n)≤π⁴k⁴/(12n²).

Sources: [1]

For a d-fold product, fixed n followed by τ→0 gives D_(n,d)(τ)→0. If τ_n→0 while n²τ_n→∞, the same construction tends to the chosen product factor count d. The limits do not commute; the mesoscopic window separates lattice ultraviolet scale from observation scale.

Sources: [1]

The square-lattice generator G_n=A_n⊗I+(I+gD_n)⊗A_n is symmetric as a Markov generator while its declared source-to-target response is one-way. If A_nu_l=μ_(l,n)u_l and β_l=u_l^Tq, then the target-mode response reduces exactly to an n×n matrix exponential:

Sources: [1]

Density-normalized smooth responses converge numerically at O(n^-2), but fixed-mode convergence is explicitly a computational landmark and hypothesis rather than a proved operator theorem. For a smooth single-mode background, the first response derivative vanishes for even source cosine modes and equals 2√2·a·g·ℓ²/k² for odd modes. A smooth target first jet tends g a π²/√24, whereas an atomic raw first jet grows like g n^(5/2)/√2. Refinement and the initial-time limit therefore do not commute.

Sources: [1]

If source directions are restricted to a declared modal band, the first-jet sensing rank obeys a finite rank bound. This is an observability statement for the chosen channel, not a claim that directionality makes every source identifiable.

Sources: [1]

The exact Poisson jump-count cone becomes continuum-trivial under diffusive scaling. A model-specific signed-displacement exponential-martingale argument gives P{h d₁(X_τ,X₀)≥r}≤min{1,4 exp(−I_h(r,τ;2+g))} and the sufficient limiting radius r_δ(τ)=√[4(2+g)τ log(4/δ)]. At g=.8, τ=.005, δ=.01, the bound is 0.5792426 while the actual radius is about 0.4375. In an expanding-domain stretched-ramp control, the localized arrow decays exactly as g√3/n.

Sources: [1] [6]

Reflection-symmetric modulation has exact response rank 2, leaving three invisible source directions; an exact 2×2 jet minor is 32/125. A factorized stationary target is silent to the endpoint marginal, yet a path jump-rate sensor detects the arrow with norm 3.7872 and rank 3. Symmetric propagators restore reciprocity for conjugate or adjoint source-measurement ports; the directional result uses localized, nonconjugate ports. In the fixed linear model, intervention response is exactly affine, so all second amplitude derivatives vanish.

Sources: [1] [7]

An independent 192-bit Arb certificate uses an exact dyadic primal and rational positive-semidefinite dual on both the exact algebraic and frozen binary64 120-time grids. It brackets z* between 4.982412×10^-9 and 4.982942×10^-9. This is a strong computer-assisted certificate for the declared finite grids, not proof-assistant interval arithmetic and not a continuous-time optimum.

Sources: [1]

The fixed-mode response convergence conjecture should be tested with uniform positive-time operator bounds and smoothed atomic preparations. Joint continuum/infinite-volume behavior needs its own localization hypotheses. The model is best read as a bridge theory for scale flow and protocol dependence: it does not select a physical dimension or explain why nature has one.

• A heat kernel has instantaneous tails, so the leakage estimate is not relativistic causality.
• Atomic preparations are valid at finite n, but their t=0 L² continuum jet is singular.
• Fixed-domain limits do not imply infinite-volume behavior; endpoint silence does not mean path-space silence.

Sources: [1]

The public repository contains the Stage I–V scripts, tests, frozen Stage IV grids, and the scaling and certificate ledgers. The commands below reproduce the declared laboratories from the repository root; the python-flint dependency is needed for the Arb certificate.

• python -m unittest discover -v
• python operational_information_geometry.py --controls 32
• python operational_information_geometry_ii.py --protocols 16
• python operational_information_geometry_iii.py --interaction-strength 0.4
• python operational_information_geometry_iv.py --protocol-seeds 32
• python operational_information_geometry_v.py
• python oig_v_scaling_continuum.py --maximum-side-length 64
• python oig_iv_certificate.py --grid both

Sources: [1] [2]