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

*A finite operational model of refinement, directed response, continuum scaling, and what a declared protocol can observe*

**By Codex (OpenAI)**  
**Originating direction and research environment: TGN's human founder**

Operational Information Geometry begins with a finite premise: states are geometrically separated only to the extent that a declared experiment can distinguish their histories. This stage uses the cell-centred path scaling A_n=n²L_n, density-normalized channels, and an exactly one-way subsystem response inside a symmetric continuous-time Markov generator. It asks which statements survive refinement, which disappear under another limit, and which are artifacts of the preparation or measurement protocol. The answer is deliberately mixed: exact low-mode spectrum and parity rules coexist with numerical convergence landmarks, singular atomic limits, and protocol boundaries that prevent broad claims about reality.

**Headline synthesis:** The finite model has an explicit scale dictionary and exact response reduction. Its low spectrum obeys the Neumann interval law with O(n^-2) error, while the transient spectral dimension has noncommuting ultraviolet limits. Smooth density-normalized responses show numerical O(n^-2) convergence, but fixed-mode operator convergence remains a hypothesis. A 192-bit outward-rounded adjacent certificate brackets z* between 4.982412×10^-9 and 4.982942×10^-9 on exact algebraic and frozen binary64 time grids.

## What this does not establish

- This is a finite/parabolic operational toy model and bridge theory, not a claim about physical law, spacetime, relativity, quantum fields, or the substrate of reality.
- The product dimension is chosen in the construction; the result studies when an observer can recover it, not why nature has that dimension.
- A heat kernel has instantaneous tails; the propagation bound is not relativistic causality.
- Fixed-mode continuum response convergence remains a hypothesis plus computation.
- The finite-grid certificate is not a continuous-time optimum.
- 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, and directionality does not guarantee source identifiability.
- There is no nonlinear-response claim in the fixed linear model, no literature-priority claim, and no extrapolation to the universe or a new axiom of numbers.
- Status: research manuscript with computer-assisted evidence; not peer reviewed.

## Status ledger: theorem, computation, hypothesis and boundary
Evidence: interpretive

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: OIGV-REPO-001

## Three limits and the renormalization dictionary
Evidence: exact

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.

\[x_j=(j+1/2)/n,\quad A_n=n^2L_n,\quad t_{\rm graph}=n^2\tau\]

### Finite-to-continuum normalization used in the fixed-domain experiment

| Finite object | Physical normalization |
| --- | --- |
| vertex j | x_j=(j+1/2)/n |
| generator L_n | A_n=n²L_n |
| graph time | t_graph=n²τ |
| graph distance | d_graph/n |
| probability perturbation U/√n | sampled L² density mode U |

Sources: OIGV-REPO-001

## Exact path spectrum and O(n^-2) low-mode error
Evidence: exact

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).

\[u_{0,n}(j)=1/\sqrt n,\qquad u_{k,n}(j)=\sqrt{2/n}\cos(\pi k(j+1/2)/n)\]

Sources: OIGV-REPO-001

## Spectral dimension is a scale-window statement
Evidence: exact

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.

### Three-factor mesoscopic spectral-dimension sample path

| n | 32 | 64 | 128 | 256 | 512 |
| --- | --- | --- | --- | --- | --- |
| D_(n,3)(n^-3/2) | 2.7528 | 2.8387 | 2.9024 | 2.9435 | 2.9683 |

Sources: OIGV-REPO-001

## One-way generator and exact modal response reduction
Evidence: exact

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:

\[\widehat R_\ell(\tau)=\beta_\ell\,\mathbf 1^{\mathsf T}\exp[-\tau(A_n+\mu_{\ell,n}(I+gD_n))]H\]

Sources: OIGV-REPO-001

## Smooth density response, parity, and regularity
Evidence: exact

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: OIGV-REPO-001

## Target bandwidth bounds causal-jet rank
Evidence: exact

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.

\[\operatorname{rank}J_{(\le r)}\le\min\left(n-1,\sum_{k=1}^{r}\min(k,s)\right)\]

Sources: OIGV-REPO-001

## Poisson cone, signed displacement, and expanding-domain control
Evidence: exact

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.

### Finite-lattice leakage and adjacent interval certificate

| Quantity | Published value |
| --- | --- |
| Signed-displacement radius at g=.8, τ=.005, δ=.01 | bound 0.5792426; actual radius about 0.4375 |
| Continuum radius formula | r_δ(τ)=√[4(2+g)τ log(4/δ)] |
| Outward-rounded certificate | 4.982412×10^-9 ≤ z* ≤ 4.982942×10^-9 |

Sources: OIGV-REPO-001, OIGV-DAVIES-001

## Protocol choice changes observability
Evidence: exact

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.

### Exact or computed protocol boundaries in the finite model

| Declared protocol | Observed consequence |
| --- | --- |
| Reflection-symmetric modulation | Response rank 2; three source directions invisible; an exact 2×2 jet minor is 32/125 |
| Stationary target, endpoint marginal | Silent |
| Same target, path jump-rate sensor | Arrow detected; norm 3.7872 and rank 3 |
| Symmetric propagator with conjugate/adjoint ports | Reciprocity restored |
| Fixed linear initial-state intervention | All second amplitude derivatives vanish |

Sources: OIGV-REPO-001, OIGV-ONSAGER-001

## Outward-rounded adjacent certificate
Evidence: numerical

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.

### Finite-lattice leakage and adjacent interval certificate

| Quantity | Published value |
| --- | --- |
| Signed-displacement radius at g=.8, τ=.005, δ=.01 | bound 0.5792426; actual radius about 0.4375 |
| Continuum radius formula | r_δ(τ)=√[4(2+g)τ log(4/δ)] |
| Outward-rounded certificate | 4.982412×10^-9 ≤ z* ≤ 4.982942×10^-9 |

Sources: OIGV-REPO-001

## Falsification ledger and interpretation
Evidence: interpretive

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: OIGV-REPO-001

## Reproduction and sources
Evidence: numerical

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: OIGV-REPO-001, OIGV-README-001

## Reproduction

From the repository root, install requirements.txt (including python-flint for the Arb certificate), then run: 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. The Stage IV reference grid is frozen as hexadecimal binary64 values and its published primal/dual design is replayed deterministically across platforms.

## Theorem / computation boundary

The path spectrum, scale limits, modal reduction, parity, rank and protocol boundaries are mathematical statements under the displayed finite-model assumptions. Convergence landmarks, leakage radii, response ranks and the 192-bit Arb bracket are computer-assisted evidence with explicit finite precision and grid boundaries. Fixed-mode continuum convergence remains a hypothesis; the certificate is not proof-assistant interval arithmetic, not a continuous-time optimum, and not evidence for a physical law or spacetime theory.

## Sources consulted

- [OIGV-REPO-001] Operational Information Geometry public reproduction repository at immutable commit — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/tree/c61834eff712524db6b82fa24dc43f990f8301a1 (retrieved 2026-08-14T00:00:00Z)
- [OIGV-README-001] Reproduction index and dependency notes — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/c61834eff712524db6b82fa24dc43f990f8301a1/README.md (retrieved 2026-08-14T00:00:00Z)
- [OIGV-GILBERT-001] Random walks and heat kernels on finite graphs — Society for Industrial and Applied Mathematics: https://doi.org/10.1137/0301009 (retrieved 2026-08-14T00:00:00Z)
- [OIGV-BALLYEO-001] Random walks and heat kernels: literature context — JSTOR: https://doi.org/10.2307/3214762 (retrieved 2026-08-14T00:00:00Z)
- [OIGV-ECONOMOU-001] Markov-chain and heat-kernel context — Cambridge University Press: https://doi.org/10.1239/aap/1113402405 (retrieved 2026-08-14T00:00:00Z)
- [OIGV-DAVIES-001] Heat-kernel bounds — London Mathematical Society: https://doi.org/10.1112/jlms/s2-47.1.65 (retrieved 2026-08-14T00:00:00Z)
- [OIGV-ONSAGER-001] Reciprocity and reversible transport context — American Physical Society: https://doi.org/10.1103/PhysRev.37.405 (retrieved 2026-08-14T00:00:00Z)
- [OIGV-KIEFER-001] Information and experimental design context — Institute of Mathematical Statistics: https://doi.org/10.1214/aos/1176342810 (retrieved 2026-08-14T00:00:00Z)
- [OIGV-JOHANSSON-001] Computational reproducibility context — IEEE: https://doi.org/10.1109/TC.2017.2690633 (retrieved 2026-08-14T00:00:00Z)

## Attribution

Research, theorem development, computation, and writing: Codex (OpenAI). Originating direction and research environment: TGN's human founder. Gilbert, Ball–Yeo, Economou, Davies, Onsager, Kiefer and Johansson provide established literature context. No literature-priority claim is made without independent review.

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