Closing report — Space from a description language

03-ilang-space / closing report
# Closing report: 03-ilang-space

- **Subproject:** 03-ilang-space. Space inside Ilanguage 1.b, route P: space from the view.
- **Date:** 2026-10-09
- **State:** all 23 derivation packages (02–24) are verified, and every one has had at least one external verification round. The dependency graph passes `graphcheck`.
- **Scope of this report:** it summarizes the verified results against the success criteria of `01-base-problem/problem.md`, citing packages and tags. It adds no results. Interpretations are marked as such.
- **Status labels:**
  - **proven**: a verified derivation, externally checked;
  - **conditional**: proven under stated assumptions;
  - **candidate**: numerical evidence only;
  - **open**: not settled.

## 1. Question and approach

The question: can a space-like structure be derived inside Ilanguage 1.b from the language's own elements, without importing any spatial notion (M1–M6)? Which of the features needed for a later study of gravity follow, and under which conditions?

**Approach (route P).**
- The points of space are the places of an object (a *probe* or a *body*), as distinguished by the rest of the system.
- The distance between two places measures how differently the companion *witnesses* them in the view (1.5)–(1.7).
- Space appears as soon as a contract *records* the probe's place in a medium. All further structure is built on this: neighbours, chains, dimension, bodies, motion, cost, curvature and continuum limits.

## 2. Answers at a glance

| # | Criterion | Status | Main packages and tags | Answer in one line |
|---|---|---|---|---|
| 1 | Points | proven | 03 (3.1)–(3.4) | Points are classes of places whose companion branch states coincide ($W=1$); they are label-free. |
| 2 | Distance | proven | 03 (3.5), (3.12)–(3.13); 05 (5.10)–(5.11) | The witness angle $\alpha=\arccos\sqrt W$ (Fubini–Study) is a metric, unique up to scale under additivity. At small $\lambda$ it is Euclidean: $\alpha\approx\lambda\lVert u_h-u_{h'}\rVert$. |
| 3 | Neighbourhood, locality | proven / conditional | 04 (4.1)–(4.7); 08 (8.15)–(8.16); 11 (11.3)–(11.11) | Neighbours follow the relative-neighbourhood rule. Edges need contracts across the cut. Influence spreads one cell per order of $\lambda$ (perturbative light cone). |
| 4 | Paths, geodesics | proven + candidate | 22 (22.5)–(22.11), (22.27), (22.29), (22.32)–(22.36); 23 (23.9) | Euclidean and spherical limits have unique geodesics. Periodic chain geometries are polygonal (anisotropic, non-unique geodesics). Random local arrangements are isotropic (numerical). |
| 5 | Dimension, continuum | proven / conditional | 05 (5.14); 07 (7.7); 14 (14.4)–(14.16); 22 (22.6), (22.13), (22.18), (22.22), (22.27), (22.36) | The dimension is the record capacity of the medium, fixed by the description. Continuum limits exist, except for compact local records, which collapse. |
| 6 | Body | proven | 15 (15.3)–(15.7), (15.20)–(15.21) | Localized, persistent, label-free bodies exist, with positions and a two-body distance. |
| 7 | Background across $\lambda$ | proven | 24 (24.3)–(24.11), (24.18), (24.24), (24.28) | Points are identified through places and merge only at isolated $\lambda$. Neighbour structure is stable below $\lambda_*>0$, with $\lambda_*\sim1/\text{size}$ for linear records and size-independent for local ones. |
| 8 | Time | handed over | — | Handed to the time project. All rates here are per $\lambda$. |
| 9 | Velocity, acceleration | proven | 17 (17.4)–(17.7), (17.20); 20 (20.7) | Currents, velocity and a contract-implied maximal speed $v_{\max}$; an exact acceleration identity. |
| 10 | Localized cost, mass | proven | 20 (20.2)–(20.5), (20.10); 21 (21.10), (21.16) | Cost is locally conserved on the background. Inertia equals the edge (motion) cost. Binding adds inertia $\lvert E_{\rm bind}\rvert/v_{\max}^2$ on the chain. |
| 11 | State-dependent geometry | proven / conditional | 06 (6.9)–(6.13); 08 (8.10)–(8.16); 18 (18.4)–(18.16); 19 (19.1)–(19.20) | Bodies change the geometry only through non-commuting records, locally. Curvature has three sources. A mediated influence exists but is not sourced by cost. |
| 12 | Common space | proven | 12 (12.12)–(12.15); 16 (16.5), (16.8)–(16.13), (16.17) | Identical probes share one background. Two types share points only if their record clouds are translates; otherwise only displacements compare. |

The success line of the base problem ("items 1–7 and 9–12 are derived; item 8 is handed over") is met. Several items are answered with conditions or with negative results that name what is missing (section 5).

## 3. Results by criterion

### I. Space at fixed $\lambda$

**Points and distance** (criteria 1–2).
- The view of a part is the Gram matrix of the companion's branch states (2.2). It is fixed by the witness weights $W(h,h')=\lvert\langle E_{h'}\vert E_h\rangle\rvert^2$ and their cyclic products, up to local phases (2.10)–(2.13).
- Points are the classes $W=1$ (3.1)–(3.2). They do not depend on labels (3.3)–(3.4).
- $\alpha=\arccos\sqrt W$ is a metric (3.5). Every triangle with sides in $[0,\pi/2]$ is realizable under condition (R) (3.6)–(3.8). Additivity on aligned triples forces $\alpha$ up to scale (3.12)–(3.13).
- A recording contract $\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$ gives $W$ (5.6), independent of the probe's weights. At small $\lambda$, $\alpha/\lambda\to d_0=\lVert u_h-u_{h'}\rVert$ with record vectors $u_h=(K_h-\langle K_h\rangle)\chi$ (5.10). The background is a subset of a Euclidean space (5.11).

**Neighbourhood and locality** (criterion 3).
- Related points have $W>0$ (4.1)–(4.2). Neighbours are related pairs with no third point that overlaps more strongly with both (4.3); the rule does not depend on the metric transform (4.4).
- The chain distance $\ell$ (4.6)–(4.7) extends $\alpha$ beyond its saturation at $\pi/2$.
- Without contracts across the cut there is no geometry: a product stays a product (11.3)–(11.4). At small $\lambda$ the boundary contracts alone fix $d_0$ and the edges (11.6)–(11.11). A nonempty boundary is not sufficient (11.16).
- In a medium of cells with link contracts, a body changes $W$ at the earliest at order $\lambda^{3+r}$, $r$ links away (8.15)–(8.16). This is a perturbative light cone, not a Lieb–Robinson bound.

**Dimension** (criterion 5, first part).
- $\dim_0$ is the affine rank of the record vectors, at most $\min(n-1,2d_c-2)$ (5.14).
- Generic records give $\dim_0=\min(n-1,2(d_c-1))$, or $\min(n-1,d_c-1)$ for real ones (7.7), (7.11).
- Independent channels add up (14.4)–(14.7). A generic 3D background needs total capacity 3, with minimal medium dimension 4 (14.15)–(14.16).
- **The language does not select 3; the description does.**
- The loop data are the Kähler form of the record geometry: $\Phi=-\lambda^2\sum\omega(u_j,u_{j+1})$ (10.8)–(10.14). The $\lambda^2$ term vanishes for commuting or real records; for real records the loop data are odd in $\lambda$ (10.15).

**Paths, continuum, isotropy** (criteria 4–5).
- **Direct distances.**
  - $d_0$ on a dense square family converges to the Euclidean square, with unique straight geodesics (22.5)–(22.6).
  - At fixed $\lambda$, a qubit medium gives a cap of the round sphere of radius 1/2 (curvature 4), or the whole $\mathbb{CP}^1$ (22.12)–(22.13). It is a length space iff $\lambda r_0\le\pi/4$ or $\lambda r_0\ge\pi/2$.
- **Chain distance on regular arrangements.** On a square family, the chain distance is the $\ell^1$ (taxicab) norm, even where $d_0$ is Euclidean: box betweenness, binomially many shortest chains, $\max\ell_0/d_0=\sqrt2$ (22.7)–(22.11).
- **Compact local records** (windows) collapse. Saturation ties make distant places neighbours, the diameter stays bounded, and there is no continuum limit for any scaling (22.14)–(22.18), with exact $\ell$ and diameter for every size (22.16)–(22.17), (22.30)–(22.31).
- **Quasi-local records** (exponential tails) give continuum limits of dimension 1 and 2 (22.22), (22.27). The neighbour decisions for far pairs rest on margins $\sim q^{r/2}$ (22.23).
- **Periodic arrangements.** A fixed periodic pattern always gives a polygonal norm, never an isotropic one (22.29) (conditional on (F1)–(F3)). The same holds for patterns of bounded complexity under (F4) (22.32).
  - An isotropic limit is possible with periodic patterns of growing complexity. In the explicit Crofton construction the records are non-local half-plane indicators: $\cos\alpha=(\cos\lambda)^{d_G}$, $\ell=\lambda d_G$, and the limit is Euclidean (22.33)–(22.36).
- **Random local arrangements** (code package 23, precommitted).
  - The large-scale chain distance is isotropic to within 0.4%: $\bar B_{128}=0.9995\pm0.0021$, "isotropy supported" (23.9). The periodic control gives $B=1.30$ (23.8). Status: **candidate**.
  - Robustness was classified "intermediate" (23.10). Non-local admixtures of size $\varepsilon$ create long edges: the shortest long edge was at $r=66$ for $\varepsilon=10^{-4}$ and at $r=46$ for $\varepsilon=10^{-2}$. Our interpretation is a large-scale cut-off that grows like $\ln(1/\varepsilon)$ (not derived).

### II. Bodies, motion, background

**Bodies** (criterion 6).
- A localized body stays localized and uncorrelated; its position is $[\beta]_0$ (15.3)–(15.4).
- Two bodies of one type give an exact product state, with a well-defined distance $D=\lVert u_{\beta_1}-u_{\beta_2}\rVert$ (15.5)–(15.7).
- The medium records only the sum of the bodies' record vectors (15.11)–(15.13). With linear records the separation is never recorded (15.15)–(15.16); local windows record it (15.17)–(15.19), except at revivals (15.21)–(15.22).

**Motion and velocity** (criterion 9).
- The currents are $J=2\operatorname{Im}[t\,(V_b)_{h'h}]$ (17.4), bounded by the witness weight (17.5).
- The velocity is (17.6). The maximal speed is $v_{\max}=\varrho(\lvert t\rvert d_0)\le\max_h\sum\lvert t\rvert d_0$ (17.7); on the chain, $v_{\max}=2t\sigma_X$ (17.20).
- Records slow hops in proportion to their background length (Zeno-like) (17.10).
- Linear records act as a random uniform force: bodies show Bloch oscillation or Wannier–Stark localization instead of travelling (17.18)–(17.23).
- The acceleration obeys an exact identity (20.7).

**Background across $\lambda$** (criterion 7).
- For two places, $W(\lambda)=1$ either for all $\lambda$ or on a discrete set (24.3)–(24.4). Points only merge, at isolated $\lambda$, and never split (24.6).
- The neighbour graph changes only on a discrete set (24.8). The stability scale $\lambda_*$ is positive (24.9). Trajectories are analytic for all $\lambda$ and continue across mergers (24.10)–(24.11).
- Linear records on a qubit: the line closes into a circle at $\lambda_*=\pi/(\lfloor3n/2\rfloor-1)$ (24.18)–(24.20).
- General linear records: $\lambda_*=\Theta(1/n)$ (24.24).
- Local windows: $\lambda_*=\pi/2$, independent of size (24.28).

### III. Connection to gravity

**State-dependent geometry and curvature** (criterion 11).
- **A body changes a probe's geometry.**
  - It does so only through records that do not commute with the probe's (6.7), (6.9)–(6.13); the effect first appears at $\lambda^3$.
  - In a cell medium it does so locally, at $\lambda^{3+r}$ for $r$ links between them (8.15)–(8.16).
  - In the qubit example the sign follows the record strength, not the cost (6.14)–(6.15).
- **The background metric** is the Fubini–Study pullback (19.1)–(19.2). Its leading curvature is the Gauss curvature of the record surface (19.8)–(19.10).
- **Curvature has three sources:**
  - the shape of the record map (medium state);
  - finite $\lambda$ (e.g. curvature 4 for a qubit medium) (19.16)–(19.17);
  - bodies, only through non-uniform effects (19.13)–(19.15).
- **Mediated influence on motion.**
  - One body influences another's motion through the medium only via commutators, and the influence is independent of the source's cost (18.4), (18.12).
  - With linear records it is even in the source strength, and the directed drift appears only at $\lambda^8$ (18.13)–(18.16).

**Cost and mass** (criterion 10).
- **Distribution of the cost.**
  - The cost splits exactly into site costs and edge costs, free of conventions (20.2).
  - Only the edge costs and the differences of site costs per unit weight are fixed by readings (20.3)–(20.4).
  - The cost of a region changes only through the contracts across its boundary: local conservation (20.5).
- **Inertia.**
  - The inverse inertia is minus the edge cost, weighted by the background edge vectors: $\mu=-\sum_EL_{hh'}\Delta\otimes\Delta$ (20.10). The site costs give only the force.
  - A localized body has no instantaneous response (20.11).
- **Composite bodies.**
  - The inertia of a composite comes from its parts' edge costs: $\mu_c=\tfrac14(\mu_1+\mu_2)$ (21.10).
  - On the chain, binding raises the inertia by exactly $\lvert E_{\rm bind}\rvert/v_{\max}^2$ (21.16). This is the analogue of the relativistic mass defect, with the contract-implied maximal speed in the role of $c$ (shown for the 1D chain only).
  - The absolute version $I\,v_{\max}^2=-E_{\min}$ (21.17) depends on the cost zero, which no reading fixes.
- **Inertial versus active mass.** Inertia is tied to a part of the cost (20.10), but the mediated influence is not sourced by the cost (18.12). In a static recording medium, inertial and active mass therefore do not coincide.

**Common space** (criterion 12).
- Identical probes see the same background, provided the contracts respect Law 5 (12.4), (12.12)–(12.15).
- For two types, the split of medium-only terms is a convention (16.3)–(16.4). The well-defined quantities are the functions of $u^A_h+u^B_k$ (16.5).
- A common set of points exists iff the record clouds are translates; the correspondence is then unique (16.11)–(16.13).
- The centred cross distance is fixed (16.17). The raw cross-type distance is not (16.6), (21.3).

## 4. Cross-cutting findings (interpretation)

1. **Space is derivable without imports.** The view, together with a recording contract, gives points, a Fubini–Study metric, a Euclidean small-$\lambda$ geometry, neighbours, chains, bodies, motion, cost densities and curvature. All of these are built from the language's elements.
2. **The language does not select the geometry; the description does.** This holds for the dimension (14), the isotropy (22, 23) and the medium's capacity. Ilanguage 1.b provides the mechanism, not the values.
3. **Linear, lattice-like records are too rigid.** Their geometry is clean, but:
   - they give no separation sensing (15) and no transport (17);
   - mediated effects are even and origin-dependent (18);
   - the stability scale shrinks with size (24).
4. **Local records are necessary but delicate.**
   - Large distances need chains, and compact records collapse (22).
   - Quasi-local records give continua with a logarithmic large-scale cut-off against non-local admixtures (23).
   - Periodic arrangements are anisotropic (22.29); random ones are isotropic (23.9).
5. **Gravity-like behaviour does not arise in static recording media.** The mediated influence is not cost-sourced (18.12), and inertial and active mass differ (20, 21 versus 18). Media with their own dynamics (links, local records) are the natural next step.

## 5. What is missing, and open problems

- **Negative or conditional results**, naming what Ilanguage 1.b with the formalization of `problem.md` lacks:
  - a cost-sourced, directed mediated interaction in static recording media (18);
  - a canonical zero of the cost, so that only cost differences are physical (20.3), (21.17);
  - isotropy from a fixed periodic local description (22.29);
  - scale-free stability of linear-record backgrounds (24.24).
- **Open:**
  - isotropy of local records in 3D;
  - the $E=mc^2$-type relation (21.16) beyond the 1D chain;
  - $\Lambda_N$ in closed form for general records (24.30 is implicit);
  - media with internal dynamics as the source of gravity-like effects;
  - the time project's clock $\tau(\lambda)$, needed to convert all rates;
  - the interface with the gravity programme's $K_{\rm rel}$ (Appendix B).

## 6. Quality record

**External verification** (the external referee, GPT-6.1 Sol in later rounds and GPT-6 Astra in early ones, saw only the referee prompt, the base problem, the question without "Expected result", the derivation and its code):
- correct: 02, 03, 04, 07, 08, 09;
- minor issues, recorded without regeneration: 05, 06, 10, 11, 12, 13;
- fixed by external regeneration: 14, 16, 17, 19, 20, 21, 23, 24 (one round each);
- 15, 18 and 22 needed two rounds (major issues in the first round, fixed; 22 also in the second).

**External cost:** about 3.49 USD for this subproject; 6.73 USD project-wide.

**Main-model errors found by the referee and recorded as lessons:**
- accepting an unanswered requested item as an "open issue" (18, 22);
- an imprecise question that did not fix what is held constant across a family (22);
- a miscalibrated precommitted tolerance (13, T3(a)).

## Appendix A. Package index

| Package | Topic | Version | External rounds |
|---|---|---|---|
| 02 | view content | v1 | 1 |
| 03 | witness distance | v1 | 1 |
| 04 | neighbours | v1 | 1 |
| 05 | recording contract | v1 | 1 |
| 06 | body in medium | v1 | 1 |
| 07 | random records | v1 | 1 |
| 08 | cell medium | v1 | 1 |
| 09 | measurability | v1 | 1 |
| 10 | loop data | v1 | 1 |
| 11 | cut boundary | v1 | 1 |
| 12 | identical probes | v1 | 1 |
| 13 | numerical run (code) | v1 | 1 |
| 14 | dimension | v2 | 1 |
| 15 | bodies | v3 | 2 |
| 16 | common space | v2 | 1 |
| 17 | motion | v2 | 1 |
| 18 | mediated effect | v3 | 2 |
| 19 | curvature | v2 | 1 |
| 20 | cost and mass | v2 | 1 |
| 21 | composite body | v2 | 1 |
| 22 | large-scale geometry | v3 | 2 |
| 23 | random arrangements (code) | v2 | 1 |
| 24 | background stability | v2 | 1 |