# RRLS v0.24 — Universal-Cycle Tomography & Resonant Collision Engine

## Status

**P = NP is not proved.**

v0.24 builds and stress-tests the string/fiber idea as two deliberately separate models:

1. a **synthetic dynamical system-identification engine** in which every explicit hair is driven by one fixed cycle;
2. an **exact finite relational-tomography engine** in which stretched SAT/CSP defect states are treated as synthetic hair endpoints and interrogated by fixed projection phases.

The first tests whether one rich cycle can recover hidden coordinates under a known deterministic law. The second tests whether higher-order collisions and same-speed temporal resonance add discriminative information about exact defect languages.

---

## 1. Synthetic universal string cycle

Each hair has a permanent base-string attachment address `a_i`, discrete coupling depth `d_i`, transverse orientation `phi_i`, and home radial length `r_i`.

Depth controls coupling by

    g(d)=exp(-alpha d).

The same two-axis multi-frequency cycle is sent to every hair.  Propagation along the string appears as an integer phase delay.  The scalar measured response is

    y_i(t)=g(d_i)[cos(phi_i) u_x(t-a_i)+sin(phi_i) u_y(t-a_i)]+noise.

No hair gets a custom probe.  The input waveform is fixed.

A one-tone scalar baseline uses only

    y_i(t)=g(d_i) cos(phi_i) sin(2pi(t-a_i)).

That intentionally exposes the amplitude/orientation ambiguity discussed in the string model.

### 700-hair recovery benchmark

| probe                     |   noise |   anchor_exact_rate |   anchor_mae_bins |   depth_exact_rate |   depth_mae |   phi_mae_deg |   tof_valid_rate |   seconds |   phi_p95_deg |
|:--------------------------|--------:|--------------------:|------------------:|-------------------:|------------:|--------------:|-----------------:|----------:|--------------:|
| single-tone scalar        |    0    |            1        |        0          |           0.445714 |     1.09429 |  nan          |       nan        | 0.02853   | nan           |
| universal multiaxis cycle |    0    |            1        |        0          |           1        |     0       |    1.0477e-14 |       nan        | 0.0285318 |   2.54444e-14 |
| radial time-of-flight     |    0    |          nan        |      nan          |           1        |     0       |  nan          |         1        | 0.0285325 | nan           |
| single-tone scalar        |    0.05 |            0.612857 |        0.93       |           0.447143 |     1.09143 |  nan          |       nan        | 0.0352775 | nan           |
| universal multiaxis cycle |    0.05 |            0.995714 |        0.00428571 |           1        |     0       |    0.303764   |       nan        | 0.0352798 |   0.905959    |
| radial time-of-flight     |    0.05 |          nan        |      nan          |           1        |     0       |  nan          |         1        | 0.0352805 | nan           |
| single-tone scalar        |    0.1  |            0.378571 |        1.85       |           0.445714 |     1.10286 |  nan          |       nan        | 0.0309249 | nan           |
| universal multiaxis cycle |    0.1  |            0.93     |        0.07       |           1        |     0       |    0.607765   |       nan        | 0.0309419 |   1.83103     |
| radial time-of-flight     |    0.1  |          nan        |      nan          |           1        |     0       |  nan          |         0.998571 | 0.0309428 | nan           |
| single-tone scalar        |    0.2  |            0.235714 |        3.28571    |           0.447143 |     1.1     |  nan          |       nan        | 0.0306461 | nan           |
| universal multiaxis cycle |    0.2  |            0.742857 |        0.275714   |           1        |     0       |    1.21397    |       nan        | 0.0306479 |   3.65449     |
| radial time-of-flight     |    0.2  |          nan        |      nan          |           1        |     0       |  nan          |         0.997143 | 0.0306488 | nan           |

At response noise 0.10:

- one-tone anchor exactness: **37.9%**;
- one-tone naive depth exactness: **44.6%**;
- universal-cycle anchor exactness: **93.0%**;
- universal-cycle depth exactness: **100%**;
- universal-cycle orientation MAE: **0.61°**.

At much larger response noise the universal cycle degrades rather than magically remaining perfect:

| probe       |   noise |   anchor_exact_rate |   depth_exact_rate |   depth_mae |   phi_mae_deg |
|:------------|--------:|--------------------:|-------------------:|------------:|--------------:|
| single-tone |     0.4 |           0.107143  |           0.43     |   1.10429   |     nan       |
| universal   |     0.4 |           0.497143  |           0.975714 |   0.0242857 |       2.83454 |
| single-tone |     0.6 |           0.08      |           0.428571 |   1.11      |     nan       |
| universal   |     0.6 |           0.378571  |           0.905714 |   0.0942857 |       4.15947 |
| single-tone |     1   |           0.0442857 |           0.392857 |   1.14      |     nan       |
| universal   |     1   |           0.241429  |           0.771429 |   0.235714  |      10.7002  |

At noise 1.0, anchor exactness falls to about **24.1%**, depth exactness to **77.1%**, and orientation MAE rises to about **10.7°**.  This is the behavior we want from a falsifiable identification model.

These recovery numbers are conditional on the **assumed synthetic law being correct and known**. They do not show that physical dimensions obey this law.

---

## 2. Time-of-flight dimensional coordinate

The radial force uses the reversible toy law

    r_i(t)/r_i(0)=exp[-k0 A exp(-alpha d_i) t]

on the inward half-cycle and the exact inverse on the outward half.

For a universal threshold `epsilon`, inward time is

    tau_d = ln(1/epsilon)/(k0 A) * exp(alpha d).

Therefore adjacent depth classes have fixed latency ratio

    tau_(d+1)/tau_d = exp(alpha).

With alpha=0.28 this ratio is about **1.323**.

Important correction: distance decreases **exponentially with time**, but multiplying the force amplitude `A` by two halves the threshold time in this first-order model; the time does not itself fall exponentially with force amplitude.

The exact reversible-cycle numerical audit was:

|   force |   max_start_error |   max_end_snapback_error |   median_mid_ratio |   min_mid_ratio |   max_mid_ratio |
|--------:|------------------:|-------------------------:|-------------------:|----------------:|----------------:|
|     0.5 |                 0 |                        0 |        0.341283    |     0.0828937   |     0.541141    |
|     1   |                 0 |                        0 |        0.116474    |     0.00687137  |     0.292834    |
|     2   |                 0 |                        0 |        0.0135662   |     4.72157e-05 |     0.0857517   |
|     4   |                 0 |                        0 |        0.000184041 |     2.22932e-09 |     0.00735336  |
|     8   |                 0 |                        0 |        3.38712e-08 |     4.96987e-18 |     5.40719e-05 |

At force 8, the median radial ratio at the middle of the cycle was about `3.387e-08`, yet the end-of-cycle snap-back error was numerically **0** in the exact model.

---

## 3. Collision + synchronization axis

Endpoints were then evolved in a two-dimensional synthetic fiber geometry.  Hairs are allowed to pass through one another.  A collision is a shared finite-resolution sensor cell.  A **resonant collision** additionally requires similar measured speed.

Only resonant groups receive synthetic Kuramoto-like phase coupling.  Thus the engine records:

- pair/triple/quadruple collision counts;
- same-position/same-speed resonant events;
- temporal interaction components;
- Kuramoto coherence `R(t)`.

Robustness runs used 6 independent populations, 260 hairs each, three sensor resolutions, five inward-force strengths, and coupling on/off.

At the middle sensor resolution:

|   cell_size |   force |   K |   cases |   median_pairs |   median_resonant |   median_Rfinal |   median_Rmax |   median_max_mult |   median_largest_component |
|------------:|--------:|----:|--------:|---------------:|------------------:|----------------:|--------------:|------------------:|---------------------------:|
|         0.1 |       0 |   0 |       6 |        13058.5 |            4241.5 |       0.0573933 |     0.060125  |               5.5 |                      254.5 |
|         0.1 |       0 |   3 |       6 |        13352   |            4113   |       0.0577737 |     0.0614489 |               6   |                      255   |
|         0.1 |       1 |   0 |       6 |        84732   |           40355.5 |       0.0573933 |     0.060125  |              21   |                      260   |
|         0.1 |       1 |   3 |       6 |        84971.5 |           40492.5 |       0.0753414 |     0.076221  |              20.5 |                      260   |
|         0.1 |       2 |   0 |       6 |       208927   |          108528   |       0.0573933 |     0.060125  |              33.5 |                      260   |
|         0.1 |       2 |   3 |       6 |       210760   |          109478   |       0.099163  |     0.099163  |              34   |                      260   |
|         0.1 |       4 |   0 |       6 |       334780   |          184440   |       0.0573933 |     0.060125  |              39   |                      260   |
|         0.1 |       4 |   3 |       6 |       335583   |          185267   |       0.130457  |     0.130457  |              39   |                      260   |
|         0.1 |       8 |   0 |       6 |       397677   |          220420   |       0.0573933 |     0.060125  |              37.5 |                      260   |
|         0.1 |       8 |   3 |       6 |       397946   |          220918   |       0.140908  |     0.142071  |              37.5 |                      260   |

For `cell_size=0.10`, median pair-collision events increased from about **13,059** with no inward force to about **397,677** at force 8 in this toy geometry.  With coupling `K=3`, median final coherence rose from about **0.0578** at force 0 to **0.1409** at force 8.

The same qualitative increase in collision counts with inward force appeared at all three tested sensor resolutions.  This is **model-dependent**, not a universal law: changing geometry, anchors, collision radius, or coupling rules can change the result.

---

## 4. Exact relational tomography

The second engine used **72 exact stretched defect languages**:

- 24 positive 1-in-3;
- 24 NAE-3;
- 24 XOR;
- defect dimensions `k=7..11`.

Every feasible defect state is a synthetic endpoint.  One universal phase uses a fixed signed projection

    y=<p,2d-1>.

States with the same `y` collide in the measurement channel.

Higher collision axes are

    C_r = sum_z binom(n_z,r).

The temporal-resonance graph adds your new condition: two states interact only when they have both the **same projected position** and the **same discrete movement speed** into the next phase.

### How much exact relation identity survived the cycle?

|   distinct_truths |   signature_classes |   ambiguous_truths |   resolved_fraction |   max_truths_per_signature | signature                                |
|------------------:|--------------------:|-------------------:|--------------------:|---------------------------:|:-----------------------------------------|
|                72 |                  46 |                 42 |            0.416667 |                          6 | static k+|D|                             |
|                72 |                  66 |                 11 |            0.847222 |                          3 | one-cycle phase: radial/weight histogram |
|                72 |                  67 |                  9 |            0.875    |                          3 | 4 probes collision C2-C4                 |
|                72 |                  67 |                  9 |            0.875    |                          3 | 8 probes collision C2-C4                 |
|                72 |                  67 |                  9 |            0.875    |                          3 | 16 probes collision C2-C4                |
|                72 |                  70 |                  4 |            0.944444 |                          2 | 4 probes full histograms                 |
|                72 |                  71 |                  2 |            0.972222 |                          2 | 8 probes full histograms                 |
|                72 |                  71 |                  2 |            0.972222 |                          2 | 16 probes full histograms                |
|                72 |                  72 |                  0 |            1        |                          1 | 32 probes full histograms                |
|                72 |                  69 |                  5 |            0.930556 |                          3 | 16 probes + temporal resonance graph     |

Static `(k, |D|)` distinguished only **41.7%** of the 72 exact truth tables uniquely.  One radial/weight projection raised this to **84.7%**.  Four full response histograms reached **94.4%**; 8 and 16 reached **97.2%**; **32 fixed full histograms distinguished all 72 finite relations**.

### Does the collision/same-speed axis add information?

|   distinct_truths |   signature_classes |   ambiguous_truths |   resolved_fraction |   max_truths_per_signature |   probes | signature                                    |
|------------------:|--------------------:|-------------------:|--------------------:|---------------------------:|---------:|:---------------------------------------------|
|                72 |                  69 |                  5 |            0.930556 |                          3 |        4 | compact collision+velocity resonance summary |
|                72 |                  69 |                  5 |            0.930556 |                          3 |        8 | compact collision+velocity resonance summary |
|                72 |                  69 |                  5 |            0.930556 |                          3 |       16 | compact collision+velocity resonance summary |
|                72 |                  72 |                  0 |            1        |                          1 |       32 | compact collision+velocity resonance summary |

The compact temporal-resonance summary resolved **93.1%** with 4–16 probes and **100%** with 32 probes on this corpus.  Low-order collision moments alone plateaued at 87.5% on the same data.

So on this finite corpus the *identity of which collisions connect across time* contained information that static collision moments discarded.

---

## 5. Random 3-SAT control

A separate corpus contained 64 nontrivial random 3-SAT truth tables at n=8..11.

|   distinct_truths |   signature_classes |   ambiguous_truths |   resolved_fraction |   max_truths_per_signature | signature            |
|------------------:|--------------------:|-------------------:|--------------------:|---------------------------:|:---------------------|
|                64 |                  32 |                 51 |            0.203125 |                          4 | static k+|D|         |
|                64 |                  61 |                  6 |            0.90625  |                          2 | one probe radial     |
|                64 |                  64 |                  0 |            1        |                          1 | 8 full histograms    |
|                64 |                  64 |                  0 |            1        |                          1 | 32 full histograms   |
|                64 |                  55 |                 15 |            0.765625 |                          3 | 32 compact resonance |

Here a single radial histogram already resolved 90.6%, and 8 full fixed-probe histograms resolved all 64.  The compact resonance summary was worse (76.6%).

That negative control matters: **synchronization is not automatically a superior representation.**  Sometimes ordinary response histograms contain the useful information more directly.

---

## 6. The biggest algorithmic wall exposed by v0.24

The exact relational experiments first enumerate the feasible defect states.  That is an oracle-like step:

    D_F = {d : exists stretched satisfying state with defect d}.

Enumerating `D_F`, or even computing exact response-bin counts for an arbitrary CNF, can hide SAT/#SAT-level work.

Therefore the fact that a 32-phase signature distinguishes all relations in a finite benchmark **does not give a polynomial SAT algorithm**.

A real wall break now requires a theorem of the following form:

> There exists a fixed polynomial-length universal cycle whose response features can be computed **directly from an arbitrary CNF in polynomial time**, without enumerating satisfying states, and whose polynomial-size response is sufficient to decide fiber occupancy exactly.

That is the proof obligation.

---

## 7. Information-theoretic caution

There are `2^(2^k)` arbitrary Boolean relations on k bits.  A polynomial number of response histograms, each containing polynomially many bounded-precision counts, has only `2^poly(k)` possible polynomial-size signatures.  Therefore no such signature can uniquely identify **every arbitrary truth table**.

This does not alone rule out polynomial-size CNF inputs: the input family itself has only exponentially many descriptions of a fixed polynomial length.  But it means the universal-cycle claim must exploit the **structured fact that the relation came from a compact formula**, not attempt to encode an arbitrary truth table.

---

## 8. What survived the experiment

The strongest supported statements are now:

- A multi-axis/multi-frequency fixed cycle can disambiguate parameters that one scalar oscillation confounds in the synthetic model.
- A reversible inward/outward cycle gives an operational latency coordinate when the coupling law is assumed.
- Stronger inward contraction greatly increased finite-resolution collision events in the tested geometry.
- Same-position + same-speed temporal collision structure can carry additional relational information beyond low-order collision counts.
- Full fixed-probe response histograms were even stronger than synchronization on some controls.
- None of this proves the physical universe is a string/fiber system.
- None of this proves P=NP, because the current exact relational responses are obtained by exponential finite enumeration.

The next technically valid target is **non-enumerative relational spectroscopy**: derive exact polynomial-cost response observables from the compact constraint description itself.