Research: add compact-space Brownian red team

docs/RESEARCH_STATUS.md

@@ -159,7 +159,7 @@ Process Geometry solver or independent adoption.
 | [`am-conformal-chart-normal-forms`](../sonnet/am-conformal-chart-normal-forms/00-problem-frontier.md) | Phase 1 exact Riccati lift, Möbius covariance, scalar gauge, cubic no-go, and eight-axis cost accounting | run oracle-isolated bounded discovery (1B) before the pendulum atlas search; no economy theorem yet |
 | [`moving-am-observer`](../sonnet/moving-am-observer/README.md) | static-observer no-go followed by blind moving-frame recovery, held-out family selection, task-equivalence quotient, blind morphism discovery, grammar stability, and dimensionful Bellman covariance | broader non-affine/multi-family or stochastic pressure, now separated into its own Sonnet |
 | [`stochastic-feedback-trap-first-passage`](../sonnet/stochastic-feedback-trap-first-passage/README.md) | bounded grammar census, exact Itô task quotient, independent BVP/Monte Carlo first-passage comparison, and reset Bellman value/policy covariance complete | calibration is closed; new discovery or API incubation requires a new contract |
-| [`brownian-scale-fibre`](../sonnet/brownian-scale-fibre/README.md) | S0/S1 initialized: blind centered finite-law scale balance, typed drift refusal, exact endpoint fibres, and concatenation pushforward | separate lattice point return, continuum neighbourhood recurrence, and singleton hitting before authorizing a Brownian/heat-kernel lowering |
+| [`brownian-scale-fibre`](../sonnet/brownian-scale-fibre/README.md) | S0/S1 plus compact red team: exact integer-to-cycle deck-fibre pushforward, typed even-cycle period obstruction, exact lazy uniform stationary law, and frozen circle/sphere boundaries | execute the sphere action/stabilizer gate and keep compact time-stationarity separate from line scale-stability before any continuum lowering |
 | [`lonely-runner`](../sonnet/lonely-runner/README.md) | exact upstream pruning transfer and bounded K4/K5 contact mechanisms; Phase 15A global audit complete | lift-aware initial search on solved cases, then frozen K13 validation; `LRC(13)` remains open |
 | [`hidden-am-noether`](../sonnet/hidden-am-noether/README.md) | bounded static-observer route reached a structural no-go; the viable moving-observer continuation is tracked separately | do not reopen static conjugation without new grammar or task semantics |
 | [`pcr3bp-history-cost`](../sonnet/pcr3bp-history-cost/README.md) | Phase 0/1 history, scale-jet, topology, and coding audit complete; Phase 2 contract frozen | execute converged return/absorbing ensemble and twisted partition audit; no numerical result yet |

sonnet/README.md

@@ -80,7 +80,7 @@ deferred until its next oracle/evidence gate is affordable.
 | [`hidden-am-noether/`](hidden-am-noether/) | static branch closed negatively | a frozen observer in the same product-affine group cannot reveal a missing stabilizer dimension | [static-observer no-go](hidden-am-noether/03-static-observer-no-go-and-schedule-split.md); continuation moved to `moving-am-observer/` |
 | [`moving-am-observer/`](moving-am-observer/) | affine deterministic phase closed | bounded blind observer discovery, task-equivalent minimum slice, and dimensionful Bellman covariance on the declared family | [study ledger](moving-am-observer/README.md); stochastic continuation moved to its own Sonnet |
 | [`stochastic-feedback-trap-first-passage/`](stochastic-feedback-trap-first-passage/) | current calibration closed | exact Itô task quotient plus independent first-passage and reset-Bellman covariance across 242 monotone charts | [Phase-4 results](stochastic-feedback-trap-first-passage/09-phase4-reset-bellman-results.md); further work requires a new Sonnet or governed extraction proposal |
-| [`brownian-scale-fibre/`](brownian-scale-fibre/) | S0/S1 active | blind centered finite-law scale balance and exact endpoint-fibre concatenation; no continuum limit or heat-kernel lowering yet | [problem frontier](brownian-scale-fibre/00-problem-frontier.md); next separate recurrence observers, then certify the continuum lowering |
+| [`brownian-scale-fibre/`](brownian-scale-fibre/) | S0/S1 plus compact red team | exact line endpoint fibres now push to finite cycles through retained deck fibres; even cycles expose period two, while lazy compact laws have an exact uniform stationary fixed point distinct from line scale stability | [compact-space red team](brownian-scale-fibre/02-compact-space-redteam.md); next execute the separate sphere action/stabilizer gate before any continuum lowering |
 | [`pcr3bp-history-cost/`](pcr3bp-history-cost/) | Phases 0–1 complete; Phase 2 frozen | lifted topology and scale-jet reconstruction separate word, clock, deck, and hyperbolic costs; no Bellman/Huffman source is yet justified | [Phase-2 contract](pcr3bp-history-cost/02-return-partition-holonomy-contract.md); next run the frozen two-gate covariance and convergence gates |
 | [`am-conformal-chart-normal-forms/`](am-conformal-chart-normal-forms/) | T0/T1; Phase 1 mechanism calibrated | exact Riccati lift, Möbius covariance, scalar-gauge invariance, cubic no-go, and eight-axis Pareto accounting; no discovery or economy theorem | [Phase-1 results](am-conformal-chart-normal-forms/02-phase1-riccati-results.md); next freeze a blind low-height grammar and run bounded recovery |
 | [`effective-scale-carrier-ladder/`](effective-scale-carrier-ladder/) | T1 / NARROW | finite syntax decision eliminates a surreal runtime; symbolic height is a real C2 obstruction, but semantic evaluation and C3/C4 separation remain open | [results](effective-scale-carrier-ladder/01-results.md), [compiler](../workstreams/carrier_ladder/compiler/), and [commit--reveal audit](../workstreams/carrier_ladder/redteam/); next implement an effective normalized hyperiteration fragment, not a general surreal runtime |

sonnet/brownian-scale-fibre/02-compact-space-redteam.md

@@ -0,0 +1,166 @@
+# Compact-space Brownian red team
+
+Status: finite cycle phase executed for
+[#162](https://github.com/mountain/process-geometry/issues/162); circle
+semantics and the sphere claim boundary are frozen.  No continuum heat-kernel
+or mixing theorem is claimed.
+
+## 1. Question
+
+The line calibration can suggest that long process time, scale normalization,
+and a Gaussian stable law are one universal mechanism.  Compact spaces force
+three notions apart:
+
+1. a locally Gaussian small-time chart;
+2. a scale-renormalized law on a noncompact space;
+3. a time-stationary invariant measure on a compact space.
+
+The native red team starts before a continuum density.  It counts finite
+nearest-neighbour histories on the integer lattice and pushes them through a
+finite deck quotient.
+
+## 2. Exact cycle quotient
+
+For horizon (N), let
+
+\[
+\pi_N(\omega)=\sum_{j=1}^N\xi_j\in\mathbb Z,
+\qquad \xi_j\in\{-1,+1\}.
+\]
+
+For the finite cycle (C_q=\mathbb Z/q\mathbb Z), the endpoint count at residue
+(r) is exactly
+
+\[
+c_N^{(q)}(r)
+=\sum_{k\in\mathbb Z}c_N(r+kq).
+\]
+
+At finite (N), only finitely many lifts have nonzero mass.  The software
+retains every pair `(integer lift, count)` rather than discarding it after the
+sum.  It then independently performs the chronological update directly on
+(C_q) and compares the two constructions residue by residue.
+
+For (q=5,N=7), both constructions return
+
+\[
+(14,35,22,22,35),
+\]
+
+with total mass (128=2^7).  The residue-zero deck fibre is
+
+\[
+\{(-5,7),(5,7)\},
+\]
+
+which makes the forgotten lift explicit.
+
+This establishes a finite exact statement:
+
+\[
+\boxed{
+\text{cycle endpoint counting}
+=
+\text{integer endpoint counting integrated along deck fibres}
+}
+\]
+
+It does not require a wrapped Gaussian, Fourier basis, transition matrix, or
+heat kernel.
+
+## 3. Winding is task-relative residual information
+
+On (C_4), the equal-horizon histories
+
+\[
+(+1,+1,+1,+1),
+\qquad
+(+1,-1,+1,-1)
+\]
+
+have integer lifts (4) and (0), but both have cycle endpoint zero.  Thus the
+cycle endpoint quotient is exact for endpoint mass and inadequate for winding,
+running extrema, or chronological path tasks.
+
+The continuum circle statement
+
+\[
+S^1=\mathbb R/(2\pi\mathbb Z)
+\]
+
+has the same structure: a circle endpoint has an integer deck fibre on the
+cover.  A wrapped law is the pushforward obtained by integrating over that
+fibre.  This continuum interpretation is recorded here but not implemented as
+a heat-kernel computation.
+
+## 4. Period two blocks naive time forgetting
+
+For the non-lazy nearest-neighbour walk on an even cycle, every step switches
+the parity class.  The clock has period two.  Consequently the terminal law
+cannot converge pointwise to the uniform law; its total-variation distance
+from uniform is at least (1/2) at every finite time.
+
+The implementation returns a typed `periodic-clock-obstruction` and names the
+three distinct repairs:
+
+- add a lazy stay event;
+- pass to a continuous-time clock;
+- use a Cesaro time observer.
+
+These repairs are not silently identified.  On an odd cycle, or under a
+strictly intermediate lazy probability, the clock is irreducible and the
+period is one.  Stay probability one is separately rejected as a reducible
+clock: the uniform law remains stationary, but every point mass is also
+stationary, so stationarity alone cannot establish mixing.
+
+For the exact lazy walk with stay probability (1/2), the uniform law is an
+exact one-step fixed point.  A bounded (C_5) audit records decreasing rational
+total-variation distances through horizon eight, ending at
+(3571/163840).  This is evidence of a bounded trend, not a proof of an
+asymptotic mixing theorem.
+
+## 5. Two types of stable law
+
+The red team uses separate machine-readable types:
+
+| Mechanism | Example | Required operation |
+| --- | --- | --- |
+| `scale-renormalized` | line Gaussian | ensemble growth followed by dynamic spatial rescaling |
+| `time-stationary` | uniform cycle/circle law | declared clock evolution on a compact carrier |
+
+The uniform compact law must not be reported as a Gaussian or as a scaling
+fixed point.  Conversely, line Brownian has no finite stationary probability
+under raw physical-time evolution.
+
+## 6. Sphere gate
+
+The next compact red team is frozen around
+
+\[
+S^2=SO(3)/SO(2).
+\]
+
+It must retain the following obstructions before any computation is promoted:
+
+- no single global additive position chart;
+- no internal global spatial dilation while the radius is fixed;
+- a tangent-chart Gaussian is not a global Gaussian law;
+- curvature, chart transition, and holonomy remain residuals;
+- the long-time stationary candidate is normalized area
+  (dA/(4\pi)), not the line Gaussian.
+
+Unlike the circle, (S^2) has no winding deck fibre from a nontrivial universal
+cover.  A successful treatment must explain rotation histories and the
+stabilizer quotient rather than reuse the circle proof.
+
+## 7. Consequence for the boundary conjecture
+
+The finite result weakens a too-simple claim that physical position must escape
+to spatial infinity.  On a compact quotient, position remains bounded while
+history depth and forgotten lifts grow.  The more plausible universal object
+is therefore the ideal boundary of the time/history fibre, whose task
+pushforward may appear as a scale-stable law, Haar measure, Riemannian volume,
+or a more detailed path boundary.
+
+The result supports further study of that distinction.  It does not yet prove
+that one arithmetic-tower boundary produces all of these measures.

sonnet/brownian-scale-fibre/README.md

@@ -16,20 +16,34 @@ Read in order:
    calculations;
 4. [`test_brownian_scale_fibre_phase0.py`](../../tests/research/test_brownian_scale_fibre_phase0.py)
    — independent certificates and source firewall.
+5. [`02-compact-space-redteam.md`](02-compact-space-redteam.md) — exact cycle
+   quotient, circle cover semantics, period-two obstruction, and the frozen
+   sphere gate;
+6. [`phase_compact_contract.py`](phase_compact_contract.py) and
+   [`compact_space_redteam.py`](compact_space_redteam.py) — machine-auditable
+   compact-space red team.
 
-Current S0/S1 result:
+Current finite results:
 
 ```text
 raw centered finite law -> active response order 2 -> scale balance a=1/2
 finite histories -> endpoint fibres -> exact concatenation pushforward
 biased law -> typed centering-required refusal
 continuum / heat-kernel lowering -> not yet authorized
+integer lattice -> exact deck-fibre pushforward to finite cycles
+even non-lazy cycle -> typed period-two obstruction
+lazy compact law -> exact uniform stationary fixed point
+circle/sphere -> time-stationary, not line scale-stable
 ```
 
 The scale result is narrower than a central-limit theorem.  The endpoint result
-is narrower than path equivalence.  S2 must next separate lattice point return,
-continuum neighbourhood recurrence, and singleton hitting before S3 authorizes
-a Brownian/heat-kernel lowering.
+is narrower than path equivalence.  Recurrence and heat-kernel continuation
+remain unauthorized while the compact sphere action/stabilizer gate is open.
+
+The compact-space red team in #162 now blocks a second conflation.  A line
+Gaussian is scale-renormalized; a compact cycle/circle law can instead be
+time-stationary.  The finite cycle result is exact, while the continuum circle
+and sphere remain semantic gates rather than heat-kernel claims.
 
 This task is the second independent representation family requested in
 [#140](https://github.com/mountain/process-geometry/issues/140).  Draft