research: initialize Brownian scale and endpoint-fibre Sonnet

docs/RESEARCH_STATUS.md

@@ -159,6 +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 |
 | [`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

@@ -80,6 +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 |
 | [`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/00-problem-frontier.md

@@ -0,0 +1,126 @@
+# Problem frontier: recalculate Brownian motion from process fibres
+
+Status: frozen S0/S1 contract for
+[#158](https://github.com/mountain/process-geometry/issues/158), downstream of
+[#140](https://github.com/mountain/process-geometry/issues/140) and the
+native-method firewall merged in
+[#157](https://github.com/mountain/process-geometry/pull/157).
+
+## 1. Primitive process before the continuum answer
+
+The first carrier is the finite history space
+
+\[
+\Omega_N=\{\xi_1\cdots\xi_N:\xi_j\in\{-1,+1\}\}
+\]
+
+with chronological concatenation.  The endpoint task is
+
+\[
+\pi_N(\omega)=\sum_{j=1}^N\xi_j,
+\qquad
+K_N(0,x)=\frac{|\pi_N^{-1}(x)|}{|\Omega_N|}.
+\]
+
+The literal history, endpoint, fibre multiplicity, and pushforward probability
+are different objects.  For the endpoint task, concatenation gives exactly
+
+\[
+K_{N+M}(0,z)=\sum_x K_N(0,x)K_M(x,z).
+\]
+
+The executable phase proves this first as an integer fibre-count identity.  It
+does not infer that two paths with the same endpoint are equivalent for
+first-passage, maximum, occupation, or physical-decoder tasks.
+
+## 2. Blind scale obligation without a supplied Gaussian
+
+For a centered finite increment law, let
+
+\[
+\kappa(s)=\log\mathbb E[e^{s\xi}].
+\]
+
+Independence makes the aggregate response additive: the log response of
+\(N\) increments at probe scale \(N^{-a}s\) is
+
+\[
+N\kappa(N^{-a}s).
+\]
+
+The native calculation reads the raw law and certifies exactly
+
+\[
+\kappa(0)=\kappa'(0)=0,
+\qquad
+\kappa''(0)=\sigma^2>0.
+\]
+
+This is not used as a truncated power series.  The exact integral identity
+
+\[
+\kappa(s)=\int_0^s(s-u)\kappa''(u)\,du
+\]
+
+shows that the first active centered response has order two.  Population
+growth and probe contraction therefore balance only when
+
+\[
+1-2a=0.
+\]
+
+The software solves that equation after discovering the active order; it is
+not passed \(a=1/2\).  For the symmetric unit law, one may independently sharpen
+the residual by
+
+\[
+\log\cosh s-\frac{s^2}{2}
+=-\int_0^s(s-u)\tanh^2u\,du.
+\]
+
+This phase certifies a necessary nontrivial scale and its local response
+coefficient.  It does **not** yet prove convergence to a Gaussian law or a
+continuum path process.
+
+## 3. Method firewall
+
+S0/S1 have no allowed lowering witness.  The native source contains only the
+finite increment law, exact centered response, chronological histories,
+endpoint fibres, and their composition.  Classical local expansion,
+transform, continuum PDE, and simulation routes are declared separately in
+`phase0_contract.py` and cannot enter a native trace without a later
+task-adequacy witness.
+
+This is deliberately stronger than asking an agent not to use an oracle in
+prose: the merged method tool rejects the corresponding event in a native lane.
+
+## 4. Held-out and red-team cases
+
+- A biased finite law returns the typed result `centering-required`; the drift
+  cannot be hidden inside a fluctuation scale.
+- An infinite-variance law is outside the first finite-law grammar.  A later
+  heavy-tail phase must enlarge the carrier and rediscover its scale rather
+  than force the diffusive answer.
+- Histories `(+1,-1)` and `(-1,+1)` share endpoint zero but have different
+  running maxima.  Endpoint fibres are task-exact only for endpoint observers.
+- Lattice return to a point, continuum return to a neighbourhood, and
+  continuum hitting of a singleton remain separate S2 claims; the 2D boundary
+  is not compressed into one word, “recurrent.”
+
+## 5. Relation to the larger programme
+
+This is the first independent scale/fibre family after the Bessel-based draft
+in PR #141.  It pressures U1, U2, U3, U4, and Effective Analysis because the
+same construction connects raw histories, scale, measured fibres, a continuum
+lowering, and later physical response.  At S0/S1 it remains T1 and leaves the
+Mathematical Core, Engineering Architecture, Theory Map, dependencies, and API
+unchanged.
+
+Surreal numbers are not needed for this finite-height task.  That is a negative
+runtime result, not an argument against later AMP or higher-height uses.
+
+## 6. Claim ceiling
+
+No new Brownian theorem, central-limit theorem, heat-kernel theorem,
+stochastic-calculus replacement, general stochastic solver, speedup, particle
+size inference, or arithmetic-universality result is claimed.

sonnet/brownian-scale-fibre/README.md

@@ -0,0 +1,42 @@
+# Sonnet — Brownian scale and endpoint fibres
+
+Research-local calibration for
+[#158](https://github.com/mountain/process-geometry/issues/158).
+
+This Sonnet recalculates the entrance to Brownian motion from raw increment
+histories.  It does not begin with the continuum process, a Gaussian density,
+the heat equation, or a supplied diffusive exponent.
+
+Read in order:
+
+1. [`00-problem-frontier.md`](00-problem-frontier.md) — primitive histories,
+   tasks, scale argument, held-outs, and claim ceiling;
+2. [`phase0_contract.py`](phase0_contract.py) — frozen executable method lanes;
+3. [`brownian_native.py`](brownian_native.py) — exact scale and endpoint-fibre
+   calculations;
+4. [`test_brownian_scale_fibre_phase0.py`](../../tests/research/test_brownian_scale_fibre_phase0.py)
+   — independent certificates and source firewall.
+
+Current S0/S1 result:
+
+```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
+```
+
+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.
+
+This task is the second independent representation family requested in
+[#140](https://github.com/mountain/process-geometry/issues/140).  Draft
+[#141](https://github.com/mountain/process-geometry/pull/141) remains an
+explicit compiler dependency; no implementation from that draft is copied
+while it remains unmerged.  The completed stochastic feedback-trap Sonnet is
+an independent downstream Itô/first-passage calibration and is not duplicated
+here.
+
+No Experimental or Public API pressure follows from S0/S1.